IC/81/S31INTERNAL REPORT

(Limited distribution}

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THEORY OF THE SURFACE MAGNETIZATION PROFILE AHD THE LOW-ENERGY,

SPIN-POLAHIZED, INELASTIC ELECTRON SCATTERING

OFF INSULATING FEHROMAGNETS AT FINITE T •

Silvia Selzer

International Centre for Theoretical Physics, Trieste, Italy,

and

Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy,

and

Horberto Majlis **

Scuola Internazionale Superiore di Studi Avanzati, Tr ies te , I t a ly .

HIRAMARE - TRIESTE

December

* Submitted for publication,

• * Permanent address: Ins t i tu to de Fis ica, Universidade Federal Fluminense,Caixa Postal 296, 2fc.21O Hiteroi , RJ, Brazil .

• * « - : * w * • =

ABSTRACT

A semi - In f in i t e Heisenberg ferromagnet with nearest-neighbour

exchange I n t e r a c t i o n s i s s tudied a t f i n i t e T. The Green functions are

evaluated by extending the HPA to consider the s p a t i a l v a r i a t i o n of the

layer magnetization, which is calculated self-consistently at all

temperatures up to T . Results for an fee lattice vith a (111)

surface turn out qualitatively similar to previous results for a sc

lattice with a (100) surface. In both cases the surface has the same Tc

as the bulk, irrespectively of the difference between surface and bulk

exchange constants. A survey is preusnted of a continued fraction repres-

entation for the Green functiont in the mixed local-Bloch basis for the

spin operators, related to the transfer matrix formalism, which allows

explicit evaluation of the diagonal Green function at each layer. The

calculation was performed by truncating the spatial variation of the

magnetization, at any T, at the third layer of the lattice.

The presently available intense sources of spin-polarized electrons

offer o possibility of obtaining quantitative information on the surface

parameters. The differential cross-section for low-energy electron scattering

is calculated for the fee lattice at zero momentum transfer. The curves

show resonances near one or both ends of the bulk magnon energy band,

depending on the"surface exchange parameters.

-1-

I. INTRODUCTION

electrons

systems.

The recent development of efficient sources of spin polarized

1) has stimulated the study of the surface properties of magnetic

Both elastic and inelastic scattering of low energy electrons off

insulating magnetic sample9 can afford very detailed information on the2)

surface magnetic structure and on the spectrum of magnetic excitations,3) h)whether surface localized or bulk ' . This is because low-energy

electrons, wita energies up to 150 eV, do not penetrate further than one or

two atomic layers into the crystal, being in consequence an invaluable probe

of the surface properties.

In this paper we present what, to our knowledge, is the first cal-

culation in which are simultaneously obtained in a self-consistent way, the

layer magnetization profile, the bulk magnetization, and the spectrum of

bulk and surface magnetic excitations for any spin S and for all temperature!

T, up to T , of an isotropic Heisenberg semi-infinite ferromagnet with

nearest neighbour exchange interactions.

We adopt the Green function formalism of Zubarev and use the

mixed local-Bloch representation, as introduced in the context of surfaceU)

magnetism by De Wames and Wolfram .

The Green functions are evaluated in the EPA by the simultaneous

self-consistent evaluation of the layer magnetization at the surface (n - 0)

plane, and the next inner plane (n » 1), assuming for simplicity that the

bulk magnetization value is reached at n - 2. We have already briefly

reported elsewhere on our KPA results for the surface magnetization in

simple cubic ferromagnets vith a (100) surface. We now present the method

of calculation of the Green functions {See.II) for the simple cubic and the

fee semi-infinite lattice structures, expressing them as continued fractions.

Since the bulk magnetization is a necessary ingredient, we calculate

i t also In the HPA, following the work of Tahlr-Kheli and Ter Haar ' .

An RPA calculation of the self-consistent magnetization profile for a 20

layer thin film, including exchange and single-ion anisotropy, was performed8)

by Diep-Th«-Hung at al. for S • * in the sc and bec structures. We

discuss our results for the magnetization profile, the surface magnetization

and th* Surface magnon dispersion relations in Sec.III.

-2-

Sec.IV is devoted to indicate how these results can te applied to the

calculation of the differential cross-section for inelastic scattering of low

energy electrons. Calculations thereof at low temperatures have already teen

performed (see Kef.9 and references therein).

In a similar calculation for a metallic ferromagnet, Saldafia and

Helman used pseudo-potential results for the bulk magnetization, although

they recognize that the surface magnetization should be used instead. We

consider, following previous treatments of this problem •"»-", that electrons

interact only with the surface spin layer, and that their interactions with

the spin system is of the contact form proposed by De Gennes and Friedel

In the fee case with a (111) surface, numerical results are

specialized to the case of EuO with "• * i* s l n c e ve * e l : t e v e i* should be

a good candidate for eventual comparison with experiments, but it is found

that no qualitative changes arise for other values of S.

