i of Uranium Dioxide and Related Phases · 2011. 8. 9. · technical reports series no. 39 thermodynamic and transport properties of uranium dioxide and related phases report of the

TECHNICAL REPORTS SERIES No. 39

Thermodynamic

and Transport Properties i

of Uranium Dioxide i • ¡i

and Related Phases

INTERNATIONAL ATOMIC ENERGY AGENCY,VIENNA, 1965

THERMODYNAMIC AND T R A N S P O R T P R O P E R T I E S OF URANIUM DIOXIDE AND R E L A T E D PHASES

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Printed by the IAEA in Austria

January 1965

T E C H N I C A L R E P O R T S SERIES No. 39

THERMODYNAMIC AND TRANSPORT PROPERTIES

OF URANIUM DIOXIDE AND

RELATED PHASES

REPORT OF THE P A N E L ON THERMODYNAMIC AND TRANSPORT PROPERTIES

OF URANIUM DIOXIDE AND RELATED PHASES HELD IN VIENNA 16 - 2 0 March 1964

INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA,. 1965

International Atomic Energy Agency. Thermodynamic and transport propert ies of

uranium dioxide and related phases . Report of the Panel on Thermodynamic . . . , held in Vienna, 1 6 - 20 March 1964. Vienna, the Agency, 1965.

105 p. (IAEA, Technical reports s e r i e s no. 39)

541 .11 546. 791. 4 -31 621. 039. 543 .4

THERMODYNAMIC AND TRANSPORT PROPERTIES OF URANIUM DIOXIDE AND RELATED PHASES,

IAEA, VIENNA, 1965 S T l / D O C / 1 0 / 3 9

F O R E W O R D

Because of the growing importance of thermodynamics to nuclear tech-nology, the International Atomic Energy Agency has init iated a project to a s s i s t in a s s e s s i n g and disseminat ing data on important nuclear mater ia l s . As a beginning, it organized a Symposium on the Thermodynamics of Nuclear Mater ia l s which was held in Vienna in May 1962. Th i s was fo l lowed by a P a n e l on the Thermodynamic P r o p e r t i e s of the Uranium-Carbon and Plutonium-Carbon S y s t e m s , held in Vienna in October 1962. The present Report i s the resul t of a further Pane l in th i s s e r i e s , convened f r o m 16 - 20 March 1964 to a s s e s s the thermodynamic and transport proper t i e s of the uranium dioxide phase and related uranium oxide phases . This Panel made a cri t ical evaluation of the available data, bearing in mind the practical a s p e c t s of the u s e of uranium dioxide a s a n u c l e a r f u e l . The f indings of the Pane l are presented by the A g e n c y in th i s i s s u e of the T e c h n i c a l R e -ports S e r i e s in the bel ief that they wi l l prove to be of value for nuc lear technology .

The Report was compiled and edited by Dr. Charles Holley of the Division of Research and Laboratories .

CONTENTS

I. INTRODUCTION 1

II. STRUCTURAL WORK 3

1. Stable phases in the U-O system 3

2. UO2 (room temperature) .' 3 2 . 1 . Lattice parameter . Dens i ty . 3 2 . 2 . Atomic posi t ions . Temperature factors 3

3. UO2 (high temperature) 5 3 . 1 . Variation of lattice parameter with temperature 5 3 . 2 . Temperature factors . Characterist ic temperatures .

Breakdown of harmonic approximation 6

4. U 0 2 + x region 7 4 . 1 . Variation of lattice parameter and density with

composit ion 7 4 . 2 . Crystal structure 9

4. 2 . 1 . Atomic positions in stat ist ical ce l l . Occupation numbers. Temperature factors . . . . 9

4. 2. 2. Interpretation of results for statist ical cel l . . . . 12

5. U 4 0 9

5 . 1 . Variation of lattice parameter with composition and temperature 15

5 . 2 . Structure 15 5 . 2 . 1 . X-ray studies 15 5 . 2 . 2 . Neutron studies 17

6. Tetragonal phases 19

7. Conclusions 20

III. THERMODYNAMICS . . . 23

1. Heat capacity measurements 23 1 . 1 . Low temperature heat capacity data 23 1 . 2 . High temperature heat capacity data 24

2. Lattice Dynamics of UO2 , 25

3. F r e e energy, enthalpy, and entropy measurements 29 3 . 1 . Chemical thermodynamics of the UO2. oo - 2.25 r e g i o n . . . 29 3 . 2 . The phase diagram, U 0 2 to U 4 O 9 38 3 . 3 . Hypostoichiometric U 0 2 39

4. Vaporization p r o c e s s e s 40

5. Theoret ica l treatment of U 0 2 + x phase 42 5 . 1 . Stat is t ical thermodynamics of interst i t ia ls and

vacanc ie s • 42 5 . 2 . Application of defect theory to UO2+ x 44

IV. SURFACE AND OXIDATION PROPERTIES 51

1. Adsorption propert ies 51 2. Oxidation p r o c e s s e s 52

2 . 1 . Low temperatures 52 2 . 2 . High t emperatures 53

V. PHYSICAL PROPERTIES 55

1. Thermal conductivity 55 1 . 1 . Low temperature thermal conductivity 55 1 . 2 . High temperature thermal conductivity 55

1 . 2 . 1 . Latt ice conductivity 57 1 . 2 . 2 . Radiant t rans fer 58 1 . 2 . 3 . E lec tronic t rans fer 59

2. E l ec tr i ca l propert ies 60 2 . 1 . Normal e l ec t r i ca l propert ies 60 2. 2. Ef fect of irradiat ion on e l ec tr i ca l propert ies 67

3. Optical m e a s u r e m e n t s 69 3 . 1 . Intrinsic absorption edge 69 3 . 2 . Defec t absorption 70 3 . 3 . Infra-red absorption 72

4. Magnetic m e a s u r e m e n t s 73

5. Diffusion p r o c e s s e s in UO^ 75 5 . 1 . Oxygen diffusion 75 5 . 2 . Uranium se l f -d i f fus ion 76 5 . 3 . Argon diffusion in calc ium fluoride as a model

p r o c e s s for f i s s i o n gas transport in uranium dioxide . . 76 5 . 4 . F i s s i o n gas r e l e a s e 78

6. Corre la t ive theory of physical propert ies 81 6 . 1 . Transport of energy 81 6. 2. Transport of matter 83

VI. PRACTICAL IMPLICATIONS OF THERMODYNAMIC AND TRANSPORT PROPERTIES 85

»

1. Interaction of fuel and can 85 1 . 1 . Thermal cracking 85 1 . 2 . Dimens iona l changes in UO2 under irradiation 86

2. Thermal conductivity 88

3. Phase equil ibria ' 88

4. Materia l transport p r o c e s s e s 89

VII. CONCLUSIONS 93

Appendix: Mathematical treatment of defect absorption 95

R e f e r e n c e s 99

Lis t of part ic ipants 103

Reports submitted to the Panel 105

I. INTRODUCTION

The h igh m e l t i n g point of u r a n i u m diox ide and i t s s t a b i l i t y under i r -radiat ion have l ed to i t s u se as a fuel in a var i e ty of types of nuclear r e a c -t o r s . A wide range of c h e m i c a l and p h y s i c a l s t u d i e s has b e e n s t i m u l a t e d by this c i rcumstance and by the complex nature of the uranium dioxide phase i t s e l f . The boundaries of this phase widen as the temperature i s increased; at 2000°K a s ing l e , homogeneous phase e x i s t s f r o m U 2 . 2 7 to a hyposto ichio-m e t r i c (UO2-X) c o m p o s i t i o n , depending on the o x y g e n potent ia l of the s u r -roundings. Since there i s often an incentive to operate a reactor at the maxi-m u m pract icab le heat rat ing and, there fore , m a x i m u m thermal gradient in the fue l , the d e t e r m i n a t i o n of the p h y s i c a l p r o p e r t i e s of the U 0 2 ± x p h a s e b e c o m e s a matter of great technological importance . In addition a complex s e q u e n c e of U - O p h a s e s m a y be f o r m e d during the p r e p a r a t i o n of powder feed mater ia l or during the s inter ing proces s ; these affect the m i c r o s t r u c -ture and p r o p e r t i e s of the f ina l product and have a l s o r e c e i v e d m u c h attent ion.

Uranium dioxide, t h e r e f o r e , provides an important example of a c o m -pound that e x i s t s as a s ing le non-s to i ch iometr i c phase at high temperatures and b e c o m e s unstable as the t emperature i s reduced , disproport ionat ing into p h a s e s of n e a r l y idea l s t o i c h i o m e t r y invo lv ing m o r e or l e s s c o m p l e x ordered s t r u c t u r e s . Ideal ly , the thermodynamic stabi l i ty and physical pro-per t i e s of U 0 2 ± x should be re lated to the s a m e atomic and e lectronic model , and i t s study should provide an opportunity for the c o r r e l a t i o n of a number of d i f ferent p r o p e r t i e s .

T h e Internat ional A t o m i c E n e r g y Agency (IAEA) t h e r e f o r e c a l l e d a panel m e e t i n g to d i s c u s s the t h e r m o d y n a m i c and t r a n s p o r t p r o p e r t i e s of the non-s to i ch iometr i c uranium dioxide phase , and this Report presents a summary of the data placed before the Pane l and of the conc lus ions reached . A con-s iderab le amount of data on the m a i n f ea tures of the phase d iagram and on the c o m p o s i t i o n l i m i t s of the v a r i o u s p h a s e s e x i s t s and X - r a y and neutron d i f f rac t ion e v i d e n c e indicate s o m e p o s s i b l e s t r u c t u r a l m o d e l s . C h e m i c a l thermodynamic va lues are known with some prec i s ion for most of the region c o n c e r n e d . S p e c i f i c d i s c u s s i o n s w e r e held on (i) the i n t e r r e l a t i o n of v i -brat ional c o n s t a n t s deduced f r o m s t r u c t u r a l work and heat c a p a c i t y data; (ii) the c o r r e l a t i o n of thermal conductivity with e l e c t r i c a l conductivity and optical data; and (iii) the calculation of entropy values by the stat ist ical treat-ment of s i m p l e m o d e l s that are cons i s tent with the s tructural , opt ical , and e l e c t r i c a l proper t i e s . The outline of a genera l i zed theory that should allow better c o r r e l a t i o n of t ransport and thermodynamic proper t i e s in the future w a s p r e s e n t e d . The i m p o r t a n c e of making al l m e a s u r e m e n t s of p h y s i c a l p r o p e r t i e s on s a m p l e s of a c c u r a t e l y known compos i t i on , which are s t r u c -tura l ly w e l l - c h a r a c t e r i z e d , w a s e m p h a s i z e d .

1

II. STRUCTURAL WORK

1. STABLE PHASES IN THE U-O SYSTEM

T h e r e are as m a n y as 16 w e l l - c h a r a c t e r i z e d uran ium oxide p h a s e s , and the ex i s tence of a dozen m o r e has been c la imed. A survey of work on the uranium-oxygen phase diagram up to 1961 has been made by ROBERTS [1]. F i g u r e 1 i s a reproduct ion of h i s phase d iagram. The work d e s c r i b e d in

Fig . 1

Por t ion of U - О p h a s e d i a g r a m . C i r c l e s d e n o t e X - R a y resul ts . (Reproduced by cou r t e sy of L . E . J . Roberts [ 1 ] )

this chapter i s r e s t r i c t e d to U 0 2 , U 0 2 + x , U4O9 and the tetragonal phases with compos i t i ons in the rangfe U 0 2 3 to U 0 2 4 . The s t r u c t u r e s of t h e s e phases are based on the f luor i te arrangement , with the additional oxygen atoms distributed e i ther at random on the f luori te la t t ice or in an ordered fashion, forming a cubic or tetragonal super lat t ice .

2. U 0 2 (ROOM TEMPERATURE)

2.1. Lattice parameter density

The s t o i c h i o m e t r i c oxide U 0 2 has a cubic s t ruc ture . The g e n e r a l l y accepted value for the lattice parameter i s a 0 = 5 .470Â (see Table I), which c o r r e s p o n d s to a theore t i ca l dens i ty , a s s u m i n g four U 0 2 units in the unit ce l l of 10.952 g / c m 3 . This theoret ical value i s c l o s e to the density of 10.950 ± 0.005 recently measured on a single crystal of vapour-grownU02 [2].

2.2. Atomic positions. Temperature factors

The crys ta l s tructure of U 0 2 was f i r s t determined by GOLDSCHMIDT and THOMASSEN [8]. The atomic pos i t ions are those for the f luori te a r -rangement (Fig . 2). Interatomic d i s tances are [9] ,

3

И - 1 2 U = 3 . 8 6 8 Â О - 6 0 = 2 . 7 3 5 Â U - 8 0 = 2 . 3 6 9 Â .

The radius rat io (cation/anion) i s 0.73; according to PAULING [10] 0.73 i s the cr i t ica l ratio, below which the f luorite structure i s l e s s stable than the tetragonal ruti le s tructure.

TABLE I

LATTICE PARAMETER OF U0 2

Parameter Reference

(A)

5.4690 ± 0. 0001 [3]

5.4704 ± 0. 0008 14]

5.4703 ± 0. 0002 [5]

5.4698 i 0. 0008 [6]

5. 4720 ± 0. 0005 [7]

5.4698 ± 0. 0002 [75]

This s t ruc ture has been c o n f i r m e d by neutron d i f fract ion [11] . The coherent nuclear scattering c r o s s - s e c t i o n of oxygen i s about one half of that for uranium; consequently the oxygen atoms can contribute appreciably to the observed neutron intens i t i es and can be located direct ly with neutrons. Moreover, the absorption c r o s s - s e c t i o n of U 0 2 for slow neutrons is at least four o r d e r s of magnitude l e s s than for X - r a y s , so that m o r e accurate measurements of the integrated intensit ies are possible by neutron diffraction.

The neutron m e a s u r e m e n t s g ive the fol lowing va lues for the i so tropic atomic temperature fac tors , В и and B 0 [11]: ,

В ц = 0.25 i 0.04 A 2 , B 0 = 0 .43± 0 . 0 5 Â 2 .

The В ' s are the quant i t ies appearing in the Debye - Wal l er fac tor e x p ( - 2 W ) = e x p { - 2 B ( s i n 0 / X ) 2 } where 0 i s the Bragg angle and X the wave length. The "R factor", giving the discrepancy between calculated and ob-s e r v e d s tructure f a c t o r s , i s l e s s than 2%. F r o m W i l l i s ' s study [11] it i s s e e n that, to a v e r y c l o s e approx imat ion , the s t ruc ture of U 0 2 at r o o m temperature i s described by the fluorite arrangement with isotropic thermal motion of the uranium and oxygen atoms.

The temperature factor В i s re la ted to the m e a n - s q u a r e d isp lacement ТГ| of the a tom in any d i r e c t i o n f r o m i t s m e a n pos i t ion by the equat ion

В = 8тг2 XjJ .

4

Fig . 2

Uni t c e l l of u r a n i u m d i o x i d e . O p e n c i r c l e s a re o x y g e n , shaded c i r c l e s a r e u r a n i u m , and o p e n squares a r e ho le s in t h e f l u o r i t e s t r u c t u r e .

Broken l i n e s c o n n e c t o x y g e n a t o m s wh ich form a c u b i c a r ray a round t h e c e n t r a l h o l e .

(The total mean-square displacement i s 3 U^. ) Inserting the values quoted above gives r m s displacements

J U^ = 0.056 Â for uranium

= 0.074 Â for oxygen.

3. U 0 2 (HIGH TEMPERATURE)

3.1. Variation of lattice parameter with temperature

BAKER andBALDOCK[12]have observed a smooth variation of latt ice pa-rameter with temperature up to 2300°C, the highest temperature of measure-ment . Table II g i v e s the l inear expansion c o e f f i c i e n t deduced by s e v e r a l

TABLE 11

LINEAR EXPANSION COEFFICIENT (X-RAYS) OF UO2

T e m p e r a t u r e range

C O

Expansion c o e f f i c i e n t Reference

20 - 20 0 0 9 . 4 X 10"6 / ° C [12]

20 - 950 1 0 . 8 X 1 0 - 6 / o C [4]

20 - 800 9. 9 X 10"6 / ° C [7]

20 - 1000 10. 5 X Ю - 6 / ° C [13]

5

authors f r o m m e a s u r e m e n t s of the la t t i ce p a r a m e t e r in the t e m p e r a t u r e range indicated. The X-ray expansion coefficient i s in reasonable agreement with the expansion coefficient obtained by dilatometry [9].

3.2. Temperature factors. Characteristic temperatures. Breakdown of harmonic approximation

It i s poss ib le to re late the temperature factors В at different tempera-tures with the c h a r a c t e r i s t i c t e m p e r a t u r e s of v ibrat ion of the so l id . The D e b y e m o d e l can be used for r e p r e s e n t i n g the v ibra t ions of the u r a n i u m atoms and the E i n s t e i n model for the l ighter oxygen a toms . The r e s u l t s , d i s c u s s e d in Sect ion III.2. (Lattice dynamics ) are

0 Debye = 182°K

в = 542°K. Einstein

A s the t emperature of UO2 i n c r e a s e s , the i n t e n s i t i e s of the neutron ref lect ions depart progress ive ly from those calculated for the fluorite model as suming i so tropic thermal motion of the individual a toms . The e f fec t i s shown in the d i f ferences in in tens i t i es of re f lec t ions occurring at the s a m e Bragg angle [14]. These d i f ferences can be accounted for in a phénoméno-logie al manner by displacing the oxygen atoms along the four (111) directions towards the adjacent holes in the structure (Fig .3) .

This d i sp lacement can be interpreted in one of two ways : e i ther the oxygen atoms are displaced at random along the < 1 1 1 ) d irect ions to give a disordered structure, or the oxygen atoms undergo anisotropic vibrations a c r o s s the ideal f luor i te pos i t ion (Fig. 4). The genera l o c c u r r e n c e of the e f fect in U 0 2 and T h 0 2 [14] and in CaF2 [15], and i ts moderate dependence on temperature, favour the second interpretation. MARADUDIN and FLINN [16] have shown that the anisotropic vibration of atoms in positions with cubic point s y m m e t r y a r i s e s from anharmonic contributions to the Debye-Waller factor; it i s c lear that the harmonic approximation cannot be used to inter-pret the temperature factors of U 0 2 at high temperature.

The neutron diffraction resu l t s for U 0 2 can be summarized as follows: (i) The uranium atoms occupy f luori te- type posit ions and execute i s o -

tropic thermal motion with a mean-square amplitude increasing with tempera-ture . At low t e m p e r a t u r e s z e r o - p o i n t mot ion l e a d s to a departure f r o m l ineari ty , but above 400°C Ü | i s proportional to the absolute temperature T and i s given by

Ü f (uranium) = 1.33 10"5 T (Â2).

At the h ighest t e m p e r a t u r e of o b s e r v a t i o n , 1100°C, there i s no ev idence of an i so trop ic v ibrat ion .

(ii) The oxygen a toms occupy fluorite-type pos i t ions but v ibrate i s o -tropical ly at low temperatures only. The shape of the "vibrating surface" can be d e s c r i b e d roughly in t e r m s of four s p h e r e s (Fig. 3). At r o o m temperature t h e s e s p h e r e s are s u p e r i m p o s e d , but at 100°C they begin to separate along the four <(111 > d irect ions joining the oxygen atom with i t s

6

F i g . 3

Vib ra t iona l su r f ace of oxygen a t o m . Above 100°C the oxygen a tom is d isplaced along the four [111] d i rec t ions as shown.

T h e l a rge c ross -ha tched c i r c l e is t he oxygen a tom with the d i sp l acemen t a long [111] shown s t ipp led . The s m a l l shaded c i rc les are uran iums and the open squares a r e holes .

Fig. 4

T w o mode l s for v ibra t ion of oxygen a toms in U 0 2 . Looking down [111 ] : (a) Oxygen a toms loca t ed s t a t i s t i ca l ly a t d i sp laced positions with e a c h a t o m vibra t ing i so t rop i ca l l y ; and (b) A n h a r m o n i c v ib ra t ion causing a s y m m e t r i c a l d i s p l a c e m e n t of e a c h a tom across t h e normal posi t ion.

surrounding te trahedron of h o l e s . At 1000°C e a c h s p h e r e i s d i s p l a c e d by as m u c h as 0.15 Â f r o m the idea l pos i t ion [14] .

4 . U 0 2 + X R E G I O N

4.1. Variation of lattice parameter and density with composition

GRpNVOLD [4] u s i n g high t e m p e r a t u r e X - r a y t e c h n i q u e s h a s s h o w n that oxygen d i s s o l v e s in a homogeneous U0 2 j . x phase above 400°C, t h e U 0 2 + x

domain extending to U 0 2 t 7 at 950°C (Fig. 1). This sec t ion deals with the structure of this U 0 2 + x homogeneous phase. X - r a y m e a s u r e m e n t s on quenched s a m p l e s show that the unit ce l l con-

t r a c t s with i n c r e a s i n g oxygen concentrat ion [3, 6, 17, 18, 19, 20] . T h e r e i s

7

_0 и

Fig . 5

V a r i a t i o n of l a t t i c e p a r a m e t e r wi th U / O r a t i o .

re la t ive ly good agreement between different X - r a y studies of composi t ions up to UO2.12 . but not for higher O/U rat ios (Fig. 5). LYNDS et al. [20] failed to o b s e r v e the U 0 2 + x s tructure between U02 .13 a n d UO2.17 when quenching from 1100°C, whereas BELBEOCH et al. [6] succeeded in quenching t h e U 0 2 + x

phase f rom a lower temperature . This d i screpancy may be due to a m o r e complex phase d iagram than i s genera l ly recognized; Be lbeoch sugges ted that for compos i t i ons higher than U0 2 .13 the s y s t e m m a y b e c o m e diphasic again above 1000°C [6], or that there i s a shift of the phase boundary between monophasic and diphasic regions towards lower O/U ratios (cf. Fig. 1)[6,20],

The quenching experiments may be vitiated by the extremely rapid s e l f -diffusion of oxygen, and it i s n e c e s s a r y to re - inves t iga te the U0 2 . io~U0 2 . 2r r e g i o n above 950°C with h i g h - t e m p e r a t u r e X - r a y techniques . E v e n then the X - r a y method, using s i l i ca capi l lar ies , i s complicated by the react ivi ty of S i 0 2 with uranium oxide at high temperature.

It i s genera l ly agreed that e x c e s s oxygen enters interstit iel l pos i t ions in the U 0 2 l a t t i c e , caus ing a contract ion of the l a t t i c e spac ing , and that the U 4 O 9 s tructure i s an ordered structure. In principle, this model could be ver i f i ed exact ly by comparing the increase in density actually measured with the i n c r e a s e calculated from the interst i t ia l model and the ce l l dimen-s ions . In pract ice , the exact comparison i s difficult for two reasons , name-l y . (a) the U 0 2 + x s t ructure i s not e a s i l y quenched-in, so that s a m p l e s at r o o m temperature tend to be mix tures of U 0 2 and a p o o r l y - c r y s t a l l i n e U4O9, and (b) many U 0 2 preparat ions have d e n s i t i e s much lower than the ca lcu lated va lue of 10.96 g / c m 3 , due to m i c r o s t r u c t u r e faul ts , and t h e s e tend to alter during Oxidation. Furthermore, the densities of powders which have been exposed to air are lower than theoretical due to the chemisorption of oxygen on the surface .

Some of the m o s t comple te s e t s of densi ty m e a s u r e m e n t s reported to date are the fol lowing:

(i) The dens i t i es of samples of composit ion U ( \ 0 2 6 t o U°2.234 which had been prepared by oxidation at t empera tures below 165°C w e r e determined

8

at the Atomic Energy Research Establ ishment (AERE), Harwell . The den-s i ty i n c r e a s e d regu lar ly with compos i t ion f r o m 10.89 to 11.21 g / c m 3 ; the density of an annealed spec imen of U 4 0 9 (U02. 240 ) w a s 11.18 g / c m 3 , giving a d e n s i t y i n c r e a s e of 0 .29 g / c m 3 f r o m U O 2 . 0 2 6 to UO2. 240 » a s c o m p a r e d with a ca lcu la ted va lue of 0 .28 g / с т з [181].

(ii) GRJÖNVOLD [4] m e a s u r e d a regu lar dens i ty i n c r e a s e on s a m p l e s that had been heated to high temperatures and cooled, but these were c e r -tainly mixtures of U 0 2 and U 4 0 9 . The measured density difference between U 0 2 and U 4 O 9 was 0.376 g / c m 3 .

(iii) The dens i t ies of s a m p l e s of U 0 2 + x which had been quenched from 1000°C were measured by LYNDS et al. [20] . According to their X - r a y m e a s u r e m e n t s , the quench w a s s u c c e s s f u l . The d e n s i t y i n c r e a s e d with oxidation, but quantitative compar i son with theory was imposs ib le because of the low dens i ty of the s tart ing m a t e r i a l , ind icat ive of c l o s e d p o r o s i t y .

(iv) Dens i ty m e a s u r e m e n t s on oxidized s a m p l e s of ( U ^ T h ^ ) 0 2 so l id so lut ions , which are s table at r o o m temperature , a lways indicated an in-c r e a s e on oxidation, but quantitative comparison i s agian vitiated by micro-structure faults [72] .

The evidence favours a model in which an 0 2 molecule d isso lves in U 0 2

to g ive a net gain of exact ly two inters t i t ia l О a toms , but the quantitative evidence i s not v e r y good except in the case of U 4 0 9 where d = 11.2 g / c m 3

[ 2 , 4 , 2 0 , 2 3 ] .

4.2. Crystal structure

When U02 i s oxidized to U0 2 + x no extra l ines appear on the X-ray photo-graphs. Examination of s ingle crys ta l s by neutron diffraction confirms that no additional re f lec t ions appear, thus the space group of U 0 2 + x i s the same (Fm3m) as for UO 2. The t e r m "space group" now r e f e r s to the s y m m e t r y properties of the "statist ical cell" obtained by superimposing all the original f l u o r i t e - t y p e c e l l s . Prov ided the extra a toms are d is tr ibuted randomly , the space groups of the U 0 2 unit ce l l and the U02 J .X s tat i s t ica l ce l l must be equivalent. Short range ordering between smal l groups of interst i t ial oxy-gens can occur, but any long range ordering, giving a different space group, i s precluded.

4.2.1. Atomic positions in s tat is t ical cel l . Occupation numbers. Temperature factors

Wi l l i s has studied s ing le c r y s t a l s of c o m p o s i t i o n s U 0 2 1 2 and U 0 2 1 3

at 800CC by neutron diffract ion [21] . Two-dimens ional Fhkk data were co l -l ec ted on U0 2 . 13 and three dimensional F ^ j data on U 0 2 1 2 . The analys is of both s e t s of data gave s i m i l a r re su l t s , but because the U 0 2 1 2 data were m o r e complete only these wil l be d i scussed below.

The general e x p r e s s i o n for the structure factor of the hkl neutron r e -f lec t ion can be written

I =n Fhki = ^ m r b r exp(2TTi) (hxr + kyr + lz r ) exp( -B r s in 2 0 /X 2 ) (1)

r = l

9

where

n i s the number of a toms in the s ta t i s t i ca l ce l l , mr i s the occupation number of the r t h atom, br i s the coherent scat ter ing amplitude of the rth atom.

хгУг2г are the posit ional co -ord inates of the r t h atom, Br i s the i sotopic temperature factor of the rth atom, в i s the Bragg angle, and X i s the wave- length.

The unknowns in this expres s ion are the number of atoms n in the (sta-t i s t i ca l ) ce l l , the posit ional co -ord inates xyz of each atom, their tempera-ture factor B, and the re lat ive proportions m of each atom in the ce l l . All these parameters can be derived by a l e a s t - s q u a r e s treatment in which the calculated and observed F ' s (38 independent hkl ' s in a l l )are matched together.

The l e a s t - s q u a r e s resu l t s are summarized in Table 111. These resu l t s wil l be d i scussed in turn for each kind of atom.

TABLE III

U 0 2 _ 1 2 : LEAST-SQUARES RESULTS FOR STATISTICAL CELL

, Atom C o - o r d i n a t e s in

s ta t i s t i ca l c e l l Cont r ibu t ion

m to fo rmula U O m

Tempera tu re factor ( Á 2 )

X У z

Uran ium

Oxygen О

Oxygen O '

Oxygen O "

Oxygen O ' "

0

u

0. 5

w

0 . 5

0

u

V

w

0 . 5

0

u

V

w

0. 5

m = 1. 87 ± 0. 03

0. 08 ± 0. 04

0 . 1 6 ± 0. 06

- 0 . 02 ± 0. 02

1 . 1 8 ± 0. 02

1 . 4 5 ± 0 . 0 4

1. 8 ± 1 . 4

2. 0 ± 1. 6

2 . 0 ( f ixed)

u = 0 .267 ± 0. 001, v = 0. 38 ± 0 . 0 1 , w = 0. 41 ± 0. 01, and the "discrepancy f ac to r " R = 3.5%.

(a) Uranium. The uranium atoms remain fixed at the equivalent positions: 000, All attempts at ref inement with the uraniums displaced from these positions were unsuccessful . The occupation number for uranium was f ixed at 1.00, in accordance with the latt ice parameter and density r e -sults; if this assumption i s wrong the f igures in the third column of Table l l l m u s t be adjusted appropriate ly . The t e m p e r a t u r e factor В = 1 . 1 8 Ä 2 i s s l ight ly higher than that (0.90 Â2) for U 0 2 at the s a m e t e m p e r a t u r e . The d i f f erence can be a s c r i b e d to a random r m s d i s p l a c e m e n t of the uranium a toms f rom the ir normal pos i t ions of about 1 / 1 0 Ä .

10

(b) N o r m a l oxygen. The oxygen atoms occupying f luor i t e - type s i t e s are labelled О in Table III. As in the case of U0 2 at 800°C, there i s an apparent d isplacement from t o ^ + ô - 5 + 6 ^ + 6 . . . where 6 = 0.017, and this d i s p l a c e m e n t can be a s c r i b e d to anharmonic v ibrat ion , which c a u s e s the oxygens to vibrate asymmetr ica l ly across the normal position. The magni-tude of this ef fect i s s l ightly greater than in U 0 2 .

The m o s t important change in О c o n c e r n s i t s occupat ion number . Whereas all the . . . . pos i t ions are f i l led in U 0 2 00 an appreciable pro-portion are empty in U 0 2 1 2 . Unfortunately, the standard deviations of the occupation numbers in Table III are high, but the proportion of empty s i t e s probably l i e s between f ive and eight per cent.

(c) Interst i t ia l oxygen. The m o s t natural place to accommodate the extra oxygens i s at the h o l e s Щ . . . . Oxygen at t h e s e s i t e s are l abe l l ed O" 1

in Table III. However, the occupation number for O'" re f ines at a negative va lue c l o s e to z e r o , and t h i s , together with the d i f f icul ty of re f in ing the temperature factor (kept fixed in Table III), discounts the possibility of these positions being occupied.

In actual fact the ex tra oxygens occupy two kinds of s i t e , l a b e l l e d O' and O " in Table III. T h e s e are shown in the s t a t i s t i c a l c e l l in F i g . 6; the O1 s i t e s are approximate ly half way f r o m the hole to the centre of the l ine joining two normal oxygen a toms , and the O " s i t e s half way f rom the hole to the n e a r e s t n o r m a l oxygen . The d i s p l a c e m e n t of the i n t e r s t i t i a l a t o m s f r o m the n e a r e s t hole i s roughly 1 Â.

Fig . 6

S t a t i s t i c a l c e l l of U 0 2 + x showing i n t e r s t i t i a l o x y g e n a t o m s i n :

(a) 0 ' s i tes ; (b) 0 " s i tes .

O p e n c i r c l e s a r e oxygens in f l u o r i t e - t y p e s i tes , sol id c i r c l e s a r e i n t e r s t i t i a l oxygens , and o p e n squares a r e ho les a t • • •

In U 0 2 + x , therefore , there are three types of defect in the oxygen sub-la t t i ce a s s o c i a t e d with the departure f r o m s t o i c h i o m e t r y , namely , in t er -stit ial oxygens át О1 s i t e s and O" s i t e s , and normal oxygen vacancies. These three defects cannot be distributed at random, as this inevitably brings s e -veral oxygen atoms too c lose together, and it i s more reasonable to suppose that the defects assoc iate together into complexes or zones.

