THE C O N N E C T I O N B E T W E E N T H E S T R U C T U R E

OF THE T H E R M A L E X P A N S I O N OF U R A N I U M ,

AND P L U T O N I U M

N. T . C h e b o t a r e v

Translated from Atomnaya ~nergiya, Vol. 10, No. 1, pp. 43-49, January, 1961 Original ar t ic le submitted April 11, 1960

AND A N I S T R O P Y

N E P T U N I U M

A connection has been established between the anisotropy of the thermal expansion and the changes occurring in the structure of a number of modifications of uranium, neptunium and plutonium on heat ing. It has been shown that in a l l structures considered the anisotropy of the thermal expansion

is due to a weakening of the four covalent bonds with increase in temperature and their nonuniform distribution in the various crystallographic directions of the l a t t i ce . The regulari t ies given lead to a better understanding of certain specific properties of uranium, neptunium and plutonium which distinguish them from ordinary metals .

I n t r o d u c t i o n

k is known that noncubic modifications of uranium, neptunium and plutonium have different values for the co-

efficient of thermal expansion in different crystal lographic directions. The anisotropy of the thermal expamion has an important effect on the properties of these meta ls . The different values of the coefficient of thermal expansion in the various crystal lographic directions indicate that during heat ing or cooling, the crystal l ine l a t t i ce does not remain unchanged. This in its turn means that the mutual positions of the atoms in the crystal l ine la t t i ce change. Therefore, for an analysis of this phenomenon it is essential to establish those relationships governing the change in the in tera- tomic distances in the crystal l ine lat t ices with change in temperature .

S t r u c t u r e a n d T h e r m a l E x p a n s i o n of a - U r a n i u m m . . . . . . . . . . . . .

The a lpha-phase of uranium has a rhombic structure with la t t ice parameters a = 2.853 A, b = 5.866 A and

c = 4.955 A. The four atoms of uranium in the unit ce l l occupy positions 4(c) of the space group D~--Cmcm:

t

0 y ~- ~ q _ ( 0 0 0 i t

where y = 0.105.

On the projection of three unit cells on the (001) plane (Fig. 1) i t can be seen that the atoms of zigzag chains threading the la t t ice in the direct ion of the c-axis are arranged in the form of layers with a distance between them of c / 2 . One of these layers is represented by the atoms ABCDEFO, the other by the atoms AIBIC1DIEIFIO1. On considering a tetrahedron with a base in the form of an isosceles tr iangle OAF and apex 01, the following character oI coordination appears: 2dl= OO i = 2.77 A; 2 d z = OB = 2.85 A; 4 d s = OA = 3.26A; 4d 4 = OC 1 = 3.31 A. The mean interatomic distance d m = 3.13 A. If the to ta l coordination number is equal to 12 then the four shortest bonds

are separated out with in tera tomic distances 2.77 and 2.85 A.

If the t r iangle OAF was equi lateral and the project ion of the atom 01 was exact ly at its center, then the struc- ture of the a -u ran ium would be an ideal hexagonal packing. Table 1 compares the values of la t t ice parameters of such an idea l packing (with the condit ion r = d m / 2 ) with ac tual values of parameters for the structure of a -u ran ium. Here we give the coefficients of thermal expansion in the range 20-500"C according to the data of [1].

4O

TABLE 1. A Comparison of the Lattice Parameters of a-Uranium and an Ideal Compact Packing

Lattice par ameters

Ideal values of parameters, A

3.13 5.42 5.11 0.16'/

Actual values of parameters, A

2.85 5.87 4.96 0.105

Extent of deviation, %

-9 .0 +8.3 -2.9

-37

Coefficient of thermal ex-

pansion along the axis, 10 r

32.9 -6.3 27.6

It can be seen from these data that under the action of the above four covalent bonds the lattice suffered con- siderable deformation, being stretched along the b-axis and shortened along the a-and c-axes.

The interatomic distances in the unit cell are determined from the following exPressions~

da = -5 V (4yb) 2 + r ;

d 2 = a; t :~ } (1)

ds --=-2- + b2 " I

d 4 = ~ l / a ~ + + c 2 -- Sub~ ( i - - 2y)

Using expressions (1) we can plot the dependence of the interatomic distances on the temperature since the temperature dependence of the lattice parameters a, b and c__ are now well known [2],

0 . ' - a,[ O-

" f l , ~ - - - i : . , I

t ~ I ~ " - '

"'1- 7"--..

