ORBITAL NEUTRON CROSS SECTION FOR HUBBARD HAMILTONIAN

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ORBITAL NEUTRON CROSS SECTION FOR HUBBARD HAMILTONIAN

S. Lovesey, C. Windsor

To cite this version:

S. Lovesey, C. Windsor. ORBITAL NEUTRON CROSS SECTION FOR HUBBARD HAMILTO- NIAN. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-573-C1-574. �10.1051/jphyscol:19711197�.

�jpa-00214021�

JOURNAL DE PHYSIQUE Colloque C 1, suppldment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1

-

573

ORBITAL NEUTRON CROSS SECTION FOR HUBBARD HAMILTONIAN

S. W. LOVESEY and C. G. WINDSOR A. E. R. E. Harwell, England

Rhumb. - La contribution orbitale B la section efficace de diffusion magnetique des neutrons est calculee et direc- tement comparee a la contribution de spin. L70pQateur d'interaction orbitale, exp(ik.r) ^K x p est exprim6 en fonction des opkrateurs de Bloch pour les electrons libres dans l'approximation des ((tight binding N. L'BlCment de matrice

<

k'

I

exp(ikf. r ) ~ x p

I

k

>

est exactement calcule et &pare de l'operateur electron. On utilise I'Hamiltonien de Hubbard pour dkcrire les effets de correlation electronique qui sont traitks dans l'approximation RPA. Puisque l'opkrateur d'interaction entre orbitales ne peut affecter les transitions entre spins opposes, la contribution correspondant 5 la section efficace contribue seulement a la susceptibilite longitudinale des spins. Contrairement au cas des electrons libres, la section effi- cace orbitale dans ce modkle ne diverge pas pour la valeur 0 du vecteur de diffusion K.

Dans le domaine paramagnktique, les contributions orbitales et de spin diffkrent seulement par leurs facteurs de forrne effectifs F s ( ~ ) . Ce dernier est symetrique dans le plan perpendiculaire a I'axe de quantification, nu1 si ^K est paral- lBle a cet axe et presente les msmes symktries que les fonctions d'onde des orbitales Blectroniques d. Pour les petites valeurs de ^K, la contribution orbitale reprksente un quart seulement de la contribution de spin.

Abstract. - The orbital contribution to the cross section for the magnetic scattering of neutrons is evaluated and compared directly to the spin contribution. exp(ik.r) ^K x ^p, the orbital interaction operator, is expressed in terms of Bloch electron operators in the tight binding approximation. The matrix element

<

k' I exp(ikf.r) K x p I k

>

isevalua- ted exactly and separates from the electron operators. Hubbard's Hamiltonian is used to describe electron correlation effects which are treated in the RPA approximation. Since the orbital interaction operator cannot effect transitions between opposite spin bounds, the corresponding contribution to the cross section contributes only to the longitudinal spln sus- ceptibility. Unlike the case for free electrons, the orbital cross section for this model does not diverge for zero scattering vector ^K.

In the paramagnetic region the orbital and spin contributions differ only in their effective atomic from factors F s ( ~ ) and FL(K). The latter is symmetric about the plane perpendicular to the quantisation axis, is zero for ^K parallel to this axis and reflects strongly the spatial characteristics of d-electron wave functions. In general the orbital contribution for small

K is about one quarter of that from the electron spin.

Introduction. - During the past few years there has been much theoretical and experimental activity aimed at understanding the magnetism of metals (particu- larly Ni, Fe and Cr) through the generalised spin susceptibility X(K, w). Efficient and accurate band structure calculations together with an increased understanding of electron correlation in metals has made possible realistic calculations of X. This work has been stimulated by extensive experimental activity, especially the use of thermal neutron scattering.

The magnetic neutron-electron interaction is the sum of two terms, the spin-spin and orbital interactions, and to date the latter has been assumed negligible in experimental analysis. Indeed Elliott argued that for small ^K the orbital amplitude is a factor mJm* smaller than the spin-spin amplitude [I]. Hebborn and Son- dheimer [2] have given a complete expression for the orbital susceptibility, but the evaluation of their formulae for a realistic model has proved difficult.

