A model calculation of susceptibility function of paramagnetic nickel

Physiea 78 (1974) 273-290 ©North-Holland Publishing Co.

A MODEL CALCULATION OF SUSCEPTIBILITY F U N C T I O N OF PARAMAGNETIC NICKEL SATYA PRAKASH and NATTHI SINGH Physics Department, Panjab University, Chandigarh, India

Received 20 May 1974 Revised 3 October 1974

Synopsis The wave-vector dependent susceptibility function of paramagnetic nickel is investigated using a noninteracting band model which is based on the APW bandstructure calculations of Hanus. The effect of exchange enhancement of the susceptibility function is also studied. The results are found in good agreement with other existing theoretical calculations. 1. I n t r o d u c t i o n . Recently,there have been some attempts to calculate

the wavenumber-dependent spin-susceptibility function o f paramagnetic nickel using multiband electronic structure. Mueller and Garland ~) calculated the wavenumber and frequency-dependent spin susceptibility of nickel in the random-phase approximation using an interpolation scheme 2,3) for wavenumber-dependent oscillator strengths and energy eigenvalues. Lowde and Windsor 4) calculated the spin-susceptibility function for nickel in the free-electron and tight'binding approximations. In the free-electron approximation, the role of the matrix elements in the expressions for the unenhanced susceptibility function is completely neglected. In the tightbinding approximation, the effects o f an s-like band are ignored. Thompson s) also discussed the spin-wave width and damping at very low temperatures using a tight-binding band structure similar to that used b y Lowde and Windsor4). Gupta and Sinha 6) reported an interesting calculation of wavenumber-dependent susceptibility function o f paramagneticchromium. They calculated the energy eigenvalues for 55 points in the (1/48)th part o f the Brillouin zone and used a so-called "spline interpolation m e t h o d " to calculate the energy values at a mesh o f 1024 points required in the numerical integration. They also calculated matrix elements for 1024 points in the Brillouin zone for all the wavevectors o f interest. It was realised and pointed out b y Evenson and Liu 7) 273

274

SATYA PRAKASH AND NATTHI SINGH

that the results for the unenhanced susceptibility function obtained using a mesh of as few as 1024 points in the entire Brillouin zone may be unreliable. Gupta and Sinha 6) extended their calculations for a finer mesh of 128 000 points b y using the above-mentioned interpolation method for energy eigenvalues. They found that the shape of the unenhanced susceptibility function x°(Q) vs. wavevector Q becomes smoother in the first zone and the magnitude of X° (Q) is also increased in the first two zones. A similar calculation was also extended by Liu, Gupta and Sinha 8) for rare earths, thorium and their alloys taking a mesh of 450 000 points in the Brillouin zone. All the above calculations involve very heavy computational efforts and require enormous amounts of computer time. In fact it is a prohibitively difficult task to include the complete band structure and actual crystal wavefunctions in the matrix elements for evaluation of the susceptibility function ×0(Q). One has to make simplifications at one stage or another. Earlier a noninteracting-band model 9) was developed to calculate the dielectric screening in transition metals and was successfully applied to investigate the phonon frequencies o f paramagnetic nickel x°). In view of the simplified calculations, we thought it worthwhile and interesting too, to apply the noninteracting-band model to calculate the spin susceptibility o f paramagnetic nickel. We compare our results with the calculations of Lowde and Windsor 4) who use a more realistic band-structure calculation in the evaluation of the susceptibility function of paramagnetic nickel. Therefore the purpose of this paper is to test the validity o f the model band structure in the investigation o f the spin-susceptibility function. However, there were some slips in the mathematical expressions in ref. 9, therefore the formalism is presented again in section 2 and the results are discussed in section 3.