II. SELF-CONSISTENT CALCULATION OF THE MAGNETIZATION PKOFItE

We consider a semi-infinite, insulating ferromagnetic crystal

described by the Heisenberg Hamlltonian

H • - (1)

where the exchange constants between nearest neighbours are: I., for bonds

in the plane of the surface, for bonds between the surface and the

second plane and I for all other bonds. We calculate Zubarev's Green

functions

The sub-indices denote lattice sites. The statistical mean ^SZ \ only

depends on the position of the plane parallel to the surface. We assume

the magnetization to be parallel to the surface in order not to consider

demagnetizing fields.

The resulting equations of motion are:

These equations are identical to those used in Kef.8 for a finite number of

layers.

2.J. Simple cubic ferromagnet with nearest neishbour interactions

' It is convenientM,6),8) to decompose the lattice vector f = (fu,f,)

to define the Fourier transform of (It) with respect to the "parallel"II' 3'

wl»«re we have taken advantage of the translation symmetry parallel to the

surface

We obtain the matrix equation

= -i8(t-t0) <[S*(t]

We use a generalization of the HPA decoupling '' ' that allows for spatial

variation of

surface:

N over an arbitrary number of planes parallel to the

- H e f f) - 2-trAft =

a i

17a)

(7b)

(3)where

- 3 - -h-

mmm - mm m

eff

+(8a)

defining the dimenalonlesa variables:

ofherwVs*

cet>)

We take for •Implielty <sm • 1 for i > , 2 ; V is the angular frequency ljj

units of 21 <S^>1> •

The matrix

- H (9)Is

4 •

0

0

O

0

J do)

- 5 -

wbere, caning ana ,we nave

(U)

A - <> ' (12)

Since 0 l i trl-fliagoilal i t is easy to calculate directly the elements of

Calling OJJ a det 0 and

a"1.

JJthe determinant obtained from

deleting the f i rs t p rows and columns, we get

(13)

°oo' V i * ho "oi(at

Vs

For p > 2 a l l determinants !>„ have the non-perturbed formt

1 2t 1 • ••

0 1 2t • ••

1 M < • t • j

«, * * • * * *

(13)

-6-

These determinants,also found by De Wames ana Wolfram in the solution of

the equations of motion for the operator S» in the Holstein-Primakoff

approximation, are the associate Tchebishey polynomials of the first kind

Using recurrence formulas (lit) one can rewrite (13) as

(16)

We substitute again D^^ in terms of DK_g and D and so on, until we

arrive at a ratio of two unperturbed determinants of the form of U 5 ) for

This ratio can now be written as

(17)

Taking now the limit (H~p) •+ « we define

Eq.(lSa) can be written in the limit H-p -• »

0M-

(18a.)

U8b)

equivalent to

(19)

- 2tt + 1 =• 0

-7-

(20)

+ e"1 - 2t (21)

Therefore, for a semi-infinite system, B

the form

» , (n" ) Q 0 and (a"1)-L1

(22a)

(22b)

In order to interpret the physical aeaning of the parameter E(t) (E<1. (l8a))

ve shall need the analytic continuation of Un(t) to the complex t plane.

For t real, and |t| < 1, we define t = cosB , and we can use the

representation

(23)

Uft(t) is an n-degree polynomial in t. All its roots are contained in \t\ $ 1.

In this interval of t, £ is complex,since from (20):

18£(t) • t t i (I - t V / 2- 1 - (2<t)

We must choose la 5 < 0, for, in this case, according to (22), (n )„_,

) . . , etc., will have a negative imaginary part, corresponding to the

correct analytic continuation, g .

If t - cos9 , for

then n > 0 if sinS < 0 and

(25)

= e19' = e i ( 9 + i n ) = e " e'"(26)

For E real, we must choose the solution l£l < 1 , to which the continued

12)fraction converges and it can be immediately verified that when

t + t + i6, E -»• £ - iC , with 5 > 0.

Fig.l represents the mapping t •+• 5(t) on the complex £ plane,

where the particular branch has been chosen for which J£(t + it)| < 1 and

Im 5(t + i6) < 0 .

The physical meaning of the parameter 5 can be visualized by con-

sidering one of the unperturbed equations (T) for a non-diagonal element

(fig) n ?! I for 1 ) 3 ;ni,

2 t (27)

If we call

8n,Jt ~ O O n B t* *n > ( 2 8 )

where the constant factor depends on I , Eq.(27) can be rewritten as

with solutions

x + 2t x + x .,n-1 n n+1

Ap

= 0 (29)

(30)

-1,

if p satisfies

(p + p"x) + 2t = 0

Comparing EQA.(31a-)with Eq. (21) we see that

(31a)

P = -5•(31*)

The quantities are the amplitudes of the eigenvectors of H at the

plane n, since Eq,(29) can be recognized as the eigenvalue equation

,ueff +x = 0,

for fixed k .