X(K, ^{W )} has been evaluated for the free electron gas but the possibility that this model allows of large electron orbits with consequently large scattering power, would in practice be precluded by electron scattering effects [3]. For 3d metals, the extreme tight-binding model is more appropriate. Here we present a calculation of the orbital contribution to the cross section for this case, and advance arguments for nickel to support its neglect.

Theory. - The partial differential neutron cross section is

Here,

I

^o

>

is the initial neutron spin state with asso- ciated probability pa,

1

I

>

the initial target state with associated probability p, and energy EL, and k is the incident neutron wave vector. A prime denotes the corresponding value in the final or scattered state, and ^K = K - K', hw ⁼ h2(k2 - V 2 ) / 2 m. The inter- action operator V(K) is given gy

W(K) ^{= n}

. C

^v e x p ( i ~

.

r,) x

where r,, s, and p, are the position vector, and spin and linear momentum operators, respectively, of the vth electron. Our aim is to evaluate the matrix element of the second operator in (2), the orbital interaction, between the (single band) tight binding states

with atomic wave functions

@(r) = f ( r ) Y;(:)

.

(4) From (3) and using extreme tight buiding approxima- tion

I?

^exp[inr

^.

^(K

^+-

^{q - q')]}

1

x 6(Aw

+

^{E , - EL,)}

.

(1) The matrix element in (5) has been evaluated by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711197

C 1 - 574 S. W. LOVESEY AND C. G. WINDSOR Johnston [4]. Thus, in the absence of spin-orbit coupl-

ing and unpolarized incident neutrons the orbital contribution to the cross section (1) is

and for d electrons

where

<

^jK

>

is a radial integral of order K.

The spin form factor FS(") which corresponds to FL(") in (7) is

F,(K) = (2 1

+

1) (- l)m (4 n)' iK

<

^jK

>

x

K

Figures la and l b show, respectively,

I

FL

1'

and

IFs

l 2

evaluated for I = m = 2 as a function of

1

K (

and various values of 6. The radial integrals used are those calculated for nickel by Watson and Freeman [5].

1

FL

1'

is seen to be very strongly anisotropic, vanishing as 6 goes to zero. The spin form factor also possesses appreciable anisotropy for the single band case.

The Application to Nickel. - For a real metal, the single band 1 = 2, m = 2 calculation examined above must be generalised to allow for all the differing values of m and the variation of the coefficients of these different orbitals as a function of the electron wave vector. A calcuIation of this type is in progress.

However, to gauge the orbital contribution in the case of nickel we have examined the spherical average

FIG. 1 . - The orbital and spin functions

I

^{FL 12 and} ] FS ¹² for a single 1 = 2, rn = 2 band. The dashed lines show the --

spherical averages Q

I

FL 121 1 FS 12.

- -

112

1

I;,

('1 1

Fs

l 2

for the I ⁼ 2, m = 2 and 1 cases and

the results are shown by the dashed lines in figure 1 . The factor half occurs because the orbital component contributes only to the longitudinal scattering. It is seen that at the values of

I

^K

I

of order 3

A-l,

typical in neutron scattering measurements, the orbital scat- tering is appreciably smaller than the spin part. At values of

I

K ( of order 10 A-I, the orbital contribution is comparable with the spin part and deserves exami- nation.

References

[I] ELLIOTT (R. J.), Proc. Roy. Soc., 1956, 235 A, 289. [5] WATSON (J. K.) and FREEMAN (A. J.), Acta. Crysf., [2] HEBBORN (J. E.) and SONDHEIMER (E. H.), J. Phys. 1961, 14, 27.

Chem. Solids, 1960, 13, 105. [6] DONIACH (S.), Proc. Phys. SOC., 1967, 91, 86.

f31 ^HEBBORN Phys., 1970, (J. E.) and 78, 175. (N. H.), Advs. P h ~ s . [7] LOWDE (R. D.) and WINDSOR (C. G.), Phys. Rev.

141 JOEISTON @. F.), Proc. Phys. Soc., 1966, 88, 37. Letters, 1967, 18, 1136.