2. Theory. For a detailed derivation o f the susceptibility function, one may be referred the reviews of Herring 11 ), Arrott 12 ) and Lowde and Windsor 4) and the references therein and the t e x t b o o k of White13). Let us consider a system o f Bloch electrons with H a r t r e e - F o c k wavefunctions t~lmo(k) and eigenvalue Elrna(k) where k is the electron wavevector,/, rn and a are the orbital, magnetic and spin quantum numbers and act as the band indices. For a weak sinusodially varying magnetic field applied to the system, the perturbation may be written in the form H1 = ~'. gtaB si'Hq exp (iq-r), l

s i is the spin of electron i, g is Lande's splitting factor, taB is the Bohr magneton, Hq is the magnetic field with wavevector q and r is the

where

(1)

MODEL CALCULATION OF THE SUSCEPTIBILITY OF NICKEL

275

position coordinate of the electron. For a paramagnetic system, Hq may be taken, say along the z axis. The total perturbation will include the self-consistent response of t h e electrons through the e l e c t r o n - e l e c t r o n interaction, which should include b o t h Coulomb interactions leading to correlation effects of the t y p e discussed by Hubbard 14) and exchange interactions. Let us postpone the discussion of these corrections to section 3.3. We can easily obtain the Fourier transform of the z component of the induced magnetization intensity without including these corrections as 6)

AM (q + G) = ~ X° (q + G, q + G') ~GG,Hq.

(2)

G'

The unenhanced susceptibility function is defined as

( ~_~ ) ~ X° (q

+ G,q + G') = -

rllrn(k! - ~?l,rn,(k') ~ ~ ~ Elm (k) El,m'(k') Ic1¢' Im

~m'

XF(q+G,q+G'),

(3)

where

F(q + G, q + G') = <~lm(k)lexp[- i (q + G)"rl I~kl'm'(k')> X

(4)

and

k' = k + q .

(5)

Here k' lies in the first Brillouin zone. G, G' are the reciprocal-lattice vectors and ~tm (k) is the Fermi occupation probability function which is taken to be appropriate to absolute zero. The spin index o has been dropped and the expression is multiplied by 2 for the spin degeneracy. In order to make the calculations tractable, we assume the model band structure that the conduction electrons are distributed in the s and five d subbands. Let Z s and ZdiS the number o f electrons per atom in the s and d-sub-bands respectively. Because o f the presence o f the perturbing field with wave vector q, the electrons themselves undergo intraband and interband transitions and redistribute themselves. Therefore we can write the unenhanced susceptibility function as

276

SATYA PRAKASH AND NATTHI SINGH

xO(q + G, q + G,) = Xss(q o + G, q + G') + X~d(q + G, q + G') -I- 0 Xds(q + G, q + G') + X~d(q + G, q + G').

(6)

Here Xss, o ×~d, ×~s and Xsd o represent the contributions due, respectively, to transitions from s band to s band, d sub-bands to d sub-bands, d subbands to s band and s band to d sub-bands. 2.1. E v a l u a t i o n o f X°ss(q + G, q + G'). In the evaluation o f × ° , we should use a wavefunction for s electrons which is orthogonal to core as well as to d wavefunctions. An orthogonalized plane wave is a suitable choice. However, it has been found that the orthogonalization corrections are very smalllS), therefore we use simply the plane waves for the electrons in the s band, i.e., ffs(k) = [ 1/(N~o) ~ ] exp (ik. r)

(7)

Es(k ) = E°s + h 2 k2 /2rns,

(8)

and

where m s is the effective mass o f the electron in the s band and E ° is the energy at k = 0. N is the number of primitive cells in the crystal and ~2o is the atomic volume. With these approximations X° becomes diagonal and reduces to the familiar form:

0,

×ssq + G ) = -

4n2h 2

" '2kFs+lq+Gl[ × m[2kFs - Iq +

1 + 4kFslq+G! )

(9)

kFs is the radius of the Fermi sphere of s electrons and is given by kFs = (3ZsTr2/~2o )~.

(10)

2.2. E v a l u a t i o n o f × ~ d (q + G, q + G'). We use the simple tightbinding wavefunctions for the d electrons, i.e., ~drn (k) = (1/N ~) • exp (ik. L) dPdm(r -- L), L

where L is the lattice vector and ~dm (r) is the d-orbital wavefunction.