The relations (28), (30) and (31b) imply

(32)

We see that p(kn,w) is the transfer matrix for this case.

Let us now return to Eqs.(7aJ and (7b) for the Green matrix g and

observe that the poles of g coincide with the zeroes of det il (defined

in Eq.(9)}.

g, for all n and 1 can be written as the ratio of two poly-

nomials. The diagonal elements of g, in particular, which are the only ones

we shall need to evaluate explicitly, can be expressed as,

(33)

where D(6 ,k..) does not depend on 4 . For complex 5 , g has a branch

cut along the semi-circumference |C| = 1 in the lower half plane, as shown

in Fig.l. For real I = X , a real root of D(5,kjJ = 0, we eliminate 2t

in terms of 5 from Eq.. (21), and substitute in Eq. (11) to obtain the

frequency at a surface magnon as

W = Ss + C" + 2 + h {1 ~ V V J ' {3k)

In Sec III we describe the results obtained for the dispersion

relation (3^), for different magnon branches and different crystal structures.

In order to evaluate the magnetization, we require the equal-time

correlation function ^S~ S *> . This can be obtained In terms of the5) a a

Green functions (2) as :-10-

135)

For the particular case S = p- , ve have

,2

(36)

Then, in terms of G defined in Eq, (5h

i* - U > 2 { f )= -i*. 2 c« x

(3T)

For a general value of S we use a result due to Hevson and Ter Haar ,

for the case of an infinite antiferramagnet:

as+i(38)

where

= -2

The labels (±) in that particular case, referred to the up or dovn sublattice.

In order to generalize (38) to apply for the case of a system with a

magnetization varying from plane to plane parallel to the surface, we mist

extend the formulae to include an infinite number of sublattices, each of

which is in fact one crystal plane. The integration over three — dimensional

-11-

k space must be substituted by an integration over the tvo-dimensional

k space, appropriate to the particular family of planes parallel to the

surface and the label ± must be substituted by the corresponding plane

index, so that we have one function 41 (S ) for each Diane in the semi-

infinite system.

In our case, then

2S+1

where I(I is:

'Ms

(U2)

We now refer to Fig.l, in order to evaluate the integraJ in C< J );

t varies from -«• to » as the frequency m does. In the range

_«. < t < -1 and 1 < t < ™, £ is real and varies from -1 to +1 .

Then in terms of the real roots {5^} v.i .n r.̂ - 0,

Substituting (1*3) into (1*1) for |t'| > X, we obtain the contribution to the

sptn deviation from the discrete spectrum of surface modes, denoted with

the index a in (U3) . The integral

-12-

in Ul) runs over the range where the Green function has a branch cut, itbeing the contribution to the spin deviation from the continuum spectrum of

•bulls, excitations.

Eqs.(ltO) for different n will give a set of self-consistent

, can b

(4/2

equations which, for S = -, can be written as

ihk)

2,2 The fee ferromawnet

We consider a till) surface. The basis vectors are chosen as

= f (1,0,1), §-(0,1,1), | (1,1,0)

The reciprocal vectors are:

B = ̂ (1,1,1), % = — (l,l,l), 31 = — (1.1,1)

(1*5)

(46)

A Reneral k vector will be written as

i = 1

or as 1? = (k[|,k,). Therefore a vector k"j| = (k-,k-) describes vave

propagation parallel to the plane (ill) although it is not itself parallel

to the plane.

In order to obtain the equation of motion for the Green function in

this case it is convenient to generalize the Fourier transform with respect

to f,, , gn by including a phase change from one plane to the next away

' ' k)from the surface

Ik) :

(US)

-13-

(fi

,)(1*9)

(|<(ki|) i s t h e phase of t he s t ruc tu r e fac tor h , defined as

h = $ e - ~ (1 + exp (2Trik ) + exp (2irlk ) ) ;

t = g- (3 + 2(cos 2ir ( t ^ - k2) + cos 2u ^ + cos 2TT kg) ;

tan (sin + sin (1 + cos + cos 2irk 2 )

Substitution of transforms (I4.8 ) eai C*9) into (1») gives just anequation equivalent to (7), and the introduction of the phase <Hk,.) (f, - g_)

effmakes the non-diagonal matrix elements of H real .