11

4.2.2 . Interpretation of resu l t s for stat is t ical ce l l

It i s tempting to interpret the re su l t s in Table III in t e r m s of the local atomic configuration in U 0 2 i X . The number of s tructures compatible with this Table i s unl imited, but can be reduced to a manageable number if we a s s u m e (a) the formation of defect zones rather than i so lated de fec t s , and (b) that each zone contains re la t ive ly few de fec t s .

The rat ios of the concentrat ions

normal О vacanc i e s : O' atoms : O" atoms

are given in Table III as 13 : 8 : 16. The model to be d i scussed f irs t i s one in which these rat ios are ideal ized as 2 : 1 : 2. Ideal rat ios suggest that an in t er s t i t i a l atom e n t e r s the la t t i ce at an O' s i t e , and thereby c a u s e s two normal oxygens to be ejected to two O" s i t e s . This situation s e e m s reason-able, as an O1 atom can be comfortably placed inside a U O i 0 0 lat t ice , pro-vided the two neares t oxygens at 1.7 A are removed (Fig. 7).

**

О

о

О О О

Fig . 7

0 ' a t o m at A e j e c t s t he two n e a r e s t oxygens at В, C .

В, С in turn a r e d i s p l a c e d a l o n g [111] towards t h e a d j a c e n t 0 " s i t e s . • - u r a n i u m a t Z = J a 0 ; • - h o l e a t Z = J a 0 ; O - n o r m a l o x y g e n a t Z = 0 , £ a 0 ; ® - 0 ' oxygen a t Z = i a .

A p o s s i b l e 2 : 1: 2 s t ruc ture b a s e d on this concept i s shown in F i g . 8. The dotted l i n e s in this d iagram outline four cubes of s ide i a 0 , and these are l abe l l ed f r o m the top l e f t -hand c o r n e r as I, II, I l landlV. In the U O 2 . 0 0

arrangement , oxygen a toms are loca ted at the c o r n e r s of each of the ^a0

cubes . In UO 2 + y an extra oxygen atom enters the lattice at the position marked E , where E i s approximate ly 1 Â along the [110] d irect ion f r o m the centre of cube II. The two oxygens at A and В are displaced along the d irec t ion [111] to pos i t ions C, D on the body diagonals of cubes I, IV r e s p e c t i v e l y .

12

Using the p a r a m e t e r s quoted in Table III the pos i t ions of the atoms in F i g . 8 are as fo l lows . The c o - o r d i n a t e s are g iven in units of a g, and the uranium atom G i s c h o s e n as the or ig in of c o - o r d i n a t e s .

U at 0 . 5 0 0 0 . 5 0

0 - 0 . 5 0 0 . 5 0

0 . 5 0 - 0 . 5 0 1 .00

0 0 1 . 0 0

0 . 5 0 - 0 . 5 0 0

0 0 0

O a t - 0 . 2 5 - 0 . 2 5 0 . 2 5

0 . 2 5 0 . 2 5 0 . 2 5

0 . 2 5 0 . 2 5 0 . 7 5

etc .

O1 at 0. 12 - 0 . 12 0. 50

O" at 0 . 0 9 - 0 . 4 1 0 . 9 1

0 . 4 1 - 0 . 0 9 0 . 0 9

The bond dis tances are:

O1 - 2U = (A). 2. 18

O' - 20" = 2. 75

O1 - 4 0 = 2. 54

O" - 3U = 2. 35

0 " - O' = 2. 75

o " - ЗО = 2. 24

0" - 3 0 = 2. 78

A di f f icu l ty with the 2 : 1 : 2 mode l i s that, although the r e l a t i v e con-centrat ions of defects are in approximate agreement with observat ion, the abso lute concentra t ions are pred ic t ed i n c o r r e c t l y . F o r U 0 2 1 2 the p r e -dicted formula i s UOj 76 O' 0.12 O" 0. 24. whereas the observed formula (Table III) i s UOx 87 О' о 08 O" 0. i6- The standard deviations of the occupation num-b e r s are high ( these standard deviat ions are ca lcu lated f r o m the i n v e r s e l e a s t - s q u a r e m a t r i x and r e f l e c t the d i s a g r e e m e n t b e t w e e n o b s e r v e d and calculated intensi t ies) , but this s t i l l fai ls to account for the discrepancy be-tween 1.76 and 1.87 for the number of normal oxygens in the formula unit. The di f f icul ty may be r e s o l v e d as indicated in F i g . 8. An extra O1 a tom can be inser ted in cube III at the posit ion F , where E and F are equivalent

13

U l i ,

Fig . 8

M o d e l for U 0 2 + x s t r u c t u r e . T h e n o r m a l oxygens , a t A and В i n UO 2 , 0 0 , a r e r e p l a c e d by i n t e r s t i t i a l a t o m s 0 " a t C , D and 0* a t o m s a t E (2 : 1 : 2 s t r uc tu r e ) ,

or a t E and F (2 : 2 : 2 s t ruc tu re ) . « Uran ium a t o m s . О N o r m a l o x y g e n a t o m s . © In t e r s t i t i a l o x y g e n a t o m s 0*. ® In t e r s t i t i a l o x y g e n a t o m s 0 " .

s i t e s related by 180° rotation about the line AB. This leads to an alternative model for U 0 2 + x , in which the numbers of O1 and O" atoms and normal oxy-gen v a c a n c i e s in the centra l de fec t c o m p l e x are' 2, 2, 2, F o r th i s 2 : 2 : 2 structure, the predicted formula , UOi. 88 O' 0.12 O" 0.12 i s in good agreement with the observed formula , U O 1 - 8 7 ± 0 . 0 3 O'0 0 8 ± 0 0 4 O " 0 1 6 ± 0 0 6-The bond lengths in the 2 : 2 : 2 s tructure are:

O' - 2U = (A)

2. 18

O' - o* = 2. 01

O1 - 2 0 " = 2. 75

O1 - 4 0 = 2. 54

O" - 3U = 2. 35

O" - 20 ' = 2. 75

O" - 3 0 = 2 .24

O" - ЗО = 2. 78

14

The О' - О' distance (E-F F ig . 8) i s short (2.01 Â), but a change of only one standard deviation in the v parameter (Table III) of O' changes this distances to 2 .16Â.

5. U 4 0 9

5.1. Variation of lattice parameter with composition and temperature

The homogeneity range of U 4 0 9 i s not wel l known. PERIC [3] and GRpNVOLD [4] cons idered that the compos i t ion i s c l o s e to U 0 2 2 5 f rom 20 to 600°C and that there i s a s m a l l spread on the oxygen def ic ient s i d e for higher temperatures . On the other hand, SCHANER [18] found a wide range of hypostoichiometry; UO2.22 a t 2 0 ° c a n d U 0 2 2 0 a t 900°C.

In the vicinity of U 0 2 25 two types of diffraction pattern, with or without super la t t i ce l i n e s , have been o b s e r v e d [6]. It appears that there are two s t r u c t u r e s ; namely , " U 4 0 9 ± y " , with super la t t i ce l i n e s and extra oxygens ordered on a 4a 0 cubic cel l , and " U 0 2 25-y with a unit cel l of about 5.445 A A homogene i ty range of 0.03 in O/U was o b s e r v e d in the U^Og^y s tructure by BELBEOCH et al. [6] . The U 0 2 . 25-y s tructure i s d i sordered, or i s or -dered in a d i f ferent way than in U 4 0 9 , without s ign i f i cant d i s p l a c e m e n t s of the uran ium a t o m s . It i s d i f f icu l t to deduce the c o m p o s i t i o n f r o m the a 0 value in the U02.25-yregion, because of the dependence of this parameter on both composit ion and quenching temperature.

The thermal expansion of U 4 0 9 has been studied by GR0NVOLD [4] and by FERGUSON and STREET [22] . The a g r e e m e n t be tween t h e s e r e s u l t s i s exce l lent . There i s a smal l contraction of the latt ice parameter between 20 and 100°C, and thereaf ter the lat t ice parameter expands uniformly with te inper ature.

5.2. Structure

5.2 .1 . X - r a y studies

For U 4 0 9 ± y superlattice l ines have been observed on X-ray [4, 23] , e lec-tron [24, 183] and neutron d i f fract ion patterns [25, 11] . S ince the s u p e r -lattice l ines are prominent at high angles on X-ray photographs, the uranium atoms must be displaced from the positions they normally occupy in the fluo-rite structure.

X - r a y s tudies of s ingle c r y s t a l s , obtained by oxidizing UO2 with U3O8 under the equi l ibrium p r e s s u r e of the mixture U 4 0 9 + U 0 2 6 [23], showed that the unit ce l l i s cubic with a = 4ao = 21.77 A. The observed extinctions for the hkl r e f l e c t i o n s indicate the l ' ÏSd s p a c e group. A mode l has been proposed by BELBEOCH, PIEKARSKI and PERIO [23]. The 64 interst i t ia l oxygen atoms are ordered within the large 4a 0 ce l l which contains 832 atoms. The ideal positions are given in Table IV.

The ex tra oxygens in th i s idea l a r r a n g e m e n t are d is tr ibuted v e r y heterogeneously (Fig. 9). There are twelve tetrahedra, each with four extra oxygens as n e a r e s t ne ighbours , and the r e m a i n i n g s i x t e e n ex tra o x y g e n s ' are equidistant f rom three of the twelve te trahedra . The complete de ter -mination of the structure would require the knowledge of 49 positional para-

15

TABLE IV

I D E A L POSITIONS O F T H E 832 A T O M S

Natu re of t h e 8 3 2 a t o m s

4 8 ( e ) pos i t ions 1 6 ( c ) 12 ( a ) and 12 ( b ) pos i t ions

( s p e c i a l pos i t ions )

24 (d) pos i t ions

(x У z) X X 0 ¿

0 . 1 8 7 5 0. 0 6 2 5 0. 0 6 2 5

0 . 4 3 7 5 0. 0 6 2 5 0. 0 6 2 5

2 5 6 U a t o m s 0 . 3 1 2 5

0 . 4 3 7 5

0. 3 1 2 5

0. 1875

0 . 1 8 7 5

0. 0 6 2 5

0. 0 6 2 5

0 . 1 8 7 5

0 . 1 8 7 5

0. 1 8 7 5

0. 125 0. 25 0 0 0 4

0. 125 0 . 1 2 5 0 0 . 1 2 5 0 4

0 . 1 2 5 0 0 0 î 8 0 i 0 . 2 5 0 4

0 . 1 2 5 0 . 1 2 5 0. 25 0. 1 2 5

5 1 2 О a t o m s 0. 125

0 . 3 7 5

0. 5

0 . 3 5 7

0. 25

0

0. 125

0. 125

0. 25

0 . 1 2 5

0 . 1 2 5

0 . 1 2 5

i 0 4

64 e x t r a - o x y g e n s 0 . 1 8 7 5 - 0 . 0 6 2 5 0. 0 6 2 5 0. 0 6 2 5

(С)

m

(Ь)

y y y

Fig . S

« 002 * 003

( d )

( a ) U 0 2 u n i t c e l l ( t h e o r i g i n is a t a n o r m a l o x y g e n a t o m ) , (b) T h e 4 i n t e r s t i t i a l s i tes a r e de s c r i be d w i t h t h e r e p e t i t i o n - l a w .

(c ) E q u i v a l e n t d r a w i n g t o ( b ) . A t o m s on t h e s a m e Z l e v e l l i e on t h e s a m e d i a g o n a l .

• I n t e r s t i t i a l s i t e О N o r m a l o x y g e n a t o m X U r a n i u m a t o m

(d) D i s t r i bu t ion of t h e 64 e x t r a o x y g e n a t o m s a t t h e i d e a l pos i t ions 4 8 ( e ) a n d 1 6 ( c ) .

m e t e r s which de termine the d i s p l a c e m e n t s of 808 a toms f r o m the ir ideal p o s i t i o n s .

For иОг.гб-у no superlattice l ines have been detected. When the compo-s i t ion U02 .25 i s reached short range order may take place among the inter-s t i t ia l oxygen a toms [31] . T h e s e ordered zones would permi t one to d i s -t inguish between the UO2.25-J phase and the U 0 2 + x phase .

From symmetry considerations based on 4a 0 ce l l s with success ive occu-pancy of the four equivalent interst i t ia l s i t e s s e v e n types of zones are pos-s ible [31] . They belong to the following space groups (one of them i s I "33d, descr ibed for U 4 0 9 ) :

Cubic Tetragonal Onthorhombic

I 43d 1 4 Cmc

I 2 г 3 I 4 2d C222

C 2 2 2 1

5.2.2 . Neutron s tudies

Neutron diffraction intensity measurements have been made on two U 4 0 9

c r y s t a l s [26]. F o r both c r y s t a l s only fundamental ( i . e . non- super la t t i ce ) r e f l e c t i o n s were m e a s u r e d . The re s t r i c t i on to fundamental r e f l e c t i o n s m e a n s that the r e s u l t s apply to the "composi te cel l" obtained by s u p e r i m -pos ing the 64 U 0 2 - t y p e s u b - c e l l s in the 2 1 . 8 Ä unit c e l l . S imi lar r e s u l t s w e r e obtained by analys ing both s e t s of data.

17

TADLE V

U 4 Og : L E A S T - S Q U A R E S R E S U L T S FOR COMPOSITE C E L L

A t o m C o - o r d i n a t e s in c o m p o s i t e c e l l

C o n t r i b u t i o n m to f o r m u l a U O m

T e m p e r a t u r e f ac to r

( A 2 )

X У z

U r a n i u m 0 0 0 - 0. 56 ± 0. 04

Oxygen 0 0. 25 0. 25 0. 25 1. 77 i 0 . 0 2 1. 57 i 0. 08

Oxygen 0 ' 0. 5 V V 0 . 2 9 ± 0 . 0 5 1. 25 ± 0. 70

Oxygen O " w w w 0. 19 è 0 . 0 4 1. 60 ± 0. 90

V = 0 . 3 7 2 ± 0. 005, w = 0. 378 ± 0. 005, R = 3 . 7 % .

The de ta i l s of the a n a l y s i s are d e s c r i b e d by ROUSE, VALENTINE and WILLIS [26] and the r e s u l t s are s u m m a r i z e d below in Table V. The oxygen a t o m s in the c o m p o s i t e c e l l o c c u p y e x a c t l y the s a m e k inds of p o s i t i o n a s the o x y g e n s in the s t a t i s t i c a l c e l l of U 0 2 + x (Table III).

T h e f o l l o w i n g two po in t s can be m a d e c o n c e r n i n g T a b l e V: (a) The 2 : 2 : 2 s t r u c t u r e a s s u m e d for the c o m p o s i t e c e l l g i v e s a for -

m u l a U 0 1 7 5 O'o 25 O " 0 25 , i n a p p r o x i m a t e a g r e e m e n t wi th U 0 1 7 7 О' 0 2g O"o ig g iven by the third co lumn. Thus the t r a n s i t i o n f r o m U 0 2 + x t o U 4 O g

probably i n v o l v e s l o n g - r a n g e o r d e r i n g of the o x y g e n z o n e s or c o m p l e x e s , wi th the c o n f i g u r a t i o n wi th in e a c h z o n e r e m a i n i n g unchanged . It a p p e a r s that z o n e s of the U 4 O g s t r u c t u r e s are a lready p r e s e n t in the d i s o r d e r e d U 0 2 + x phase .

(b) The data w e r e r e c o r d e d at r o o m t e m p e r a t u r e and there i s no e v i -d e n c e of a n h a r m o n i c contr ibut ions to the D e b y e - W a l l e r f a c t o r s . T h i s i s s i m i l a r to the behaviour of U 0 2 at r o o m t e m p e r a t u r e , but the t e m p e r a t u r e f a c t o r s of both u r a n i u m and o x y g e n a r e h igher than in U 0 2 . E a c h s i t e in the c o m p o s i t e c e l l i s occupied by 64 a t o m s , e a c h of which m u s t be s l ight ly d i sp laced in d i f ferent d irec t ions from the ideal posit ion. Since the t empera -ture fac tors are re la ted to the mean-square d isplacement of the atoms from their average pos i t ions , it i s not surpr i s ing that they are higher in U 4 0 9 .

A s a f i r s t approx imat ion WILLIS [26] a s s u m e d that the uranium a t o m s a r e not d i s p l a c e d f r o m the f l u o r i t e - t y p e p o s i t i o n s , a l though the e x i s t e n c e of super lat t i ce l i n e s in X - r a y photographs indicates that there must be s m a l l uranium d i s p l a c e m e n t s . It i s l ike ly that d i sp lacements of both uranium and oxygen a toms occur and that the magnitude of these d i s p l a c e m e n t s can only be found by the interpretat ion of neutron super la t t i ce data.

Both X - r a y and neutron s tud ie s a g r e e that the unit c e l l i s 4 a 0 and the s p a c e group i s I ' Î S d .

18

6. TETRAGONAL PHASES

A number of tetragonal phases have been descr ibed in the range U 0 2 3 to U 0 2 4 [1].

The a - U 3 0 7 phase o c c u r s in the e a r l y s t a g e s of oxidat ion of U 0 2 at temperatures below 135°C. There i s d isagreement whether c / a i s l e s s than or g r e a t e r than unity; publ ished v a l u e s are 0 .99 [3 , 27, 28] and 1.01 [29] . It i s possible that the intensit ies are reversed, indicating c / a < 1, bv a smal l quantity of U 0 2 s u p e r i m p o s e d on the te tragonal phase with c / a > 1. The O/U ratio of ( Ï - U 3 O 7 i s certainly l e s s than 2.33 and probably l e s s than 2.30.

At 180°C, for O/U > 2 .3 , there i s a t rans i t i on to the 7 a ( o r ß . U 3 0 7 ) phase . The c o n v e r s i o n i s comple te for O/U = 2 .33 . The y i phase ( c / a = 1 .033, с = 5.556 Â) can be p r e s e r v e d indef in i t e ly be low 350°C, but above th i s t e m p e r a t u r e it t r a n s f o r m s to у 2 + U 0 2 6 :

T l - T 2 + U 0 2 - 6 . (2)

The у2 phase probably e x i s t s over a range of compos i t i ons , with c / a varying from 1.017 at 350°C to 1.010 at 650°C. According to Eq. (2), y, has a higher oxygen content than 7 2 . The probable composi t ions are:

U 0 2 33 for уг

UO2 3 0 for у2 at 350°C.

U O < 2 . 3 0 for y 2 at 650°C.

Above 600°C react ion (3) occurs:

7 2 - * U 4 0 9 + U 0 2 6 . (3)

The r e v e r s i b i l i t y of reac t ions (2) and (3) has not been conf irmed, and the thermodynamic stabil ity of the tetragonal phases has not yet been deter-mined; it i s not even known whether these phases are stable or metastable . There i s good agreement concerning the y, and y2 phases in recent studies [22, 27] on the b a s i s of X - r a y diffraction s tudies .

WESTRUM and GRÇiNVOLD [28] have reported the existence of an oxide U 0 2 . 3 7 having a s tructure of l ower than te tragonal s y m m e t r y , w h e r e a s BELBEOCH et a l . [29] have interpreted the c h a r a c t e r i s t i c f e a t u r e s of the d i f fract ion l i n e s as due to a s tra ined tetragonal s t ruc ture .

A brief s u m m a r y of the var ious observat ions on the tetragonal oxides i s g iven in Table VI, u s ing the p s e u d o - f l u o r i t e c e l l a s the b a s i s for d e s c r i p t i o n .

The tetragonal phases are ordered s tructures based on the f luorite ar-rangement . ANDRESEN [25] has pointed out the s i m i l a r i t y in the neutron diffract ion patterns of U 4 O 9 and a - U 3 0 7 . The neutron diffract ion pattern of y i shows super lat t i ce peaks [30] and the X - r a y di f fract ion pattern of y 2

shows n u m e r o u s super la t t i c e l i n e s , but they are too c l o s e to be r e s o l v e d into their components [31]. The true unit c e l l s of the tetragonal phases are

19

TABLE VI

TETRAGONAL OXIDES

C o l l e d g e c / a C o m p o s i t i o n o/u

Stab i l i ty r ange

( U O , ) 5. 470 1 2. 00 -

< x - U 3 0 7 5. 4 6 0. 99 or 1. 0 1 2. 26 <250°C

У1 a = 5. 371 - 5. 384 1. 030 - 1. 033 2 . 3 3 <460"C

Уг a = 5. 394 - 5 . 4 0 8 1. 010 - 1. 017 « 2 . 30 <600°C

at l e a s t as large as that of U 4 0 9 . No structure determination has yet been published of any of the tetragonal phases .

7. CONCLUSIONS

In UGj + x and the ordered phases based on the f luorite structure the ex-c e s s oxygen occupies interst i t ia l positions in the latt ice .

In the U 0 2 + x domain the lat t ice contract ion with i n c r e a s i n g O/U ra t io has been de termined by different w o r k e r s and the a g r e e m e n t i s fa i r ly r e a s o n a b l e .

At l e a s t up to 900°C there i s general agreement on the posit ions of the phase boundaries . It i s only at higher temperatures that the recent X - r a y work [6] i s at var iance with the thermodynamic data.

T h e r e i s a g r e e m e n t between X - r a y s and neutron r e s u l t s c o n c e r n i n g the space group and the 4a0 cubic cell of U 4 0 9 . A model of the U 4O g super-ce l l has been proposed which gives the "ideal positions" of uranium and oxy-gen atoms.

Neutron di f fract ion work has been confined to the phases U 0 2 , U 0 2 + x

and U 4 O 9 . Two main conclusions have emerged from these studies, namely: (a) At room temperature the "vibration surface" of both uranium and oxygen

a toms i s spher i ca l , but at higher t emperatures this s ta tement i s true only for uranium. F o r oxygen the sphere b e c o m e s d i s tor ted at r e l a t i v e l y low t e m p e r a t u r e s (100°C) to a more complex s u r f a c e of the type shown in F i g s . 3 and 4. This distortion ar i se s from anharmonic contributions to the Debye-Wal ler factor.

(b) U 0 2 i s oxidized to U 0 2 + x by the formation of defect zones rather than i s o l a t e d point d e f e c t s . E a c h zone contains oxygen a toms at two d i f ferent kinds of in ters t i t i a l s i t e , label led O' and O". The uranium sub- la t t i ce i s unaffected by the presence of these zones , but vacanc ies appear in the sub-latt ice of normal ( f luorite-type) oxygens. The most l ikely configuration of atoms in the zone i s descr ibed by the 2 : 2 : 2 structure (Fig. 8) with two each of normal oxygen vacanc ies , interstit ial O1 and interst i t ial O" atoms. This zone p e r s i s t s in the o r d e r e d U 4 0 9 s t r u c t u r e , and the t r a n s i t i o n to U 4 0 9

20

involves s imply an ordered linking together of these zones . Further specu-la t ion about the U 4 O 9 s t r u c t u r e m u s t await the in terpre ta t ion of neutron s u p e r l a t t i c e data.

Major problems, requiring further investigation by diffraction methods, are:

(i) Determinat ion of the homogene i ty range of U 4 0 9 , par t i cu lar ly at low t e m p e r a t u r e s , and the inves t igat ion of U 0 2 25-y s t ruc tures ;

(ii) Examination of U 0 2 + x above 900°C to r e s o l v e the doubts r a i s e d by BELBEOCH et al. [6] about the val idity of the phase d iagram shown in F i g . 1.

(i i i) Determinat ion of the crys ta l s t ructures of al l the ordered phases with large unit c e l l s .

21

III. THERMODYNAMICS

1. HEAT CAPACITY MEASUREMENTS

1. 1. Low temperature heat capacity data

In a report on isoperibol ( isothermal jacket ca lor imeter) heat capacity m e a s u r e m e n t s on U 0 2 by JONES, GORDON, and LONG [32] the ex i s tence of a relatively blunt lambda-type transformation at 28.7°K is indicated. Their suggestion that the thermal anomaly originated in the conversion from anti-f e r r o m a g n e t i c to paramagnet ic order ing has been supported by magne t i c susceptibi l i ty studies [33-36] and by neutron diffraction analys is [37] which have shown that only the l ines c h a r a c t e r i s t i c of the f luor i te s tructure a r e observed at 77°K but that at 4. 2°K additional l i n e s cons i s tent with ant i fer-romagnet ic ordering appear.

In the course of a s e r i e s of invest igat ions on the heat capacity of both s table and m e t a s t a b l e uranium oxidé c o m p o s i t i o n s by WESTRUM et al . [28, 38-40], the presence of a very smal l , 0. 09 ca lor ie per gram formular m a s s °K ( c a l / g f m °K) trans i t ion in 0 - U 3 O 7 at 30. 5°K was revealed . Since this metas tab le f o r m i s prepared by oxidation of UO2 at 50 to 1350C, the poss ib le presence of a res idue of UO2 in the preparation could not entirely be excluded [28, 40]; but fa i lure of the observed trans i t ion to co inc ide in t emperature with that reported by JONES, GORDON and LONG [32] w a s without explanation, s ince sol id solution format ion between the p h a s e s i s not expected. The reported ana lys i s (99. 3% U0 2 , 0. 7% U0 3) for the Jones et al. sample could probably be better interpreted in t e r m s of a U 0 2 + x and U4O9 - y mixture, (see Fig . 1).

F o r t h e s e r e a s o n s and b e c a u s e of the rather unusual shape reported f o r the heat capacity curve in the trans i t ion reg ion [33], fur ther study of the thermal properties on well characterized UO2 was undertaken by WESTRUM and HUNTZICKER [41]. Their measurements were made by precision adia-batic ca lor imetry on two w e l l - c h a r a c t e r i z e d s a m p l e s deviating f r o m stoi -ch iometry by l e s s than ± 0. 1% and contaning only a few parts per mi l l i on of meta l l i c e l e m e n t s . One s a m p l e c o n s i s t e d of U 0 2 c y l i n d e r s p r e s s e d by the Mallinckrodt Chemical Works, and the other was f l a m e - f u s e d mater ia l prepared for these studies by the Uranium Division of the same f irm. These s a m p l e s wi l l be r e f e r r e d to herea f t er as the "cylinder" and "f lame-fused" sample , respec t ive ly .

The heat capacity curves in the transition region are presented in Fig. 10 for both the above s a m p l e s a s we l l a s for that of J o n e s et al . V a l u e s f o r this and other c o m p o s i t i o n s a r e shown in F ig . 11.

It i s immediately evident that the transition temperature for the cylinders i s 30. 4°K (in contrast to the 28. 7°C value of Jones et al. ) while that for the

f l a m e - f u s e d m a t e r i a l m a y be a few tenths of a d e g r e e l ower . M o r e o v e r , the heat capacity r e a c h e s a maximum value in the vicinity of 400 cal /gfm °K or about forty t i m e s the previously reported values .

T h e s e new data now accord with the sugges t ion of WESTRUM and GRONV0LD [28] that the trans i t ion observed in the ir o?~U02e33 sample i s indeed caused by the presence of the U 0 2 phase, and it has been pointed out by BELBEOCH et al. [29] that the p r e s e n c e of U 0 2 may affect the apparent

23

Fig . 10

T h e h e a t c a p a c i t i e s of U 0 2 s a m p l e s i n t h e r eg ion of t h e a n t i f e r r o m a g n e t i c - p a r a m a g n e t i c t r a n s i t i o n .

JONES, GORDON, and LONG [32] - o - o - WESTRUM and HUNTZICKER [41] , c y l i n d e r s ;

- ® - B WESTRUM and HUNTZICKER [ 4 1 ] , flame f u s e d .

c / a ratio, ( see Chapter II). The r e a s o n f o r the intermediate behaviour of the f l a m e - f u s e d mater ia l i s not evident.

Thermal functions based on data taken on the cy l indr ica l s a m p l e s are summarized for selected temperatures in Table VII. The thermal properties of the transi t ion are summarized in Table VIII. The thermal behaviour i s a lso consistent with the magnetic data a s considered in the sect ion on mag-netic propert ies .

1. 2. High temperature heat capacity data

New enthalpy increment determinat ions by CONWAY et al. [42] over the range 900 to 2000°C (with respect to 25°C) do not accord wel l with those of K E L L E Y et al . [43] o v e r the c o m m o n range 900 to 1200°K. The heat capacity equation of RAND and KUBASCHEWSKI [44] i s therefore adopted for the present purposes above 298°K and the data of WESTRUM et al. [41] for the va lues below that temperature .

The des i rab i l i ty of obtaining high t e m p e r a t u r e heat c a p a c i t i e s by the adiabatic techniques rather than by enthalpy increment de terminat ions i s obvious. Such data would permit the resolut ion of m any ambiguit ies in the present analys i s .

24

Т, 'К

F i g . 1 1

Heat c a p a c i t y d a t a on s e l e c t e d u r a n i u m ox ides . Sources for a l l t h e d a t a a r e as g i v e n in r e f e r e n c e [40]

e x c e p t tha t t he d a t a on U 0 2 b e l o w 60°K a r e f rom WESTRUM and HUNTZICKER [41] .

TABLE VII

THERMAL PROPERTIES OF URANIUM DIOXIDE (Units: cal, gfm, °K)

T,"K C p S* H - - H Ô - ( G " - H » 0 ) / T

10 0. 064 0 . 0 1 7 0 . 1 3 0. 004

25 1. 836 0 . 4 9 7 9. 77 0 .106

35 2. 367 1. 953 53. 38 0. 428

50 3. 270 2. 938 95. 22 1. 034

100 6. 830 6. 304 348. 3 2. 821

200 12. 32 12. 90 1332. 5 6. 242

350 16. 23 20. 93 3512 10. 900

2 9 8 . 1 5 15 .20 1 8 . 4 1 2696 9. 37

2. LATTICE DYNAMICS OF U 0 2

This section descr ibes the correlation of data on characteristic tempera-tures , derived from neutron diffraction studies, with experimental measure-

25

TABLE VIII

THERMAL PROPERTIES OF THE U 0 2

ANTIFERROMAGNETIC-PARAMAGNETIC TRANSITION

Observer Sample T t "K C p m a x AS t

JONES e t a l . [ 3 3 - 28. 7 ~ 9 0 . 8 7

WESTKUM and HUNTZICKER[41] cyl inders 3 0 . 4 ~ 4 0 0 0. 86*

1 . 1 0 * *

WES TRUM and HUNTZICKER[41] f l a m e - f u s e d 3 0 . 1 ~ 11 0 . 4 7

* Based on resolut ion of t he da t a with a somewhat arbi trary e s t ima t ion of the l a t t i c e con t r i -but ion .

* * Based on the e s t ima t ion of t he l a t t i c e h e a t c a p a c i t y of U 0 2 as given in the l a t t i c e d y n a -m i c s sec t ion of this report .

m e n t s of heat c a p a c i t y . T h e p r e s e n t a n a l y s i s s u p e r s e d e s that g i v e n by WILLIS [14] .

WILLIS 111 ,14] and WILLIS, LAMBE and VALENTINE [45] have p r o -vided data that should be one of the bes t s o u r c e s of informat ion on the cha-r a c t e r i s t i c t e m p e r a t u r e s of s t o i c h i o m e t r i c U 0 2 . S ince t h i s i s a d ia tomic c r y s t a l there m u s t be a separat ion of the frequency distribution into at l eas t two degenerate branches . The m a s s ratio between the a t o m s i s 15:1; hence it i s a s s u m e d that the v ibrat ions of the oxygen ions can be represented by an E i n s t e i n op t i ca l branch and the v i b r a t i o n s of the uran ium i o n s by a D e b y e accoust i c branch. While such a complete separation i s , at the present t ime, unsupported by theoret ica l analys i s , it has led to usefu l va lues for characte-r i s t i c t em pera tures .

T h e quantity B D , to be a s s o c i a t e d wi th the t e m p e r a t u r e weaken ing of the neutron d i f f r a c t i o n l i n e s f r o m a m o n a t o m i c cubic l a t t i c e wi th a D e b y e f r e q u e n c y d i s t r ibut ion , i s g iven by WEINSTOCK [46] in the equation:

- X w h e r e x = 0 D / T , m i s the m a s s of the s c a t t e r i n g c e n t r e and ф(х) = ( l / x ) / ( ß / e ß - l ) o ß .