(11)

MODEL CALCULATION OF THE SUSCEPTIBILITY OF NICKEL

277

Assuming that the d-orbital wavefunctions on different atomic sites do not overlap appreciably, we write with the help of eqs. (11) and (3) X~d (q -t- G, q + G') = -- (g/~B/2) 2 ~ ~ ~ k

m

Adm, d m' (q + G)

m

X Z~rn,dm' (q 4- G') ~ rldm (k) - rldm'(k q- q)~.:~ (1 2)

Edm (k) - Edm'(k + q J'

where Adm,dm, (q + G) = f dm (r) exp [-- i (q + G). r] ~bdm'(r) dr.

(13)

In ref. 9 the argument ofr/d m, and Edm' was k + q + H where H is the reciprocal-lattice vector. It should be k + q as m e n t i o n e d in eq. (12). A consistent correction has been introduced in all the following expressions. We use the parabolic band approximation for d electrons16), i.e., (14)

Edm (k) = E~m + h z k2/2mdm .

Here mdm is the effective mass of the electron for the m t h d subband and E°dm is the value of E~lrn (k) for k = 0. Here it is to be noted that the parabolic band approximation is fairly good representation for s-like electrons but for tightly bound d electrons, this approximation is very crucial. However, this makes the analytical integration over k possible. Using (14) in (12) and replacing the sum over k by integration, we get X~d(q + G , q + G') = - (g/aB/2) 2

1

2 ~r2-h2

~ m d m k F d m Adm,dm(q +G ) m

× A~lm,dm (q + G') X

( 1 + 44--~Fdmamq: ~ In

~

,

for m = m',

(15)

2mdm' m m1 rr2~"---'T-Adm'dm '(q -t- G)

= _ (g/aB/2)2~ ~

X A~tm,dm'(q + G') [I'(q) + I" (q)] f o r m =/=m'.

(16)

278

SATYA PRAKASH AND NATTHI SINGH

Here

kFdm is the Fermi m o m e n t u m

kFdm "k~dm q ( I'(q) - ~ 4q at \1 /"(q) =-if- 1 +

for the ruth d sub-band and

+ 2]]ln 12kFdmq--k}dm ~ + q21

~]] ]2kFdmq + k~drn ~ -- q2 I' (17)

1 . [In

[-2

+ 2 kF m +

+ lnl 2q + 2~kFdm -- (--~k)' ) 2q + 2~kFdrn + (--~)½

-q2(l+¼)-~[tan-X( -~

i f ~ < 0,

X½ ]

(18)

'

J

i f X > O,

X = - 4q 2 (~ + 1),

(19) (20)

= (mdm'/mdm) -- 1.

(21)

The analytical expressions for Adm,dm' (q + G) are the same as evaluated in ref. 9. Eqs. (15) and (16) correspond to intra and inter sub-band transitions respectively. 2.3. E v a l u a t i o n o f X~s (q + G, q + G'). Using (7) and (11) in (3) we get

X~s(q+G,q+G')=

(g~B) 2 1 ~k ~m rldm(k)--rls(k+q) X f ~bd*m (r) e i(k-

G)'r dr fdpdm (r) e -i(k- G')'r dr. (22)

In the corresponding eq. (27) of ref. 9, only the diagonal part was retained and the argument o f r/s and E s was k + q + H. The corrected form is given in eq. (22) and the corresponding corrections are used in all the following expressions. In the evaluation of eq. (22) the transitions to be considered, are from d subbands to the s band. Therefore the sum over the initial states k is to be carried out over all the occupied states in the ruth d subband. The states k + q are in the s band and should be unoccupied for transitions to take place. The theoretical calculations of

MODEL CALCULATION OF THE SUSCEPTIBILITY OF NICKEL

279

Lowde and Windsor*) for the susceptibility function are for the diagonal part. Therefore we perform the calculations only for the diagonal part of dielectric matrix, for the first instance. Using (8) and (14) and replacing the sum over k by integration eq. (22) simplifies to X~s(q + G, q + G) = - (g/aB/2) 2 (8ms/~0ze/~ 2)

N~

ffFdm

dkk 2

m0

ff 2;

[F2(lk-

0 0

GI)] 2 IY~ (Ok-G,qSk-G)l 2 a -- b cos Okq

× sin OkdOkddPk,

(23)

where y~n are the spherical harmonics, 0/c_ G and ~ k - G are the zenith and the azimuthal angles respectively, of the vector k - G, Okq is the angle between the vectors k and q and a =