The exchange constants for bonds linking spins on the (111) surfaceplane and betveen the surface and the next inner plane are eventuallyassumed different from the volume constants. We define

1 - + Ml - (1 -

The matr ix il = \ ) i l - He

O

oo

oo

can be written as:

-4o

- l i t -

O0o

ooo

(50)

* • •

(51)

I t turns out convenient to divide a l l elements of S]

so that in terms of the new variables t ' • t/3$ , a,L = »„a j j = ci.,,/3(t> i e t c . , we obtain a matrix ft'

as a of Eq.(7) for the sc l a t t i ce . In particular, making the substitutions

by 3* (£.,),

u00 " "00'which has the same structure

t? °00 a00' e t c ' ' t o sinlP1:i-fy t n e notation we have

In terms of the new variable t,

v = 3(2*t + 2 + - Yn

(52)

(53)

and the self-consistent system of equations for a as defined in (8b) i sthe same as (1*0) and (1+1) with the argument of the Bose function modifiedto

a1(T,S) 2 + Ml - T1(£t

where(55)

III. RESULTS FOR THE MAGNETIZATION AND THE DISPERSION RELATIONS

We show in Figs.2 and 3 the curves for ^ s o ^ T E - T f o r

values of g^, with £,| = 1 and S = £ . Fig.h shows the dispersion

relations of the different surface magnon branches for the fee lattice

(the corresponding results for the sc lattice were already published in

Bef.6). The limits of the continuum are also indicated in 7Ig.it. As a general

trend, we note that increasing T is equivalent to decreasing £ and £.a >

since in both cases the total field on surface spins decreases.

As T increases, therefore, acoustic surface branches tend to appear

or to become more separated from the continuum from below, while on the

contrary,optical surface 'branches tend to disappear, or to approach the upper

limit of the continuum. & i ̂ 1 favours (°^ l c a. ) branches, respectively.

As it has already "beenfound earlier for the sc lattice, the bulk

critical temperature Tc also drives the transition to the paramagnetic pha3e

at the surface, in fee case.

From Figs.2 and 3 we observe that as £, decreases, the surface shows

a tendency to cross-over to the two-dimensional behaviour T = 0 .

As regards the spatial variation of the magnetization (5"ig. U) in the

present paper we have limited ourselves, for simplicity, to consider only

variations of ^ S / over the first two planes.

The results, which coincide qualitatively with those obtained

previously for the simple cubic case , indicate that the convergence to

the bulk magnetization is very fast indeed, supporting our assumption that

only the first few planes would differ appreciably from the 'built. Other

results, obtained with Monte Carlo calculations, indicate that a small

deviation from the bulk can extend to greater distances . We can however

he confident that this correction will affect our results by Just a few

percent after the second plane, since in two different calculations for the

sc structure, involving, respectively, one and tvo altered planes,

the surface magnetization differed by only 155S.

It must be remarked that as g,̂ increases above 1.5, the magnetization

of the second plane becomes slightly greater than the bulk value at high T.

This result may De due to the restriction to only two varied planes. Also,

the critical behaviour of ^ S^ y for T ^ T is presumably dependent on

this approximation, since the effect of imposing ag = 1 is to increase

artificially the self-consistent values ^ 3 ^ ̂ and <S* ̂?

Further work to extend the present calculations to o = 1 is

already in progress.

We estimate that the present results will be unaltered for T ^ 0 . 9 T c

and for P -$ 2.0.

-15- -16-

IV. INELASTIC SCAOTERMG OF LOW-ENERGY ELECTRONS

The interaction between the incident electrons and the spin lattice

is described by a contact potential introduced by de Gennes and Friedel :

(56)

where G is the coupling constant, SI is the volume of the unitary cell,

S(r) the spin operator of the incident electron, S-j- the spin operator in

t and,-Kposition of the lat t ice, / p(rj the electronic spin density of the

ion.

The differential cross-section in the Born approximation is:

m is the free electron mass; | k,s "^ the incident electron state;

|k',s'^ the scattered electron state; i and f denote, respectively,

the initial and final states of the target. W(i) = e~^Ei/Tr(e"^H) is

the probability of finding the magnetic system in the initial state i,

and P(s) is the probability that an incoming electron has spin s. We

assume small energy transfer E, so k' ~ k. The wave function of the

incident electron is assumed of the form

Let us calculate the matrix element

' (59)

The lth term of che Sum is (a « +, -,

- 1 7 -

r

We c a l l

<a'|sais> =

Now we calculate the integral

(60)

(61)

where

^AktrrH .(63b)

Here, we assume for simplicity that the electron wave functions ao

not depend on spin. Then, (57) becomes

••>> - »

s

, * , • " '

*£- (63)

The terms in C63) involving S are substituted in the BPA by the averages:

S Z —*t / s z S and are subsequently independent on time, so they do not contri-n ^ n '

bute to the inelastic scattering. They will be excluded in what follows.