The corresponding quantity BE for an Einste in model with a s ingle characte -r i s t i c f requency can be der ived f rom Weinstock1 s paper a s the equation:

_ h j l + e x p ( - e F / T ) E mkeE 1 - exp(- e E / T ) ' w

w h e r e E q s . (4) and (5) r e l a t e the c h a r a c t e r i s t i c t e m p e r a t u r e s 0 D , 6e wi th Bd , Bg f o r a m o n a t o m i c lat t ice . -

26

The exper imenta l quantit ies are the t emperature f a c t o r s f o r uranium and oxygen in thé diatomic UO2 latt ice , Bu and Bo respect ive ly , which must be related to B 0 and B E . There i s no theoretical guidance on this point, but r e s u l t s in good agreement with the spec i f i c heat data are obtained by using Eq. (4), with the right-hand side multiplied by 1Д/3, for Вц and Eq. (5), with the right-hand s ide mult ipl ied by 2Д/3, f o r B 0 . The equations to be used , t h e r e f o r e , are:

(6 )

В 2 h i _ l + e x p t - f t / T ) , , а о / З т 0 к е Е 1 - e x p ( - fc / Т ) K '

о If the B1 s are e x p r e s s e d in (A)2, these become

26. 857 /хп , , , Л (8)

and B r

69. 081, f t a n h i f ' (9)

where xd = 0D/T and xE = 0e / T . The range of observat ions i s such that Eqs . (8) and (9) can be rewri t ten a s

and

x e \ 2 _ 69. 081 2 J BqT

1 - 0 . 325 6 9 . O 8 I V 1

B 0 T (11)

in both of which W i l l i s ' s reduced temperature (T1) i s to be used for (T). The resul t s of these calculations are given in Table IX. The consistency

of the va lues of 0D , and 0 , derived independently at different t emperatures over a range exceeding 1000°C, i s striking. The average values for charac-ter is t ic temperatures are

0D = 182°K,

0E = 445°K.

27

TABLE XII

CHARACTERISTIC TEMPERATURES OF U 0 2

T'(°K) BD(A2) V K ) В0(Аг) e ^ K )

293 0 .31 163. 0 0 .49 442.5

492 0 .38 190.3 0. 75 439.3

609 0. 51 182.6 0 .88 446.8

748 0. 61 185.0 1 .05 450 .1

915 0. 79 179. 7 1.26 452.4

1086 0. 85 188. 8 1 .55 443.7

1249 1 .02 184. 8 1. 78 442.6

1348 1 .13 182.4 1 .86 449.5

1455 1. 21 183. 1 2. 09 440.3

Average^ 182. 2 Average = 445. 2

The v a l u e s of B o in Tab le IX have b e e n c a l c u l a t e d by W i l l i s wi th the a s s u m p t i o n that the oxygen a t o m s a r e v ibrat ing i s o t r o p i c a l l y about t h e i r n o r m a l f l u o r i t e p o s i t i o n s . He h a s shown [14], h o w e v e r , that the o x y g e n a t o m s r e l a x in <111> d i r e c t i o n s a s the t e m p e r a t u r e i s r a i s e d ; if t h i s r e -laxat ion i s taken into account , B o at 1000°C i s reduced f r o m 1. 87 to 1. 26. Wi l l i s has not given va lues of B o at each temperature of observat ion ca lcu-lated on the b a s i s of re laxat ion , and so the f ina l va lue of 0g r e c o m m e n d e d h e r e i s the p r e v i o u s va lue mul t ip l i ed b y ^ l . 8 7 / 1 . 26. T h i s f i n a l r e s u l t i s eE = 542°K. 0D r e m a i n s unchanged at 182°K.

These va lues for the character i s t i c t emperatures are now used to calcu-l a t e the l a t t i c e v i b r a t i o n a l contr ibut ion to C v [182] , u s i n g the equat ion

C v = 3R ( 1 2 / :

XD e x - l ) i - d x - 3 x D / ( e D - 1) + 6 R x l e E / ( e E - 1 ) 2 , (12)

and to c o m p a r e the ca l cu la ted C v v e r s u s T c u r v e with the o b s e r v a t i o n s of JONES et al. [32]. The r e s u l t s are shown in F ig . 12. Curve a has been ca l -culated f rom Eq. (12) and curve b shows the portion aris ing from the acousti-ca l branch only; the c i r c l e d points represent the exper imental observat ions . T h e o b s e r v a t i o n s a r e of Cp, but the d i f f e r e n c e b e t w e e n C p and Cv i s not important in th i s t e m p e r a t u r e range. The a g r e e m e n t i s grat i fy ing . Only the acoust ica l branch i s excited at very low temperatures , whereas the prin-c ipa l contr ibut ion c o m e s f r o m the opt ica l branch at h igher t e m p e r a t u r e s ; h e n c e the d i f f i cu l ty in obta ining a g r e e m e n t b e t w e e n D e b y e c h a r a c t e r i s t i c t e m p e r a t u r e s d e r i v e d in o ther w a y s [9].

An interes t ing application of the heat capac i t i e s evaluated f rom Eq. (12) i s the reso lut ion of the s e v e r a l contributions to the t h e r m a l capacity in the

28

Fig. 12

Heat capac i ty versus tero pe r a tme for U 0 2 . Circles denote expe r imen ta l points. Full curve a is c a l cu l a t ed from Eq.(12);

curve b shows ca lcu la ted contribution for acous t ica l branch only.

reg ion of the magnet i c t rans i t ion at 30°K. A s m a y be s e e n f r o m F i g . 13, good accord obtains between the m e a s u r e d heat c a p a c i t i e s of WESTRUM and HUNTZICKER [41] above 30°K and those evaluated f rom the latt ice dy-namica l data. Below the trans i t ion the agreement i s s t i l l adequate. Using the l a t t i ce dynamica l v a l u e s a s an e s t i m a t e of the l a t t i c e entropy be low 45°K, we obtain an entropy of transition of 1 .10 ca l /g fm °K. This i s in better accord with the t h e o r e t i c a l value R In 2 (1. 38 c a l / g f m °K) than the v a l u e s obtained by est imat ing the latt ice contribution in U 0 2 by the usual approxi-mation methods.

It i s des irable to obtain data for est imating the (total) latt ice heat capa-city of the i sos tructural oxides thorium dioxide (diamagnetic) and plutonium dioxide (paramagnetic). Although measurements of the heat capacity of Pu0 2

have been reported recent ly by SANDENAW [47] they are compl ica ted by the r e l e a s e of s e l f - i r r a d i a t i o n s tra in energy.

3. F R E E ENERGY, E N T H A L P Y , AND E N T R O P Y MEASUREMENTS

3. 1. Chemical thermodynamics of the UO2. 00-2.25 region

Although a cons iderab le number of t h e r m o c h e m i c a l m e a s u r e m e n t s on the uranium ox ides has now been made , it w i l l be s e e n below that the in-formation available i s neither suff ic iently complete nor suff ic iently p r e c i s e to provide a real ly re l iab le and accurate s e t of t h e r m o c h e m i c a l propert ies of the s y s t e m . F o r th i s reason , and because of the s h o r t n e s s of t i m e for

29

О 10 20 30 40 50 60 70 80 90 ТСК)

Fig. 13

Heat capac i ty versus t empera tu re for U0 2 near transition at 30°K

the evaluat ions , it w a s decided to b a s e the a s s e s s m e n t of the data on that previously carr ied out by RAND and KUBASCHEWSKI [44], and to res tr ic t it to the region x= 2. 00 to 2. 25 in U0 2 + x .

F o r the partly d i s o r d e r e d U 0 2 + x s ing le phase reg ion at high t e m p e r a -tures , RAND and KUBASCHEWSKI suggested the following values (Table X) for the partial entropies and heats of dissociat ion in the reaction

2[Oluo2+x = (0 2) , (13)

from which values of ДН and AS at other composit ions can of course be inter-polated. The t emperature -dependent t e r m s imply a constant value of the

30

TABLE XII

PARTIAL MOLAR ENTROPIES AND ENTHALPIES FOR THE REACTION 2 [ 0 ] u o = 0 2 [44]

x in U 0 2 + x д н 0 г

Temp/range

CK)

0. 05 60. 5 - 16. 1 log T 73400- 7T 1000-1750

0. 10 67. 3 - 16. 1 log T 7 6 900 - 7T 1000-1750

0 . 1 5 74. 0 - 1 6 . 1 log T 8 0 4 0 0 - 7T 1000-1750

0. 20 80. 8 - 16. 1 log T 82 700- 7T 1290-1750

0. 22 83. 5 - 16. 1 log T 8 3 4 0 0 - 7T 1355-1750

0. 24 86. 1 - 16. 1 log T 83 700 - 7T 1390- 1750

increment in heat capacity of the react ion shown in Eq. (13), ДСР= - 7, inde-pendent of temperature and composition. This i s an est imate because actual measurements of heat capaci t ies have not been made.

To arr ive at such a tabulation it i s usually des irable to start the evalu-ation with e i ther the enthalpies or entropies of format ion which are , how-ever, here available only for the l imiting composit ions at room temperature. The evaluation was there fore based on s e v e r a l s e r i e s of equil ibrium m e a s u r e -ments and their temperature coeff ic ients . Fortunately the resul t s of ARONSON and BELLE [48] (EMF: U02 .0-2.2, 1150-1350°K), of BLACKBURN [49] (ef-fusion method: UO 2.15-2.63. 12 2 0 - 1 4 2 0°K) and of ROBERTS and WALTER [50] (direct p r e s s u r e m e a s u r e m e n t s : UO2.1-2.3, 1325-1700°K) agree very wel l , and the evaluat ions of the part ia l enthalp ies and en trop ie s of d i s s o c i a t i o n were based by Rand and Kubaschewski on these re su l t s in the concentration range U02.05-2.24 .

More recent equilibrium measurements by HAGEMARK [51, 52](thermo-g r a v i m e t r i c : UO 2 . 0 - 2 . 2 5 , 1173-1773°K) and by MARKIN and BONES [53] (EMF: UO2.0-2.19, 800-1300°K) have produced temperature coeff ic ients that lead to part ia l en trop ie s e s s e n t i a l l y in a g r e e m e n t with t h o s e s t i m a t e d by Rand and Kubaschewski .

It w a s t h e r e f o r e decided to accept the part ia l entropy curve proposed by Rand and Kubaschewski and to relate all the recent f ree -energy measure-ments in the region under rev iew to these entropies . In addition to the two reports mentioned in the preceding paragraph measurements have been car-r ied out r ecen t ly by KIUKKOLA [54] (EMF: UO2.014-2.67, 1073-1473°K), by GERDANIAN and DODÉ [55] ( thermogravimetric: UO2.00s-2.15, 1180°K) and by ANTHONY et al. [56] ( C 0 / C 0 2 equilibria: U02 .15-2.6i, 1500-2000°K). The value found for ДН by these las t authors ( - 9 7 kcal) i s a lmost certainly too high; the technique they used was to s t r e a m g a s - m i x t u r e s over a heated sample and analyse the product af ter equilibration. It s e e m s poss ib le that the equil ibrium p r e s s u r e s dif fered f r o m those assumed. To compare con-venient ly the recent r e s u l t s with p r e v i o u s o n e s , the interpolated par t ia l

31

entropies above, together with the temperature-dependent terms , have been used to calculate enthalpy t e r m s correspondingto those in column 3 of TableX, according to

ДНог = AG 0 j + T ASÓ, + 7T - 16. l T l o g T . (14)

If the agreement w e r e perfect the enthalpies so calculated should all l i e on a s ingle l ine when plotted against x for UO2+X.

The actual resu l t s are shown in Fig. 14 together with the curve a s s e s s e d by Rand and Kubaschewski. It may be seen that in the range U O 2 . 0 3 - 2 . 2 0 the agreement of the new resul t s among themse lves and with the a s s e s s e d curve i s v e r y sa t i s fac tory , that i s , within ± 1 kcal . However , s o m e adjustment of the curve in the range U O 2 . 2 0 - 2 . 2 5 , s e e m s to be needed.

It may a l s o be s e e n f r o m F ig . 14 that, with increas ing O / U ratio, the va lues derived f rom the resu l t s obtained at the lower temperatures become

E — 3 80

2.00 2.05 2.Ю 2.15 2.20 2 .25

« . 0 / U RATIO

Fi g . 14

Д H q versus X for U 0 2 + x . O - Mark in and Bones, 1000°K; • - Markin and Bones, 1300°K; 2 Д - K iukko la , 1173°K; A - K i u k k o l a , 1373°K; û - Aukrust , e t a l . , 1473°K;

• - Aukrust , e t a l . , 1673°K; V - G e r d a n i a n and Dodé , 1180°K¡

solid l i n e - Rand and Kubaschewski .

32

consistently higher than those pertaining to higher temperatures of measure-ment. Th i s s e e m s to imply that the heat c a p a c i t i e s u s e d f o r the react ion shown in Eq. (13) are too negat ive for higher va lues of x, while they are of the right order of magnitude at lower va lues of x. ACpthus b e c o m e s l e s s negative a s x i n c r e a s e s . It wi l l be difficult , however , to make reasonably r e l i a b l e e s t i m a t e s , and f o r th i s and other r e a s o n s m e a s u r e m e n t s of t rue heat c a p a c i t i e s in the UO2+X s i n g l e p h a s e r e g i o n a r e v e r y d e s i r a b l e ( s e e Sect ion III. 1. 2).

A l inear extrapolat ion of Rand' s enthalpy and entropy c u r v e s into the region U O 2 . 0 0 - 2 . 0 3 i s not justifiable, either on theoretical or on experimental grounds. The shape of the entropy curve in this reg ion has been d i s c u s s e d theore t i ca l l y on the b a s i s of a s i m p l e a tomic o r d e r - d i s o r d e r m o d e l ( s e e Sect ion III. 4. 1. ). In the ideal c a s e , that i s , if the s to i ch iometr ic compo-sit ion contains a toms in lat t ice pos i t ions only, and no la t t i ce posi t ions are vacant, ASoj should r i s e to infinity. However , the ev idence s u g g e s t s that this ideal c a s e i s not approached even at room temperature, and consequently the partial entropy should pass through a maximum.

O/U Fig . 15

ASo 2 versus x for U 0 2 + x in low x r eg ion .

Exper imenta l m e a s u r e m e n t s in the low O / U region have been carr ied out by HAGEMARK [51] and by MARKIN and BONES [53]. The part ia l molar entropies, Д§о2 , derived from the temperature coeff ic ients are shown in Fig. 15. It may be seen that the two se ts of resul ts disagree below x= 2.01, the maximum in Hagemark' s curve presumably occurring at a much lower x than that in Markin and B o n e s ' s curve. A s the s to ichiometr ic composit ion i s approached, the f r e e energy measurements become increasingly difficult,

33

as i s demonstrated by the shaded area of Hagemark1 s re su l t s indicating the systematic experimental e r r o r s and by the spread of the experimental values reported by Markin and Bones. Never the le s s , the discrepancy between the two s e t s of m e a s u r e m e n t s i s probably rea l . It cannot be explained on the grounds of the apprec iable d i f f erence in the t e m p e r a t u r e of the m e a s u r e -ments because the maximum should shift towards U O 2 . 0 0 0 and become lower a s the temperature d e c r e a s e s .

At th i s s tage , it d o e s not appear p o s s i b l e to r e s o l v e the d i f f e r e n c e s . HAGEMARK [51] made equil ibration exper iment s us ing C 0 / C 0 2 m i x t u r e s at 900 to 1500°C, c h a n g e s in c o m p o s i t i o n be ing m e a s u r e d by m e a n s of a quartz spring microbalance; correc t ions for buoyancy and for vaporization w e r e applied. MARKIN and BONES [53] studied this composit ion range by galvanic ce l l techniques , us ing a coulometr ic t itration technique to vary the composit ion by smal l amounts. The measurements were made from 700 to 1000°C. A standard state, arbitrari ly defined a s U02.ooo<was taken as that compos i t ion which i s in equil ibrium with a 1 0 / 1 mixture of CO and C 0 2 at 850°C. F o u r c o m p o s i t i o n s between U O 2 . 0 1 and U O 2 . 0 S w e r e a l s o analysed

-70-

Ud-, , , .

- 4 - 3 - 2 l o g i

Fig. 16

AGq versus x for U O ; + x in low x reg ion . Mark in and Bones; O - H a g e m a r k .

34

by an abso lu te method invo lv ing d i s s o l u t i o n in ac id and e s t i m a t i o n of U(VI) by polarography; in t h r e e c a s e s out of four , va lues of O / U agreed to within ± 0, 001. It w a s t h e r e f o r e a s s u m e d that the s t o i c h i o m e t r i c s tate w a s known to t h i s a c c u r a c y . M o s t of the m e a s u r e m e n t s w e r e m a d e on two v e r y d i f -f erent m a t e r i a l s : (i) a f lake f r o m a s in tered pel le t , containing 100 to 200ppm F e and o ther i m p u r i t i e s to the 10 to 50 ppm l e v e l , and, (ii) a f l a k e f r o m a f u s e d c r y s t a l of UO2 w h i c h w a s v e r y pure , the h i g h e s t i m p u r i t y r e c o r d e d be ing 10 ppm of Al . The r e s u l t s on t h e s e two s p e c i m e n s w e r e in e x c e l l e n t agreement , and the s a m e r e s u l t s w e r e obtained on oxidation and on reduction.

Actua l ly , part ia l f r e e e n e r g i e s d e r i v e d f r o m the two s e t s of m e a s u r e -ments agree fa i r ly we l l , a s may be s e e n f r o m Fig . 16, the d iscrepancy e x i s -t ing m o s t l y in the t e m p e r a t u r e c o e f f i c i e n t s . They n e c e s s a r i l y a l s o appear in the Д Н о , - х d iagram which wi l l not be reproduced here but can, if required, be constructed f r o m F i g s . 15 and 16. It i s diff icult to suggest a ca lor imetr i c m e t h o d f o r t h i s t e m p e r a t u r e range su i tab le f o r t e s t i n g the equ i l ibr ium r e -s u l t s ; q u a n t i t i e s of t h e o r d e r of 200 c a l would h a v e to be m e a s u r e d wi th good accuracy at re la t ive ly high t emperatures . The h igh-temperature calori-m e t e r developed at the National Phys i ca l Laboratory may be suitable, where-a s a new Calvet c a l o r i m e t e r for high t e m p e r a t u r e s might be m o r e accurate . Moreover , the d i f f e r e n c e s can not be r e s o l v e d by integration of partial thermo-d y n a m i c q u a n t i t i e s and c o m p a r i s o n wi th the a v a i l a b l e i n t e g r a l q u a n t i t i e s , because the d i f f e r e n c e s involved are too s m a l l and within the e r r o r s entering th i s type of evaluat ion. N e v e r t h e l e s s th i s in tegrat ion h a s b e e n c a r r i e d out in o r d e r to have an approx imate check on the ava i lab le informat ion . Us ing the part ial entrop ies of Rand and Kubaschewski at an arbi trary temperature of 1400°K f o r the range UO2.03-2.24 and t h o s e of H a g e m a r k and Markin r e -spect ive ly for the range UO2.00-2.03 , the integral entropies between the l imi t s x= 2. 00 and 2. 25, us ing the equation

X A S = ^ j A S o 2 d x (15)

0

have been obtained a s f o l l o w s : ASi4oo= 19. 35 e. u. / m o l e 0 2 u s i n g H a g e m a r k ' s data, ASi400 = 19, 9 e. u. / m o l e O2 us ing Markin1 s data. T h e entropy change of the r e a c t i o n

2 = 8 + ( 0 2 ) , (16)

obtained f r o m the l o w - t e m p e r a t u r e heat c a p a c i t i e s of JONES, GORDON and LONG [32] f o r U 0 2 and OSBORNE, WESTRUM and LOHR [39] f o r U 4 0 9 i s :

A S 2 9 8 = 32. 7 e . u. / m o l e 0 2 . Us ing the s o m e w h a t doubtful a v e r a g e va lue , ДСР= - 7, d i s c u s s e d above and the entropy change at r o o m t e m p e r a t u r e to c a l c u l a t e the entropy change at 1400°K, one obtains ASuoo = 26. 3 e. u. which i s 6. 95 e. u. and 6. 4. e. u. r e s -p e c t i v e l y h i g h e r than the in tegra ted v a l u e s . The d i f f e r e n c e m a y be due to the entropy of disproportionation of U4O9 into иОг.244 (disordered) andU02.6. Th i s d isproport ionat ion m a y be r e g a r d e d e s s e n t i a l l y a s an o r d e r - d i s o r d e r transformation of U 4 O 9 . An entropy of transformation of 3. 5 or 3. 2 e .u . /mole

35

U 4 O 9 which fol lows from the above vaàues i s commensurate with that of com-parable meta l oxides.

Thus , the entropy v a l u e s f o r the r e a c t i o n shown in Eq. (16) a r e con-sistent , but further calculations should be postponed until experimental heat capacit ies of U 4 O 9 and UO2+X are available.

The enthalpy i n c r e m e n t in Eq. (16) i s not v e r y w e l l known. We m a y accept the enthalpy of formation of UO2 to be ДН29р = - 258. 7± 0. 6kcal /mole as obtained by HUBER and HOLLEY [57], (from the enthalpies of combustion of uranium, U O 2 . 0 1 4 and U O 2 . 0 0 6 extrapolated to the s to ich iometr ic compo-sition together with a recalculated value for U3O8), in preference to the pre-l iminary value of - 259. 9 ± 2 . 3 k c a l / m o l e a s obtained by GERDANIAN, MARUCCO and DODÉ [58] f r o m d irec t m e a s u r e m e n t s in a Calvet ca lor i -m e t e r . No enthalpy of combust ion of U 4 O 9 i s known. If one a s s u m e s that it i s f o r m e d f r o m neighbouring ox ides with no evolution of heat, a value of ДНг98= + 75. 2 kcal r e s u l t s for equation (16). It i s , of course , l ikely to be m o r e pos i t ive . F r o m heats of solution of U O 2 and U 4 O 9 in nitric acid [59], one m a y compute + 8 9 . 6 kcal . The value a s s e s s e d by RAND and KUBA-SCHEWSKI [44], and exac t ly c o n f i r m e d by MARKIN and ROBERTS [60] wi l l be re ta ined , n a m e l y , Д Н 2 9 8 = 84. 0 kca l .

It i s more rewarding to derive the data for equation (16) from free ener-gy m e a s u r e m e n t s ; and in th i s concluding d i s c u s s i o n the U 4 O 9 - U 3 O 8 two-phase r e g i o n m a y be included.

R e s u l t s for the two-phase reg ions tend to be in better agreement than those f o r s i n g l e - p h a s e r e g i o n s , because the compos i t ion d o e s not have to be known accurate ly . RAND and KUBASCHEWSKI [44] tabulated average v a l u e s f o r the ЩОд-у-иОг+х reg ion f r o m 800 to 1350°K and extrapo la ted v a l u e s at 298°K. Later r e s u l t s by MARKIN [53] and by KIUKKOLA [54] are in exce l l ent a g r e e m e n t up to 1373°K.

RAND and KUBASCHEWSKI [44] accepted the r e s u l t s of ROBERTS and WALTER [50] which showed the U4O9 phase decomposing peritect ical ly at 1396 ± 5°K to U O 2 . 6 and U O 2 + X . The only la ter work in which the reg ion c l o s e to U 4 O 9 has been studied i s that of Kiukkola, who interprets his results as showing the U 4 O 9 phase persist ing to 1488°K. However, Kiukkola's results for compos i t ions between UO2.2 and UO2.25 do not a g r e e we l l with those of other authors, e. g. BLACKBURN [49], and resul t s at one temperature only a r e reported f o r the c o m p o s i t i o n s 0 / U = 2. 245 and 2. 248. The resu l t f o r O / U = 2. 245 f a l l s on the l ine prev ious ly a s c r i b e d to 0 / U = 2. 239, w h e r e a s that for 0 / U = 2. 248 actually agrees with previous values for the U 4 O 9 - U O 2 . 6

two-phase region. The previous resu l t s [50] can probably st i l l be accepted. Kiukkola p laces the upper l imit of the U 4 O 9 phase at 0 / U = 2. 250 because

the r e s u l t s at th i s c o m p o s i t i o n and at 0 / U = 2. 300 w e r e ident ica l , and in excel lent agreement with Roberts and Walter, who measured identical pres -s u r e s for the composit ions 0 / U = 2. 250, 2. 257, 2. 279 and 2. 303. The rather scattered resu l t s for this phase limit determined by interpolation of effusion-ce l l measurements [49] give higher values.

T h e r m o d y n a m i c v a l u e s for the U 4 O 9 - U O 2 . 6 r e g i o n g iven by Rand and Kubaschewski a r e at

298-1395°K (U4O9-UO2.6)» AS(02) = 3 5 . 2 e. u. ; ДН(Ог) = 72. 8 kcal ,

36

and at

1395-1750°K (UO2+X-UO2.6)» AS(02) = 36. 7 e. u. ; ДН(0 2 ) = 74. 9 kcal .

Markin's va lues w e r e combined with e a r l i e r va lues [60] to g ive

ДН(С»2) = 78. 6 kcal f rom 873 to 1395°K

and

AS(0 2 )= 39. 2 e . u . ,

which are in exce l lent agreement with K1UKKOLA [54], who f inds

ДН(0 2 ) = 77. 1 kcal f rom 1073 to 1473°K

and

Д§(0 2 ) = 38. 5 e. u . .

At temperatures above 1395°K, the resul ts of Roberts and Walter and earl ier resu l t s combined to give,

ДН(02)= 81. 0 kcal

and

Д§(0 2 ) ='40. 2 e. u. .

Recalculat ion y i e l d s

ДН(Ог) = 79. 8 kcal

and

AS(0 2 ) = 4 0 . 1 e . u . .

The numerical va lues given by Rand and Kubaschewski in this region there-f o r e s e e m low, but al l such va lues w e r e chosen to give the bes t integrated values for ДН and ДБ, and it may be that l inear extrapolation i s not justified f r o m 870 to 298°K. A r e - a s s e s s m e n t of the ent ire range i s d e s i r a b l e but the task could not be a c c o m p l i s h e d by the P a n e l in the t i m e a v a i l a b l e .

The shape of the c u r v e obtained when AG i s plotted against T f o r the two-phase reg ion indicates that the compos i t ion of the phase boundaries i s not changing appreciably, but Roberts and Walter did record a smal l increase in the O / U ratio of the phase l imit of the U O 2 + X phase f r o m 1350 to 1700°K. Further r e s u l t s in exce l l ent a g r e e m e n t have been reported by ANTHONY, KIYOURA and SATA [56] and by AUKRUST, F0RLAND and HAGEMARK [52]. T h e s e r e s u l t s a r e plotted in F i g . 17. The l o w e r l i m i t of the UO2.6 phase Jound by t h e s e authors w a s 0 / U = 2. 605 to 2. 610, in a g r e e m e n t with many

37

RATIO О / U

Fig. 17

Phase d iagram of the U - 0 system for the tempera ture range 1000 to 1700"C. X - Anthony, Koyoura and Sato; A- Blackburn; O - Roberts and Walter; + - Hagemark .

other measurements [60]; it s e e m s theft this phase l imit i s not much affected by changes in temperature from 870 to 2000°K.

3. 2, The phase diagram, UO2 to XJ4O9

The present a s s e s s m e n t of the thermodynamic data has given no reason to doubt the calcualtion of the phase boundaries made by Rand and Kubaschewski. These are shown as ful l l ines in Fig. 18. Phase boundaries deduced by f ive other authors are also shown on this f igure. GR0NVOLD's resul ts [4] were deduced from high-temperature crystallography; ROBERTS and WALTER [5C§ u s e d d irec t p r e s s u r e m e a s u r e m e n t s , and BLACKBURN [49] an e f fus ion ce l l ; MARKIN and BONES's [53] EMF m e a s u r e m e n t s have been d i s c u s s e d above, and ARONSON, RULLI and SCHANER [61] deduced phase boundaries f r o m b r e a k s in e l e c t r i c a l conductivity v e r s u s t e m p e r a t u r e plots . Not a l l the exper imenta l points could be plotted on the f igure; the dotted curve ex-p r e s s e s the mean r e s u l t s fa ir ly wel l , though there i s cons iderable scatter . It should be noted that there must be a point of inflection in the U O 2 + X boundary around 0 / U = 2. 15. Crysta l lographic data which are at var iance with th i s phase diagram above 1000°C have been discussed in Chapter II of this Report and it m a y be noted again that KIUKKOLA [54] shows the U 4 O 9 phase ex -tending to s ign i f i cant ly h igher t e m p e r a t u r e s .

The oxygen deficient boundary of the XJ Og-y phase at high temperatures i s taken from the concurrent resul ts of Blackburn, and of Roberts andWalter.

38

Fig. IS

Phase boundary between U 0 2 + x and UO z + x + U 4 0 , . y . • - Gr^nvold; X - Roberts and Walter; о - Blackburn; Д - Aronson, Rulli , and Schauer;

® - Markin and Bones. Full l ine - Rand and Kubaschewski.

The r e a s o n s for placing the o x y g e n - r i c h boundary of the U4O9 phase at UO2.25, and f o r r e j e c t i n g the h igher v a l u e s obtained by Blackburn, have been g iven in the prev ious sec t ion . The reg ion of the phase d iagram above 1123°C has been plotted in F i g . 17 and h a s a lready been d i s c u s s e d .

3. 3. Hypostoichiometric UO2

The behaviour of UO2 at t e m p e r a t u r e s above 1800°C has been the subject of s o m e confus ion in the l i t e r a t u r e . The format ion of a hypos to i ch iometr i c u r a n i u m o x i d e , a s d e t e c t e d by the p r e s e n c e of f r e e u r a n i u m in UO2 a f t e r c o o l i n g of the s a m p l e , h a s been ques t ioned s i n c e contaminat ion (e . g. wi th carbon) m a y have i n f l u e n c e d the r e s u l t s .

Recent i n v e s t i g a t i o n s on uncontaminated s a m p l e s by ROTHWELL [62] ind ica te that U 0 2 _ x can b e f o r m e d a b o v e 1800°C at s u f f i c i e n t l y low p a r t i a l p r e s s u r e s of o x y g e n p r o v i d e d that the v a p o r i z a t i o n of UO2 i s not t h e p r e -dominant p r o c e s s :

 = + | x ( 0 2 ) .

A f t e r c o o l i n g in an iner t a t m o s p h e r e , i n c l u s i o n s of m e t a l l i c uran ium have been found in the UO2 matr ix . Indirect evidence that th is uranium may have precipi tated f r o m a homogeneous high temperature , UO2-X- phase on cool ing.

39

a r i s e s f r o m the fact that a rhenium w i r e in contact with UO 2_x at this high temperature showed no tendency to me l t and break, which should have oc -curred in the p r e s e n c e of f r e e uranium [63].

AITKEN et al. [63] obtained a l imi t ing value f o r the solubil i ty of U in U O 2 - X by vaporization studies in hydrogen and found 0 / U = 1. 89. (An earl ier ratio of 1. 96 by ANDERSON et al. [64] s e e m s doubtful due to the difficulty of determining the compos i t ion f r o m the lat t ice parameter ) . The mel t ing point of the congruently vaporizing U O 2 - X was found to be 100 to 150° higher than that of the s to ichiometric oxide [65].

M e a s u r e m e n t s of the e l e c t r i c a l conductivity and of the Seebeck c o e f -f ic ient of U O 2 - X at 1800°C [65] and 2050°C are consistent with an n-type metal e x c e s s semi-conductor . The large values of the Seebeck coeff ic ient (- 2500 to 5000 ßV/°С) cannot be attributed sole ly to an electronic contribution f rom the f r e e uranium at room temperature. An increase of thermal conductivity on irradiat ion has been ascr ibed to the format ion of U O 2 - X [66]. B e c a u s e the formation of the иОг-х phase may have appreciable effects upon the trans-port phenomena and perhaps on f i s s i o n gas r e l e a s e , furtheir s tudies of i t s stabi l i ty and proper t i e s a r e des irable .

