ORBITAL CONTRIBUTION TO GENERALIZED SUSCEPTIBILITY AND MAGNETIC SCATTERING OF NEUTRONS IN TRANSITION METALS

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ORBITAL CONTRIBUTION TO GENERALIZED SUSCEPTIBILITY AND MAGNETIC SCATTERING

OF NEUTRONS IN TRANSITION METALS

Y. Obata, K. Sasaki, N. Mori

To cite this version:

Y. Obata, K. Sasaki, N. Mori. ORBITAL CONTRIBUTION TO GENERALIZED SUSCEPTIBIL- ITY AND MAGNETIC SCATTERING OF NEUTRONS IN TRANSITION METALS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-812-C1-813. �10.1051/jphyscol:19711286�. �jpa-00214315�

JOURNAL DE PHYSIQUE Colloque C 1, supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 8 12

ORBITAL CONTRIBUTION TO GENERALIZED SUSCEPTIBILITY

AND MAGNETIC SCATTERING OF NEUTRONS IN TRANSITION METALS

Y. OBATA and K. SASAKI

Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki-ken, Japan and N. MORI

Shibaura Institute of Technology 9-3 Shibaura, Minato-ku, Tokyo, Japan

Rbume. - On donne l'expression de la contribution orbitale d'un syst5me d'8lectrons itinkrants a la diffusion des neutrons. On obtient une relation entre la sectlon de diffusion neutronique et la susceptibilitk orbitale g8n8raliske dans l'approximation des klectrons libres et dans l'approximation des liaisons fortes. Dans cette derniere approximation la contribution est calculk pour des mktaux aux bandes p et d. Pour un modele a bande p la contribution orbitale peut Ctre 8valuee avec prkcision de mCme que celle des spins, quand le vecteur de diffusion se confond avec un des axes principaux.

Par contre pour des mktaux a bande d une ktude plus approfondie sera nkcessaire pour obtenir une bonne prkcision dans les rksultats.

Abstract. - The orbital contribution to the neutron magnetic scattering by itinerant electron system is formulated.

Remarks are given on the relationship of this cross section to the generalized orbital susceptibility, in the limits of free electron and tightly bound electron approximations. In the latter approximation the orbital contribution is calculated for p-and d-band cubic metals. For p-band model both orbital and spin contributions can be evaluated with accuracy when the scattering vector lies in one of the principal axes, while for d-band metals accurate evaluation awaits further study.

1. Introduction. - It has been generally recognized, both theoretically and experimentally, that the orbital paramagnetic contribution to the static susceptibilities of the transition metals is comparable to, and in some cases greater than, the spin paramagnetic contribution [I]. The magnetic scattering of neutrons via the inter- action between the neutron magnetic moment and the orbital current of the electrons has been calculated by several authors [2,3] for the localized electron systems.

The importance of the orbital contribution to the magnetic scattering of neutrons in the itinerant electron system has not explicitly been discussed.

In the present article we start from the magnetic scattering formula of the itinerant electron system and discuss the relation of the scattering cross section t o the generalized orbital susceptibility. Using the tight- binding approximation, the orbital and spin contribu- tions to the scattering cross section are discussed for metals with partly filled p- or d-bands.

11. Magnetic scattering formula and relation to generalized susceptibility. - In the one-electron ap- proximation the differential cross section for inelastic scattering of a n unpolarized neutron beam is given by

+ < i l k ~ f > < ~ I M # I ~ > ] , (1) where Q and h o are the transferred momentum and energy, respectively, ( i > and / f > are the initial and the final one-electron states, and f (8) the Fermi-Dirac

distribution function. For the orbital scattering, the operator M is given by,

and for the spin scattering,

M a = eiQr Sa , ₍₃₎

and 2 is the Hermite conjugate to My that is

Mu = - V, e-iQ.1 I Q 1 (orbital) . ₍₄₎

More generally the scattering cross section can be written in the form

and since the fluctuation-dissipation theorem [4] relates this quantity t o

the cross section could be related to the generalized susceptibility. That is, one may define the <<genera- lized susceptibility ^>) as given by eq. (6), starting from the scattering cross section formula (5). This procedure is justified for the spin part. For the orbital part [5], however, some reservations should be made. The real part of the generalized susceptibility thus derived from the cross section does not tend to the correct orbital susceptibility of the Bloch electrons in the limit of w = 0 and Q ^-, 0. Indeed for the free-electron system lim x(Q, w = 0) tends to infinity as l/Q2. This is because the term proportional to the square of the vector potential is neglected. Exact calculations of

x (Q) for the free electron system are given by Balten- sperger [6j and Hebborn and March [7].