-18-

We introduce the mixed loeal-Bloch spin operators

{6k)

Due to angular momentum conservation the part of (63) vhich contri-

butes to the matrix element (59) is

(65)

Substituting {65) into (63) and writing

we find

(66)

We shall assume that electrons do not penetrate farther than the first

plane , n = 0.

The correlation functions

related to the Green functionsA±t..,n OK||9n /

can be

X (67)

-19-

We assume that P(s), the polarization fraction of the incident electrons,

has the form

(68)

The matrix elements

We assume f- is parallel to the magnetization. For s' = t 9 corresponding

to absorption of energy by the electron beam,

>; BCE)(€9)

For s ' = ̂ (Corresponding to a loss of energy by the electrons

x

^11^ (70)

where B(E) = (efE - I)" 1 .

V. RESULTS FOH THE SCATTEEING CROSS-SECTIOH

Our results show that the spectral density of magtion states atthe surface plane, vhich i s the main ingredient entering expressions(6'9 ) and (TO) for the inelastic scattering cross-section, is stronglydependent upon the surface parameters. Curves for a a/(dEd£J} (Figs.6and 7) at small momentum transfer |Aki.| 'v-0, and for £ < 1 show Justa strong maximum near the low-energy edge of the continuum and as energyincreases, they decrease unti l near the upper edge ft = 1 Jim gQ0 vanishes

-20-

1/2like (1-t) . For £.*• 1, on the contrary, ve observe that, besides the

strong maximum close to the lower energy edge, there appears another peak

at the upper edge.

It is not difficult to Interpret this qualitative behaviour of

the local density of magnons. For £,< 1, as Fig.l* shows, there is an

acoustic surface magnon branch, of energy very close to the lower edge of

the continuum. This shov3 itself as a strong surface resonance in the bulk

magnon wave functions of low energy at n = 0, and as a concentration of bulk

states therein. As gj^ increases, we still find, as can be seen in Figs.6

and 7 for £, = 1.5> an acoustic surface magnon, but at the sarae time an

optical surface magnon branch appears at energies above the continuum,

aad this produces a similar resonant effect upon the bulk states vith energy

close to the upper continuum edge, which manifests itself in the smaller

peak at the high energy side.

The effect of temperature, apart from an overall scaling due to

the liulk magnetization factor in Im g (Eq.(7b))is to reduce appreciably

the surface magnetization as compared to the bulk value, as discussed in

Sec.II, and shown in Figs.2 and 3- This reduction of KSQ} a s T

increases, is in a vay equivalent to a reduction tioth of £ and £ ,

and in this respect the present results are different from those ofo)

Harriague et al. for the bcc ferromagnet at T fc 0, where only £

was modified.

Another obvious effect of a finite T is the possibility of

magnon absorption by the incident electron.

The calculations of Sec.Ill are performed with a view to interpreting

a scattering experiment with spin polarized electrons.

The elastic scattering differential cross-section will depend,

approximately (that is, within RPA) upon ( S S*! S ) as we mentioned in

Sec.IV. This fact has been applied by Wolfram et al. to obtain

experimentally1 the surface magnetization of anti-ferromagnetic HiO.

Therefore, through simultaneous measurement of the elastic and inelastic

spin polarized scattering cross-section, one can obtain \ S Q ^ and the

local spectral density of magnons, Im gQ0(4k|. ,m).

VI. COHCLUSIOHS

This paper extends a series of previous works on the spectrum of surface

magnetic excitations based on the Heisenberg exchange model to the whole range of

temperatures up to the critical temperature, incorporating into the formalism

a self-consistent procedure for the calculation of the non-uniform

magnetisation which is relatively simple and can be applied with small

modifications to a variety of situations. The limitations of the present

treatment, particularly the simplication that only two planes were assumed

to be different from the bulk, can easily be overcome in principle and

work to generalize the formalism to an arbitrary number of varied planes

is already In progress. Other short coinings of the present formal ism are

inherent to the KPA and must be overcome by performing a better treatment

of the longitudinal fluctuations, that Is by considering the effect of the

difference S z - ^ S * S and eventually also the effect of finite lifetimen n *

upon the magnon states. Some of these improvements ought to be definitely

incorporated into the theory before toe critical behaviour of the surface

magnetization can be more confidently described. Our results indicate,

at any rate, that the critical exponent of the surface magnetization is

certainly less than one, and that it seems to depend on £ . These

conclusions, provisional as they are on the basis of the preceding criticisms,

agree qualitatively with the Monte Carlo calculations of Binder and

Hohenlaerg , with renmrmallzation group calculations by several authors

and are also consistent with recent experimental measurements by

Alvarado et al. of the critical exponent of the (100) surface

magnetization of NI.