E~, E

Oi@ O- @

Fig. 1. The projection of three unit cells of a-uranium on the (001) plane. Height in fractions of a parameter c_: e) 1 /4 ; 0 ) 3 / 4

The results of the calculations are shown in Fig. 2. This figure also gives values of the parameter.y which (taking into account the absence of noticeable changes with temperature in the difference in interatomic distances d4-da) can be determined from the following expression:

=02 0_ 0 (2)

With increase in temperature the short bonds (see Fig. 2,a) in- crease considerably, whereas the long bonds (see Fig. 2, b) do not change noticeably. The value of y also increases from 0.105 at room temperature to 0.112 at 640~ which is in good agreement with the experimental data given in [1].

From equations (1) it follows that a = d~; b = V - ~ ] _ d~ ;

~ / d 2 / ,]2 AS 2 1 - [ - ~ 3 - - ~ 4 e = 2 d l - - 4d]_dl ,

i .e. , that an increase in the short bonds d 1 and d z with temperature in the absence of noticeable changes in d s a n d d 4 leads to an increase in the parameters a and c_ and a decrease in the parameter b.

The effect of short bonds on the anisotropy of thermal expansion can be seen from a consideration of their ori- entation with respect to the axes of the latt ice. The values of the direction cosines for these types of bonds are as

follows, t ~ ~" Type of bon cos a a cos a b cos %

t 2d l=2 .77A I 0 0,44 0.90 I 112d2=2:85A ' 1 0 0

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It can be seen from these data that short bonds act mainly in the direction of the a-and c-axes, causing an increase in the coefficient of thermal expansion in these directions with respect to its mean value. On the other hand, in the diection of the b-axis, the coefficient of thermal expansion is much less than its mean value.

y d,I 0.I12 2.91 2.86 .,:

.2.,85 2.90 0. I I I ";• i �9 .~2.89 2.8~/

, . i O.lOg ~2.83 2,83 i x ~2.87 2.82 • / I

o IO~0'10~ lZSS= Sl .~. . I / "2,~ t l ~ ~ i I 0.1O8

o , , o 5 _ r . . . ~ 2 ~ 7 ~ , . . . . " - ~,~,K

0.103 " l ' ""L78

"" 2.75 ! I

-200 -100 0 100 200 300 400 500 ~00

Temperature, *C

3.3r x

,~ 230

3.22 3.2r da

-2017 -100 0 iO0 200 300 400 500 600 b Temperature, ~

Fig. 2. Temperature dependence of interatomle dis- tances in the structure of a -u ran ium and the param- eter y: a) short bond and parameter Z; b) long bonds.

r

_1#

8 ,

o o

J

X

Fig. 3. Projection of four unit cells of a-neptunium on the (001) plane

This analysis shows that the strong anisotropy of the thermal expansion of a-uranium is caused by the consider- able weakening of the four covalent bonds with increase in temperature and their uneven distribution along the axes of the crystalline lattice.

S t r u c t u r e a n d T h e r m a l E x p a n s i o n o f a ~

According to the data of [3], a -neptunium has a rhombic lattice, the temperature dependence of whose dimensions is given in Table 2. The structure corresponds

,4 s _ Preen. In each unit cell there to the space group ~ h are eight atoms occupying the following positions:

( 4 l , i t , z , , 4 N p i - - 4 ( c ) : 3= Y l z G T 5 - - Y I , •- t - , j

where Yl = 0 .208 3= 0.006; z 1 = 0.036 4- 0.006:

, , , ) 4 N p = - - 4 ( c ) : _ _ + y~z,; -T , - ~ - - y~, y + z2 ,

where y~ = 0 .842 5: 0.006; z 2 = 0 .319 4- 0.006, Figure 3 shows the projections of four unit cells of

a-neptunium on the (001) plane. From a consideration of the diagonal cells ABCD, which consist of eight cells of the type EFGH and FIKG with dimensions a ' = 3.399 A, c ' = 3.332A (c ' / a ' = 0.98) and an apex angle equal to 88 ~ (the cells are on two levels) it can be seen that the structure of a-neptunium is the result of deformations of a body-cen- tered cubic lattice,

Like the previously considered structure, in the a - neptunium structure four covalent bonds appear clearly with distances from 2.60 to 2.63 A. The mean interatomie

distance is d m = 3o10 A.