(24)

k 2 [(ms/mdm ) -- 1 ] -- q2,

b = 2kq, and

(25)

4 F2(Ik-GI) = 481k-GI 2 ~

i=1

aioti (t~? + I k - GI2) 4 "

(26)

a i and t~i are the parameters of the 3d radial wavefunction and are taken to be the same as in ref. 9. The analytical evaluation of (23) for a general direction o f q is not possible. However, restricting q along the z direction the analytical integration over ~bk can be done straightforwardly using the method of partial fractions 17). These expressions are different for m = 0, + 1 and + 2 and are too lengthy to be presented here. The integrals over Ok and k are done numerically. However, if we take G = 0, then the angular integral over 0g also becomes possible and it can be done exactly as discussed in ref. 9 for the general direction o f q. For this case we get (g_~B) 2 8ms ~ ( X~s(q, q) = ~207rh2 m ( - 1)m

kFdm

dk [F2(k )] 2k2

0 X [O~rnO2o- mlo +(O]rn o2-

1-m

+O2-1mO21-m)ll

-4- (D22mD2_2_ m + D2_2mD~_m) I2 ~ .

l

(27)

280

SATYA PRAKASH AND NATTHI SINGH

D~m' are the elements of rotation matrices with argument ( - % -/5, - o0 where a,/5, 3, are the Euler angles and

Io = ~(½Ino - 3In2 + 9 In4) , 11 = !~ (_ in 2 + in4) ' I z = ~(½Ino - Inz + ½In4),

[b - a[,, b\b + In°-

ln gT-S

'

1 ( ~ b + 2a3 a4 ]b-al) -~-g + - ~ In ~

b

.

(28)

2.4. E v a l u a t i o n o f Xs°d (q + G, q + G'). We proceed exactly in the same manner as in the evaluation o f ×~s (q + G, q + G'). However, in this case the transitions are from the s band to d subbands. The diagonal term of Xs°d can be written as

Xsd q + G , q + G )

=-

--

~2orr~2 m mdm

dkk 2 0

X ~f 2; [F2(lk+q+Gl)12ly~n(Ok+q+a,r~k+q+a)12 F o o a - b c o s Okq × sin OkdOkdePk,

(29)

where a' = [(mdm/m s) -- 1] k 2 - q2 and Ok+q+G,~k+q+G are the zenith and the azimuthal angles respectively of the vector k + q + G. Restricting q along the z direction, the analytical integration for q~k can be done completely but the expressions are again t o o lengthy to be presented here. The integral over Ok and k is to be performed numerically. However, if we take G = 0, the integral can be evaluated for the general direction o f q as discussed in ref. 9. For this case (29) reduces to X~d(q, q) = --(g/aB/2) 2 ( 8 / ~ o 7r~2) ~ mdm (-- 1) m m

MODEL CALCULATIONOF THE SUSCEPTIBILITY OF NICKEL

281

kF~ , k 2 X [DomDo-m 2 2 f I o dk 0

kFs

+(DZ_1mD21_ m +DUlmD2_l_m) f l~lkZdk 0

+ (DZ--2mD~'m + O~m D2-2-m) k;s 1'2 k 2 dkl,

(30)

0

where 4

4

~ aiajaiaj

fo =~(48) 2 ~ i= 1

j=l

+1

( 2 q 2 - k2 + 3k2t2 + 4qkt)2 at, s(t)

X I

(31)

with

s(t) = ( ~ + k 2 + q2 + 2kqt)4 ( ~ + k 2 + q2 + 2kqt)4 (a' - bt), 4

1'1 =

-

~

(48) 2 ~

(32)

4

~ aiaj~ia j

i= 1 j=l +1

×

~ [(q + kt)2 (k 2 + q2s(t) + 2kqt) - (q + kt) 4 dt

(33)

-1

and 4

1~= ls(48)2 ~ i= 1

4

~aiaj°ti°tj j=l

+1

~ k ' ( t 2 - 1 ) 2 dt.