In the tight-binding approximation, however, the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711286

ORBITAL CONTRIBUTION TO GENERALIZED SUSCEPTIBILITY C1 -813 limit lim x (Q, w = 0) reduces to the orbitalparama-

gnetic susceptibility given by Kubo and Obata [S], because the Peierls' technique is used implicitly. In this approximation the matrix elements of M with respect to the Bloch states $ ~ ' s are reduced to those between the atomic states 9,'s as follows,

< ^{vk'l M,} I pk > ⁼

= A@', k+Q) C U$'(k') U,"(k) < m' I M, I m > , (7) where

A@,, k,) = 1 , if k1 - k, = reciprocal lattice vec- tor

= 0, otherwise and U F Q ' s are defined by

111. p-band model. - The p-electron system in the simple cubic lattice is a simple and physical example where the magnetic scattering cross sections can be calculated with accuracy. The matrix elements of M between Bloch states do not depend on k, the wave vector of the Bloch state. When the scattering wave vector Q lies in one of the crystal axes, say in the (001) direction, and at the absolute zero of tempera- ture, the integral of the following form

x d(&,@ + ^{q) -} &,(k) - ho) (9) is reduced t o the two-dimentional integral which can be calculated numerically to within an accuracy of

+

Moreover, the differential scattering cross sections become singular when w + w,(q), if the scattering

vector lies in one of the crystal axes. The critical frequencies o,(q)'s are given by

spin :

hw,,(q) = 2 B sin (qaJ2) , (10) hoc2(q) = 2 A sin (qaJ2) ,

and (1 1)

orbit :

fiw,(q) ⁼ A

+

&,(k) ⁼ A cos k, a - B(cos k,, a + cos k, _a) . (13) The cross sections diverge as

spin 191 :

l / J w c - w w ^-+ wc -

(14) 0 0 -+ 0,

+

orbit :

l n / w , - w l . (1 5 ) Details will soon be published elsewhere.

IV. d-band metals. - The matrix elements of the M operator depends on k, except the very special case where both k and Q are parallel to one of the crystal axes. In general the orbital matrix elements can be expressed in the fifth order spherical harmonics, while the spin matrix elements in the fourth-order harmonics.

We have done a very preliminary calculation but the results are far from satisfactory in view of the difficulty of the numerical calculations involving the 6-type singular function [9]. At this stage we can only say that for the degenerate d-bands the orbital and the spin contributions are in general comparable with each other (I).

(1) We have neglected the many-body effect. The spin contri- bution is enhanced by this effect, while the orbital part is not much affected.

References

[I] See, for instance, GLADSTONE (G.), JENSEN (M. A.) [6] BALTENSPERGER (W.), Phys. Kondens. Materie, 1966, and SCHRIEFFER (J. R.), c( Superconductivity 5, 341.

in the Transition Metals ⁾⁾ sec. 111 B, in : ⁽⁽ Super- [7] HEBBORN (J. E.) and MARCH (N. H.), Phys. Letters, conductivity ⁾⁾ vol. 11, edited by R. D. Parks 1969,29 A, 432.

(Marcel Dekker 1969). [8] KUBO (R.) and OBATA (Y.), J. Phys. Soc. Japan, [2] TRAMMELL (G. T.), Phys. Rev., 1953, 92, 1387. 1956, 11, 547.

[3] LOVESEY (S. W.) and RIMMER (D. E.), Rep. Prog. Phys., [9] IZUYAMA (T.) and KURMARA (Y.), J. Quantum Chem.,

1969, 32, 333. 1967, 1 S, 651.

[4] IZUYAMA (T.), KIM (D. .I.) and KUBO (R.), J. Phys. [lo] LIPTON (D.) and JACOBS (R. L.), J. Physics. C , 1970,3,

Soc. Japan, 1963, 18, 1025. 1388.

[5] SCHNEIDER (T.), Solid State Comm., 1970, 8, 279.