In accordance with the data given in Table 2, the values of the coefficients of thermal expansion for a - neptunium in the range 20-275~ are equal to:

% = 2 4 . lO ~" i / ~

% = 2 5 . lO -s l/~ % = 3 4 . iO -6 i / ~

, a m = 2 8 , 6 . 1 0 -6 I /~

It can be seen from these data that the anistropy of ther- mal expansion for a-neptunium is expressed rather weakly. Since the anistropy is connected with nonuniform distribu- tion of short bonds along the axes of the lattice, as was ob- served for the structure of a-uranium, we will consider the projections of these bonds for the o~-neptunium strueture (Table 3).

It can be seen from these data that the covalent bonds are relatively uniformly distributed along all three axes of the latt ice, which is in good agreement with the

42

absence of c lea r ly expressed anisotropy of the thermal expansion.

At a temperature of 278 ~ 5~ a te t ragonal structure of 8-neptunium forms [3], the result of deformation of the

body-centered cubic l a t t i ce , but with a degree of distortion much less than for s -nep tun ium. Of the eight bonds in this structure, four correspond to a min imum in tera tomic distance 4d =2.79 A and the other four to 4dmax = + 3.26 A.

m'n Hence it can be seen that in the 8 -nep tun ium structure, four covalent ~onds appear, although in correspondence with the increase in tempera ture the difference d m - dmi n here becomes much less than in the a -nep tun ium structure (0.23 and 0.48 A respect ively) . At temperatures of 550-570~ a body-centered cubic la t t ic of ~,-neptunium forms

with eight ident ica l bonds at a distance 2.97 A.

S t r u c t u r e a n d T h e r m a l E x p a n s i o n o f y - P l u t o n i u m

According to the data of [4] 7"p lu t~ has a rhombic la t t i ce , the temperature dependence of whose d imen-

sions is given in Table 4.

TABLE 2. The Temperature Dependence of the Latt ice

Parameters of a -Nep tun ium

Latt ice parameters , A Temp. ,

~ a b c

20 4.723=[=0.001 4,88710.002 6.663+0.003 212 4.746+0,002 4,909~0.003 6.704+0.004 275 4.752• 4.920• i6 .722 i0 .004

TABLE 4. The Tempera ture Dependence of Latt ice

Parameters of 7"Plu t~

Temp. , oC

213 233 258 312

Latt ice parameters . A

3.16052 3,15909 3.15622 3.15397

5.76275 5.76769 5,77371 5.78574

t0.1442 t0 . t6 t5 10.t834 10,2290

TABLE 3. a -Nep tun ium Latt ice

Type of bond

Projections of Short Bonds on the Axes of the

Projections on the axes of the l a t t i ce , A

td1=2.598 A td2=2.632 A 2d3=2.634 A

y,

da

0 0 4.72 4.72

db

1.79 2,20 t .30 5.29

de

1.87 1.55 1.94 5.36

TABLE 5. The Temperature Dependence of the Inter- a tomic Distances in the Structure of Y-Plutonium

Temp. , ~

2t3 233 258 3t2

Interatomic distances, A

4d1 2d2

3.022 3,160 3.026 3,t59 3,03t 3.t56 3,052 3.t54

4da

3.286 3.288 3.290 3.295

The structure corresponds to the space group D R - F d d d . In the unit ce i l there are eight atoms occupying the

position 8 (a):

000 } ( , , t ~ ' ' )

t I t + 0 0 0 ; 0 ~ 2 ; 20-ff; -g-g0 4 4 4

The structure of 7 -p lu t~ is an al ternat ion of several distorted pseudohexagonal layers p laced perpendicu- lar to the c-axis. Figure 4 shows the project ion of three unit cei ls on the (001) plane and one unit ce l l on the (100) plane.