(34)

-- 1

In all the corresponding equations of ref. 9, iq + G I appears in place of q, which is an error. 2.5. E v a l u a t i o n o f x(q + G,q + G) w i t h o u t m a t r i x e l e m e n t s . A common approximation has been to neglect entirely the effects of matrix elements and if it is done here, the expression for the unenhanced susceptibility function can be written as

k k' tm i'm' E l m ( k ) - E f m ' ( k ' ) "

(35)

Eq. (35) is equivalent to a use of the free-electron approximation for sand d-electron wavefunctions. The expressions for 2°ssand X~cl are

282

SATYA PRAKASH AND NATTHI SINGH

obtained from eqs. (9) and (15) respectively replacing q + G by q in eq. (9) and putting Adm,dm'(q + G ) = 1 ineq • (15)" Using (8) and (14) in eq" (35) we get for d -~ s transitions

[ldm q2 [ 1 ) ( ~ ~ 11 ) ~ ln2r/kFdm - 2q - (-/a 1 )I1 + r/ 1 1 + r/ 2r/kFd m 2q + (--/a I )½ - -

--

1

]2rlkFdm +2q--(--lal)t2) ] ~ ,

+

forgl~0,

(36)

lnl2~kFd m + 2q + (--~x .,7r'~h2~ {Idm q - @ ( 1 + t a n -~

+

[tan-i

l ) gtli

(2q;2~kVdm)]} ,

,

ldl~

] (37)

for~>0,

where

4,7q(' IJln

2kFdmq -- rlk}dm + q2 2kFdmq + rlk~dm _ q2

,

(38)

7"1= (ms/mdm) - 1,

(39)

~1 = - 4q2(~ + 1).

(40)

Similarly for s-d transitions X~(q)=E

mdm [

m

+T

1

i

~(_-~2)~

\

]2~'kFs + 2q + (--~2)½

11]

2~'kFs -- 2q -- (--/.t2)____~. + In 1 2 ~ F s 2q + ( - . 2 ) ~

,

for P2 < 0,

=~m~--i-~2mdm{ Ism +q2(l+-~-)2-~[tan-a( /a22 [

(41)

~2~"

]

MODEL CALCULATIONOF THE SUSCEPTIBILITYOF NICKEL + t a n - l ( 2q+2~'kFs~]}g2½ ], ,] ,

forta2 > 0 ,

283 (42)

where I

kFs ~"

[k~s 4q

4~" q (1 +~-)] In 2kFsq -- ~'k~s + q2 [ 2kFs q + ~-k~s z-q--2 ['

(43)

= (mdrn/ms) -- 1

(44)

/az = - 4q2(~" + 1).

(45)

and

Since the matrix elements are ignored in eq. (35), ~0 (q) will deviate considerably from the correct value at q = 0, unless there is only one band at the Fermi level and the expression in eq. (35) involves only that band. However, in that case the neglect of interband matrix elements will lead to errors at finite q. Here ~o (q) is periodic in the reciprocal lattice but the periodicity of ~0 (Q) is not guaranteed because of the presence of the matrix elements. 3. Results and discussion. 3.1. M o d e l f o r b a n d s t r u c t u r e . We have used the noninteracting band model as discussed in ref. 9 for the calculation of the susceptibility function. As pointed out by HeinelS), in the absence of the s - d hybridization, if we choose the direction of k as the axis of quantization, the d subbands will consist predominantly of rn = 0, + 1, + 2 atomic orbitals respectively in ascending order of energy and only the first will hybridize strongly with the plane waves. But because of the s - d hybridization and some overlapping between the atomic d orbitals which is taken into account in the calculations of the energy band structure, the ordering of the d subbands with respect to magnetic quantum number rn does not remain the same even in the noninteracting band model as shown in fig. 1 of ref. 9. In such a situation it is too difficult to assign the magnetic quantum number m to all the d subbands at all the points of the Brillouin zone. Therefore we assign m to different d subbands on the basis of the d components of the basis functions of high symmetry points along the principal symmetry directions [ 100], [ 110] and [ 111 ]. However, such an assignment is not rigorously correct for the general symmetry directions in the Brillouin zone. But we assume it to be nearly valid in the whole Brillouin zone to make the calculations tractable Is ). With this approximate m assignment, the isotropic energy band is