4. VAPORIZATION PROCESSES

In the vapour phase in equilibrium with the dioxide phase above 2000°C three gaseous oxides , UO, UO2 and UO3 have been observed. The standard f r e e e n e r g i e s of f ormat ion of al l t h e s e ox ides have been de termined with varying degrees of accuracy. The thermodynamic properties of the gaseous dioxide have been ca lculated f r o m vapour p r e s s u r e m e a s u r e m e n t s [75] of the congruent ly subl iming dioxide , the compos i t i on of which i s v e r y near the s t o i c h i o m e t r i c compos i t ion . The r e s u l t i s that

AG? (U0 2 , g) = - 125 400 + 6. 22 T c a l / m o l e . (17)

The gaseous trioxide i s observed under oxidizing conditions and i ts thermo-dynamic propert ies have been obtained [76] f rom a study of the equilibrium.

Í + Í ( 0 2 ) = ( U O 3 ) . (18)

The s t o i c h i o m e t r y of the g a s e o u s m o l e c u l e w a s d e m o n s t r a t e d to be that corresponding to the trioxide. Reasonable es t imates of the absolute entropy of g a s e o u s U 0 3 compared with known v a l u e s of m o n o m e r i c and p o l y m e r i c tungsten trioxide indicate that uranium trioxide i s monomeric. The standard f ree energy of formation i s

AG° (UO3, g) = - 198 500+ 1 9 . 0 T c a l / m o l e . (19)

De MARIE et al. [77] have reported a m a s s spectrometr ic study of vapori-zing c o m p o s i t i o n s y ie ld ing g a s e o u s UO, U 0 2 and U 0 3 . The g a s e o u s mon-oxide w a s f i r s t o b s e r v e d by CHUPKA [78]. B e c a u s e of uncer ta in t i e s in

40

i on i za t iona l c r o s s - s e c t i o n s the p r o p e r t i e s of the m o n o x i d e a r e not a s w e l l known a s t h o s e f o r the d ioxide and the t r i o x i d e .

AGf (UO) = (- 16 800 - 10. 0 T) c a l / m o l e . (20)

1900 °K

Fig. 19

Suggested description of the var ia t ion of to ta l pressure with composi t ion

in the u ran ium-oxygen system be tween uranium and uranium dioxide :

(a) Vapour pressure of pure uranium by Rauh and Thorn;

(b) Effect of oxygen observed by Rauh and Thorn;

(c) System studied by d e Maria e t a l . ; and

(d) Vapour pressure of uranium dioxide measured by Ackermann et a l . ,

(Reproduced by courtesy of R.J. Ackermann , E .G. Rauh and R.J. Thorn [79])

A l l of the u r a n i u m o x i d e s e x c e p t UO2.00 v a p o r i z e i n c o n g r u e n t l y , i . e . the c o m p o s i t i o n of the vapour i s c o n s i d e r a b l y r i c h e r in oxygen than i s the condensed phase . The vapour i s predominantly oxygen with a s m a l l amount of g a s e o u s t r i o x i d e . Hence , any of the o x i d e s with O / U > 2 w i l l u l t imate ly y i e l d UO2.Û0 in vacuo . The s c h e m a t i c phase d i a g r a m (Fig . 19) of p r e s s u r e v e r s u s c o m p o s i t i o n i l l u s t r a t e s the g e n e r a l behaviour [79]. At c o m p o s i t i o n s with a ratio of oxygen to uranium l e s s than two, the vapour i s predominantly g a s e o u s UO. It has been es tabl i shed by absolute effusion m e a s u r e m e n t s [80] and by m a s s - s p e c t r o m e t r i c observat ions [79] that smal l amounts of d i s so lved oxygen at the l o w e r t e m p e r a t u r e s s u p p r e s s the vapour p r e s s u r e of uranium. H e n c e , t h e r e a p p e a r s to be a congruent ly v a p o r i z i n g c o m p o s i t i o n n e a r the uranium r i c h end a s indicated in F ig . 19. The genera l c h a r a c t e r of the dia-g r a m at h igher t e m p e r a t u r e s i s not ye t known. But above about 2100°C the vapour p r e s s u r e of the uran ium contaminated with oxygen e x c e e d s that of

41

pure uranium so that the oxygen contaminant can be removed. At all t e m -peratures the minimum at UO2.00, however, appears to remain . If th is i s true, then the production of UO2-X i s accompl i shed under conditions where there i s a preferent ia l l o s s of oxygen out of the s y s t e m so that the compo-sit ion of the so l id can "ride" up on the indicated boundary.

5. THEORETICAL TREATMENT OF U0 2 + x PHASE

5. 1. Statistical thermodynamics of interstitials and vacancies

The mode l used in th i s sect ion i s e s sent ia l ly that developed by ANDERSON [67]* with the fol lowing exceptions: (a) a dist inct ion i s made between the contributions made to the v ibrat ional partit ion function of the c r y s t a l by an ion in a regular lat t ice posit ion and one in an inters t i t ia l po-sition; and (b) it i s a s s u m e d that the meta l sublatt ice i s f ixed and per fec t and that the defects are in the oxygen sublattice. Further, the calculations have been extended h e r e to the actual evaluation of partial thermodynamic quanti t ies us ing r e c o g n i z e d s o u r c e s for whatever auxi l iary quant i t ies and constants are required.

The relat ionship between the activity, Xo, of an oxygen atom in the gas phase and the (partial) p r e s s u r e of oxygen gas i s

Po2 = XjjkTQ exp(D/RT). (21)

The act ivi ty i s to be equated separate ly to that a s s o c i a t e d with inters t i t ia l oxygen ions and to that as soc ia ted with vacanc ies in the f luori te oxygen po-s i t ions in the condensed phase. Thus the numbers of those defects in equi-l ibrium with the same pressure of oxygen can be found and the corresponding value of x in U O 2 + X found from their di f ferences . Then the partial quantities are found directly from the following equations:

AGot - RT In P 0 j

= 2 RT In X0 + RT ln(kTQ) + D, (22)

-AS Ö ! = 2 R ^ l n X 0 + T ^ ^ / x ^ + R ( / l n Q + T ^ ^ ) + R + R l n k T , (23)

ДНо2= Дбо г + ТДН0 ! . (24)

The semigrand parti t ion function for the de fec t s , f r o m which the two e x p r e s s i o n s f o r the ac t iv i ty in the condensed phase a r e deve loped , i s

-E r ( T , N u , X o H p ^ N u . N t . N J q N ' q ^ eRT . (25)

N f N v

* Anderson 's or ig ina l equat ions a r e n o t log ica l ly cor rec t , s ince d i f f e r en t i a t i on was pe r fo rmed under

t h e incor rec t assumption tha t t h e number of l a t t i c e sites was f i xed .

42

In this equation Nu i s the number of uranium atoms present , N¡ i s the number of inters t i t ia l ions, Nv i s the number of regular oxygen lat t ice va-canc i e s , q¡ and qv are the contributions of an interst i t ia l ion and a regular lattice ion, respect ively , to the crystal partition function and E is the exces s energy introduced into the crystal by the presence of the defects. This energy i s given by

E = N v E v - N 1 E i - EvvNv2/(evNu) - E„ N2/(ÚÍNu), (26)

where E v i s the energy required to remove a regular lattice ion, as an atom, to a posit ion of re s t far f rom the crystal , E¡ i s the s a m e for an interst i t ia l ion, E v v i s related to the energy decrease brought about by the formation of a pair of v a c a n c i e s and correspondingly f o r Ej¡. Fo l lowing Anderson, no account has been taken of e l ec tronic d i sorder . The distr ibut ion function, П, of Eq. (25) i s g iven by

(gj Nu)! (o-yNu): ~ Ni !(a¡Nu - Nj)'. N v i { e v N 0 - N v ) l * '

in which ai i s the number of poss ib le interst i t ia l s i t e s per uranium ion and a y (s 2) i s the poss ib l e number of oxygen vacanc i e s per uranium ion.

Equation (25) i s developed by the usual procedure [67] to give

i \ i 1 - 0y Ev , 2 6V Ew . /nn\ l n X o = l n - g RT RT 4v* * '

and

i n T E¡ 2 0i Eii mn\ l n X 0 = l n J T g ~ ~ i y r ~ RT " l n q i ' ( 2 9 )

in which the convenient quantities

e v = Nv/(o„Nu)

and

6i = N¡/(eiNu)

have been introduced. The evaluation of p a r a m e t e r s such a s Ev , E i , e t c . , that i s n e c e s s a r y

before computations can be carr ied out, was made in the fol lowing manner. F o r the present work the pr inc ipa l i n t e r e s t w a s at s m a l l v a l u e s of x.

Even though interst i t ial oxygen i s the principal defect over most of the range of x, th is cannot be the only defect a s x-> 0 because the entropy would then i n c r e a s e without bound. Further , the occurrence of vacanc ie s can be used to explain the formation of a lower phase boundary near x = 0 v ia the prec i -pitat ion of m e t a l l i c uranium. Hence E v and Evv have been d e t e r m i n e d by setting x = 0 exactly at the lower phase boundary at about 1250°K and, in ad-dition, by requir ing that the optical ana lys i s (Section V. 3. ) be sa t i s f i ed at

43

960°C. The energy Eu has been set by requir ing that a two phase reg ion via the formation of a higher oxide be no longer possible above 1125°C. This i s the temperature observed by ROBERTS and WALTER [68] above which U4O9 can no longer ex i s t . B e c a u s e of the in t ere s t in s m a l l v a l u e s of x, a¡ has been se t equal to unity s ince this would be expected to be (nearly) the f i l l ing dens i ty when the actual density of i n t e r s t i t i a l s i s s m a l l .

V a l u e s of the q1 s have been ca lcu lated f r o m

with an appropriate c h a r a c t e r i s t i c t emperature 0. Since Eq. (30) i s inde-pendent of any defect density, i ts effect i s as though в were an Einstein tem-perature r e g a r d l e s s of what might be thought to be the t rue nature of the f requency dis tr ibut ion. Since , in fact , it i s the light a tom d e f e c t s which are being d i scussed , this i s not ser ious . It was found however that, given a cho ice of two c h a r a c t e r i s t i c t emperatures , making a supposedly proper choice of the higher for both q¡ and q v y ielded entropy resu l t s nowhere near the observat ions at l a r g e r va lues of x ( ~ 0 . 05). By introducing a separate q¡ and q v a "proper" 0 can be used at l e a s t f o r the regu lar oxygen l a t t i c e and st i l l allow a c l o s e r agreement with observation than i s otherwise possible.

F inal ly , E¡ w a s chosen by requiring that the value of - Д 0 о , at 0 = 0. 03 and T= 800°C be that observed by MARKIN and BONES [69] at this tempera-ture and x= 0. 03. The i l lustrat ive calculat ions of - Д § 0 г and - ДЙо 2 shown in F i g s . 20 and 21 w e r e m a d e with the p a r a m e t e r s g iven in T a b l e XI.

The entropy maximum shown in Fig. 19 bears a qualitative resemblance to that observed by MARKIN and BONES [69] and i s reminiscent of s imi lar behaviour in a l loys [70]. However, the s i z e and locat ion of the calculated maximum i s sharply dependent on temperature, contrary to the observations of Markin and Bones.

In a l loys , a min imum can be observed [70] at what would here c o r r e s -pond to hypos to i ch iometry and, in fact , it can be shown by di f ferent iat ing Eq. (23) that ex trema should be expected at va lues of x given to a good ap-proximat ion by

Since v/x2=± x, an entropy extremum (a minimum) i s a l so to be expected in the hypos to i ch iometr i c reg ion . Its actual observat ion could be prevented by the intervention of a lower phase boundary, determined by the value of Evv. The appearance of such a cr i t i ca l temperature for the ex is tence of multiple va lues of x with the s a m e partial entropy would be an interest ing repetition of the phenomenon of a cr i t ical temperature for two phase formation a s s o c i -ated with Eqs. (28) and (29).

5. 2. Application of defect theory to UOo+x

The f o r m a l theory of a nonstoichiometric oxide containing both oxygen vacanc ies and interst i t ia ls has been outlined above in Section 5. 1. We point

(30)

(31)

44

о/и Fig. 20

Ca lcu la t ed par t ia l molar entropy of oxygen in U 0 2 + x

o/u

Fig. 21

Ca lcu la ted par t ia l mola r enthalpy of oxygen in Ю 2 + х

TABLE XII

PARAMETERS USED IN CALCULATION OF PARTIAL MOLAR ENTROPIES AND ENTHALPIES

j V i

Oij 2 1

9j(°K) 870* 200

E ; ( k c a l / m o l e ) 132.887 94 .723

E j j ( k c a l / m o l e ) 15. 036 5. 56

* Although al l the present calculat ions were done prior to development of the analysis for ©E shown in Section III 2 . , iis use would not a l te r the results appreciably

out here three complications that affect the application of the theory to U02 + X(

deal ing f i r s t with the reg ion UO2.01 to UO2.25» w h e r e the concentrat ion of v a c a n c i e s can be neg lec ted .

Presumably some account must be taken of the electronic disorder ari-sing because of the oxidation of a portion of the U atoms. Although the theory of the e lectr ica l propert ies of U0 2 + x i s not yet sufficiently developed to give posi t ive guidance, it s e e m s that a reasonable approximation at the present t ime i s that the extra charges are loca l ized on U ions. (Section V).

ARONSON and CLAYTON [71] have applied a model involving local ized e lec tronic d i sorder to both U0 2 + x and (U, Th)0 2 + x sol id solut ions. The r e -action a s s u m e d was

0 2 (gas ) + 4 U4 + + 2 interst i t ia l s i t e s = 2 O2" + 4 U 5 + . (32)

The choice of U0+ rather than U°+ was made on the basis of the crystallo-graphic data on (U, Th)02 + x [72], which shows a regular but non-linear lattice contracting with increas ing x up to a mean U valency of 5. 0, and thereafter a lattice expansion. There i s also fragmentary data on paramagnetic resonance spectra which indicate that U5+ i s present in UO:+x and (U, Th)02+x [73],though it d o e s not prove that U 5 + i s f o r m e d exc lus ive ly ; magnet ic ev idence a l s o lends some support (see below), though it is , in part, conflicting. The model i s a l so in accord with later measurements of the e lectr ica l conductivity and thermoelec tr ic power on sintered plates [61].

Aronson and Clayton deduced the fol lowing express ion for the configu-rational entropy contribution:

к In W = к In Nil Nç [_(N¡ - N0¡)'. NQ¡ '. NU S:NU 4:

(33)

where N¡, N 0 i , Nc , Nu s , and Nu< are the numbers of interst i t ial s i tes , inter-st i t ia l oxygen ions , cation s i t e s , U5+ and U4+ ions re spec t ive ly . They find

46

__ v 2v A S ( 0 2 ) = - 2 R l n y f - ^ - 4 R I n j ^ + Q. (34)

where Q includes a t erm for the decrease in entropy on converting one mole of gas to the sol id and a t e r m for the contribution to the vibrational entropy of the solid of the added interstit ial oxygen ions. Eq. (34) i s plotted on Fig.22 with Q = - 49 e. u . , and the term expressing electronic disorder on this model of l oca l i zed U5 + ions i s a l s o plotted; it can be s e e n that near ly the ent ire entropy change can be accounted for in this way. It i s worth noting that at low values of x, the t e r m for the entropy Д§(Ог) b e c o m e s - 6R In x, which predicts a too rapid change of AS with x.

/ / / „ //S

EO. <34—yf t r i « 5

EO. (35a)

•

/ *

/

s i ^ x *

/ ^ ^ EXPE OMENTAL CURVE

EO. (36)

W , ' I I I /

III/

1 . I 1 • i i i I . L I

2.0 2.1 2.2 o/u

Fig. 22

Curves from theore t ica l equation for A S n

A di f ferent approach w a s outl ined by HAGEMARK [74]. Taking the v iew that inters t i t ia l oxygen occupied ( i , s i t e s in the f luor i te c e l l , he noted that, in the l imi t , only one per unit c e l l was occupied, but that they must all be equally available when x-> 0. This i s compatible with the assump-tion that, when x i s small , each 0 ¡ wil l block the occupancy of the 12 nearest in ters t i t ia l s i t e s , but, a s x i n c r e a s e s , the a v e r a g e number of in ters t i t ia l s i t e s blocked per Oí ion must decrease owing to neighbouring Oi ions blocking s o m e of the s a m e s i t e s . He w r i t e s f o r the conf igurat iona l entropy t e r m

te № H (a Nu - N ^ ) ' (Not)I

w h e r e a i s the number of ava i lab le in ters t i t i a l s i t e s / U ion, and a = -—^-r— ' 1 + 12 x

47

Then,

AS(0 2 ) = - 2R In a - x x (35a)

and this equation also i s plotted in Fig. 22; the value assumed for Q= 21.4e.u. Recently WINSLOW [184] has shown that the express ion evaluated for Д§о г

in Eq. (35a) i s not cons i s t ent with the defect s tructure mode l proposed by Hagemark. When ex - a(No¡ ) the partial m o l a r entropy of oxygen i s

The extra t erm in Eq. (35a) is due to the incomplete express ion for the con-f igurat ional t e r m W. Us ing the a(x) g iven above, Winslow has ca lcu lated values for ASo2 in Eq. (35b) (Q set equal to - 24. 7 e. u. ). The result i s shown in Fig. 22.

A l l p r e v i o u s t r e a t m e n t s have a s s u m e d that t h e r e i s only one type of interst i t ial position. We must now consider the implications for the thermo-dynamic treatment of the actual structure of U02 + x as determined by neutron diffraction. The structure assumed i s the one involving "complexes" of O', O" and latt ice vacanc ies d i scussed in an ear l ier sect ion of this Report. The unit of the structure, on this interpretation, involves two e x c e s s О ions and six U ions and could be written Uß014 (see Section II. 4. 2. 2. ). If all U atoms took part in units of U 6 0 1 4 , the formula weight would correspond to U02 .33, the composit ion of the most stable tetragonal oxide, and the limiting compo-sition of (U, Th)02+x solid solutions.

The important point i s that it i s not neces sary to allow for the indepen-dent introduction of О', O" and latt ice vacanc ies , but rather for the occur-rence of "zones" or "complexes" of s o m e type, whose number i s governed by the number of O' ions , equal to x. A conf igurat ional t e r m Zx mus t be added to al low f o r the Z equivalent conf igurat ions of a "complex" around any g iven O' s i te .

The prob lem then b e c o m e s f o r m a l l y s i m i l a r to that a lready treated: the number of "zones" i s equal to half the number of O1 ions if we consider that the O1 ions enter in pairs. The basic problems remain the same,namely, (i) the calculat ion of the number of avai lable s i t e s a s a function of number occupied and (ii) the contribution of the e lectronic d i sorder to the entropy.

A s one e x a m p l e of a v e r y s i m p l e t rea tment , c o n s i d e r the c a s e of O' ions entering in pa irs , with the occupancy of each pair blocking 36 s imi lar s i t e s . Further , take the loca l i zed charge model with two U5 + ions trapped on the two U ions neares t to the two O' ions, and the other two U5* ions f ree to move in the latt ice. We then have, for an approximate treatment,

AS 0 , = - 2R In — + Q. 1 / V I V I - Y (35b)

and

Д3(0 2 ) = Q - г l o g z - R In x - 2R In , X„ + In ( 1 - х ) (36) 12 - 36x 1 - 2x

48

Equation (36) i s a l so plotted in Fig . 22; the experimental r e s u l t s are f itted fa ir ly we l l up to x= 0. 15, with Q - \ log z= - 32. 4 e. u.

The conclus ion of this sect ion i s that the variat ion of enthalpy and en-tropy in this range can be understood on the b a s i s of r e a s o n a b l e m o d e l s , but that the exper imenta l data are not p r e c i s e enough to al low a def in i t ive choice between severa l models . Confirmation i s needed of the exact values of the part ia l quanti t ies at the e x t r e m i t i e s of the compos i t ion range. The mos t important consideration, however, i s the proper treatment of the d is -order due to e lectronic e f fec t s , which must be related to the mechani sm of conduction and charge local izat ion in UO2+X at high temperatures .

The range U O 2 . 0 0 to UO2 .01 . The d iscrepancy between the two s e t s of e x p e r i m e n t a l r e s u l t s ava i lab le f o r th i s r e g i ó n [53, 74] h a s been noted in Section III. 3 . 1 . The resu l t s obtained by Hagemark are in reasonable agree-ment with a mode l in which the p r o c e s s i s s t i l l the incorporat ion of in ter -st it ial oxygen with large posit ive values of AS due to the behaviour of t e r m s such as l n x as x-»0. By contrast , Markin's re su l t s indicate a rapid change in ДЙ to more negative values as x-»0; a new p r o c e s s i s occurring, and the obvious hypothesis i s that oxygen vacancies are being f i l led.

The f o r m a l treatment of a model involving v a c a n c i e s and in ters t i t ia l s given in Section III. 4. 1. above shows the same qualitative features as found by Markin, ( s e e F i g . 15). T h e r e i s one important d i f f erence . Any mode l based on a vacáncy concentration which changes with temperature wil l pre-dict that the posit ion of the maximum in the AS v e r s u s x curve wil l change with temperature , and this was not found by Markin and Bones . Their r e -sul ts are bet ter f itted by an assumption of a static concentration of oxygen vacancies , independent of temperature. Since they found the same behaviour on studying a number of s a m p l e s of d i f ferent impuri ty concentrat ion , an explanation based on i m p u r i t i e s of l ower valent ca t ions p r e s e n t in UO2 i s unlikely. It i s poss ib le that the oxygen vacanc i e s are a s s o c i a t e d with ura-nium v a c a n c i e s quenched- in f r o m high temperature treatment , though the s l ender exper imenta l ev idence avai lable does not support this . It may be noteworthy that the two s e t s of r e s u l t s [53, 74] w e r e obtained in v e r y di f -ferent temperature ranges , though the theoret ical model in Section III. 5. 1. would actual ly predict a m a x i m u m in the AS curve at h igher va lues of x at higher t empera tures .

No further advance can be made in the understanding of the r e s u l t s in this reg ion at present . What i s cer ta in i s that AG(0 2 ) v a r i e s very rapidly with compos i t ion below 0 / U = 2. 01, ( see F ig . 16) and th i s e m p h a s i z e s the need for very carefu l control of ambient conditions if meaningful va lues of any phys ica l propert i e s are to be obtained.

49

IV. SURFACE AND OXIDATION PROPERTIES

The actual state of UO2 powder feed material used for preparing pel lets can af fect s inter ing ra te s and, through the m i c r o s t r u c t u r e of the final compact, even important propert ies during use . One of the most important parameters i s the state of aggregation of the powder i t se l f - the degree of crystal l ini ty , crysta l l i te s i ze , type of crystal l i te agglomerate, and porosity p r e s e n t . T h e s e m a t t e r s a r e r e f e r r e d to b r i e f l y in a l a t e r s e c t i o n of the report but were not spec i f ica l ly d iscussed by the Panel. Further important variations can be due to chemical changes caused by the presence of adsorbed s p e c i e s on the s u r f a c e and a concentrat ion gradient of oxygen through the part ic les , with the poss ible formation of shells of different stable or metast-able s t ruc tures , occurr ing whenever UO2 has been exposed to a ir . T h e s e c h e m i c a l cons iderat ions are summar ized here . Although the inf luence of the surface i s of greates t ef fect when the part ic le s i ze i s smal l , it i s worth noting that many of t h e s e e f f e c t s have been shown to be quite g e n e r a l and the compos i t ion of sur face l a y e r s of even a large s ingle c r y s t a l wil l d i f fer from that of the bulk.

1. ADSORPTION PROPERTIES

The adsorption propert i e s of U 0 2 are brief ly summarized in Table XII below [81] ; the amount of gas reacting is recorded in units of Vm, the volume of 0 2 neces sary to form one physical ly-absorbed monolayer on the UO^ surface.

The low temperature chemisorpt ion of CO and of 0 2 mutually interfere , while the prior adsorption of H2 has little effect on subsequent 0 2 adsorption. A s i m p l e hypothes i s i s that O2 and CO adsorption o c c u r s on U a t o m s , but H2 reacts with О ions, possibly forming OH". The decomposition of CO [82], which o c c u r s on a fully reduced surface of U0 2 above 500°C, proceeds to a l imit which depends on the surface area but not on the temperature. There i s no deposition of bulk carbon, but rather the formation of a type of surface carbide, which i s accompanied by some loss of oxygen from the U0 2 surface; the chemisorpt ion of oxygen at -183°C i s inhibited by this "carbide" l a y e r . The deposit ion react ion stops if a few per cent of CO2 are added to the CO.

A c a l o r i m e t r i c study of the adsorption of O2 on U 0 2 at -183°C showed that the enthalpy of adsorption fe l l from ~ 5 5 k c a l / m o l e to ~ 6 k c a l / m o l e as the surface was progress ive ly covered [81] . These observations agree with others that indicate that some, but not all, of the oxygen chemisorbed on the surface i s removed from the surface during evacuation at room temperature [83, 84] .

Since U0 2 powders are commonly prepared by reduction of higher oxides at 500 to 1000°C, it i s worth noting that such preparat ions may contain ab-sorbed hydrogen o r a s table sur face 'carbide 1 . The format ion of s u r f a c e carb ide during reduct ion by CO can be prevented by the addit ion of a f ew p e r cent of CO2.

The bulk of the hydrogen can be removed by pumping above 600°C, but no measurements have been reported of the residual hydrogen content. Simi-larly, CO^ and H¿0 adsorbed at low temperatures can be removed by pumping above 200°C. Adsorbed hydrogen can react with oxygen subsequent ly ab-

51

TABLE XII

ADSORPTION P R O P E R T I E S O F U 0 2

Gas T e m p e r a t u r e CC)

React ion Gas r e a c t i n g

(V/Vm)

0 , - 195 to - 183 Chemiso rp t ion 0 . 3 - 0 . 6

- 138 to 50 Su r f ace ox ida t ion s 2 .2

> 80 Bulk ox ida t ion

H2 - 183 Chemiso rp t ion ~ 0 . 3

20 - 0

> 400 Chemiso rp t ion 1. 0 - 1 . 6

CO - 183 to 20 Chemisorp t ion 0. 7 - 0 . 8

275 Chemisorp t ion ? 0. 08.

> 500 Carbon deposi t ion ~ 2 . 0

C 0 2 20 to 200 Chemisorp t ion > 0 . 4

sorbed and be desorbed a s water . A cons iderable portion of the oxygen chemisorbed on the surface i s mobile at room temperature and above; since the maximum enthalpy of adsorption i s 55 kca l /mole , it might be predicted that all the chemisorbed oxygen should be mobile at 700°C. However, con-tinual oxidation of the surface may proceed at all temperatures , s ince the oxygen partial p r e s s u r e s of a normal "vacuum system" or of "pure argon" are far higher than those reported in Section III. for the equilibrium pres -sures of O2 over UO2+X c lo se to stoichiometry; the extent of oxidation will , of course , be l imited by the maximum quantity of oxygen available.

2. OXIDATION PROCESSES

2. 1. bow temperatures

Oxygen i s not ordinarily mobile in the UO¿ lattice at temperatures below about 70°C, but more than one chemisorbed layer i s absorbed at tempera-tures above about -100°C. Between these temperature l imits , if isothermal conditions are maintained, the amount of O2 absorbed at any temperature and pres sure i n c r e a s e s as the logarithm of the time of exposure to O2 [85] to a l imit of about 2.2 Vm.

Density and X-ray evidence on the low-temperature oxidation of larger part ic les of U 0 2 suggest the penetration of the lattice by oxygen and the formation of a surface skin of one of the tetragonal oxides 185, 29) . Infra-red l ines character i s t i c of amorphous UO3 have been observed when very s m a l l (IOOÂ) part i c l e s of U 0 2 are oxidized at room temperature [86, 87] . There i s a l so evidence that higher oxides are formed on l a r g e r p a r t i c l e s (surface area 2 m 2 / g ) from enthalpy of wetting experiments, which indicate

52

a different dependence on oxygen content for heat of wetting in water and in organic liquids [88] and from the behaviour of some of the tetragonal oxides in the atmosphere [29] . Slow transformation to higher oxides on the surface may then be a general phenomenon.

2. 2. High temperatures

Small par t i c l e s oxidize exothermical ly on exposure to a ir or O2 giving part ia l or complete convers ion to U3O8. Complete oxidation to UO2 can be effected at 1 atm O2 for smal l particle preparations and at higher p r e s s u r e s for large particle preparations. Preparations with the usual range of particle s izes .oxidize at low temperatures in two stages, the first to about U O 2 . 3 3 , the next step, to U3Oe, commencing at about 220°C or higher. Oxidation occurs above 80°C by a p r o c e s s which f o l l o w s a d i f fus ion law with an ac t ivat ion energy of 20 to 25 k c a l / m o l e [1] . The v a r i o u s s t r u c t u r e s that a r e found at moderate t e m p e r a t u r e s have been s u m m a r i z e d in Sect ion II. The f i r s t product i s normally " a U A " formed as a layer which thickens as oxidation p r o c e e d s , with eventual c o n v e r s i o n to Y1U3O7 . The f inal compos i t ion and rate of oxidation depends on the O2 p r e s s u r e , so the amount of oxygen dif-fusing inwards i s control led by surface p r o c e s s e s . The s tructures formed must be in a state of strain, s ince the contraction in the volume of the unit ce l l which takes place i s not accompanied by any particle break-up or change in surface area. The rate of diffusion i s determined by diffusion through the product layer , and s t ra in at the interphase boundaries i s indicated by the p r o f i l e s of the X - r a y l i n e s [29] . The s t ruc tures in i t ia l ly f o r m e d may be thermodynamica l ly m e t a s t a b l e a s we l l as being in a s ta te of s tra in . The oxidation p r o c e s s i s thus even more complex than has been previous ly r e -ported and the s t r u c t u r e s actual ly p r e s e n t in a g i v e n s p e c i m e n a r e not a s i m p l e function of O / U rat io; they w i l l depend a l s o on the p a r t i c l e s i z e , temperature and O2 p r e s s u r e , rate of oxidation and duration of the exper i -ment . F u r t h e r c o m p l e x changes take p lace on anneal ing, s o m e of which have been s u m m a r i z e d in Sect ion II.

F o r the present purpose it i s only n e c e s s a r y to note that s a m p l e s of UO2+X which have been quenched or cooled from high t emperatures usual ly c o n s i s t of a mix ture of p h a s e s . Approximate ca l cu la t ions of the ra te of migration of oxygen through the U O 2 + X s tructure can be made using the dif-fus ion r e s u l t s c o l l e c t e d e l s e w h e r e (Section V) and a g r e e with the e x p e r i -mental observation that an extremely rapid quench i s necessary for the high-temperature structure to be preserved.

It has frequently been found that the f inal product of oxidation of UO2 be low200°Chas a O / U ratio above 2.33. The explanation may be that a new structure of composition UgO^ i s formed with lower symmetry than tetragonal [28 ] , but the X - r a y ev idence has a l s o been d i s c u s s e d in t e r m s of l a t t i ce strain [29] and the poss ib le formation of thin layers of higher oxides noted above.

53

V. PHYSICAL PROPERTIES

1. THERMAL CONDUCTIVITY

The thermal conductivity of uranium dioxide at e levated temperatures has been studied m o r e intens ive ly than that of any other oxide [ 9 / 89] b e -cause of i t s pract ica l importance . Only recent ly have s ingle c r y s t a l s suf -f ic iently large to permit accurate conductivity measurements become avail-able . T h e s e s ingle c r y s t a l data have proved very interest ing and form the backbone of the work to be d i scussed which, for convenience, i s divided into two temperature ranges .