The in tera tomic distances in the 7 -p lu t~ structure and their temperature are given in Table 5. The mean in tera tomic distance at 213~ is d m = 8.16A.

From these data it can be seen that , s imilar to the previously considered structures, four covalent bonds appear c lear ly in the 7 -p lu t~ structure, the in tera tomic distances in the direct ion of which increase considerably (from 3.022 to 3.052 A) with increase in tempera ture . The in tera tomic distances in the direct ion of the long bond change much less, the distance d 2 even being reduced somewhat with increase in tempera ture .

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The values of the direction cosines for all three types of bond are as follows:

Type of bond

4dl~3.022A 2dz=3.160A 4da=3,268A

C05 @a

o

O. 48

COS ~b

0.88

COS ~c

0

In accordance with the data given in Table 4 the coefficients of thermal expansion of 7"plut~ in the range 213-312~ are equal to:

~ = - - 19.7. l 0 -" ! / ~ ]

~ = 39 .5 . i 0 -6 l / ~ } , ~ m = 34 .5 . l 0 -e I /~

ct c = 8 4 . 3 . 1 0 -6 I /~ J

On comparing these values with the direction cosines given above, it can be seen that the strong anisotropy of the thermal expansion is due to the short bond 4d 1, mainly acting in the direction of the c-axis. In accordance with this, the coefficient of thermal expansion in the direction of the c-axis has a maximum value, in the direction of the b-axis it is much less,and in the direction of the a-axis it has a negative value.

�9 8 r .

T_ T _ _ _ l O- 5.7g

a - / /

~ig. 4. Projections of three unit cells of 7-plutonium on the (001) plane (a) and one unit cell on the (100) plane (b). Height in fractions of a parameter c_: o) 0; Q) 1/4; O)1/2; �9 3/4.

S t r u c t u r e a n d T h e r m a l E x p a n s i o n o f a - P l u t o n i u m

According to the data of [5], a-plutonium has a monoclinic lattice, whose parameters at 21"C are:

a = 6 .1835 i 0 .0005 A ; b ~- 4 .8244 • 0 .0005 A;

c = t0 .973 ~k 0.001 A; ~ = 101.81 ~ 0 .02 ~

The structure corresponds to the space group C ~ - P21/m. In the unit cell there are 16 atoms, occupying

eight double positions:

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2(e): • .

The approximate values of the x and_z parameters are as follows:

Parameters PUl PUll Pul i I Pulv

x 0,332 0,767 0.138 0;65t z 0.152 0.169 0.337 0.456

Parameters Puv ~uvz Puv I I PUVlll

x 0.013 0,459 0.335 0.885 z 0,6t7 0.642 0.924 0.897

Figure 5 shows the project ion of four unit cel ls of or-plutonium on the (010) p lane . The numbers near the a t - oms indicate the number of the position, the atoms with a minus sign are p laced at a height equal to half of the

pa ramete r with respect to the remain ing atoms ly ing in the p lane of the diagram. The structure of the a - p h a s e can be represented as an a l te rna-

t ion of very strongly distorted pseudohexagonal layers p laced perpendicular to the b -ax i s at a

dis tance equal to a half of the parameter .

, - - ~ 7

TABLE 6. In tera tomic Distances in the a -P lu ton ium Structure

Short bonds Long bonds

.~ ~ in te ra tomic o ~ " distances

in te ra tomic ,.Qo~ ~-~ distances ~,.C3

'xzJ,.~

L) E

l I I

I t i IV

VI I V I I I

5 2,52--2.69 4 2.62--2.69 4 2.54--2.67 4 2.54--2.70 4 2.54--2.73 4 2.58--2.73 4 2.52--2.70 3 2.68--2.70

7 t0

t 0 t0 10 t0 10 13

3.40--3.51 3.33--3.62 3 20--3.67 3.20--3,51 3.30--3.56 3.33--3.62 3.34--3.56 3.30--3.67

t2 t4 t4 14 14 t4 14 16

TABLE 7. Projections of d-d m on the Main Crys ta l lo- graphic Directions of the a - P l u t o n i u m Latt ice

Direct ion Projections of d -d , A

m

for short bonds for long bonds

. L c

fl b II c

--8.8 --7.3 --27.4

8.8 7.3 9.4

Fig. 5. Project ion of four unit cel ls of plutonium on the (010) plane.