284

SATYA PRAKASH AND NATTHI SINGH

constructed as discussed in ref. 9 and shown in fig. 2 therein. The isotropic parabolic band model may be seen to be a good approximation for the s-type energy bands but it gives only a rough picture of the nonisotropic behaviour of the d bands. However, it includes all the essential features of the band structure while the computational efforts are greatly reduced. In view of the approximations involved in the derivation of the susceptibility function as discussed in section 2, the simplifications in the noninteracting band model may not be t o o serious. 3.2. U n e n h a n c e d s u s c e p t i b i l i t y f u n c t i o n . ~O(q) is isotropic and it is shown in fig. 1 for q along the P ~ X direction for the (3d)9(4s) 1 and (3d) 9"4 (4s) 0"6 configurations. The values for the later configuration are larger than the values for the former configuration because a large number of itinerant d electrons are available in the configuration (3d) 9"4 (4S) 0"6. ×0 (q) is anisotropic and it is calculated along the three principal symmetry directions [ 001 ], [ 110] and [ 111 ] but the anisotropy is found to be quite small except in the vicinity of q -- 0. Therefore the results for the diagonal part, i.e., for x ° ( q + G) are shown only for q along the [001 ] direction in figs. 2 and 3 for the configurations (3d) 9 (4s) I and (3d) 9"4 (4s) °'6 , respectively. The contributions of intraband and interband transitions are also shown there separately. Here intraband contributions include Xs°s + X~d with rn = rn' and interband contributions include X~d with m 4= m' + ×~s + X~d. It is to be noted that the expressions for the interband contributions do not reduce to the correct limit at Q -- 0 because of the nonorthogonality o f the s and d wavefunctions. These values are taken to be zero at Q = 0 and calculated explicitly for finite values of Q. The interband contribution is of oscillatory nature and is responsible for the broad features of ×O(q + G). A broad maximum at IQ[ = 0.8 for the configuration (3d) 9 (4s) 1 is consistent with the calculations of Lowde and Windsor4). Both contributions drop o f f smoothly as Q increases which is the correct qualitative behaviour. Comparing the results for ~o (q) and X° (q) it is noticed that the inclusion of the matrix elements greatly reduces the magnitude, and the shape of the curve is also changed. In fig. 2, the results for the present calculations for the configuration (3d) 9 (4s) 1 are also compared with the calculations of Lowde and Windsor. Our results are 20% higher for q in the first Brillouin zone because Lowde and Windsor completely neglected the contribution of s electrons. For the sake of comparison, the results for chromium due to Gupta and Sinha are also presented there. It is noticed that the qualitative behaviour except for a small peak in the low q region, is the same. That peak corresponds to the spin-density wavevector for chromium which is not expected in paramagnetic nickel. However, a structure present in the

MODEL CALCULATION OF THE SUSCEPTIBILITY OF NICKEL

285

I000

x\ \ 900

800

?

700

600 >,

500 I-I,-

4OO

300

200

IOO o

o.o g

I

O.

2

i

o'.,

i

o'.6

I

I

o.B

I

Fig. 1. ~0(q) vs. q for paramagnetic nickel. The solid and dashed lines represent the results for (3d) 9 (4s) I and (3d) 9"4 (4s) 0"6 configurations, respectively, x°(q) is measured in units of - (0.5 g~B)2 and q is measured in inverse bohr units. susceptibility function due to Lowde and Windsor 4) and due to Gupta and Sinha 6) is not found in our calculations. This is a consequence o f the effective-mass approximation for d bands in which the slope of the bands at the Brillouin-zone edge is not zero. It is exactly the fiat part o f the bands that leads to the structure in the susceptibility function. x°(Q) becomes - (0.5 gpB)2D (E F) and Q = 0 where D (E F) is the density of states on the Fermi surface. In the present calculations, the values of ×O(Q) at Q = 0 are 48.3 and 59.5 in units of - (0.5 g/.tB)2 for the configurations (3d) 9 (4s) 1 and (3d) 9"4(4s) °'6, respectively. The experimental value is 80 in the same units i f D ( E F) is determined from the electronic specific-heat data 19 ). From figs. 2 and 3 it is also evident that the results o f the present calculations are closer to the experimental value than the results o f Lowde and Windsor at Q = 0.