Another conclusion of our work is, that "by measuring the elastic and

inelastic differential cross-section for low energy, spin-polarized electrons,

a great deal of information can be gathered on the exchange interactions at

the surface plane of insulating ferromagnets, and we expect that formulas

(69) and (70) of Sec.IV provide a useful model for interpreting such

experiments. In particular, the effect of changing the temperature and/or

the surface parameters can be traced to the appearance of resonance peaks

In the differential scattering cross-sections, which, in turn, is simply

related to the surface spectral distribution function of the magnetic

excitations.

-21- -22,-

ACKNOWLEDGMENTS

During the whole course of this work, we have enjoyed invaluable

advice on numerical computation methods from Prof. Sergio S, Makler and

many stimulating discussions with Profs. Enrique V. Anda and Jos£ E. Ure.

We acknowledge also several useful comments from Prof. Leo Fallcov. Special

thanks are due to Prof. A. Balderesehi for having provided us with the

necessary data for applying a fast numerical integration method in k-space.

Prof. Fernando A. de Oliveira we thank for his careful reading of the

manuscript. We are grateful to Professor A"bdus Salaa, the International

Atomic Energy Agency and UNESCO for hospitality at the International Centrtt

for Theoretical Physics, Trieste. The present work has "been partially

supported by CHPq. and FIHEP of Brazil(and by Seuola Internazionale Superiore

di Studi Avansati (SISSA), Trieste.

- 2 3 -

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16) T. Wolfram, R.E. De Wames, W.F. Hall and P.W. Palmberg, Surf.

Sci. 28_, 1*5 (1971).

17) J.C, Le Guillou and J . Zinn-Justin, Phys. Rev. Bgl, 3976' (i960);

J .S . Reeve and A.J. Guttmann, Pnys. Rev. Letters k$_, 1581 (I960).

18) S. Alvarado, M. Campagna and H. Gopster, to be published.

- 2 U ,

FIGURE CAPTIONS

Fig-?

o 1/2The mapping £ = t + (t -1) . We shov the branch corresponding to

the analytic continuation of t in the upper half plane. The lower

semi-circumference is the image in the complex I plane of the branch

cut -1 * t * 1 where the'continuum lies. Long arrows

Indicate the sense in which £ travels along its trajectory in the

lover half circumference as t varies in the whole real axis

-» < t < » -

Surface magnetization for the sc lattice as a function of

temperature: T = kT/(6M ). The bulk magnetization is also shown.

The c r i t i ca l temperature in our units isa (100) plane; S = 5- .

0.33- The surface is

Fig.3 Surface magnetization for the fee lattice with a (111) surface as

a function of reduced temperature T

The reduced c r i t i ca l temperature is

indicated values of £. .

kT/(12K ) and S I2

T • 8.0h, for £ = 1 and t he

Dispersion relations for surface magnons in the fee l a t t i c e . The

dashed region is the bulk continuum. Along the horizontal axis Is

plotted A(k ) = 1 - Y-.'ki.) (see Eq.(53)). The different curves

correspond respectively to : A : T = 2, £. • 1.5; B : T = 2, £JL = 0 .1;

C : T « 5, £j_= 0-1; D : t = 2; £ x = 0 .1 ; E : T = 5, £j_= 0.1 and

Magnetization profile for the fee l a t t i ce , T = 6, £,|= 1,

£x= 1.5 (A), 1(B), 0.5(C), O.l(D). aa = < S ^ / B. % = < S v O l > / S

Plot of the differential scattering cross-section for inelastic

scattering of slow electrons for magnon emission. All dimensional

and kinematical factors have been omitted, so that the quantity

along the vert ical axis is Im gQ0(l + Bf-E^for the fee la t t i ce

for T = 2, £j_ * 0.5, 1.5 and 2.0, g,, = 1.

Flg.T The same as in Fig .6 for T = 5, fj_= 0 . 5 , 1.5 and 2 . 0 , = 1.

2 6

-25 -

0

0 0.15 0.30 0.45 0.60 075

A(i\> c . , . , .

crn

5

U

3

2

1

0

fee

T-6.-

i -

/ /

- /

i

1

n

.5

crV

2 9 3O

50

40

30

a•a

s20

10

-1 -0.5

T=5

FlG-GFIG.7

TC/fll/i 55

IC/Bl/156

IC/81/157IHT.REP.*

IC/61/153IOT.R.-T.*

IC/81/159INT.REP.•

IC/81/160IRT,SEP.» .

J .