45

The interatomic distances in the a-plutonium structure for atoms at different positions are given in Table 6.

The top part of Figure 5 shows the short bonds for atoms in all positions, the continuous lines representing bonds lying in one layer and the dotted lines representing bonds between the atoms of neighboring layers (each dotted line corresponds to two bonds).

Atoms at the positions I and VIII (for example, atoms M and N) are joined by a short bond. Considering these atoms as a single whole, we find, as at the other positions, a mean number of bonds per atom of 4, and a coordina- tion number of 14. Therefore, in the a-plutonium structure, as in all the structures considered above, four covalent bonds clearly appear.

TABLE 8. Dependence of the Coefficient of Thermal Expansion on the Main Crystallographic Directions of a-Plutonium and the Temperature

Coefficient of thermal expansion

a~ j_c a2 [I b az II c

Temperature range, *C

21--10~ --196--20

66 49 73 53 29 t4

Since the short bonds in the a-plutonium structure nevertheless vary over certain limits (from 2.52 to 2.73 A) in order to characterize their distribution in three mutually perpendicular directions in the lattice, calculations were

made for the projections of the difference d-d m in these directions. The results of the calculations are given in

20-i00 Table 7.

Table 8 gives values of the coefficient of thermal 68 expansion along the directions shown in Table 7 in the 77 35 temperature range 21-104"C [6], -196-20, 20to 100~

It canbe seen from the data of this table that the pro- jection of d-d m both for short and for long bonds has a consid- erable value along all three directions of the lattice. There- fore, for a-plutonium thereisnot such a clearlyexpressed anisotropy of the thermal expansion as for 7 -plutonium.

- to a-neptunium, the difference in the distances d m - d m i n increases from 0.23 to As in the transition from 0.48 A, on transition from 7 - to a-plutonium this difference increases from 0.12 to 0.56 A. In agreement with this, the covalent character of the bond on transition from high temperature modifications of plutonium to the lower tem- perature modifications increases considerably and the crystalline structure deviates to a greater extent from the struc- ture of ideal metall ic lattices assembled on the principle of compact packing of the atoms.

The saturation of covalent bonds in the low temperature region and their relatively uniform distribution along the various crystallographic directions means the the anisotropy of the thermal expansion for a-plutonium is expressed

comparatively weakly.

With increase in temperature, the weakening of the covalent bonds is accelerated and their distribution in the various crystallographic directions becomes more uneven, which finds expression in the strengtheuingof the anisotropy of the thermal expansion. In fact, the anisotropy of the thermal expansion for/~-plutonium, as follows from [6], is expressed more strongly than for a-plutonium,and the anisotropy for y-plmonium is expressed more strongly than for t~-plutonium. The problem of the thermal expansion of 5- and ~-plutonium is more complex and requires a special

study.

The author is very grateful to S. T. Konobeevskii for discussing the results of this work and for his valuable com-

ments.

LITERATURE CITED

1. S.T.Konobeevskii et al, Proceedings of the Second International Conference on the Peaceful Use of Atomic Energy (Geneva, 1958). Report of Soviet Scientists, Vol. 3, Nuclear Fuel and Reactor Metals [in Russian] (Moscow),Atomic Energy Press, 1959, p. 396.

2. J. Bridge, C. Schwarz and D. Vaughan, J. Metals, 8, No. 10, 1282 (1956). 3. W. Zachariasen, Acta crystallogr., 5, No. 5, 660 (1952). 4. W. Zachariasen and F. Ellinger, Acta erystallogr., 82 No. q, 431 (1955). 5. Ibid., 10, No. 12, 776 (1957). 6. A. Coffi'--nberry e t al, Proceedings of the Second International Conference on the Peaceful Use of Atomic Energy

(Geneva, 1958). Selected Reports of non-Soviet Scientists, Vol. 6, Nuclear Fuel and Reactor Materials [in

Russian] (Moseow),Atomic Energy Press, 1959, p. 157. 7. N . T . Chebotarev and A. V. Beznosikova, Atomnaya gnergiya, 7, 6, 68 (1959).

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