1.1. Low temperature thermal conductivity

Thermal conductivity measurements in the low temperature range from 4 to 300°K have been made by BETHOUX et a l . [90] on a s intered poly-crys ta l l ine mater ia l of apparent density 9.97 and by PENNINCKX [91] on a s ingle c r y s t a l ( see F ig . 23). The general shape of both curves i s the same; in part i cu lar , a m i n i m u m in the t h e r m a l conduct ivi ty c o r r e s p o n d i n g to a paramagnêt ic-ant i ferromagnet ic transit ion has been observed in both deter-minations at the s a m e temperature, 30°K (see Section III. 1. 1.). Between 4 and 200°K the s ing le c r y s t a l v a l u e s co incide to a good approximation with those given by Bethoux et a l . for the s intered polycrystal . From 200°K the two values start to diverge and at 300°K the thermal conductivity of the single crysta l i s about 20% l e s s than that of the polycrystal . In both measurements the t h e r m a l conductivity d e c r e a s e s again be low 12°K. It i s somewhat surpris ing that in the very low temperature range (4 to 30°K) the curves for a s ingle c r y s t a l and for a po lycrys ta l correspond so we l l . This s u g g e s t s that the grain s i ze i s not the factor limiting the mean free path of the phonons at the se very low t empera tures . The s ingle c r y s t a l had an oxygen content cer ta in ly l a r g e r than 2 .001, a s could be deduced f r o m i t s e l e c t r i c a l con-ductivity {<7= 1.7 X 10"3 ohm"1 cm'1 at 300°K). Transmiss ion electron micro-scopic observations have shown that in such a crystal small U 4 O 9 precipitates are homogeneously distributed in the UO2 matrix, their mean distance being about 1000 Â . One can expect that such prec ip i ta tes w e r e a l so present in the s intered material . It i s poss ible that these precipitates are responsible for the scattering of the phonons at very low temperatures .

Some data on the thermal conductivity of a s ingle c r y s t a l are shown in Table XIII.

1.2. High temperature thermal conductivity

Earl i er experimental work on measurements of thermal conductivity of UO2 and UO2+X have been adequately summarized in several publications [9,89]. These studies showed that s intered s to ichiometric UO2 obeys a relationship of the type К = A / ( B + T ) , where К i s the t h e r m a l conduct iv i ty o v e r the temperature range 200 to ~1300°C. Factors found to affect the conductivity were e x c e s s oxygen, which had a large and deleterious effect, and the shape and distribution of the porosity, which had a minor effect .

55

"К

Fig. 23

T h e r m a l conduct ivi ty of U0 2 . © Single crystal . ® Polycrystal.

TABLE XIII

T H E R M A L CONDUCTIVITY OF SINGLE CRYSTAL U 0 2

T ( ° K ) К ( m W / ' K cm)

300 83

77 40

30 9

12 22

4 11

More recent exper iment s have been concerned with m e a s u r e m e n t s of the t h e r m a l conductivity of U 0 2 s ing le c r y s t a l s and of hypos to ich iometr ic po lycrys ta l l ine mater ia l . The work on s ingle c r y s t a l s was large ly s t imu-lated by Hanford workers who advanced the hypothesis that energy transfer by photons was l ikely to make an appreciable contribution to overal l conduc-tivity under certa in conditions [93] . Determination of the thermal conductivity of s ing le c r y s t a l s showed that there was a minimum in the plot of conduc-tivity against temperature at about 600°C [94], and a maximum in the curve around 1200°C fo l lowed by a continuously fal l ing conductivity with further i n c r e a s e of t emperature [95] .

Attempts to ver i fy this hypothesis in the practical case of a fuel element by workers at the Atomic Energy of Canada, Limited (AECL) failed to reveal any enhanced thermal conductivity occurring as a result of in-pi le columnar

56

grain growth [66] . These invest igators re-conf irmed the Battelle Memorial Institute (BMI) resu l t s [94] on enhancement in thermal conductivity of single c r y s t a l s of fused U0 2 ; however they a lso showed that the improvement in thermal conductivity was los t by oxidation, which a l s o e l iminated the f ree uranium or ig inal ly p r e s e n t . They then demonstra ted that p o l y c r y s t a l l i n e hypostoichiometric s in t er s a l so displayed the increased conductivity shown by single crysta l hypostoichiometric material . The AECL workers [66] sug-ges ted that the i n c r e a s e in thermal conductivity i s due to the inclus ion of a l a r g e e l ec tron ic conduction component at e l evated t e m p e r a t u r e s when the e x c e s s uranium disso lved to form hypostoichiometric uranium dioxide. The thermal conductivity of hypostoichiometric mater ia l at ~100°C i s the same as that of s to ichiometr ic mater ia l [96] . Examination of the rods that Scott u s e d (ROTHWELL [62]) in h is t h e r m a l conductivity d e t e r m i n a t i o n s showed s o m e f r e e uranium to be presen t , so his r e s u l t s , which a g r e e with other workers for stoichiometric U 0 2 , were obtained on slightly hypostoichiometric mater ia l . F u r t h e r exper imenta l work on hypos to i ch iometr i c m a t e r i a l i s obviously necessary . Stoichiometric and near-stoichiometric U02 are semi-conductors . Contributions to thermal conductivity wil l a r i s e f rom energy transfer by phonon/phonon interactions, by photons and by current carr i er s . These modes of t rans fer can interact with one another and, if strong, such interact ions can be of importance . However, s ince exper imenta l data on such interactions are l imited, the approach adopted will be to consider each of the modes of energy t r a n s f e r separate ly and deal with interact ions a s a perturbation. This approach i s essent ia l ly that adopted in the most recent and comprehens ive ana lys i s of the exist ing data [97] .

1 . 2 . 1 . Latt ice conductivity

Over the major portion of the temperature range of pract ica l in teres t and hence of exper imenta l m e a s u r e m e n t s , the t h e r m a l r e s i s t a n c e should a r i s e from phonon/phonon interact ions . Exis t ing theory shows the latt ice contribution to the thermal conductivity (K) to be inverse ly proportional to t e m p e r a t u r e (T) w h e r e a s in fact К a l / ( T + constant ) . At 1300°C the d i s -crepancy may be a s much a s 15%.

It must be remembered that the existing theory i s based on smal l devi-at ions from harmonic v ibrat ions in a per fec t l a t t i c e . KLEMENS [98] has pointed out that "there ex i s t s , as yet , no full treatment of highly imperfect latt ices". In Section II. it i s shown that at room temperature both uranium and oxygen can be treated as harmonic osc i l la tors but that at temperatures above 100°C the harmonic approximation breaks down for oxygen. Under t h e s e c i r c u m s t a n c e s one may query whether ex i s t ing la t t i ce conduct ivi ty theory i s applicable to UO2 at high temperatures and if not, whether a great dea l of e m p h a s i s should be p laced on the o ther p o s s i b l e contr ibut ions to t h e r m a l conductivi ty . Theory developed e x c l u s i v e l y for phonons wi l l not indicate at present qualitatively in which direction the enhanced anharmonic behaviour of the oxygen a t o m s wi l l af fect t h e r m a l conduct ivi ty; however , one of the e f f e c t s of anharmonic t e r m s i s to br ing about equipart i t ion of energy and thereby to increase the contributions of other modes to the trans-f e r of heat (Sect ion V. 6. ). In th i s connect ion the s t ruc ture of UO2-X ob-vious ly should be invest igated with respect to the 'vibration surface' of i ts

57

oxygen atoms. It will be of great interest to examine whether the anharmonic contributions to the v ibrat ion sur face are g r e a t e r in UO2-X than in UO2.00.

1 . 2 . 2 . Radiant t r a n s f e r

The radiant t r a n s f e r contribution (Kr) to conduct ion i s g iven by [99]

16 к n2 T 3 Kr = — , 3 a

where к i s the Stefan-Boltzmann constant, n the index of refract ion, T the temperature in °K, and a the absorption coe f f i c i ent for incident radiation. Although this equation predic ts the increase in t ransmis s ion of energy with decreasing a, it does not say anything about the equipartition of energy. Con-sequently, even though a large value of a resulting from equipartition would predic t a s m a l l radiat ive t r a n s f e r , the net e f f e c t which a c c o m p a n i e s the large a would be to increase the total transfer of energy. For U0 2 the value of n ( ~ 2 . 2 ) i s virtual ly independent of temperature, but as normal stoichio-metr ic UO2 i s an extr ins ic semi -conductor up to 900 to 1100°C [100], f ree carr i er absorption i s to be expected so that a wil l be a function of tempera-ture a s we l l a s wave - l eng th . Bates and s e v e r a l other inves t i ga tors have demonstrated that s ingle crys ta l UCb i s transparent over a large portion of the i n f r a - r e d (3-13 ц т ) with minimum va lues of the absorption coef f ic ient at c i r c a 6 cm"1 [101] and this transparency has been invoked to explain the large d i f ference in thermal conductivity observed with UO2 s ingle crys ta l s [93] compared to normal s in tered m a t e r i a l . It has been o b s e r v e d that a s ingle c r y s t a l thin sec t ion which appeared bright red at room temperature b e c a m e opaque at 600°C [102] but this e f fect could be due to broadening of an absorption peak.

CHRISTENSEN [971 has der ived va lues for absorpt ion c o e f f i c i e n t s for m a t e r i a l deviat ing s l ight ly from s t o i c h i o m e t r y in the d irec t ion of e x c e s s meta l and in addition extrapolated to the behaviour of trulv stoichio-metric material (defined as being an intrinsic conductor at room tempera-ture). F o r UC>2.ooothe radiation contribution would peak sharply at 600°C, 0.016 wat ts /cm°C, for UO2.003 at 900°C,0.008 watts /cm°C, and for UO2.03 at ~1500°C, 0.001 w a t t s / c m ° C . In this temperature range the latt ice contr i -bution would vary f rom 0.04 w a t t s / c m ° C at 600°C to 0.02 w a t t s / c m ° C at ~1500°C. According to these calculations there will be little effect of radiant transfer on UO2.003 but truly stoichiometric material would show an appreci-able improvement of about 35% at 600°C. This i s to be compared with the o b s e r v e d i m p r o v e m e n t of thermal conductivity for s ing le c r y s t a l s of UO2 compared with s in tered po lycrys ta l l ine mater ia l of ~90% at 1200°C. The actual ex is tence of a "UO2" which i s intrinsic down to room temperature has yet to be demons tra ted . Slightly h y p e r s t o i c h i o m e t r i c U 0 2 i s reported to be p- type extr ins ic at r o o m - t e m p e r a t u r e w h e r e a s hyposto ichiometr ic ma-ter ia l shows n-type meta l e x c e s s conductivity [65] . Data on the behaviour of hypostoichiometric material at higher temperatures are not yet available. In the light of the interpretation of e lec tr ica l conduction in t erms of a smal l polaron transport p r o c e s s (Section V . 2 . ) the val idity of such ca lculat ions

5 8

m a y be quer ied , but no quantitat ive a l ternat ive t r e a t m e n t c a n at p r e s e n t be o f f e r e d .

A further factor which increase s the absorption coeff icient in the trans-parency region i s grain boundary scattering: the coefficient for polycrystal-line UO2 at wave-lengths l e s s than 8 ц т i s reported to be an order of magni-tude greater than that for single crystal material [101] .

The genera l s ituation with regard to poss ib le radiation contribution i s s e e n to be confused . D i r e c t m e a s u r e m e n t s of absorpt ion c o e f f i c i e n t s a s functions of temperature , wave-length, and composi t ion in the range UO2-X to UO2.005 both for s ing l e and p o l y c r y s t a l l i n e m a t e r i a l a r e needed b e f o r e further advance i s p o s s i b l e . In a s e n s e , such m e a s u r e m e n t s a r e d i r e c t e x p e r i m e n t a l d e t e r m i n a t i o n s of the in terac t ion Hami l ton ian funct ion (Sect ion V. 6. ).

1 . 2 . 3 . E l e c t r o n i c t r a n s f e r

In a semi-conductor exhibiting mixed conduction the conductivity Kei due to m i x e d hole and e l e c t r o n conduct ion i s g i v e n in [103] , by the equat ion

w h e r e k= B o l t z m a n n ' s constant , e = c a r r i e r c h a r g e , T = t e m p e r a t u r e °K, On = e l e c t r o n i c conduct iv i ty , crp = hole conduct ivi ty , and total conduct iv i ty a = <7n + ap , E g = ac t ivat ion energy for conduction.

F o r extrinsic semi-conduct ion either crn >>0P or сгр» crn and in either c a s e the second term in brackets in Eq. (37) will be very smal l compared with the f i r s t . Near ly s to ich iometr ic UO2 has a conductivity of about 10"1 i î crn 1 at 1200°K so that Kei i s ~ 2 X 10"tí wat t s / cm°C. To r a i s e this contribution to s ign i f i cance a 103 to 104 i n c r e a s e in extrinsic conduction i s needed. It has been stated that the high temperature e lectr ical conductivity of hypostoichio-metr ic mater ia l i s greater than that of stoichiometric material , but no sug-gest ion has been made that the increase i s several orders of magnitude. An explanation of the enhanced conductivity of UOo-x must be sought e l sewhere . If e l e c t r i c a l conduction in this range i s due to smal l polaron transport this approach may not be valid but no alternative approach can be offered at present .

The t erm стп стр/ст has a maximum value of cr/4 when стп = crp, i . e . when in tr ins i c s e m i - c o n d u c t i o n o c c u r s . If Eg i s of the o r d e r of 1 e V then the s econd t e r m in b r a c k e t s in Eq. (37) wil l dominate the f i r s t by about two o r d e r s of magnitude.

There has been no thorough investigation of the intrinsic semi-conduction behaviour of nearly s to ichiometr ic polycrystal l ine UO?. WOLFE [100] has tentatively suggested that intrinsic conduction starts in such material at about 900°C with an act ivat ion energy of 0.95 eV and that the band gap i s double this , i . e . 1 . 9 eV. BRIGGS [104] has d e t e r m i n e d the t h e r m a l ac t ivat ion energy (Eg) for i n t r i n s i c conduct ion a s 1.26 eV s tart ing at 1100°C.

Thus in the c a s e of intr ins ic conduction, to a f i r s t approximation K e i =2T (k /e ) 2 I O O C J . At 1200°K where <R~0.1, K E I ~ 2 X 10^ w a t t s / c m ° C and extrapolated to 1400°K it would be ~ 3 X 10"3 wat t s / cm°C which i s becoming

(37)

5 9

s ign i f i cant . Data for the e l e c t r i c a l conduct ivi ty of s ing l e c r y s t a l UO2.000 (Sect ion V. 2. ) show that it b e c o m e s in tr ins i c at ~ 8 0 0 ° K with E g ~ 2 . 9 e V . Specific conductivities are 0 . 5 - 1 X 10"2 ohm"1 cm"1 at 800°K and 1-2 ohm"1 cnr1

at 1000°K. These v a l u e s g ive K e J ~ 0 . 4 X 10"4 wat t s / cm°C at 800°K and ~10"2 watts /°C at 1000°K. This latter value represents a contribution which i s 20% of the exper imenta l value for polycrystal l ine ' s to ichiometr ic ' UO2 at this temperature (1000°K). If therefore hypostoichiometric material showed a lower temperature of transit ion to intrinsic conduction and greater than an order of magnitude i n c r e a s e in intrinsic conduction, the e lectronic contri-bution to thermal conduction could be significant in polycrystal l ine material and even more so in s ingle c r y s t a l s . It should be remembered that these latter calculations as sume equal mobi l i t ies for the two c a r r i e r s but this has not been shown to be true.

To summarize , it would seem at present that insufficient data are avail-able to a s s e s s a c c u r a t e l y the radiant and e l e c t r o n i c contr ibut ions to the t h e r m a l conduct ivi ty of e i ther s t o i c h i o m e t r i c o r h y p o s t o i c h i o m e t r i c UO2 whilst the application of standard phonon scattering theory to UO2 could bear c l o s e r study. The enhancement of conductivity of s ingle crys ta l fused UO2 (actually UO2-1O i s of considerable practical interest and s imilar studies need to be made on p o l y c r y s t a l l i n e U 0 2 - x t o e s t a b l i s h the t h e r m a l conduct iv i ty behaviour of this m a t e r i a l m o r e f i r m l y .

2. ELECTRICAL PROPERTIES

2. 1. Normal electrical properties

The e lectr ica l propert ies of UO¿ have been studied by several investiga-tors . Adequate reviews have been written by MEYER [105] for the work up to 1940, and by WILLARDSON and MOODY [106] for the work up to 1961. From these reviews, it i s c lear that UO2 is a p-type extrinsic semi-conduc-tor be low ~800°C, i t s conductivity ar i s ing from the pos i t ive h o l e s due to deviat ions f rom the s t o i c h i o m e t r i c compos i t ion . Above ~800°C in tr ins ic s e m i - c o n d u c t i o n has been o b s e r v e d . The r e s u l t s of WILLARDSON et a l . [107] , which w e r e the m o s t s igni f icant b e f o r e 1960, w e r e in terpre ted by them on the assumption that UOs i s a c la s s i ca l semi-conductor and that a band picture i s val id. It has been shown, however, by HEIKES and JOHNSTON [108] that band theory i s inadequate in descr ibing the propert ies of a large number of ionic semi-conductors , such as the transition metal oxides. These authors explain the conduction mechanism by a jumping of electrons (or holes) f r o m one cat ion to a neighbouring cation, the act ivat ion energy being a s -soc iated with the mobility rather than with a c a r r i e r production. The e l ec -trons are thus not free to move through the latt ice, but are local ized at the metal ions. The conduction by this hopping mechanism should be considered [108, 109] as a thermally activated diffusion process , with a mobility

ß = e d ~ V Q exp ( - E / k T ) ( 3 8 )

in which d i s the jump distance, v0 the jump frequency and E the activation energy . The mobi l i ty of the charge c a r r i e r s i s v e r y low («1 c m 2 / V s ) in

6 0

al l the se compounds . The conductivity can be e x p r e s s e d as :

a = W С const T X exp ( - E / k T ) ( 3 9 )

in which W i s the probability that an electron, upon attempting a jump will find a s i te unoccupied.

In a study of the e l e c t r i c a l propert ies of U O 2 + X , ARONSON et al . [61] used the ideas of Heikes and Johnston in order to explain the electronic con-duction in UO2. They a s s u m e d an activated jumping of loca l i zed holes be-tween U5+ and U4 + in UO2+X and thus introduced the concept of the hopping m e c h a n i s m of conduction in UO2 .

Recent m e a s u r e m e n t s , both on s intered p e l l e t s [100] and on s ing l e crys ta l s [110] have not revealed any Hall-effect , and thus indicate very low mobi l i t i es .

The r e a s o n that the usual concept of energy bands cannot be used i s that the electron-phonon interaction in these ionic crys ta l s i s so strong that polarons of s m a l l radius are created . The treatment of the smal l polaron motion that i s used to descr ibe the current transfer in these compounds has led to a hopping model at suff iciently high temperatures (the c a r r i e r s move by random jump), character ized by an activated mobility:

It therefore provides a theoretical bas i s for the current transfer mechanism of Heikes and Johnston.

Measurements of the e lec tr ica l properties of UOjmade before 1960 are i n c o m p l e t e and made on poor ly c h a r a c t e r i z e d s a m p l e s . HASIGUTI and KIYOURA [ l l l l for instance , observed irreversible hysteres is when measuring the conductivity and stated that this was accounted for by the absorption of e x c e s s oxygen during the measurements because of the poor vacuum. Such e f f ec t s have a l s o been observed by WILLARDSON et al . [107] , who noted that during m e a s u r e m e n t s on ox id ized s a m p l e s changes in t h e s e s a m p l e s occur, due to the appearance of U 40 9 . y and U 3 0 7 . z : the conductivity decreases and the materia l becomes n-type. These authors thus related the magnitude and the type of conductivity found to the phase re la t ionsh ips in the UO2 to U 4 O 9 sys tem. They further pointed out that caution must be exercised in any quantitative interpretation of the conductivity data because of the e f fec t s of grain boundaries in sintered UOc pel lets . In fact, the e lectr ical conductivity data of poly crysta l l ine s a m p l e s have only a l imited s ignif icance as the con-ductivity i s at least partly determined by the res i s tance of the intergranular boundaries.

The m e a s u r e m e n t s of the e l ec t r i ca l conductivity and of the thermo-e l e c t r i c power by ARONSON et al . [61] at t e m p e r a t u r e s of 500 to 1150°C have resulted in a much better understanding of the e lec tr ica l propert ies of иОг+х. They showed that the e l ec tr ica l conductivity in the single phase region can be represented by

It* e"E/kT •

a = (3.8 X 10 6 /T) X 2x (1 - 2x) exp ( - E / k T ) . (40)

6 1

A plot of log aT against l / T was shown to be a straight l ine for samples of a constant composi t ion, the activation energy E being 0.30 eV. Breaks in these plots appeared to correspond to the transition of the single-phase region to the two-phase UO?+x - U4O9.J. region.

ARONSON et al. showed that their thermoelectric power measurements as a function of oxygen e x c e s s can be represented by the equation:

in which a i s the Seebeck coef f ic ient . M e a s u r e m e n t s of the conductivity and the Seebeck coe f f i c i en t in the

t e m p e r a t u r e range f r o m 25 to 1100°C on s a m p l e s with l a r g e gra in s i z e s (60 ц т and 150 ц т have recen t ly b e e n reported by WOLFE [100] . The crys ta l s he used had composit ions deviating only slightly from stoichiometry (O/U = 2.003). In F ig . 24, the e l ec tr i ca l conductivity i s plotted against 1 / Т . From this f igure it i s evident that the grain s i ze has a large influence on the magnitude of the e lectr ica l conductivity. From the slope of the log oT versus l / T l ines an activation energy of 0.20 eV is calculated, being somewhat lower than the value obtained by ARONSON et al . (0.30 eV). The low mobi l i t i e s ca lcu la ted f rom the conductivity data by Wolfe are in a g r e e m e n t with the proposed mechanism of jumping of loca l ized holes and the observation that no Hall coef f ic ient could be measured. Between 400 and 800°C the Seebeck coeff ic ient has a constant value (Fig. 25), which means, to the f irst approxi-mation, that there i s a constant number of carr i er s in this region. Further, from the fact that the Seebeck coefficient i s positive in the whole temperature range invest igated by Wolfe, it i s concluded that the conductivity i s p- type . Above about 800°C a marked d e c r e a s e in the thermoe lec tr ic power was ob-s e r v e d which may be interpreted a s a trans i t ion to in tr ins ic conduct ivi ty . This i s a l s o c o n f i r m e d by the change in s lope of the conductivity c u r v e s . From the slope of the plot of log ст versus l / T an activation energy of 0.95 eV i s calculated, corresponding to an energy gap of 1.9 eV. Evidence for the ex i s tence of intrinsic conduction, found by Wolfe at high temperatures , has a l so been found previously by WILLARDSON et al . [107] .

M e a s u r e m e n t s of the e l e c t r i c a l conductivity of UO2 s ingle c r y s t a l s of d i f ferent compos i t i on a s a function of t e m p e r a t u r e have been reported by NAGELS et al. [110] in Table XIV. Interpretation of their results was done on the bas i s that the charge carr i ers in UO2 are small polarons.

H a l l - e f f e c t m e a s u r e m e n t s p e r f o r m e d between 200 and 950°K on the ir s ingle crys ta l s y ie lded no measureable response , indicating a Hall mobility smal l er than 0.06 c m 2 / V s in the whole temperature range (ß< 0 .015cm 2 /V s at room temperature) . The low mobility in UO2 can only be explained if the ef fect ive m a s s of the polaron i s very large . Nagel1 s e l ec tr ica l conductivity measurements between 90 and 800°K for crys ta l s with composit ions varying between U02 .oooand иОг.мбаге plotted in F ig . 26 as InaT v e r s u s l / T . Ac-cording to the theory of s m a l l polarons, which in i t s most s impl i f i ed form l eads to

(41)

<r» y exp ( - E / k T ) , 1 (42)

6 2

— - ю3/т Fig. 24

Electr ical conduct ivi ty of U02 003 as a function of reciprocal t empera ture

the c u r v e s should be straight l ines with s lopes determined by the activation energy. The activation energy from 0.34 - 0.19 eV with increasing conducti-vity i s shown in Table XV. This e f f ec t could be explained on the b a s i s of Nagaev ' s theory of s m a l l po larons [112] which p r e d i c t s a d e c r e a s e of the mobility activation energy with decreas ing degree of de fec t iveness . As can be s e e n from Fig . 26 the conductivity c u r v e s of the s a m p l e s with dif ferent oxygen content display a remarkable d i f ference at low temperatures . F o r the spec imens with a very low oxygen content a l inear relationship between I n a T and l / T ex i s t s over about eight decades . However, when the oxygen e x c e s s increases the InaT curves bend off at low temperatures. The complete

6 3

temperature (°c)

Fig. 25

Seebeck coefficient of U02 ш

data for these s l ight ly oxidized U 0 2 s ingle c r y s t a l s can be separated into two components:

E E (тТ = А1 X e x p ( - p j r ) + A 2 X e x p ( - j ^ ) . (43)

The v a l u e s of E 2 (high t e m p e r a t u r e ) v a r y f r o m 0 .19 - 0 .22 e V and Ej ( low t e m p e r a t u r e ) f r o m 0 .065 to 0 .080 e V .

TABLE XIV

S P E C I F I C E L E C T R I C A L C O N D U C T I V I T Y O F U02 .ooo*

Temperature CK) о(П cm)"1

298 5. 6 x l 0 " 7

350 5. 0X10"6

400 2. 0X10"5

450 6. 3 x l 0 " 5

500 1. 6X10"4

550 3.1X10"4

600 5. 4X10"4

* Lowest O/U - ratio.

6 4

T Fig. 26

E l e c t r i c a l conduc t iv i t y of s ing le crysta ls of U 0 2 . Results of Nagels e t a l .

The conduct iv i ty of s i n g l e c r y s t a l s with O / U rat io s m a l l e r than 2 .001 have values ranging from 6 X lO"? to about 2 X 1 0 - 3 ( П е т ) - 1 at room tempera-t u r e . The l a t t e r va lue i s obtained for s a m p l e s having an O / U rat io equal to 2 .001. No further change in the conductivity occurs a s the e x c e s s oxygen content i n c r e a s e s s t i l l f u r t h e r to UO2. об- T h i s f e a t u r e i n d i c a t e s that the upper l imi t of the UO2+X phase i s U O 2 . 0 0 1 at room temperature , c l o s e to the l imi t indicated by the recent thermodynamic data. It should be pointed out that the m e a s u r e m e n t s by Wolfe and by Nage l s d i f fer markedly , e s p e c i a l l y for the e l e c t r i c a l conductivi ty .

Measurements of the thermoelec tr ic power of a s e r i e s of single crys ta l s [110] are shown in F ig . 27. F r o m the sign of the Seebeck coeff ic ient it could be deduced that the conduct ion i s p - t y p e for t h e s e c r y s t a l s . F r o m 120 to 300°K the var ia t ion of the t h e r m o e l e c t r i c power with t empera ture i s s m a l l indicating a rather constant number of charge c a r r i e r s . In the same tempera-ture range the conductivity i n c r e a s e s exponentially over about four decades .

6 5

TABLE XV

SPECIFIC E L E C T R I C A L CONDUCTIVITY O F URANIUM DIOXIDE A T ROOM T E M P E R A T U R E (290°K)

Observer Sample

о ( П е т ) " 1

Activat ion energy (eV)

Observer Preparation Grain s ize

(jim) ОД] - ra t io

о ( П е т ) " 1

Activat ion energy (eV)

ARONSON et a l . [61] sintered pe l le t 60 2. 003 2. 4 X l o " 4 0. 30 (98-99%)

WOLFE [100] sintered pel le t 60 2. 003 7. 8 XIО"4 0. 20. (98.5%) 150 2 . 0 0 3 9. 3 X Ю"4 0. 20

NAGELS et al. [110] single crystal - < 2. 001 6. 1 0 - 7 - 2 . 1 0 ' 3 * 0. 34 - 0 . 1 9 2. 003 2 . 1 0 " 3 0. 22

* Due to variations in composit ion between UO2,00!l and U02.ooi •

_ 700 gl

600-> a. û: 500' ш S о о- 400-а СЕ о 300-UJ ш g 200-о: U1 Í 100

0

- 1 0 0

F i g . 2 7

Seebeck c o e f f i c i e n t of s ingle crysta l U0 2 as a func t ion of t e m p e r a t u r e

This behaviour can be explained if one assumes that the mobility is thermally act ivated. However , h e r e the quest ion a r i s e s whether the formula of the t h e r m o e l e c t r i c power deduced for band s e m i - c o n d u c t o r s may be u s e d f o r semi-conductors where the current transfer occurs by small polarons. Above 300° the thermoe lec tr i c power d e c r e a s e s ; this probably corresponds to an i n c r e a s e in the number of charge c a r r i e r s , which can be expla ined by a change in the ratio U O 2 / U 4 O 9 , the formation of UC>2+X being favoured at in-c r e a s i n g temperature .

2 . 2 . Effect of irradiat ion on e l ec tr i ca l propert ies

The influence of f i s s i o n fragment damage on the e l ec tr ica l conductivity of UO2 s ingle c r y s t a l s of different composi t ion af ter s u c c e s s i v e per iods of reactor exposure has been studied by NAGELS, [113] . The main effect ob-s e r v e d on the e l e c t r i c a l behaviour of UO2 i s an appreciable d e c r e a s e in conductivity for n o n - s t o i c h i o m e t r i c U 0 2 s a m p l e s (Fig . 28). A constant value i s reached at an integrated thermal neutron flux of about 4 X 1010 n/cmS (or 4 X 1015 f i s s i o n s / c m 3 ) , suggest ing a saturation of damage in UO2. The e l ec tr i ca l conductivity of a s to ichiometric crys ta l remained practical ly un-changed after exposures up to 1.6 X 1016 f i s s i o n s / c m 3 .

The fact that the conductivity of the non-s to ich iometr ic c r y s t a l s drops markedly shows that the defect centres produced by irradiation are mainly ho le - t raps , which reduce the init ial f ree hole concentration.

A s s tated above, the conductivity of the n o n - s t o i c h i o m e t r i c c r y s t a l s r e a c h e s a constant va lue at about 4 X 1Ö15 f i s s i o n s / c m 3 . Th i s e f f e c t can eas i ly be understood if one considers that there i s some overlapping of tracks,

CD о

о CT . —— Д © x. x x"x" —-о 1 » >-"X-

ф — ъ sï. © / X ©

-X Л CURVE о/и

© -

© -

© 2.002 © 2.00 2 © 2.11

350 250 150 Ó -50 ' "150 © TEMPERATURE (°C)

6 7

Fig. 28

Decrease in e lec t r i ca l conduct ivi ty for non-s to ichiometr ic U02 during i rradiat ion

e s p e c i a l l y at ra ther l arge d o s e s . The frac t ion of vo lume a f f ec ted by the f i s s i o n t racks , taking over lapping into account , i s g i v e n by the formula :

f v o l = l - e " v o N ( 4 4 )

where N i s the number of tracks per cm3 and v0 the volume of a track. In-troducing v0 =4.5 X 1СГ16 (d= 100 Â, 1 =6 цт) , one finds that 97% of the volume i s af fected by f i s s i o n tracks for N = 8 X 1 0 l 5 / c m 3 ( o r 4 X 1015 f i s s i o n s / c m 3 ) , and 100% for N = 10 1 6 /cm 3 . There i s a good agreement between the exper i -mental ly observed and the calculated value of the number of f i s s i o n tracks at which saturation of damage occurs .

Recovery exper iments , performed in the temperature-range from -150 to 850°C using the method of BALARIN and ZETZSCHE [92] , showed the e x i s t e n c e of two dist inct annealing s t a g e s . A r e v e r s e annealing p r o c e s s occurs between 350 and 500°C. It has been observed by transmission electron microscopy that the superlattice structure of the U 4 O 9 precipitates i s destroyed by the pas sage of f i s s i o n fragments leading to a homogenization of the pre -cipitated U 4 O 9 phase in the UCb matrix [113] . The further decrease in con-ductivity, o c c u r r i n g upon subsequent heating to 350°C, probably ind ica te s that the e x c e s s oxygen i s rearranged in the ordered U 4 O 9 s u p e r s t r u c t u r e . A s a c o n s e q u e n c e of th i s reprec ip i ta t ion , e l e c t r i c a l l y ac t ive o x y g e n i o n s a r e r e m o v e d .