J , t:;;:- T.\-r I.OP!:,; - i -a lr inc i: . hruir™ r i r j i ' l u r e ,

C.C. >LA"=A and ',f..Y.. PAFirA - FRCtovi?::!.! or. h;'poV.i:;in and o:;ti:-aticnof pE±l-Ity violr-M;!,* p;*r:ir ••* er" i n iniu1] -;ir f o r c o .

cf the

A.H. !!AYYAR arvi <i. K1JRTA".A - Motion of i\ r.ner.p' ; ? col i ton aV.nutl a t t i c e aol i ton in a !!eir>e:ihorg chain.

J .A. MiGPASTAY - iii,~j-s phenomenon vi'i rau^e i n v a r i a n t s .

1ST.REP.•

IC/81/162IHT.REP.*

IC/81/163IHT.REF.»

IC/8I/I6I4INT.REP.*

XC/Bl/165INT.flEP.*

IC/ei/l66IST.REP.

IC/Bl/167INT.REP."

IC/81/168

IC/81/I691ST.HEP.•

IC/Ol/170

IKT.REP."

IC/81/171

IW.RRP.*

IC/fil/173

R.V. F!!ARMA. - A note on the hard sphere rcodel of surface entropy ofliquid metals.

A. SRFSIJ'I and G. OLIVIER - Self-consistent thpe-ry of localisat ion Inbinary al loys.

S. CRATTERJEE and 5.V r-!J3THMANYAN - Radiation induced Spinning of a.charge density vnve systen.

S.I . HUSAIN and Z. AKSAN. - A differentiable structure on space-tinemanifold corresponding to to ta l pure radiation.

ZHANG IJ-YUAK - An approximate method for calculating electron-phononmatrix element of a disordered t ransi t ion metal and relevant commentson superconductivity.

ZHANG LI-YUAN and YIN DAO-LE - Energy band theory of heterometal super-posed film and relevant comments on superconductivity in heterontetalBysterns.

D. KUMAR - Proposed for ft non-linear reactor.

S. NARISON - On the Laplace transform of the Voinberg sum ru les ,

F. CLAEO - Local fields in ionic crys ta ls .

ZHANG LI-IUAH - Pe ie r l s i n s t a b i l i t y ajj-J superconductivity \n aub-stitutionally disordered paeudo o;ie-di-'jn:;ior^U. conductors

S.G, KAMATH - Penpuia e f fec ts in 43 -f 0 E de-euvs in a five flavour() ()

G. SICfATORE and M.P. TOPI - Opt in i rc i r p d o n j'.Jjr.ie appro>:lr.iit ioa forthe s t ruc tu re of l iqu id iiLHili KC'.UIC r.s L-ICC*.!-. :i-i<its pltiSrsis.

Z. FRK2THI and 0 . OLIVTKi( - '.:i\i<\y or i:,c- r o t i J • ry frtf.e in a Cn;;leyt r e e .

L . KAT (Uli C . CMKTO - b a

IC/71/JY3 A. r.'Nvy - F ^ r'•;•-; pec tru-' n;-u!::M :.•'•••-f!i'«'. : -.r?c in a r.-ni-lin;----

K:/3l/17-..

local 'i r.iclov elocrrcn liqul-i.

IC/81/177Iirr.P.EP.*

IC/81/178

IC/Bl/179INT.REP.•

IC/Bl/180

IC/61/181INT,REP.•

IC/81/1B2INT.KEP.*

IC/81/183INT.REP.•

IC/81/18U

IC/81/1S5

IC/Bl/186

IC/61/167

IC/81/188

IC/81/189

IC/81/190

IC/81/191

IC/81/192

n. rAL"..".': !'i:;! K.I. li'.'JT.'i - C'f-i-o;;itioii correct ion to i>lan -̂.on d ispers ion:I-'fbuttsJ oC a r t - ' l / ^Jp Ichin:iru, Tot3'j,ll , Viinse and !"it:CS <

AlCffiD 0~"..v; - E las t i c and renrrM.-eR.cnt 'ic-nttering L-et.veen tvo- I r t ter-sioting aoutcronc as 11 four-bouy j:roblgr.,

S.P. <3e ALVIS and J .E . KIM - Siipcrsymnetrie SU( l l ) , the i nv i s ib l eaxion and tvoton

IC\/S]/19>s

T. CARIMI, G. KALMATI and K.I . GOEJJEH - ExR.ct dynamicftl p o l a r i z a b l l l t yfor one-conpo:ient c lnsa ica l plasr.as.

LEAH MIZHACKI - Coments on ''dyon3 of charfie e6/£ir".

V. PAAR and S, BRAIiT - Signature effect in SU(6) model for quadrupolephonons.

Z.G. KOINOV an* I.Y. YASCHEV - Path-integral calculation of the densityof states in heavily doped strongly cotspensated semiconductors in amagnetic r i e ld .