286

SATYA PRAKASH AND NATTHI SINGH 10.0

4!

9.0

40

8.0

v"

z

35

q,

5o

o

7.0

\

:£ __ :£ o -I-

u

o o

O

6.0

o

o

O I--

s.o

25

,~

INTRABAND

,.J

4.0

20

3.0

TOTAL

I0

2.0

5 ~- INTERBAND

1.0

o ~

o.o

'

i

I.o

- -

i

2.0

3.0

1o.o

4.0

-ff 0/"o) Fig. 2. x°(Q) vs. Q and q along the [0011 direction for paramagnetic nickel. Solid lines represent the results for the (3d) 9 (4s) 1 configuration. Open circles represent the calculations of Lowde and Windsor (ref. 4). The crossed solid line represents the results for chromium on the right-hand side scale. The units are the same as in fig. 1.

3.3. E x c h a n g e - e n h a n c e d s u s c e p t i b i l i t y . The accurate calculation o f exchange enhancement is, however, very difficult even within the spirit of the random-phase approximation. For a realistic multiband system, containing Bloch electrons, one has to simplify the form o f the e l e c t r o n - e l e c t r o n interaction and will have to make further approximations in solving the coupled set of integral equations, describing the self-consistent linear response of the system to a weak applied magnetic field. Generally, a fi-function t y p e e l e c t r o n - e l e c t r o n interaction is assumed and the enhanced susceptibility function is divided by an exchange-enhancement denominator. Recent attempts to obtain a better solution for the purpose of discussing spin waves and magnetic scattering of neutrons from the itinerant electron system have been made b y

287

MODEL CALCULATION OF THE SUSCEPTIBILITY OF NICKEL

72I 64 56

'~o 4B<

"~"

40

32

INTRABAND

<

~

24

%

le

t;"

TOTA~

-81

o.o

,:o

2:o

3'.o

I

~" CIla,3 Fig. 3. x°(Q) vs. Q for q along the [001 ] direction for paramagnetic nickel for the configuration (3d)9"4(4s) ~6. The units are the same as in fig. 1.

several authors 2°-22) using Wannier representations of the electron wavefunction and a hamiltonian similar to that of Hubbard. They arrive at a result in which the exchange enhancement is taken into account by the formal inversion of a matrix involving Coulomb and exchange integral between Wannier functions in different bands. Yamada and Shimizu 21) evaluated the dynamical susceptibility for a multiband system and calculated it in the two-band model scheme for ferromagnetic nickel. Sokoloff22) investigated the case of chromium in detail using a simplified free-electron like band structure. Very recently Cooks and Wood 23) have calculated the susceptibility function of ferromagnetic nickel using a realistic band-structure calculation. In the present nonintracting-band scheme we consider only the diagonal part of the susceptibility matrix. Since it is not possible to

288

SATYA PRAKASH AND NATTHI SINGH

separate the d-like states from s- and p-like states because o f hybridization effects in transition metals. We, therefore, assume that Hubbard-type correlation interaction operates between all occupied s,p and d states of the system. Further we represent the exchange interaction with a 6 function, which is somewhat crude; however, screening of the exchange due to correlation effects as well as the averaging effects over all the occupied electron states lends some justification for representing this as a short-range interaction. Let I be the total strength o f exchange and correlation interactions; we can write the exchange-enhanced susceptibility function as

x°(Q) x (Q) - 1 - Ix ° (Q) '

(46)

However, a more accurate expression for exchange-enhanced susceptibility is ×O(Q) x ( Q ) -- 1 - I ( Q ) x(Q)"

(47)