R e c o v e r y of the conductivity s t a r t s only at about 650°C, with an an-nealing peak at 725°C. This second annealing p r o c e s s obeys second order kinet ics , the a s soc ia ted activation energy of defect migration being 2.5 eV. By comparing the energy of motion determined in this way with the activation energy for uranium and oxygen se l f -di f fus ion (3.8 to 4.7 eV for U and 1.3 eV

6 8

for О), one finds that the most plausible explanation for the 2.5 eV p r o c e s s and thus for the recovery stage at 725°C i s uranium point defect migration. This sugges t s that the defect centres , which act as hole traps, are uranium vacanc ies .

3. OPTICAL MEASUREMENTS

Optical propert ies , correc t ly interpreted, can throw much light on the e lectronic structure of a mater ia l . There have been few determinations of such propert ies for uranium dioxide, possibly because i ts high optical density n e c e s s i t a t e s the u s e of thin s p e c i m e n s with attendant handling p r o b l e m s . When the optical propert ies of UO5 were last reviewed, far from light being thrown on i t s e lectronic structure, it was pointed out that a large difference ex i s t ed between the band gap derived from optical absorption data and that derived from e lec tr i ca l conductivity data [9] . In the following sect ions , an attempt i s made to r e s o l v e this difference; exist ing data are interpreted in t e r m s of defect absorption and new data on the optical propert ies of s ingle c r y s t a l s are rev iewed.

3. 1. Intrinsic absorption edge

Uranium dioxide i s generally recognized as a semi-conductor which can become intrinsic at suff ic ient ly high temperatures . Despite the d i scuss ion of the recent e l ec t r i ca l data above in t e r m s of polaron theory, the conventional language of semi-conductor theory will be used here . Values for the energy gap of 2.09 eV [101], 2.18 eV[114] , 3.0 eV[107] , 1.9 eV[100] , 1.26 eV[104] and 5.25 eV [ 104, 115] , have been derived from different types of exper i -mentation, in part icular f rom m e a s u r e m e n t s of opt ical obsorpt ion and e l e c t r i c a l conductivity.

Of the optically observed energ ies , the highest i s most l ikely tobe correc t s ince absorption due to other s o u r c e s can be so large in this material as to appear to be the intrinsic absorption unless f i lms are used or the crysta l i s ex tremely thin. It has been observed by RALPH [102] and by WILLIAMS [116] that it i s poss ible to prepare single crysta ls sufficiently thin to appear ye l low by transmitted l ight. Hence, even in s ingle c r y s t a l s , the intr ins ic gap, when observed optically, cannot be as low as ~2 eV and the film results [104, 115] are accepted. Thus it would appear that this edge, when observed optically, i s not l e s s than 5.25 eV.

It i s not to be expected that there should be direct numerical agreement between e l e c t r i c a l and optical observat ions of the intrinsic band gap. F o r a polar s e m i - c o n d u c t o r it has been pointed out that the optical and thermal act ivat ion e n e r g i e s should not co incide s ince the opt ical trans i t ion o c c u r s without re laxat ion w h e r e a s the thermal trans i t ion c o r r e s p o n d s to the di f -f erence in equil ibrium configurations [117], ( see Section V. 6 . 1 . ) , Under such c i r c u m s t a n c e s the ratio of optical to thermal activation energy i s ap-proximate ly that of the s tat ic to high frequency d ie l ec tr i c constants . F o r U0 2 this latter ratio i s 24:5.8 [104] so that the thermal activation energy for in tr ins ic conduction corresponding to the 2400 Â edge i s 1.25 eV. It has b e e n pointed out in Sect ion V. 2. that band theory i s inappropiate for U 0 2

6 9

and that smal l polaron theory should be applied. Correlation of the intrinsic absorpt ion edge with the e l e c t r i c a l p r o p e r t i e s a s in terpreted in t e r m s of s m a l l po larons does not appear poss ib l e at p r e s e n t .

3. 2. Defect absorption

In addition to providing an est imate of the energy gap between the valence band and the conduction band, the optical observations made by ACKERMANN et a l . [115] on thin f i l m s of uranium dioxide show two part ia l ly r e s o l v e d peaks with c e n t r e s n e a r 300 and 400 nm. Both w e r e present in f i l m s an-nealed at ~960°C for two w e e k s at an ion gauge reading of 10~6 m m Hg or l e s s . When these f i l m s w e r e heated in s u c c e s s i v e l y h igher p r e s s u r e s of oxygen the optical density at the shorter wave- length peak grew and that at the longer wave- length peak diminished.

Since the absorption coeff ic ients in the region of these peaks are c n r 1 , ver i f i ca t ion of the situation for bulk mater ia l i s very diff icult and detailed numerica l ver i f i ca t ion a lmost imposs ib le . The re f lec tance measurements on uranium dioxide powders made by COMPANION and WINSLOW, [118] how-ever , do v e r i f y the c h a r a c t e r of these two peaks to the extent p o s s i b l e by this method. This ver i f i ca t ion , plus c o n s i s t e n c y with m e a s u r e d t h e r m o -dynamic propert ies , indicate that the resul t s of the thin fi lm measurements are appl icable to bulk m a t e r i a l , that the short w a v e - l e n g t h peak i s to be a s s o c i a t e d with inters t i t ia l oxygen ions and that the long wave- length peak i s to be a s s o c i a t e d with the p r e s e n c e of v a c a n c i e s in the regu lar f luor i te oxygen lat t ice . Actual numerical as soc ia t ions were used in the application of the defect theory of nonstoichiometry. The details of the determination of these numbers wil l be given here and in the Appendix to this Report.

A relat ionship, g iven by Eq. (8) of the Appendix has been der ived b e -tween the sum of d is tr ibut ions of o s c i l l a t o r s of two types , e a c h of which g ives r i s e to s e v e r a l absorption bands, and the resulting index of refraction and extinction coeff ic ient . The resolution actually observed i s a partial, but definite , one into two peaks . If only two (Gaussian) distributions are used with the v a l u e s of the optical constants determined for the ir we l l annealed f i l m s by ACKERMANN et a l . [115] the fol lowing constants for the d i s t r i -butions g ive the fit shown in F i g . 29.

.i N j f j / N u Ej(eV) Wj(eV)

V 0 .00866 3.25 0.603

i 0 .0439 4 .25 1.098

Here, V means "vacancy," i means "interst i t ia l ," Nj/Ny i s the ratio of the number of de fec t s of the jth type to the number of uranium atoms, fj i s the o s c i l l a t o r strength, Ej i s the centre , and Wj a m e a s u r e of the width of the jth type of o s c i l l a t o r .

An object of this analys is was a direct ver i f icat ion of the defect theory of nonstoichiometry by a direct determination of a coincident pair of inter-stit ial and vacancy densi t ies . If this i s to be done it i s necessary to know the osc i l la tor strengths which, in turn, requires knowledge of the actual nature

7 0

E ( E L E C T R O N V O L T S )

Flg. 29

Resolution of observed opt ica l absorption by thin f i lms of U0 2 in to two broad bands. T h e circles are the observations cast into a form required by Eqs. (52) and (53),

and the l ine shows the fit obta ined. The ordinate is (volunie) / (polar izabi l i ty) per uni t energy ca lcu la ted in (eV)" 1 .

3 6 m En(E)K(E) 0 e 2 h 2 [ n 2 ( E ) + 2 ]

rtilî • Г LL_ e j = lNu/irW j

of the absorbing c e n t r e s . A s far a s actual knowledge deduced from inde-pendent determinat ions i s concerned, there i s none, of c o u r s e . In o r d e r to determine if these re su l t s are within reason, then, cer ta in further sup-posi t ions , which must be reasonable, must be made. Since the assumption of local ized centres was made, a vacancy must contain at least one electron. B e c a u s e of the rather s trong ionic c h a r a c t e r of U 0 2 i t has b e e n supposed here , indeed, that it contains two e lectrons so that the e l ec tr i ca l neutrality in the reg ion of the vacancy i s maintained. It should be noted that a weak absorpt ion at about 3 eV has been o b s e r v e d in ThO s [120] where the s a m e 2 -e l ec tron centre at an oxygen vacancy, but no interst i t ial centres , could be expected. It i s further argued in the Appendix that Wv i s e x c e s s i v e for it to be descr ipt ive of a s ingle transit ion.

7 1

In the c a s e of the interst i t ia l peak the width i s e v e n g r e a t e r . The s implest argument s e e m s to be that an interstit ial entity i s also doubly charged (nega-tively) and the wide band contains many unresolved transitions derived from the s ix 2p e lectrons of the oxygen ion.

It i s c lear , of course , that all these transit ions which are suggested as contributing to the width of two peaks must be of energ ie s in the neighbour-hood of 2 - 5 eV. They must be from the ground s ta tes of th'ese c e n t r e s to high exc i tat ion l e v e l s , o r to the conduction band, o r from the full band to (empty) high lying l e v e l s of the "impurity" c e n t r e s , It i s quite l ikely that s o m e of the i n f r a - r e d l e v e l s [101] could be trans i t ions between c l o s e l y neighbouring exci tat ion l e v e l s of these c e n t r e s , for each of which there i s an o s c i l l a t o r s trength . N e v e r t h e l e s s , p r e s e n t c o n s i d e r a t i o n s have b e e n made on the b a s i s that the total o s c i l l a t o r strength to be a s s igned to these two supposedly complex peaks i s equal to the number of e lec trons assumed to be in the centre . Thus f v = 2 and f¡ = 6, which would mean a composi t ion of the se f i l m s of U02.oo3-

The arguments given here and in the Appendix s e e m to lend reasonable support to this ana lys i s and interpretat ion of these two peaks . Substanti-ation would require, however, much work in the way of measurements of the optical absorption at low temperatures , measurements of photoconductivity, e lec tron spin resonance analys is , and so on.

A comparat ive ly low absorption peak has been observed [115, 121, 122] at about 665 nm, corresponding to an energy of 1.86 eV. KIKUCHI andNASU [122] have suggested i ts assoc iat ion with interst i t ia l oxygen. COMPANION and WINSLOW [118] however, found it to be most pronounced at that oxygen concentration where one would expect the greatest humber of vacancies . The magnitude of the energy involved would a lso make it eas ier to associate with the oxygen vacancy absorption centre as proposed ear l i er . It i s difficult to draw conc lus ions on this aspect of the problem from GRUEN'S [121] work s ince his plates of sol id solutions of T h 0 2 - U 0 2 were, apparently, of variable th ickness and he only gave optical density in arbitrary units , and s ince he u s e d pure T h 0 2 p lates as blanks.

In either c a s e , such an associat ion would be another factor which would reduce the total osc i l la tor strength of one or the other of the defect peaks by s o m e unknown amount below those ass igned in that analysis .

3.3. Infra-red absorption

B A T E S [101] has o b s e r v e d many i n f r a - r e d peaks in U 0 2 in an in i t ia l study. He has a s s o c i a t e d them in a genera l way with uranium meta l inc lus ions and impurity centres in addition to those to be expected in pure well annealed uranium dioxide of varying stoichiometry. As he recognizes, further sorting and refinement of these observations i s required.

An important development has been the d i scovery in this region of the absorpt ion s p e c t r u m of U 0 2 of a window in the range 3 -13 jum [101] , ( see Sect ion V. 1 .2 .2 . ).

7 2

4. MAGNETIC MEASUREMENTS

There i s reasonable qualitative agreement between three se t s of measure -ments of magnetic susceptibi l i t ies of (U, Th)C>2 solid solutions 1123, 124, 180] . The C u r i e - W e i s s law i s obeyed with the magnetic moment p of the U(IV)ion fall ing from 3.1 to 3.2 Bohr magnetons (BM) in U 0 2 to 2.85 to 3.0 BM at in-finite dilution, and the Wei s s constant Д d e c r e a s e s from 210 to 230 for U 0 2

to 26 for 2% UO2 in T h 0 2 ; some residual value would be anticipated for Д at inf inite dilution. The magnet ic moment for the g a s e o u s U4+ ion in the 5f 2

configuration i s 3.58 BM (L-S coupling) o r 3.84 BM (j-j coupling). The ex-perimental resu l t s were therefore interpreted as showing that the U(IV) ion had the 6d2 conf igurat ion, the suscept ib i l i ty being c l o s e to the ' sp in-only ' value for two unpaired e l e c t r o n s (=2 .83 BM) with the orbi ta l contribution near ly , but not quite, quenched by the c r y s t a l f i e ld .

The suscept ib i l i ty r e s u l t s for U 0 2 fo l low the C u r i e - W e i s s law w e l l from ~100°K upwards. A maximum in the susceptibi l i ty occurs about 30°K with a corresponding maximum value of [d(XT)/dt] between 30 and 31°K. This t emperature i s in exce l l en t accord with the o b s e r v e d anomaly in the heat capacity reported to this Panel [41], (cf. Section III.1.1. ) as indeed it should be according to the theory of FISHER [125] . The two se t s of data [36, 126] presented in F ig . 30 a s a plot of XT against T are in good accord . Be low 24°K, the suscept ibi l i ty i s reported [35, 36] to be constant and equal to 1.59 X 10"5/g o r 3900 X 10" 6 /mole w h e r e a s pre l iminary data on another sample [126] r e v e a l a m o r e complex dependence on t empera ture .

A different interpretation of the resu l t s for U02 and (Ü, Th)C^ was pro-posed by HUTCHINSON and CANDELA [127]. They calculated the crysta l -line field energies of an ion surrounded by eight oxygen ions in a cubic array, as in UOs, and showed that the observed values above 100°K were consistent with a 5f2 configuration and a triplet state being energetically favoured. The magnetic susceptibi l i ty was calculated as X= N{(g3)-/KT} x (25 /6 ) , with g = 4 / 5 , equal to the spin only value g = N ( ß 2 / 3 K T ) 4S(S + 1), with S = 1.

DAWSON and LISTER [180] studied a s e r i e s of o x i d e s made by low temperature oxidation. The suscept ibi l i ty d e c r e a s e d as oxygen was added but a Curie-Weiss law was obeyed fairly well with large positive values of Д. The results were compared with predictions made by assuming that the U(IV) was oxidized to U(VI) which was assumed tobe diamagnetic. After correcting for the diamagnetism of U(.IV) and 0 = , and neglecting any contribution due to U(VT), the susceptibil ity of the remaining U(IV) in each oxide was calculated. Effective moments calculated from цец = 2.83\/YU(IV) (Т + Д) were as follows:

Composition 2.00 2.06 2.11 2.18 2.25 2.30 Meff 3.20 3.33 3.09 3.01 2.97 2.94

This decrease in jueff was attributed to increased quenching of the orbital angular momentum. It i s probable, however, that the compos i t ions of the s a m p l e s u s e d w e r e somewhat in e r r o r (i. e . low in O / U by ~ 0 . 0 5 ) and the concentrat ion gradient of oxygen through the s a m p l e s would probably a l ter at the higher t emperatures of measurement (563°K).

The suscept ibi l i t ies of UO2.1, U 4 O 9 and U O 2 . 3 3 have been measured from 1.5 to 44°K[36] . Al l the se s p e c i m e n s showed a maximum in the X - T plot

7 3

т C K )

Fig. 30

Magne t i c measurements on U0 2 . x - Leask, et a l . ; O - Westrum, et a l . T h e in f lec t ion at 30°K occurs a t t he m a g n e t i c transition point .

at 6.4°K, the magnitude of which was l a r g e r the higher the oxygen content; the behaviour of the sample of UO;,i between 0 and 20°K was not much affected by the thermal h i s tory and state of annealing. Extrapolated va lues at 0°K w e r e a s fo l lows:

X/g 105

UO2.00 1.59 UO2.1 2.7 U4O9 3.5 UO2.33 ' 4 .5

The resul ts of ARROTT and GOLDMAN [35] are marred (as they recog-nized) by uncertainties regarding the composition and phases present in their s a m p l e s , but they too report a magnetic anomaly at about 4°K for many of the oxidized samples . They could not decide bétween a model based onU(VI) o r U(V) on the b a s i s of the ir r e s u l t s . The i r c o n c l u s i o n that a t w o - p h a s e

7 4

region e x i s t s between UO2.31 and UO0.43 i s certainly incorrec t and probably r e f l e c t s the unsat i s factory nature of their s a m p l e s .

Some r e s u l t s on oxidized (U, Th)C^+x so l id so lut ions w e r e reported by DAWSON and ROBERTS [129]. Solid solutions containing 14.8, 27.9and43.1% of UO2 w e r e oxidized to mean U v a l e n c i e s between 4.3 and 5.46. The s u s -cept ibi l i ty at 300°K d e c r e a s e d approximately l inear ly with i n c r e a s i n g ox i -dation until the U oxidation number was 5.0, and m o r e s l o w l y t h e r e a f t e r . The r e s u l t s w e r e in terpreted a s favouring the introduction of U(V) in the ear ly s t a g e s of the oxidation, but it was noted that the e a r l i e r r e s u l t s f o r UO2+X did not fa l l into the s a m e pattern. Since oxidation of (U, Th)0 2 so l id solutions offers the possibil ity of preparing homogeneous ЛЮп+х phases which are quite stable at al l t emperatures , these measurements could wel l be ex-tended, and s u s c e p t i b i l i t i e s m e a s u r e d o v e r a w i d e r t e m p e r a t u r e r a n g e .

A ra ther s u r p r i s i n g a s p e c t of the c o m p a r i s o n of magnet ic behav iour and thermal properties i s the absence of any appreciable magnetic heat capa-city contribution in the data reported on a-U3O7, ß - U 3 0 7 , U 4 O 9 , and U 3 0 8

below 10°K. Fur ther work i s n e c e s s a r y to conf i rm th is o r to r e s o l v e the apparent discrepancy. It appears that the magnetic transition at 6.4°Kmight be assoc ia ted with local magnetic ordering within the complexes or "zones" of the ЩОд structure , which occur in the U0 2 + x and other s tructures , ( s e e Section II).

5. DIFFUSION PROCESSES IN U 0 2

In this sect ion the data available on the diffusion of oxygen, uranium and rare gas f i s s i o n products through U 0 2 are d i scussed . It i s evident that the m e c h a n i s m of r e l e a s e of f i s s i o n product g a s e s depends on the s tate of the U 0 2 la t t ice fol lowing irradiat ion, which i s br ie f ly d i s c u s s e d in Sect ion VI; it may therefore be misleading to make too c lose comparisons between f i ss ion product migrat ion and se l f -d i f fus ion p r o c e s s e s . Of the latter, recent ev i -dence sugges t s that the se l f -d i f fus ion of uranium through the bulk latt ice i s s l ower than has prev ious ly been reported and, indeed, that re l iable va lues of diffusion coe f f i c i en t s have not been establ ished yet .

We include also brief mention of a detailed study of the diffusion of argon in c a l c i u m f luor ide c r y s t a l s which has shown that in this c a s e the la t t i ce diffusion of a rare gas atom i s not affected by the concentration of vacancies and i n t e r s t i t i a l s p r e s e n t , but much a f fec ted by radiat ion e f f e c t s ; i . e . by defects of higher order. The strong s imi lar i t ies between Ca F2 and UO> with regard to rare g a s d i f fus ion make it s e e m l ikely that the s a m e holds true a l s o in the la t ter c a s e .

5.1. Oxygen diffusion

AUSKERN and B E L L E [128] studied the exchange of O18 between CO^ and s a m p l e s of иОг and U0 2 + x at t e m p e r a t u r e s from 400 to 800°C. At the lower temperatures studied, some of the UOo+x samples should have been in the t w o - p h a s e reg ion , but no account was taken of t h i s . A n a l y s i s of the samples before and after exposure to СОг indicated no appreciable oxidation. This result may indicate that the oxidation equilibrium between CO? and U0 2

7 5

i s not readily established at temperatures below 800°C although the exchange of Ö18 between gas and solid surface is rapid; however, the CO content of the gas was not measured and it i s imposs ib le to calculate how much oxidation should have occurred .

The resul ts indicated that diffusion i s a s s i s t ed by the presence of inter-st it ial oxygen. A sample analyzing as U0 2 wi (average particle radius = 2.65 д т ) showed D = 1.2 X 10"3 exp (-65 300/RT). D values for the non-stoichiometric ox ides w e r e higher , r i s ing with x, and the act ivat ion energy g iven i s 29 700 c a l / m o l e , though a lower value might be p r e f e r a b l e . Va lues for two s a m p l e s w e r e :

0 / U = 2 .004 , a = 0.5 ц т ; D = 7. 0 X 10"6 exp ( - 2 9 7 0 0 / R T ) 0 / U = 2 .063; a = 1.3 цш; D = 2.06 X 10"3 exp ( - 2 9 7 0 0 / R T ) .

The i n c r e a s e of D with x (at low values of x) i s compatible with an "interst i t ia lcy" (indirect interst i t ial) mechan i sm of diffusion, a s had been sugges ted on the b a s i s of e a r l i e r exchange exper iments [129] . Values for the diffusion of oxygen down a concentration gradient in UO^+x cannot readily be obtained from oxidation experiments at low temperatures because of the format ion of tetragonal s tructures under these conditions, ( see Section II).

It i s worth noting that Q.e]f reached 10"12 cm'2 /s at 750°C for O/U = 2.002 and at 420°C for O / U = 2 .063. The resu l t s are shown in F ig . 31.

5. 2. Uranium self-diffusion

AUSKERN and BELLE [131] measured the diffusion of U : 3 3 in sintered pe l l e t s of U 0 2 of > 98% theoret ica l density held in a H2 s t r e a m , es t imat ing the penetrat ion by a - e m i s s i o n from the sur face exposed a f ter s u c c e s s i v e grinding operations. They found D = 4.3 X 10"* exp (-88 000/RT) from 1450 to 1850°C, but they noted s o m e d e c r e a s e in the apparent dif fusion coef f i c i ent with time of anneal, and the activity profile differed somewhat from the calcu-lated prof i le for volume diffusion. Similar, but rather higher, values were reported by Lindner for dif fusion anneals carr i ed out in H2 [132]; he found D = 0.23 exp [ -104 600/RT] . In later work ALCOCK and Me ÑAMARA [133] found D = 1.18 exp [ -108 0 0 0 / R T ] , but they a lso examined two single crysta l s p e c i m e n s and found much l e s s U penetrat ion at 1600°C, the di f fus ion c o -eff ic ient being ~1X10" 1 5 c m 2 / s ; the corresponding value for polycrystal l ine UO2 i s ~2X10" 1 4 c m 2 / s . This i s direct evidence that grain boundary diffusion accounts for the major part of the t racer penetration on polycrystal l ine ag-gregates and the variabil ity of the results obtained by different investigators may be understood on this bas i s .

5. 3. Argon diffusion in calcium fluoride as a model process for fission gas transport in uranium dioxide

After introducing argon atoms into calc ium fluoride s ingle crys ta l s by reactor irradiation, by which the argon i s created from calcium in an (n, a) react ion , one can study a p r o c e s s analogous to that of xenon d i f fus ion in UO2I134].

7 6

TEMPERATURE (°C)

Fig. 31

Sel f -d i f fus ion of oxygen in uranium dioxide (Reproduced by courtesy of A. B. Auskem and J. Belle [128])

The r e l e a s e of Ar from CaF2 has been extensively studied with respect to the following tasks:

(i) To t e s t the assumpt ion that a dif fusion p r o c e s s i s re spons ib l e for the g a s r e l e a s e ; to e s t a b l i s h quantitat ively the condi t ions under, which the re l ease i s due to diffusion alone, or to diffusion combined with evaporat ion of the s p e c i m e n s u r f a c e , and to account f o r the e f f e c t of evaporat ion [135] . (cf . Sect ion V. 5. 4 . );

(ii) To d e t e r m i n e the a tomic m e c h a n i s m of the dif fusion; (iii) To establ ish the connection between radiation damage in the crysta ls

and the rate of the gas re l ease . The diffusion mechanism was investigated on crysta l s which were doped

with sodium and yttr ium respec t ive ly in order to vary the concentrat ion of F r e n k e l - d e f e c t s in the f luor i te la t t i ce [136] (Fig . 32). It turned out that l a t t i ce d i f fus ion p r e d o m i n a t e s be tween 600 and 1400°C with argon a t o m s jumping from interstit ial to interstit ial (enthalpy of migration 67.4 kcal/mole), whereas the transport below about 600°C occurs along dis locations (enthalpy of migration 8 kca l /mo le ) .

7 7

т[°с] 1200 1000 800 600 400 300 200

— • f V ' ]

Fig. 32

Argon diffusion in CaF z as a funct ion of t empera tu re . The branch with the highest slope was obtained on crystals with a su r f ace - to -vo lume rat io s / v <10 c m " 1 .

The crystals used in the low tempera ture region are charac ter ized by s / v к 2 х 1 0 2 c m " 1 . o = C a F 2 / N a + ; O ? C a F 2 / Y + 3 ; - = CaF z .

Even a moderate irradiation has the effect that measured diffusion co-eff ic ients are not representative of the real diffusion process . The apparent D-value always l i e s below the real one, the difference exceeding a factor of 1000 at high neutron doses , (Fig. 33). Since this di f ference i s temperature dependent, a l l important transport data obtained with highly irradiated crysta ls are distorted, e . g . activation energ i e s and D 0 - v a l u e s . The resu l t s are in reasonable agreement with analogous observat ions on UO2 which have r e -cently been published by MacEWAN and STEVENS [137] and may indicate a trapping of rare gas atoms in defect c lus ters due to radiation damage.

5.4. Fission gas release

A great dea l has been wri t ten about f i s s i o n gas r e l e a s e from U 0 2 and i s s u m m a r i z e d in two recent r e v i e w s [138, 139] . Gas r e l e a s e may occur by r e c o i l , l a t t i ce dif fusion, gra in growth and evaporation. Gas r e l e a s e d by r e c o i l cons t i tu tes only a v e r y minor contribution to gas r e l e a s e in the practical case . Some results attributed to recoi l l o s s e s for in-pile measure-ments could a l so be interpreted by diffusion mechanisms with very low acti-vation energy (diffusion along dislocations) [136] . Gas re l ease by diffusion after light irradiation has been extensively studied. The values for the acti-vation energy fal l into three groups centred at 49, 71 and 129 kca l /mol , the value of 71 being dominant in mos t c a s e s [140, 141] . However it has been shown that the r e l e a s e kinet ics due to continuous evaporation of the surface i s very s i m i l a r to that caused by diffusion [135] . With the aid of a s imple m o d e l for the compet i t ion be tween d i f fus ion and evaporat ion the c o r r e s -

7 8

F»ST NEUTRONS/cm 2 (E »O .SMeV)

FAST NEUTRONS/cm2 ( E n > 0 . 5 MeV)

F ig .33

Argon diffusion in CaF2 as a function of fast neutron dose

(a) "Macroscopic" diffusion coeff ic ient for argon diffusion in CaF2 a t 750°C as a funct ion of the fast neutron dose,

(b) Fraction of a l l argon a toms which are f ree to d i f fuse in the l a t t i ce at 750°C as a funct ion of the fast neutron dose.

ponding contributions to the r e l e a s e have been accounted for quantitatively for different m e t e r i a l s [135] . The resu l t s show that in the c a s e of U 0 2 the re l ease i s probably mere ly due to evaporation at temperatures above 1500°C. But one has to b e a r in mind that at about 1600°C equiax ia l g r a i n - g r o w t h beg ins to o c c u r in s i n t e r e d UO2 and this change in g a s r e l e a s e behaviour may be a l so as soc ia ted with such structural changes . However, at s l ightly higher temperature (~1800°C), in a fuel rod much more marked structural changes occur, v iz . columnar grain growth, accompanied by virtually com-plete gas re l ease [150] . The only likely hypothesis so far advanced to explain these phenomena i s based on the continued movement of macroscopic lenti-cu lar vo ids up the t emperature gradient to the ax i s of the rod by a vapour transport proces s [151] . It may be significant that at temperatures approxi-mating that which m a r k s the l ower l imit of co lumnar gra in format ion UO2

7 9

b e c o m e s appreciably p las t ic and ÚO2-X i s f o r m e d by heating at low oxygen part ia l p r e s s u r e s . In i rrad ia ted U02+x the c o l u m n a r g r a i n s extend m u c h farther towards the c i r c u m f e r e n c e of the fuel rod [152] but it i s not c l e a r whether columnar grain growth occurs at a lower temperature in such ma-ter ia l which has a lower thermal conductivity than sto ichiometric U 0 2 [89] . A l s o found in this reg ion a r e second p h a s e s which have b e e n repor ted a s uranium for i rradiated UO2 c lad in Z irca l l oy [163] and as (mo lybdenum+ ruthenium) or (barium + cer ium) containing phases of U 0 2 c lad in s t a i n l e s s s tee l [159] . The presence of liquid phases can lead to columnar grain growth of the type descr ibed. Phase relationships in the UOo.x /U02 ranges as well a s the UO2/ f i s s ion oxide are obviously relevant to an understanding of this situation.

F o r temperatures below 1600°C the agreement for actual values of dif-fus ion c o e f f i c i e n t s f rom po lycrys ta l l ine s in tered m a t e r i a l i s not quite a s good as for act ivat ion e n e r g i e s , and D va lues have genera l ly been derived from exper imenta l di f fusion c o e f f i c i e n t s (D1) by the equivalent sphere con-cept , the s p h e r e ' r a d i u s having been der ived from sur face a r e a m e a s u r e -ments . A di f fus ion coe f f i c i en t (D) of about 10"14 c m 2 / s at 1400°C i s fa ir ly typ ica l and if one a s s u m e s un i form r e l e a s e f rom the whole s u r f a c e f o r a year, this would correspond to re lease from a layer only about 10"6 cm thick. Slight departures from sto ichiometry due to surface oxidation are known to g ive r i s e to a cons iderably enhanced heating burst [143, 148] , and g a s i s known to be re l eased more rapidly in UO=ix [144], which may be attributable to an evaporation p r o c e s s [135] or to the effect of oxidation on micros truc -ture. The importance of adequate characterization and control of the physico chemical nature of the surface during such experiments must be emphasized.

A rapid physical non-destruct ive test ing method for determining minor departures from stoichiometry might therefore prove a very useful practical tool . The e f f ec t of departure f r o m s t o i c h i o m e t r y on the m e t a l - r i c h s i d e upon dif fusion behaviour has as yet rece ived l itt le attention apart from ob-servat ion of cool ing burs t s which never the le s s could constitute an appreci -able amount of f i s s i o n gas [146] . This behaviour i s not repeated in hypo-stoichiometric single crys ta l s [147] . Heating bursts may be due to a number of c a u s e s and e v i d e n c e has b e e n advanced that f i s s i o n xenon may be i m -m o b i l i z e d by c l o s e d poros i ty in the in i t ia l m a t e r i a l [137, 136] .

Diffusion coef f ic ients for s ingle crysta l and fused material appear to be lower than those for s intered mater ia l s but recent work has shown the rate of gas re l ease to be a function of dose [137, 136] as well as type of UO?. The d i f ference in behaviour between the s ingle crys ta l and s intered mater ia l in the la t ter invest igat ion has been explained in t e r m s of trapping by var ious types of l ine de fec t s , e . g . grain boundaries which would vary from s ingle c r y s t a l to po lycrys ta l l ine m a t e r i a l . However , fused po lycrys ta l l ine m a -t e r i a l i s s i m i l a r in behaviour to s ingle c r y s t a l s [149] so a m o r e sophis t i -cated explanation must be sought; it i s suggested that vacancy c lus ters may be responsible [137] .

To s u m m a r i z e , in the temperature range below 1600°C r e l e a s e i s pre-dictable although the m e c h a n i s m of r e l e a s e must s t i l l be regarded as i m -perfectly understood. From a practical viewpoint it s e e m s desirable to have UÓ2 very c l o s e to s to ichiometric composit ion, of low surface area and pos-sibly having c losed porosity or vacancy c lus ters as traps for f i ss ion product

8 0

g a s e s . The only statement that can be made about gas r e l e a s e for the temperature range above 1600°C i s that the mechanisms of structural changes occurring are insufficiently understood to know which physical properties are involved and hence to enable any prediction of how f iss ion gas release may be minimized. The obvious practical answer to this problem i s to'l imit centre temperatures as far as practicable. This in turn means limiting fuel ratings. The bes t method of removing th i s rating l imitat ion would at present s e e m to be by increas ing the thermal conductivity of the oxide.