C. PAITAGIOTAKOPOULOS - Repularisation of the quantum field theory ofcharges and monopoles.

A. SADIQ and M. AHSAN KHAH - Entropy of percolating and random inf in i tec lus ters .

A.H, BASSYOUHI - Stark broadening of isolated U.V.-lines of U I , Alland Si I l>y a D.C.-ARC plasma without and vith Cs.

A.H. BASGYOUNI - Imprisonnient of resonance radlatiqn in atomicabsorption spectroseopy. i

B. BUTI - Interaction of turbulent solar vind vith cometary plasma t a i l s .

M.K, PARIDA and A, RAYC11AUDHURI - Low nass parity restorat ion, vcaXinteraction phenomenolOBy and rrand unif icat ion.

J , PRZYSTAWA - Syior.etry and phase t rans i t ions .

P. BUDINICH - On "eonformal apir.or geometry": An .attempt to 'Vuidersitnnd"internal symmetry.

Y. FUJ1H0TO - The axlon, foraion raas ' ln SU(9) O.U.l-h.

U. KAGPJS - Hemarks on the non-un!q;tenen'.-. of the c(uionic*L5eaerKy-r.o^entuntensor Of c lass ics ! field theories .

J . TARSK1 - Path iritcgr&l ove>- plvsaeapacp, their d^Mirltloo' an4 d l ^ l eproperticr-.

!!.?. rTV.Z - CliUislc-i!-i-ocliunic- ; !" a hrr-J ' i;: ,1 top.

K.O. AKrr:."!7 - Oil cl-.:>nlcal stlut'lona of Giirf.ey'a eiv."foi-aal-inVEriiint

! ' ! ( ! • • - • : : :•• • f ' T V . / •.::••• A V A i ; •• : i K! : '< • . "

IC/81/196-IHT.REP.*

P. KIKOI-.OV - On the correspondence between infinitesimal and integraldescription of connections.

IC/81/197 A.B..C1ICUDKARY - On an improved effective range formula-

IC/81/199IHT.REP.*

ic/ei/199

IC/Bl/200

IC/81/201IHT.REP.*

IC/81/202IHT.BEP.*

IC/81/203

i

IC/8l/2Ol»

K.H. PAIKTl'R, P.J. GHOUT, H.H. MARCH and H.P. TOSt - A model Of theiee-d-electrbn metal interface.

TEJ SRIVASTAVA - On the representation of generalized Dirac (Clifford)algebras.

R. JENGO, L. HASPERI and C. CMERO - Variational discussion of theHamiltonian Z(H) spin model in 1+1 and 2+1 dimensions.

D.A. AKYKAMFONG - Supel-symnetry breaking in a magnetic field.

D.K. SRIVASTAVA- Integrals involving functions of the type

A. BULCAC, P. CABSTO1U, 0. DUMITRESO) and1 S. HOLAB - Fermi liquifl modelof alpha decay.

ANDRZEJ COZDZ - Equivalence of the L-diminiahing operator and the ..Hill-Wheeler projection technique and its application to the five-dimensional harmonic oscillator.

IC/81/205 M. CIAFAL0N1•- Soft gluon contributions to hard processes.

IC/61/206 Summer Workshop on High-Energy Physics (July-August 1981) (CollectionIKT.REP.» of lecture notes).

IC/81/207 E. BOBULA and Z. KAUCKA - Sufficient condition for generation ofINTREP.* multiple solidification front in one-dimensional solidification of

fcinary alloys.

IC/81/209INT.REP.•

ic/ai/210

IC/81/212IKT.REP.#

IC/8l/?13

IC/8l/£l!t

IC/81/215

IC/8l/.?l6

' TC/&i/.-:17

A. SADIQ - Cultural isolation of third-World scientists.

E.W, MIELKE - Beduction of the Polncare gauge field equations fcymeans of a duality rotation.

IC/Sl/Sll ABDU3 SAIAM and J. STRATHDEE - On Ksluia-Klein theory.

B,H. HARCH and H.P. TOSI - Saturation of Debye screcnine length andth* free energy of cuperionlc FbPg .

H.R. KAMDAVI - Anatomy of erand «nify<in« croupn.

A.R. Cl'.G'JWiARY - A study of 3-vave n-p scattering uainj; a nev effectiverange foraula.

S. O1B?.FWSK1 - Kartree-Fock approxlontton for the one-dlaenaionalelectron |T̂ n.

T. P.HTVIU'TAVA - A two-cop.Kar.i-nt wavc--t'<iwiticn for V"irt.iclc» of tvln-^anil nr >i"7.oro t't*:1! r,̂i:'n (Tavl 1),

H.V, :'.l!Ai;>fA, C,, {^I^TOHK -i:.,] y..r. Tei'.in t r l :-.•'•;;! w -.. !: r.

l-r!-.!-.̂ ? ionic f w i v