Lowde and Windsor have recently reviewed the static and m o m e n t u m dependent values o f L They have also extracted out the m o m e n t u m dependence of I from their neutron-scattering measurements. But all those expressions for 1are valid only for a small region of Q. Singwi et al. 24 ) calculated I(Q) self-consistently in the free-particle approximation which is hardly adoptable for d electrons. In view of these uncertainties, the present calculations for the exchange-enhanced susceptibility function are performed for the static value o f / , which has been taken to be 0.66 eV as suggested by Hodges et al. 19), using the band-structure calculations o f Hanus, which have also been used here for constructing the noninteractingband model. The results are shown in the fig. 4 for both configurations for Q along [001 ] direction. The susceptibility function is enhanced by the factor of 3.8 to 1 and 10 to 1 for the configurations (3d) 9 (4s) 1 and (3d) 9"4 (4s) °'6, respectively. A c k n o w l e d g e m e n t s . The authors wish to acknowledge stimulating discussions with Professor S. K. Joshi, Dr. K. N. Pathak and Professor S. K. Sinha. Financial support from CSIR and UGC is also acknowledged.

289

MODEL CALCULATION OF THE SUSCEPTIBILITY OF NICKEL w

I

200

600

180

540

160

480

14G

420

120

360

0

'~ o I,<[

300

ioo

I

I

~

l

80

240 ~

l l

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60

180

I

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40

120

20

60

0

0.0

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i 2.0

0

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-¢ O/a.) Fig. 4. x(Q) vs. Q for q along the [ 001 ] direction for paramagnetic nickel. The solid line represents the results for the (3d)9(4s) 1 configuration on the left-hand side scale and the dashed line represents the results for the (3d)9"4(4s) °'6 configuration on the right-hand side scale. The units are the same as in the fig. 1.

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

Mueller, F. M. and Garland, J. W., Bull. Am. Phys. Soc. 13 (1960) 58. Mueller, F. M., Phys. Rev. 153 (1967) 659. Mueller, F. M. and Phillips, J. C., Phys. Rev. 157 (1967) 600. Lowde, R. D. and Windsor, C. G., Adv. Phys. 19 (1970) 813. Thompson, E. D. and Myers, J. J., Phys. Rev. 153 (1967) 574. Thompsor/, E. D., Int. J. Quantum Chem. 1S (1967) 619. Gupta, R. P. and Sinha, S. K., Phys. Rev. B3 (1971) 2401. Evenson, W. E. and Liu, S. H., Phys. Rev. 178 (1969) 783. Liu, S. H., Gupta, R. P. and Sinha, S. K., Phys. Rev. B4 (1971) 1100. Prakash, S. and Joshi, S. K., Phys. Rev. B2 (1970) 915. Prakash, S. and Joshi, S. K., Phys. Rev. B4 (1970) 1770.

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SATYA PRAKASH AND NATTHI SINGH

11) Herring, C., in Magnetism, vol. IV, G. T. Rado and H. Suhl, eds., Academic Press (New York, 1966). 12) Arrott, A., in Ref. 11, vol. 2B. 13) White, R. M., Quantum Theory of Magnetism, McGraw-Hill Co. (New York, 1970) chs. 3, 4 and 5. 14) Hubbard, J., Proc. Roy. Soc. A276 (1963) 238. 15) Hanke, W. R., Phys. Rev. B8 (1973) 4585, 4591. 16) Hayashi, E. and Shimizu M., J. Phys. Soc. Japan 26 (1969) 1396. 17) Gradshteyn, I. S. and Ryzhik, I. M., Tables of Integrals, Series and Products, Academic Press (New York, 1965) p. 56. 18) Heine, V., in The Physics of Metals. Part I, Electrons, J. M. Ziman, ed., Cambridge University Press (London, 1969). 19) Hodges, L., Ehrenreich, H. and Lang, N. D., Phys. Rev. 152 (1960) 505. 20) Englert, F. and Antonoff, M. M., Physica 30 (1964) 429. 21) Yamada, H. and Shimizu, M., J. Phys. Soc. Japan 22 (1967) 1404; 25 (1968) 1001. 22) Sokoloff, J. B., Phys. Rev. 180 (1969) 613; 185 (1969) 770; 185 (1969) 783. 23) Cooks, J. F. and Wood, R. F., Phys. Rev. B7 (1973) 893. American Institute of Physics Conference Proceedings, No. 10, p. 1218 (1973). 24) Singwi, K. S., Sj~51ander, A., Tosi, M. P. and Land, R. H., Phys. Rev. B1 (1970) 1044.