6. CORRELATIVE THEORY OF PHYSICAL PROPERTIES

It i s helpful to summar ize here some of the corre lat ive theories which should s e r v e to coordinate many of the physical propert ies of the uranium dioxide phase in the future. In particular, we outline here general theories concerned with the transport of energy and of matter.

6.1. Transport of energy

The transport of energy within a so l id l ike the uranium dioxide phase (not at equilibrium) i s accomplished via several mechanisms such as phonons, excitons, polarons, f ree charges , etc . These coextens ive subsys tems can be coupled to each other to various extents and consequently can be isolated from each other, partially isolated, or equally partitioned energy wise . To obtain an insight into the problem one must adopt a perspec t ive o r f r a m e -work which r e c o g n i z e s this p o s s i b l e partial equipartit ion. The fo l lowing treatment s k e t c h e s the theore t i ca l a s p e c t s .

Although the problem of the electron-phonon interaction in this particular c a s e of ionic or covalent bonds has not been treated, to our knowledge, the equivalent one in a metal where the continuum of the free e lectrons tends to e s t a b l i s h equipart i t ion has been inves t iga ted . An exce l l en t s u m m a r y i s p r e s e n t e d in ZIMAN's [154] book on transport phenomena in s o l i d s . In the c a s e of meta l s , one ini t iates the study-by constructing the Hamiltonian function for a s y s t e m composed of ions and e lec trons . The form, however, i s e s s e n t i a l l y the s a m e a s that for a molecu le , so that the quest ion i s the s a m e as that involved in the separat ion of the vibrational part of the wave function f rom the e lec tronic part .

The Schrödinger equation for the molecu lar s y s t e m i s

in which i r e f e r s to the co-ordinates of the nuclei, j re fers to the electronic c o - o r d i n a t e s , and V(X4Xj) i s the potential energy which depends upon the pos i t ions of both the nucle i and the e l e c t r o n s . In the B o r n - O p p e n h e i m e r procedure of handling th i s equation, one a s s u m e s that the mot ions of the e lec trons are adiabatically i solated from the motions of the nuclei . To ac-

(59)

= e^XiXj) ,

8 1

compl i sh this formal ly , one wri tes the wave function a s the product of two functions. Thus

(MX¡Xj) = u (XjXj)v(Xj). (60)

Substituting the right s ide of this equation into the Schrödinger equation, one can then separate out the equation,

{"Iâ;vf+E}v(Xi)=e,v(Xi)' (61) i

in which

E - + ñpñrtf ^ ¿ x t + v Л Е ? + u ( 6 2 )

If the f i r s t two t e r m s in the sum are identif ied as

i 1

and if it i s set equal to zero, then substitution of e from Eq. (62) into Eq. (59) with ф = uv y ie lds the results:

J + v t X ^ j u f X ^ H E ^ M X ^ ) ; (64) j

and e1 = e .

Hence the original Eq. (59) has been divided via He= 0 into two equations, one involving only the nuclear co-ordinates , Eq. (61), and another involving both nuc lear and e l ec tron ic co -ord ina te s , Eq. (64). However , the la t ter contains the kinetic energy of only the electronic co-ordinates and therefore p e r m i t s Х ; to be treated as a parameter . Consequently i t s solutions y ie ld eigenvalues, E(X¡), which contain X t as a parameter. The value of E(X ;) so obtained can be employed in the equation containing the kinetic energy of the nuclei , Eq. (61). It i s apparent that this procedure has separated the two kinetic energ i e s so that it contains no mechanism for effecting the transfer of kinetic energy between phonons and e lec trons . The means for the inter-change i s contained in the interactional Hamiltonian function, Eq.(63) .

This e x p r e s s i o n i s an extremely important one with re spec t to the question of t rans fer of energy via radiation because it i s the one describing the part i t ioning of e n e r g y . It i s the equation which should be s a t i s f i e d in express ing the extent to which the e lectrons "follow" the nuclear motion and to which the nuclei "follow" the e lectronic motion. Under r e v e r s i b l e con-ditions these two would be equal. However, there i s no assurance that the p r o c e s s i s revers ib le because of the large difference in the m a s s e s . As the nuclei move, the e lectrons can probably a lso move, but the electronic motion

8 2

can be so much f a s t e r than the nuc lear motion that the nucle i cannot keep up with the e l ec t rons . Consequently, although the t rans fer of energy from phonons to e l e c t r o n s i s probably accompl i shed during one per iod of o s c i l -lation, that in the r e v e r s e direct ion i s not. The p r o c e s s may therefore be an i r r e v e r s i b l e one, un le s s continua such as free e lectrons are available to es tabl i sh equipartition.

6. 2. Transport of matter

Because of the extreme diff icult ies of the experimentation, the data ob-tained by Auskern and B e l l e , ( s e e Section V.5. ) are not p r e c i s e , but they do never the les s character ize reliably the general behaviour and thereby are in a c c o r d with the o b s e r v e d thermodynamic p r o p e r t i e s . A qualitat ive de-scr ipt ion can be e f fec ted by r e f e r e n c e to the s tat i s t ica l model for diffusion developed by RICE [155] who has evaluated the frequency Г, with which an atom jumps a d i s tance A x in the e x p r e s s i o n for the d i f fus ion c o e f f i c i e n t :

D = | T ( A X ) 2 . ' (65)

To evaluate Г Rice d e s c r i b e s a s ituation in which a centra l atom v ibra te s about i t s equilibrium posit ion while the surrounding shel l of atoms vibrates a s a group. When the two f r e q u e n c i e s are in phase the c e n t r a l atom can p a s s over a saddle point into the avai lable vacant s i t e s ( lattice vacancy or in ters t ia l cy ) . Thus

r = S p . . ( 2 j P({6}), (66)

in which p^ty i s a distribution function describing the probability of the co-e x i s t e n c e of an atom at s i t e i and a vacancy at s i t e j , and P({6}) i s the f r e -quency with which there occurs a vibrational amplitude sufficiently great for the atom to surmount the saddle point and a s imultaneous d is tort ion of the surrounding lat t ice so that the atom can pass over the col . The product of these functions should be summed over all nearest pairs. The function рЛ2) , however, i s approximated by the site fraction of vacancies so that

. Рц(2) = в. (67)

If U0 i s the energy which an atom central ly located within a s h e l l of neigh-bours m u s t acquire to o s c i l l a t e with a c r i t i c a l ampl i tude (q0 ), Uj i s the energy needed to expand the she l l suff ic ient ly , and gkl(2) i s a function c o r -relating pa irs к and 1, P ({6}) can be evaluated in t e r m s of these quantities. The resu l t i s

D = ra 0i/ exp ^ J R T J exp ( - ^ j ^ ) (68)

k>l

in which r i s a g e o m e t r i c a l p a r a m e t e r such that r = z (Дх) 2 /2а 2 where z i s the number of n e a r e s t ne ighbours , a i s the la t t i ce p a r a m e t e r , and v i s a

8 3

"weighted mean frequency" given by a normal mode ana lys i s . The function Wki i s the f ree energy as soc ia ted with the pair corre la t ion function. Thus

Ww = - RT In gwl2> = ДНк1 - TASk l . (69)

Since it has b e e n a s s u m e d that the addition of oxygen to the uranium dioxide phase does not change the lattice dynamics (see Section III.2.) , one i s just i f ied, to the s a m e order of approximation, in a s suming that U 0 , Uj, and W|d are independent of composit ion. Consequently, the general course of the diffusion coeff ic ient i s dictated by the variation of the function в or the probability of finding an atom at one s i te and a vacancy at an adjacent s i te , summed over all poss ible pairs , which i s a function of the energies Ev, E v v , E¡ and EH ( see Section III.4.1. ). Although numerical values are not available for Nv and N;, at l eas t va lues at s to ichiometry are indicated by the optical p r o p e r t i e s ( s e e Sect ion V . 3 . ) , and the g e n e r a l trend i s sugges ted by the v a r i a t i o n of the part ia l m o l a r enthalpy and entropy, ( s e e F i g . 13). N e a r s to ichiometry the diffusion i s controlled by the smal l number of oxygen ions avai lable for migrat ion. To migrate the ion must c o m e from a lat t ice s i te and move into e i ther a vacancy or in ters t i t ia l posit ion, m o s t probably the latter because there i s an extremely large number of them available. Hence the act ivat ion energy for d i f fus ion at c o m p o s i t i o n s n e a r s t o i c h i o m e t r y i s determined e s sent ia l ly by the energy needed to create a vacancy. With in-creas ing e x c e s s oxygen the number of ions available for diffusion i n c r e a s e s so that the diffusion rate i n c r e a s e s . As the number of occupied interst i t ia l s i t e s approaches the total number avai lable , the d i f fus ion c o e f f i c i e n t un-doubtedly attains a maximum value and thereafter d e c r e a s e s . The exper i -menta l r e s u l t s c l e a r l y show an i n c r e a s e d di f fus ion rate a s x in U 0 2 + x in-c r e a s e s , but r e s u l t s at l arge va lues of x have not yet been reported , ( s e e Sect ion V. 5. ).

8 4

VI. PRACTICAL IMPLICATIONS OF THERMODYNAMIC AND TRANSPORT PROPERTIES

The l imitations of uranium dioxide as a reactor fuel are now fairly well known as a l s o are the f a c t o r s repons ib le for t h e s e l i m i t a t i o n s . A b a s i c understanding of the propert ies detërmining these latter factors should en-able an a s s e s s m e n t to be made of whether the pract ica l l imi tat ions might be amel iorated. In the following sect ions such an a s s e s s m e n t i s attempted based upon the thermodynamic and transport p r o p e r t i e s which have been d i s c u s s e d in prev ious chapters .

The l imi ta t ions of UO2 mani fes t t h e m s e l v e s p r i m a r i l y a s d i f f i cu l t i e s with the cladding material and ar i se because of various modes of interaction of the fuel with the clad. After discuss ions of these interactions the relevant thermodynamic and transport properties are examined.

1. INTERACTIONS OF FUEL AND CAN

1.1. Thermal cracking

The can of a UO2 - f u e l l e d e l ement has to act a s a s tructural m e m b e r a s we l l a s a phys ica l b a r r i e r between coolant and fuel b e c a u s e of the i n -abil ity of UO2 to accomodate the thermal strain imposed in s e r v i c e which in turn resul t s in cracking. This strain i s equal to the product of the l inear coe f f i c i ent of thermal expansion X and the t emperature d i f ference a c r o s s the fuel (0i~02) and reaches a maximum value at the cool surface in the case of a fuel rod. The strain can be accomodated e i ther e las t i ca l ly or p l a s t i -cal ly . The maximum l inear strain that can be accomodated will be the ratio of ultimate strength (a ) to Young's modulus (E); thus the maximum tempera-ture d i f ference which can be accomodated will be this ratio divided by the thermal expansion coeff ic ient . Using typical values of X, E and a, for U 0 2

at r o o m t e m p e r a t u r e (0i~02) has a value of ~ 7 0 ° C , which has been c o n -f i rmed experimental ly . T h e s e va lues for a, E and X vary l i t t le with t e m -perature up to 800°C. Above 800°C there are few data available on the e last ic propert ies of UO2, but the strength var ies little up to 1300°C as does a l so the coeff ic ient of thermal expansion, so that up to 1300°C when plast ic f low mani fe s t s i tse l f in s to ich imetr ic UO2, no major i n c r e a s e in the value of (01-02) can be accomodated. At typical g a s - c o o l e d reactor fuel ratings, a s u r f a c e / c e n t r e t e m p e r a t u r e d i f f erence would be about 1000°C and t h i s would r i s e to ~ 1600°C for a m o r e highly rated fuel in a liquid cooled r e -actor . Thus at l eas t an order of magnitude i n c r e a s e in ст/ЕХ i s needed to approach the possibil ity of eliminating thermal cracking of U0 2 . Appreciable changes in X and E are not l ikely to be effected by minor compositional changes and no improvements in strength of U 0 2 suff iciently great to influ-ence s ignif icant ly the thermal cracking of UO2 in the e la s t i c range of b e -haviour can be expected.

Accomodat ion of s tra in by p las t i c deformat ion b e c o m e s of prac t i ca l interest at temperatures above ~ 1300°C in UO2 [153] and above ~ 8 5 0 ° C in UO(2+x) [156, 157]. The factor of practical interest i s how much strain can be accomodated plast ical ly before cracking occurs . A strain of ~ lO"3 has

8 5

to be a c c o m m o d a t e d for each 100°C t e m p e r a t u r e d i f f e r e n c e . F a i l u r e in UO2 s e e m s to have occurred by grain boundary parting in both types of oxide but at a much lower strain in UO2 ( ~ 0 . 2%) for comparable flow rates than in U O 2 + X • The m e c h a n i s m of f low in t h e s e two m a t e r i a l s i s not c l e a r l y e s tab l i shed .

To summarize , under a steady thermal gradient it i s unlikely that any-thing can be done to el iminate cracking in U0 2 at temperatures below which it behaves e la s t i ca l ly . It might be p o s s i b l e to extend the range of p las t i c behaviour and the extent to which plast ic strain could be accommodated by minor composit ional changes .

Unfortunately , even if une racked m a t e r i a l e x i s t s under s teady s tate condit ions, on reactor shut-down this c o r e wil l be subjected to quenching s t r e s s e s , the magnitudes of which cannot be a s s e s s e d in a general manner. To a certain extent, the core would be insulated by the cracked peripheral steady state case . However, as the temperature i s continually decreasing, the mater ia l wil l certainly be unable to support this e las t ic s train without cracking.

The last way in which the problem of thermal cracking could be a l l ev i -ated i s by reduct ion of the s tra in , i . e . by reducing (0i~02) by i n c r e a s i n g thermal conductivity. An order of magnitude increase would be neces sary to eliminate thermal s t r e s s cracking completely under a steady temperature gradient and i s obviously out of the question. Minor i n c r e a s e s might how-ever be s ignif icant in preventing cracking of a p las t ic core during s t r e s s r e v e r s a l on cool ing.

All in all there would seem to be no possibil ity of making improvements in the r e s i s t a n c e to thermal cracking of UO2 other than minor ones which might re l ieve the can of a small part of its function as a structural member.

1.2. Dimensional changes in UO2 under irradiation

Dimensional ly , U 0 2 i s a re lat ive ly stable mater ia l under irradiation, but it does show á def inite , increas ing m a c r o s c o p i c change of d imens ions with increas ing burn-up and may thus exert a p r e s s u r e on the can. Most of the evidence relating to this topic has come from a s e r i e s of irradiations on plate type spec imens carried out by the Westinghouse Atomic Power De-velopment Corp. (WAPD) [158]. Surface t emperatures were about 300°C and centre temperatures were calculated to be normally lower than 1000°C. Up to about 12X10 2 0 f i s s i o n s / c m 3 a rate of volume increase of ~ 0 . 2% per 1020 f i s s i o n s / c m 3 i s obtained for UO2 of 95°o theoretical density; for greater burn-ups the rate of expansion rapidly increase s . The effect of irradiation temperature , spec imen shape, and rate of burn-up are as yet i l l -def ined . Thus cyl indrical spec imens irradiated at the Atomic Energy Research E s -tablishment, Harwell (AERE), to ~ 5 X 1 0 2 0 f i s s i o n s / c m 3 at centre tempera-tures of up to 1600°C showed no major dimensional changes [159], however th icker p lates irradiated by WAPD to about 6X10 2 0 f i s s i o n s / c m 3 showed quite marked volume i n c r e a s e s [158]. Restraint, which is c lose ly a s s o c i -ated with spec imen shape, might therefore be very important. Such large swel l ings were accompanied by the presence of large bubbles in theUC>2 [158].

8 6

This m a c r o s c o p i c instabil i ty i s not re f l ec ted on the s u b - m i c r o s c o p i c scale by changes in lattice parameter. At a burn-up of — 3. 5 X 10'2i f i s s ions / cm 3 and a cen tre t e m p e r a t u r e of ~ 950°C, UO2 i s reported to re ta in i t s c r y s t a l l i n i t y with l i t t l e change in la t t i ce p a r a m e t e r [179]. E x p e r i m e n t s using UO2 in both polycrysta l l ine and single crysta l .form [160] have shown that at 65°C the increase of lattice parameter reaches a maximum of ~0.1% a t ~ 6 X 1 0 l 6 and 5X1017 f i s s i o n s / c m 3 r e s p e c t i v e l y . With increas ing dose there i s a d e c r e a s e in latt ice parameter to that represent ing 0. 05% expan-sion, which value remains unchanged up to the maximum exposure examined (3X10 2 0 f i s s i o n s / c m 3 ) . At 400°C there is no difference in behaviour between s ingle and polycrystal l ine material , the maximum value of 0. 05% occurring at 4X10 1 7 f i s s i o n s / c m 3 , the saturation value being ~ 0 . 0 1 % . Pos t i r rad i -ation annealing studies on irradiated UO2 single crysta ls show that the struc-tural changes have been e l iminated af ter 24 h at ~ 900°C whilst there i s s t i l l s o m e res idual damage ( — 0. 01%) in the po lycrys ta l l ine mater ia l even a f t er 24 h at 1000°C. The i n f e r e n c e to be drawn f r o m the d i spar i ty in s u b - m i c r o s c o p i c and m a c r o s c o p i c behaviour i s presumably that the so l id f i s s i o n products are re jected complete ly from the UO2 la t t ice , as are the g a s e o u s f i s s i o n products .

This being so, there must be an unavoidable expansion, which for the so l id f i s s i o n products wi l l be determined by the ir s tate of c h e m i c a l c o m -bination, poss ibly modified by minor compositional changes in the UO2. No pract ical lead on this latter point i s available s ince comparisons of the m a c r o s c o p i c and s u b - m i c r o s c o p i c dimensional changes during i so thermal irradiat ion of U O 2 and U O 2 + X have yet to be made .

The posit ion with respect to f i s s ion gas agglomeration i s somewhat dif-ferent s ince there i s now a body of information as soc ia ted with swel l ing of uranium metal [161, 162]. In s imple t e r m s it s e e m s desirable to retain the f i s s i o n g a s e s in the f o r m of bubbles suf f ic ient ly s m a l l for the internal gas p r e s s u r e balanced by the ( 2 > / r ) force to be high. (Where y i s the surface tens ion and r i s the radius of the bubble. ) This problem reduçes to one of controlling the nucleation and initial growth of the bubble, which in the metal case has been shown technologically to be brought about by metallic additions, init ial ly present in the uranium micros t ruc ture a s compound prec ip i ta tes . The p r e c i s e role of these precipitates with respect to nucleation and growth i s not yet c l e a r l y defined although their pract i ca l e f f i c a c y i s not in doubt [161]. However in drawing analogies it must be borne in mind that f i s s i o n g a s r e l e a s e f r o m uranium meta l does not b e c o m e apprec iab le until quite l arge amounts (~10%) of swel l ing have occurred in marked contrast to the UO2 c a s e . It t h e r e f o r e needs to be substantiated that the growth of inert gas bubbles in UO2 provides an appreciable contribution to the macroscopic growth of UO2 on irradiation. Within the range of linear growth this appears doubtful. Recent work on xenon diffusion in UO2 has been interpreted on the bas i s of the trapping of this gas at two poss ible s i tes , one of which appeared to be c losed porosity in the original material and the second "small vacancy c l u s t e r s " [137] . Evidence has been acquired for the trapping of argon in such c l u s t e r s in irradiated CaF2 [136]; other evidence on gas r e l e a s e from UO2 has been cons idered in a previous chapter. Electron m i c r o s c o p e studies of irradiated UO2 revealed a very l imited number of bubbles at r e g i o n s which had been at t e m p e r a t u r e s above 1400°C and be low 1800°C

8 7

where columnar growth c o m m e n c e s [162]. T r a n s m i s s i o n e lectron m i c r o -scope s tudies of i rradiated and annealed m a t e r i a l a r e s t i l l at too ear ly a s tage to be def init ive , but nothing comparable to the uranium c a s e has yet appeared. Whether this absence of macroscopic growth i s due to a naturally m o r e homogenous bubble nucleation in UO2 (because of e i ther the inherent p r o p e r t i e s of the UO2 in this t emperature range or b e c a u s e of the lack of plast ic i ty of the UO2), i s a matter for speculation. If the latter i s true then further speculat ion must revolve around poss ib l e d i f f erences in the behaviour of g a s a g g l o m e r a t e s at gra in boundar ies and in the body of the c r y s t a l .

At h igher t e m p e r a t u r e s , within the gra in growth ( 1600 -1800°C) and columnar growth (> 1800°C) ranges , gas r e l e a s e i s so great that it s e e m s unlikely that bubble growth as previously conceived can contribute appreciably to macroscop ic swel l ing.

At burn-ups w h e r e g r o s s swel l ing o c c u r s th i s i s a s s o c i a t e d with the p r e s e n c e of large bubbles in the UO2 and also high gas r e l e a s e [158]. B e -haviour in this region i s as yet insufficiently well-defined to permit discussion although "irradiat ion induced plast ic i ty" or s o m e such equivalent phrase i s often encountered in repor t s deal ing with this subject .

To summar ize it s e e m s poss ib le to define some areas in which a bas ic understanding of the p r o p e r t i e s or p r o c e s s e s involved might enable s o m e of the p r e s e n t l imi ta t ions of UO2 as a fuel to be e a s e d . T h e s e a r e a s a r e t h e r m a l conductivity , phase equi l ibria and m a t e r i a l t ransport p r o c e s s e s .

2. THERMAL CONDUCTIVITY

An increase of thermal conductivity, should, for a given fuel rating and geometry reduce the centre temperature and hence e a s e practical problems assoc ia ted with f i s s i o n gas r e l e a s e and possibly thermal cracking.

The r e s u l t s quoted for s ing l e c r y s t a l s [95] t r a n s l a t e d into p r a c t i c a l t e r m s mean that the rod-power (Jkd0) for the temperature interval 500 to 1600°C i s 50% g r e a t e r than for po lycrys ta l l ine mater ia l . Compared with s in tered m a t e r i a l th i s i s a v e r y d e s i r a b l e i m p r o v e m e n t , of c o n s i d e r a b l e practical s ignif icance particularly if polycrystal l ine material having s imi lar p r o p e r t i e s can be deve loped. The b a s i c understanding of the p r o c e s s of thermal conduction in иОг at e levated t empera tures i s not suff ic ient ly ad-vanced to enable predictions of practically possible improvements tobe made.

Having stated this , the probable e f f ec t s of radiation induced damage on the p r o c e s s e s which contribute to thermal conduction need to be a s s e s s e d . The e a s i e s t way of sett l ing speculation on this point involves direct in-pi le m e a s u r e m e n t s of thermal conductivity of s ingle and polycrysta l l ine U02-x-The r e s u l t s of such e x p e r i m e n t s may a l s o throw further l ight on the c o n -duction p r o c e s s e s in unirradiated m a t e r i a l .

3. PHASE EQUILIBRIA

The importance of minor deviations from stoichiometry, in the direction of oxygen e x c e s s or d e f i c i e n c y , upon the prat ica l p e r f o r m a n c e of UO2 a s a fuel has been made c l e a r in the p r e c e e d i n g s e c t i o n s . Interes t at p r e s e n t

8 8

centres on hypostoichiometric material largely because of the enhanced ther-mal conductivity which it i s reported to have [66, 95], but a l s o b e c a u s e of i t s p o s s i b l e p r e s e n c e in high t e m p e r a t u r e r e g i o n s of i rradia ted m a t e r i a l [163]. The general def ic iencies in knowledge of phase equilibria in the region U O 2 - U O 2 - X are such as to render the interpretation of experimental obser -vations very doubtful. Minor deviations from stoichiometry in the direction of oxygen e x c e s s may g ive r i s e to quite marked d i f f e r e n c e s in f i s s i o n g a s r e l e a s e . Thus non-des truc t ive methods for the c l o s e charac ter i za t ion of the 0 : U ratio in the range 2 -х to 2 . 0 1 , pr ior to irradiat ion, are needed to render irradiat ion exper iments m o r e meaningful .

A detailed knowledge of phase equilibria i s a l s o e s sent ia l for adequate control of the preparative and fabrication proces se s leading to the production of a c l o s e l y charac ter i zed UO2 body. T h e s e a s p e c t s are d i s c u s s e d in the next sec t ion .

4 . MATERIAL TRANSPORT PROCESSES

The importance , for a variety of r easons , of obtaining a high density low surface area sintered UO2 was realized at an early stage of its develop-ment . Technolog ica l a n s w e r s w e r e arr ived at we l l in advance of a b a s i c understanding of the s intering p r o c e s s . An understanding of the transport p r o c e s s e s l ikely to be involved i s , at this point in t ime, only of interest in-asmuch as they may throw light, in a general manner, on the sintering pro-c e s s . An understanding of material transport under the influence of s t r e s s at elevated temperature does st i l l have some immediate interest in the con-text of strain relaxation and bubble growth.

Sintering theory for oxides i s not on a part icularly f i r m footing at present . The case for material transport during sintering by a bulk diffusion mechanism has been strongly urged and widely accepted over the past decade. Models for sintering based on diffusion p r o c e s s e s have been subject to con-s i d e r a b l e study, and theore t i ca l development has reached a s tage w h e r e quantitative comparison of calculated and observed sintering behaviour has become poss ib le . A s the volume diffusion coef f ic ients for anion and cation genera l ly di f fer it has been normal to a s s u m e that the movement of the ion having the lower dif fusion coef f i c i ent would be rate control l ing. This a s -sumption has led to s e r i o u s d i screpanc ie s between theory and experiment , with sintering being more rapid than would be predicted from measured dif-fusion coeff ic ients [164]. Thus the apparent diffusion coeff ic ients calculated from data on the later s tages of alumina sintering are f ive orders of magni-tude greater than the measured oxygen diffusion coeff icient in single crystal alumina [165]. This discrepancy can be reduced by two orders of magnitude if the di f fus ion coe f f i c i en t typical of po lycrys ta l l ine m a t e r i a l i s used . A rat ional izat ion of the situation has been attempted based on the suggest ion that during sintering movement of the s lower volume diffusiong ion may not be rate-control l ing a s this ion may choose to move more rapidly by a grain boundary p r o c e s s [164]. A s BURKE and COBLE [164] state, "the final e x -planation of this puzzl ing d i screpancy in apparent dif fusion coe f f i c i en t s i s one of the most important problems in sintering theory today". This s tate-ment r e f e r s to the behaviour of relat ively uncomplicated oxides . The p r e -

8 9

vious sect ions have demonstrated that "uranium dioxide" i s a very complex mater ia l . Oxygen in e x c e s s of s to ichiometry will invariably be present in p o w d e r s be fore s inter ing: it wi l l be non-un i formly distr ibuted at r o o m -temperature and this distribution will vary in a complex manner during the sintering cycle as a function of temperature and atmosphere. Thus the sur-face area of a "UO2" powder has been found to vary c lose ly with change and d irec t ion of change of i t s oxygen content [167]. By varying t e m p e r a t u r e and oxygen pressure independently, it was found that there was a pronounced decrease of surface area during reduction but a much smaller change during oxidation. The s t ruc tures and proper t i e s of the crys ta l l i t e a g g l o m e r a t e s wil l vary according to the method of chemica l preparation and are difficult to charac ter i ze [166]; var iat ions in these proper t i e s can result in the d e -ve lopment of opt imum compact ing p r e s s u r e s which wil l d i f fer f r o m p r e -paration to preparat ion [167, 168, 169]. B e c a u s e of inadequacy of c h a r a c -terization of starting material with respect to all these factors, comparison of the resul ts of different invest igators i s virtually impossible .

However , in s tudies of the s inter ing of "UO2" an e f fec t has béen ob-served which may ultimately throw further light on the sintering process . De-parture from s to ich iometry has a benef ic ia l e f fec t on s intering behaviour over the composit ion range UO2-UO2.02; increase of oxygen content beyond UO2.03 has no further benef ic ia l e f fect [170, 171]. Both enhancement and saturation e f fects have to be explained. It must however be remebered that the ef fect of m i c r o a g g l o m e r a t e state upon this optimum 0 : U rat io has not been examined. Much more needs to be known about self-diffusion processes in иОг-ЩОд before d i scuss ions of these phenomena can be raised above the leve l of pure speculation.

With respect to the creep of polycrvstall ine oxides at elevated tempera-tures the situation again i s far from sat is factory. Polycrysta l l ine alumina [172, 173] and beryll ia [174] are found to deform in a way which i s in accord with the Herr ing-Nabarro model for diffusional creep . However, the di f -fusion coef f ic ients calculated from experimental creep data are comparable to the m e a s u r e d v a l u e s for the cation which d i f fuses in these c a s e s m o r e rapidly than the anion by severa l orders of magnitude [175, 176]. The s i tu-ation i s virtual ly identical with that for s intering theory and the same s o l -ution of the diff iculty has been proposed.

No ser ious attempt has been made to interpret the sintering behaviour of U 0 2 and UO2+X in t e r m s of the Coble m o d e l s due l a r g e l y to the known complexity of the p r o c e s s e s . Attempts have been made to interpret the creep behaviour of UO2 and UO2+X on the bas i s of the diffusion model [156]. The act ivat ion energy for the deformat ion of UO2 i s ~ 9 1 k c a l / m o l e [153, 156] which i s to be compared with ~ 8 8 kca l /mo le [9] for uranium se l f -di f fus ion and ~ 65 kcal /mole for oxygen [177]. This suggests that the rate-controlling ion i s uranium but the diffusion c o e f f i c i e n t s calculated from experimental data using the Herring-Nabarro model are greater than the measured values by about three o r d e r s of magnitude [156]. Th i s s i tuation i s analogous to that already descr ibed for alumina and it i s tempting to take refuge in the hypothesis that uranium might in fact be diffusing via grain boundaries. How-ever the recent e x p e r i m e n t s of ALCOCK and McNAMARA [178] suggest that the uranium diffusion coeff icient used in these comparisons was a grain

9 0

boundary value, and that the volume diffusion coef f ic ient could be an order of magnitude lower . An alternative explanation to diffusional creep i s ob-viously n e c e s s a r y and a l i t t le qualitative information in favour of grain boundary sliding has been obtained. Fa i lure o c c u r s in UO2 under f lexural creep by grain boundary separation after about 0. 2% maximum fibre strain.

Non-sto ichiometric oxide deforms much more readily and at lower t e m -p e r a t u r e s than the s t o i c h i o m e t r i c m a t e r i a l with an act ivat ion e n e r g y of ~ 65 k c a l / m o l e [156, 157] which i s to be compared with ~ 30 k c a l / m o l e for oxygen di f fus ion [131]. No wel l substantiated value for uranium se l f d i f -fusion in U O 2 + X e x i s t s at present although an activation energy of ~ 5 2 kcal has been reported for U 0 2 heated in argon and hence probably non-stoichiometric [132]. Grain boundary sl iding def inite ly o c c u r r e d in UO2 + X and m a x i m u m fibre strains of up to 2% in bend have been observed prior to grain boundary fa i lure . If swell ing due to gas p r e s s u r e occurs by a Herr ing-Nabarro m e -chanism, on the evidence presented, the behaviour of a bubble within a grain could be o r d e r s of magnitude di f ferent f rom that at a gra in boundary and markedly inf luenced by spec i f i c impur i t i e s . The importance to an under-standing of these creep p r o c e s s e s of establishing a comprehens ive qualita-t ive p ic ture of cation and anion diffusion in both UO2 and U 0 2 + x in s ing le and po lycrys ta l l ine form, i s apparent.

Mater ia l transport via a vapour phase appears to be of pract ica l i m -portance only in the columnar growth p r o c e s s . Adequate agreement ex i s t s between experiment and calculation using U 0 2 vaporization data [150, 151]. However the overwhe lming technolog ica l importance of th i s p r o c e s s with respect to f i s s ion gas r e l e a s e has been commented upon already. Suppres-s ion of co lumnar growth would lead to substantial reduction in f i s s i o n gas r e l e a s e at centre t emperatures above 1800°C, so that a very c lear and de -tai led understanding of the bas ic m e c h a n i s m s involved appears des i rab le .

9 1

VII. CONCLUSIONS

It w i l l appear f r o m the deta i l ed d i s c u s s i o n of the e v i d e n c e in the body of the Report that a great deal r emains to be done before the various proper-t i e s can be corre la ted into a s ingle , coherent model of the nature of the non-s t o i c h i o m e t r i c uranium dioxide phase . However , an encouraging start has been made . It i s not d e s i r a b l e to attempt to s u m m a r i z e the c o n c l u s i o n s of the p r e c e e d i n g s e c t i o n s h e r e , s i n c e they would be m i s l e a d i n g u n l e s s qua l i -f i ed by a detai led d i s c u s s i o n of much of the exper imenta l ev idence , and this has been given already. Instead, we shall l i s t again the immediate quest ions which s e e m to demand further attention and which w e r e e m p h a s i z e d during our d i s c u s s i o n s . T h e s e t o p i c s w e r e the fo l lowing: (i) T h e r e i s a need for X - r a y or neutron di f fract ion work at high t e m p e r a -

t u r e s (> 1000°C) part i cu lar ly in the reg ion 2. 15< 0 / U < 2. 25, in o r d e r t o r e s o l v e the q u e s t i o n s r a i s e d by X - r a y w o r k on quenched s a m p l e s in t h i s r e g i o n .

( i i ) S o m e f u r t h e r work on s t r u c t u r a l t o p i c s i s d e s i r a b l e , in p a r t i c u l a r : (a) neutron d i f fract ion data on c o m p o s i t i o n s for which 2 . 1 5 < 0 / U < 2 . 2 5 ,

to r e f i n e the v a l u e s f o r s i t e occupancy and hence the m o d e l f o r U 0 2 + x ;

(b) the c o m p l e t e s t r u c t u r e f o r U 4 O 9 ;

(c) neutron d i f f r a c t i o n s t u d i e s of the t e t ragona l p h a s e s ; and (d) a t t e m p t s t o d e c i d e the s t a t u s of the U O 2 . 2 5 p h a s e w h i c h d o e s not

show the s u p e r s t r u c t u r e l i n e s c h a r a c t e r i s t i c of w e l l - c r y s t a l l i z e d U 4 O 9 .

( i i i ) The h y p o s t o i c h i o m e t r i c UO2-X reg ion i s not w e l l c h a r a c t e r i z e d . It i s n e c e s s a r y to def ine: (a) p r e p a r a t i v e cond i t ions , (b) the var ia t ion of x with t e m p e r a t u r e and oxygen p r e s s u r e , (c) the phase l i m i t s , and (d) s tructural and phys ica l p r o p e r t i e s of w e l l - c h a r a c t e r i z e d s a m p l e s . T h e frfee e n e r g y of f o r m a t i o n of g a s e o u s UO i s not w e l l e s t a b l i s h e d .

(iv) More accurate v a l u e s for the partial molar enthalpy and entropy of oxy-gen in UO2« a r e required for a r igorous t e s t of theore t i ca l m o d e l s . It i s unl ikely that equi l ibr ium m e a s u r e m e n t s at high t e m p e r a t u r e s can be made with suff ic ient accuracy . Enthalphy m e a s u r e m e n t s in an adiabatic c a l o r i m e t e r at high t e m p e r a t u r e s might be p r e f e r a b l e . It i s p a r t i c u -lar ly n e c e s s a r y to m e a s u r e AS values c l o s e to the s to ich iometr ic c o m -p o s i t i o n by a new t echn ique . M e a s u r e m e n t s of heat c a p a c i t i e s in the U O 2 + X s i n g l e - p h a s e r e g i o n should be m a d e .

(v) Magnetic susceptibi l i ty m e a s u r e m e n t s on U 4 O 9 and U 3 O 7 show anomal ies at 6°K, which a r e not apparent in s p e c i f i c heat m e a s u r e m e n t s . M o r e m a g n e t i c or t h e r m a l s tud ie s a r e required to r e s o l v e t h i s .

(vi) M o r e e x p e r i m e n t a l data on t h e r m a l conduct iv i ty at high t e m p e r a t u r e s i s required; s ingle crys ta l and polycrystal l ine material should be studied in the s t o i c h i o m e t r i c and hypos to i ch iometr i c s t a t e s . The e x p e r i m e n t s should be conducted in -p i l e a s wel l as o u t - o f - p i l e and the ef fect of p r o -longed irradiat ion better es tab l i shed .

(vii) M e a s u r e m e n t s of the e l e c t r i c a l conduct iv i ty of both s i n g l e c r y s t a l and po lycrys ta l l ine s p e c i m e n s should be made at t e m p e r a t u r e s up to 1200°C

9 3

at least ; the compos i t ions should be careful ly controlled and extend to the hypostoichiometric range if poss ib le , in view of poss ible important contributions to the thermal conductivity.

(viii) There i s much scope for work on optical and infra-red absorption a s a function of temperature and composition. These measurements are also relevant to the theory of thermal conductivity.

(ix) Se l f -d i f fus ion coe f f i c i en t s of U and О in UO2+X need to be better e s tab-l ished, espec ia l ly at higher values of x. More work i s required on the mechanism of foreign gas diffusion in UO2 and иОг+х-

(x) The e f f ec t s of thermal gradients both on m a s s and heat t r a n s f e r were d i s c u s s e d only br i e f l y f r o m a theore t i ca l s tand-point . Th i s subject does not s e e m to have rece ived the attention it d e s e r v e s in v iew of i t s great technological importance . Finally, we s t r e s s that this Panel was not constituted to d i scuss irradi-

ation e f f ec t s in any detail , though the major variat ions in propert ies due to irradiat ion e f f e c t s have been noted in the foregoing sec t ions . Our d i s c u s -s ions do serve to draw attention to the potentially variable nature of the un-irradiated mater ia l and to the need for the very complete character izat ion of s a m p l e s before irradiat ion.

9 4

APPENDIX

MATHEMATICAL TREATMENT OF D E F E C T ABSORPTION

The complex polar izabi l i ty , at energy E (or frequency v, where E=hi / ) , of an o s c i l l a t o r of m a s s m w h o s e r e s o n a n c e e n e r g y i s Ej and which , b e -c a u s e of damping , h a s a natural l i n e width, y¡ , i s

e2h2

m E ? - E 2 + i7jE

w h e r e fj i s the t rans i t ion probabi l i ty or o s c i l l a t o r s trength . The fo l lowing d e v e l o p m e n t f r o m Eq. (1) of an e x p r e s s i o n which wi l l r e l a t e the n u m b e r s of a b s o r b i n g c e n t r e s t o o b s e r v e d opt ica l d e n s i t y i s an e x t e n s i o n of a r g u -m e n t s m a d e by D E X T E R [119] .

In a so l id , b e c a u s e of the t h e r m a l mot ion of the a t o m s of the c r y s t a l , the a b s o r p t i o n by a g i v e n type of c e n t r e wi l l not be sharp; t h e r e w i l l be a d i s t r ibut ion of v a l u e s of Ej . If, f o r e x a m p l e th i s n o r m a l i z e d d i s t r ibut ion i s g i v e n by s o m e funct ion of energy , Gj(E), then an o s c i l l a t o r which g i v e s r i s e to a s i n g l e a b s o r p t i o n band in the s o l i d w i l l h a v e a p o l a r i z a b i l i t y

ttj(E)= — f ] f 2G ^ 2

E ' ) d E ' . (2) m

JJ E'2 - E 2 + Í7j E

If, on the other hand, t h i s o s c i l l a t o r has s e v e r a l l e v e l s which g ive r i s e to m o r e than one absorpt ion band, i t s po lar i zab i l i ty wi l l be

e 2 f i 2 V . Г ' G* (E')dE' . ai(E)- TZA/ E'2 -E« + íy'E ' (3) J о J

i . e . t h e r e w i l l be a d i s tr ibut ion of energy , due to the t h e r m a l mot ion , f o r each band. If, in addit ion, o s c i l l a t o r s of m o r e than one type are p r e s e n t , t h e r e be ing N¡ of type j, the po lar i zab i l i ty of the s y s t e m b e c o m e s

< * ( E ) = ^ N j 0 j ( E ) . (4)

i

F o r the p r e s e n t appl icat ion t h e r e a r e two types of absorb ing c e n t r e s to be cons idered , those due to randomly distributed vacanc ies in the regular cubic oxygen lat t ice and those due to randomly distributed interst i t ia l oxygen ions . F o r th is c a s e , then, Eq. (4) b e c o m e s

a ( E ) = N v a v ( E ) + N i a i ( E ) . (5)

Many of the arguments and a s s u m p t i o n s to be made in the fol lowing d e -v e l o p m e n t a r e t h o s e of D E X T E R [119] . M o r e o v e r , m a n y of t h e s e p l u s the ones added for the p r e s e n t appl icat ion are not jus t i f i ed a pr ior i ; the p r i n c i -pal j u s t i f i c a t i o n i s the c o n s i s t e n c y of the r e s u l t with o ther known or s u r -m i s e d f a c t s . H e n c e they wi l l m e r e l y be s ta ted .

9 5

It i s assumed that the absorbing centres are well local ized and that the L o r e n t z l o c a l f i e ld c o r r e c t i o n i s appl ied. Then the re la t ion be tween the complex index of refraction, (n-iK), and the complex polarizability, as given by Dexter, i s

j

where n i s the index of refraction and К is the extinction coefficient at energy E.

Equation (6) i s a s g iven by D e x t e r , who a s s o c i a t e s no with the index of re fract ion of a pure, ideal, transparent host crys ta l . His e i s inserted to correc t the polarizabil ity of the host for the poss ible neces sary removal of i t s a toms to allow (substit ional) impurity a toms to be present . F o r his application, where the values of Nj can be treated as smal l compared to the, numbers of atoms in the host, he can a lso as sume n to be c lose to no.

In the present c a s e no e i s to be used s ince no direct comparison with an unadulterated crystal i s to be made and, indeed, cannot be made. Instead of the n0 , the equation a s u s e d h e r e would contain n e , the v e r y high f r e -quency index of re frac t ion of the actual c r y s t a l .

If one chooses to work with the imaginary parts of Eq. (6),. the result i s

6nK _ 4 T e 2 - h 2 V V . Г TfEGf(EMdE' (n2 + К 2 ) 2 + 4(n2 - K 2 ) + 4 ~ 3 m L ¡ L j J ( E - 2 - E2 )2 + (Т . {Е)2 '

i c o J

where n and К on the le f t are , of course , functions of the energy, E. Here it i s argued, with Dexter , that the natural widths, 7J, are many orders of magnitude l e s s than the actual band widths so that each integrat ion can be p e r f o r m e d by sett ing

TjE , ( E | 2 — E 2 ) 2 + ( T ¿ E f 2 F 6 ( E ' " E ) '

б being the Dirac deltá function. Fur ther , the p r e s e n t a n a l y s i s has been made by neglect ing K2 compared to n2 . When th i s s impl i f i ca t ion i s made, the integrat ions are per formed and the concentrat ions of the absorbing c e n t r e s are written as fract ions of the uranium atoms present , Nu , so that Eq. (7) b e c o m e s

6n(E)K(E) e 2 h2 N T I V /N¡ N V f l p t . .„.

If only two distributions are ass igned ( s e e text), it i s implied that one can write

. ^ f j G j t E ) - G,IE). (9)

1

9 6

An analys is has been made on this basis by using the same Gaussian dis tr i -bution a s proposed by Dexter:

dt

Gj(E) = (•JwWj)'1 exp [ - (E - Ej)2/Wj2], (10)

which i s normalized to the degree that the width measure, Wj , i s small com-pared to Ej. That i s

. / E - E j Г «

i M — = Г . 17 J e W. J

0 E j /2 —1 L

Wi

and the integral on the right i s only 10'4 for Ej/Wj a s s m a l l a s 2. 57. Some further cons iderat ion to deta i l s of substitution (9) must now be

made . If G®(e) i s Gauss ian , of the form of Eq . (10 ) , it can be shown that

the integration corresponding to normal izat ion g i v e s the resul t Ef? , and i '

that the mean energy i s

Si-'Í 'W. i

where

'l-I'i-t

Thus Ej would be a s s o c i a t e d with Ej of the text , and the la t ter cannot be d irec t ly a s s o c i a t e d with any l e v e l spacing.

The width m e a s u r e of one of these composi te peaks can be found from the average of 2(E-Ë)2 to be

W2 ^ f / t W » )2 + 2^Tf | (Ej - E)2

If it i s a s s u m e d that every W?, the width m e a s u r e for a s ingle transit ion, has the s a m e value, than

W2= f j tWf) 2 + 2 ^ f ' ( E ® - E ) 2 .

t

If i s es t imated by using resu l t s for an F centre in an alkalihalide, it i s

found to be ~ 0 . 3 eV at room temperature. If, in this c a s e , it i s assumed

that all values of E ? - E are small compared to the widths and fy= 2, f¡ = 6 are

u s e d , one f inds the c o m m o n W® to be 0 . 4 3 eV and the c o m m o n W¡ to be

0 . 4 5 eV.

9 7

R E F E R E N C E S

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of non-stoichiometry in U 0 2 + x " . Thermodynamics of Nuclear Materials, IAEA, Vienna (1962) 713. 53] MARKIN. T . L . and BONES, R.J. , U . K . A . E . A . Rpt AERE - R 4042 (1962). 54] KIUKKOLA, K . , Acta. chem. scand. 16 (1962) 327. 55] GERDANIAN, P. and DODÉ, M . , C .R. Acad. S e i . , Paris 255 ( 1962) 665. 56] ANTHONY, A . M . , KIYOURA, R. and SATA, T . , J. nucl . Mater . 10 (1963) 8. 57] HUBER, E .J . , HOLLEY, C.E. and MEIERKORD, E .H . . J. Amer. chem. Soc. , 74 (1952) 3406; FARR, J.D.,

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U02 and U3Og, " Thermodynamics of Nuclear Materials, IAEA, Vienna (1962) 693. 61] ARONSON, S . , RULLI, J.E. and SCHANER. B.E., J. chem. Phys. 35 (1961) 1382. 62] ROTHWELL, E . J . , J. nucl. Mater. 6 (1962) 229. 63] AITKEN, E . A . , BRASSFIELD, H.C. and McGURTY, J . A . , ANS Trans. 6 (1963) 153. 64] ANDERSON, J .S . , SAWYER, J .O. , WARNER, H. W. , WILLIS, E.M. and BANNISTER, M . J . , Nature,

London 185 (1960) 915. 65] DUNCAN, R.N. and FERRARI. H . M . , ANS. Trans. 6 (1963) 154. 66] MAY, J .E. , NOTLEY, M . J . F . , STOUTE, R.J. and ROBERTSON. J . A . L . , AECL - 1641 (1962). 67] ANDERSON, J . S . . Proc. roy. Soc. A 185 (1946) 69. 68] ROBERTS, L.E.J , and WALTER, A.J . , U . K . A . E . A . Rpt AERE - R 3345 (1960). 69] MARKIN, T . L . and BONES, R . J . , U . K . A . E . A . Rpt AERE - R 4178 (1962). 70] ÖLANDER, A. , Z . Meta l lk . U (1937) 361. 71] ARONSON, S. and CLAYTON, J . C . , J. c h e m . Phys. 32 (1960) 749. 72] ANDERSON, J .S . , EDGINGTON, D.N. , ROBERTS, L.E.J, and WAIT, E. , J. chem. Soc. (1954) 3324. 73] HUTCHINSON, C . A . , private communicat ion. 74] HAGEMARK. K . , Kjeller Rpt. KR-48 (1963). 75] ACKERMANN, R . J . , GILLES, P. W. and THORN, R.J. , J. chem. Phys. 25 (1956) 1089. 76] ACKERMANN, R . J . , THORN, R.J. , TETENBAUM, M. and ALEXANDER, C . , J. phys. Chem. 64

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London (1957) 218. 82] ROBERTS, L.E.J . , WALTER, A.J . and WHEELER, V . J . . J. c h e m . Soc.(1958) 2472. 83] FERGUSON, J . F . and McCONNELL, J. D. M . , Proc. roy. Soc. A 241 (1957) 67. 84] McCONNELL, J . D . M . , J. c h e m . Soc. 1958. 947. 85] ANDERSON, J . S . , ROBERTS, L.E.J. and HARPER, E. A . , J. chem. Soc. (1955) 3946. 86] HOEKSTRA, H . R . . SANTORO, A. and SIEGEL, S . , J. inorg. nucl . C h e m . 18 (1961) 166. 87] JAKËS, D. and SEDLAKOVA, L. , Proc. Prague Conf. 1 (1963) 155. 88] KOLAR, D. , Croat ia . C h e m . Acta 35 (1963) 123, 280; NUS - R - 426 (1964). 89] ROBERTSON, J .A .L . , ROSS, A . M . , NOTLEY, M . J . F . and MacEWAN, J.R. , J. nucl . Mater. 7

(1962) 225. 90] BETHOUX, D . , THOMAS, P. and WEILL, L . , C .R. Acad. S e i . , Paris 253 (1961) 2043. 91] PENNINCKX, R. and NAGELS, P . , private communica t ion . 92] BALAR1N,M. andZETZSCHE, A. , Phys. Stat . sol. 2 (1962) 1670. 93] BATES, J.L. , Nucleonics 19 (1961) 83. 94] DANIEL, J . L . , MATOLICH, J . , Jr. and DEEM, H. W. , HW-69945 (1962). 95] DeHALAS, D . R . , Nucleonics 21 (1963) 92. 96] RÜSSEL, L. E. , U . K . A . E . A . Rpt AERE, Harwell, private communica t ion . ;97] CHRISTENSEN, J . A . , WCAP - 2531 (1963). 98] KLEMENS, P. G . , Solid State Physics, 7, Academic Press, London (1957) 1.

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[99] GENZEL, L . , Glas tech. Bei ich. 26 (1953) . [100] WOLFE, R . A . , WAPD-270 (1963). [101] BATES, J . L . , H W - 79033 (1963). [102] RALPH, J. E . , Kings Co l l ege , London, pr iva te c o m m u n i c a t i o n . [103] SMITH, R . A . , Semi -Conduc to r s , C a m b r i d g e University Press (1959) 169. [104] BRIGGS, A . , U . K . A . E . A . Rpt AERE, Harwel l , p r iva te c o m m u n i c a t i o n . [105] MEYER, W . , Z. E lek t rochem. 5 0 ( 1 9 4 4 ) 274. [106] WILLARDSON, R.K. and MOODY, J . W . , in Uranium Dioxide (J. BELLE, Ed.) , USAEC, Washington, D . C

(1961) 243. [107] WILLARDSON, R.K. , MOODY, J .W. and GOERING, H. L . , J. inorg. nucl. Chem. 6 (19S8) 19. [108] HEIKES, R.R. and JOHNSTON, W . D . , J. chem. Phys. 26 (1957) 582. [109] VanHOUTEN, S . , Phys. Chem. Solids 22 (1960) 7. [110] NAGELS, P . , DEVREESE, J. and DENAYER, M. , J. appl. Phys. 35 (1964) 1175. [111] HAS1GUT1, R.R. and KIYOURA, R . , "Fundamenta l researches on the physical meta l lurgy of nucl'ear

fuels in Japan , " Proc. 2nd UN Int. Conf . PUAE 6 (1958) 34. [112] NAGAEV, E . L . , Fiz . Tverdogo Tela 4 (1962) 2201; Eng. Trans. Soviet Phys. Solid State 4 (1963) 1611. [113] NAGELS, P . , private communica t ion . [114] AMELINCKX, S . , Physical Properties of U 0 2 Single Crystals , Euratom Quarter ly Rpt N o . 2 (1962) 21 . [115] ACKERMANN, R . J . , THORN, R.J . and WINSLOW, G. H. , J . op t . Soc . A m e r . 49 (1959) 1107 . [116] WILLIAMS, J . , p r iva te c o m m u n i c a t i o n . [117] MOTT, N .F . and GURNEY, R. W . , Electronic Processes in Ionic Crystals, Oxford Univ. Press, Oxford

(1950) 160. [118] COMPANION, A. and WINSLOW, G. H . , J. opt. Soc. Amer . 50 (1960) 1043. [119] DEXTER, D . L . , Phys. Rev. П 1 (1958) 119. [120] BODINE, J. H. and THIESS, F.B. , Phys. Rev. J98 (1955) 1532. [121] GRUEN, D . M . , J. Amer . c h e m . Soc. 76 (1954) 2117. [122] KIKUCHI, T . and NASU, S . , pr ivate communica t ion to BATES, J .L . [101] . [123] TRZEBIATOWSKI, W. and SELWOOD, P. W. , J. Amer . c h e m . Soc. 72 (1950) 4504. [124] SLOWINSKI, E. and ELLIOT, N . , Acta Cryst. 5 (1952) 768. [125] FISHER, W.E. , Phil. Mag. 7 (1962) 1731. [126] WESTRUM, E . F . , J r . , HACKER, H. and LIN, M . , pr ivate communica t ion . [127] HUTCHINSON, C . A . and CANDELA, G . A . , J. c h e m . Phys. 27 (1957) 707. [127a]DAWSON, J . K . and ROBERTS, L . E . J . , J. c h e m . Soc. (1956) 78. [128] AUSKERN, A. B. and BELLE, J . , J. nucl . Mater . 3 (1961) 267. [129] ROBERTS, L . E . J . , Far . Soc. Disc. N o . 2 3 (1957) 156. [130] COMPAAN, K. and HAVEN, Y. , Far. Soc. Disc. No. 23 (1957) 105. [131] AUSKERN. A.B. and BELLE, J . , J. nucl . Mater . 3 (1961) 311. [132] LINDNER, R. and SCHMITZ, F. , Z . Naturf . 16a (1961) 1373. [133] McNAMARA, P . , Thesis, London (1963). [134] LAGERWALL, T . , Nukleonik 4 (1962) 158. [135] LAGERWALL, T. and SCHMELING, P. , HMI-B27 (1963). [136] LAGERWALL, T . , Nukleonik 6 (1964) 179. [137] MacEWAN, J .R . and STEVENS, W. H. , J. nucl . Mater . 11 (1964) 77. [138] CHILDS,' В . G . , J. nucl . Mater . 9 (1960) 217. [139] LINDNER, R. and MATZKE, H . , Atomkernenerg ie 9 (1964) 2 . [140] LONG, G . , U . K . A . E . A . Rpt A E R E - R - 4 0 9 0 , 106. [141] BRENNER, A . , FELIX, F . . LAGERWALL, T . , SCHMELING, P. and ZIMEN, K .E . , EUR 1729d (1964). [142] BARNES, R .H. , KANGILASKI, M. , MELEHAN, J .B . and BOUGH, A. , USAEC Rpt SMI- 1533 (1961). [143] STEVENS, W. H. et a l . , TID - 7160 (1961) 7. [144] LINDNER, R. and MATZKE, W . , Z . Naturf . 14a (1959) 582. [145] HA WES, R. , U . K . A . E . A . Rpt AERE - R 3865 (1961). [146] ROTHWELL, E . , J. nucl . Mate i . 5 (1962) 241. [147] ROTHWELL, E . , pr ivate c o m m u n i c a t i o n . [148] MÖLLER, P . , WAGNER, K. and ZIMEN. K . E . , to be published. [149] DAVŒS, D. and LONG, G . , U . K . A . E . A . Rpt AERE-R4374 (1963).

[150] DeHALAS, D.R. and HORN, G . R . , J. nucl . Mate r . 8 (1963) 207.

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[151] MacEWAN, J .R . and LAWSON, V . B . . J. Amer , c e r a m . Soc. 45 (1962) 1. [152] MURRAY. P . . PUGH, S . F . and WILLIAM, J. Т Ш - 7 5 4 6 (1957). [153] ARMSTRONG, W. M . , IRVINE, W.R. and MARTINSON, R. H. . J. nucl . Mater . 7 (1962) 133. [154] 2,IMAN, J. M . , Electrons and Phonons, Oxford Univ. Press, London (1960) 175. [155] RICE, S . A . , Phys. Rev. 112 (1958) 804; RICE, S . A . and FRISCH, H. L . , Annu. Rev. phys. C h e m .

11 (1960) 187. [156] S C O T T , R . , HALL, A.R. and WILLIAMS, I . . I . nuc l . Ma te r . 1 (1959) 39. [157] ARMSTRONG, W . M . and IRVINE. W.R. , J. nucl . Ma te r . 9 (1963) 121. [158] BERMAN, R. M . , BLEIBERG, M. L. and YENISCAVICH, W . , J. nuc l . Ma te r . 2 (1960) 129. [159] BRADBURY, B. T . , p r iva te c o m m u n i c a t i o n . [160] WAIT, E. , e t a l . . U . K . A . E . A . Rpt AERE - R-4268 (1963). [161] BELLAMY, R. G . , Inst. M e t . , Symposium on Uranium and Graph i t e . Paper 8 (March 1962) . [162] BARNES, R .S . , U . K . A . E . A. Rpt AERE-R-4429 (1963) . [163] ROAKE, W . , H W - 73072 (1962) . [164] BURKE, J . E . and COBLE, R. L. , Progress in C e r a m i c S c i e n c e , 3 Pe rgamon Press (1963) 199. [165] COBLE, R. L . , J. appl . Phys. 32 (1961) 787, 793. [166] WILLIAMS, J. , S c i e n c e of Ce ramics 2 (1964) . [167] P O D É S t , M. , JAKÈS, D. , Rpt UJV 7 2 6 / 6 3 (1963) .

[168] PODÉáf , M. andJADESOVA, L., "Theory of the fabr ica t ion of c o m p a c t e d uranium d iox ide pel le ts by

. powder m e t a l l u r g i c a l me thods , " (in Russian) New Nuclear Mater ia ls including N o n - m e t a l l i c Fuels

1 IAEA, Vienna (1963) 117.

[169] BEL, A. , DELMAS, R. and FRANCOIS, B. , J. nuc l . Mate r . J1 (1959) 259.

[170] WILLIAMS, J. , BARNES, E. , SCOTT, R. and HALL, A. , J. nucl . Mate r . 1 (1959) 28 .

[171] JAKEâOVÀ, L . , S i l ica tes (in press).

[172] FOLWEILER, R . , J. app l . Physics 32 (1961) 773.

[173] WARSHAW, S . L . and NORTON, F. H . , J. A m e r , c e r a m . Soc. 45 (1962) 479.

[174] CHANG, R . , J. nuc l . Mater . 2 (1959) 174.

[175] PALADINO, A. E. and KINGERY, W. D. , J. c h e m . Phys. 37 (1962) 957.

[176] AUSTERMAN, S. B . , N A A - S R - 3170 (1958).

[177] AUSKERN, A.B. and BELLE, I . , J. c h e m . Phys. 28 (1958) 171.

[1781 ALCOCK. C. B. and McNAMARA. P . . Doctoral Thesis (McNamara ) , London (1963).

[179] DANIEL, R . C . , BLEIBERG, M.L. , ME1ERAN, H.B. and YENISCAVICH, W. , WAPD-263 (1963).

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Australia (1958), Australia AEC (1958) 588.

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[183] NAGELS, P . , private communicat ion .

[184] WINSLOW, G .H . , private communica t ion .

1 0 2

LIST OF PARTICIPANTS

The m e m b e r s attending the Panel Meeting, held on 16 - 20 March 1964 at the Headquarters of the International Atomic Energy Agency in Vienna to a s s e s s the thermodynamic and transport propert ies of the uranium di-oxide and related p h a s e s , were:

ROBERTS, Dr , L . E . J . (Chairman)

BELBEOCH, D r . B e l l a ,

CORDFUNKE, D r . E . H . P .

F E L I X , D r . F .

HAGEMARK, D r . K.

JAKES, D r . and M r s . D .

KUBASCHEWSKI, D r . O.

KOLAR, D r . D.

LAGERWALL, D r . T .

C h e m i s t r y Div i s ion Atomic Energy R e s e a r c h Es tab l i shment Harwel l , Didcot , B e r k s . England

C o m m i s s a r i a t à l ' energ ie atomique C . E . N , de Saclay B . P . No. 2, G i f - s u r - Y v e t t e Seine et Oise , F r a n c e

Reac tor Centrum Nederland Pet ten ( N . H . ) Nether lands

Hahn-Meitner-Inst i tut für Kernforschung G l i e n i c k e r s t r a s s e 100 B e r l i n 39 F e d e r a l Republic of Germany

Institute of S i l icate Sc ience Techn ica l Univers i ty of Norway Trondheim, Norway

Nuc lear R e s e a r c h Institute of the Czechos lovak A c a d e m y of S c i e n c e s R e í u Prahy Czechos lovak Soc ia l i s t Republic

National P h y s i c a l Laboratory Teddington, Middlesex United Kingdom

N u c l e a r Institute "J. Stefan" Jam ova 39 Ljubljana, Yugos lavia

Hahn-Meitner-Inst i tut für Kernforschung G l i e n i c k e r s t r a s s e 100 B e r l i n 39 F e d e r a l Republic of Germany

1 0 3

Studiencentrum v o o r Kernenerg ie Laborator ie van het S. C . K . t e Mol-Donk Be lg ium

Argonne National Laboratory 9700 South Cass Avenue Argonne, 111. United States of A m e r i c a

Hahn-Mei tner-Inst i tut für Kernforschung G l i e n i c k e r s t r a s s e 100 B e r l i n 39 F e d e r a l Republic of Germany

WESTRUM, D r . E . F . Department of C h e m i s t r y U n i v e r s i t y of Michigan Ann Arbor , Mich. , United States of A m e r i c a

WILLIAMS, D r . J. Metal lurgy Div i s ion Atomic Energy R e s e a r c h Es tab l i shment Harwel l , Didcot , B e r k s . United Kingdom

WILLIS, D r . B . T . M . Metal lurgy Div i s ion Atomic Energy R e s e a r c h Es tab l i shment Harwel l , Didcot , B e r k s . United Kingdom

WINSLOW, Dr . G . H . Argonne National Laboratory 9700 South Cass Avenue Argonne, 111. United States of A m e r i c a

NAGELS, D r . P .

THORN, D r . R . J .

WAGENER, D r . K.

SECRETARIAT

Sc ient i f i c S e c r e t a r i e s : HARA, D r . R. \ D i v i s i o n of R e s e a r c h HOLLEY, D r . C . E . J and L a b o r a t o r i e s , IAEA

1 0 4

REPORTS SUBMITTED TO THE PANEL

The fol lowing reports were submitted to the Panel and the a s s e s s m e n t contained in this publication i s based on these reports and d i s c u s s i o n s held among the Pane l m e m b e r s :

[1] BELBEOCH, B.

[2] CORDFUNKE, E . H . P .

[3] HAGEMARK, К.

[4] JAKES, D.

[5] KUBASCHEWSKI, О.

[6] LAGERWALL, T.

[7] NAGELS, P .

[8] ROBERTS, L . E . J .

[9] THRON, R . J . and WINSLOW, G . N ,

[10] WILLIAMS, J.

[11] WILLIS, B . T . M .

[12] MÖLLER, P . WAGENER, К , and ZIMEN, K . E .

U 0 2 - l i k e s tructures

The e l e c t r i c a l propert ies of urani-um dioxide

The partial f ree enthalpy, enthalpy and entropy of oxygen in the U 0 2 + x

phase at higher temperatures

About the poss ib i l i ty of appreciat-ing the influence of non-sto ichio-metr ic oxygen on structure, s inter-abil ity and some other propert ies of U 0 2

P h y s i c o c h e m i c a l propert ies of uranium dioxide

Argon diffusion in calc ium fluoride as a model p r o c e s s for f i s s i o n gas transport in uranium dioxide

Preparat ion and physical propert ies of U 0 2 s ingle crys ta l s

Thermodynamic , chemica l and magnetic propert ies of the non-s to ich iometr ic uranium dioxide phase

Some aspec t s of the thermo-dynamic and transport propert ies of the U 0 2 phase

Notes on the pract ical s igni f icance of certa in thermodynamic and transport propert ies of uranium dioxide

Neutron and X - r a y diffraction studies of U 0 2 , U0 2 + x and U 4 O 9

Thermograv imetr i c t e s t s of urani-um dioxide and interpretation of the non- ideal rare gas re l ease on annealing of irradiated samples

1 0 5

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Documents

i of Uranium Dioxide and Related Phases · 2011. 8. 9. · technical reports series no. 39 thermodynamic and transport properties of uranium dioxide and related phases report of the