Phys. kondens. Materie 14, 1--6 (1971) �9 by Springer-Verlag 1971

Orlglnalarbeiten / Travaux orlglnaux ] Original Papers

A Determination of Resonance Amplitudes for a Theory of Crystalline Field Parameters

in Heavy Rare Earth Metals

J . M. DIxo~r

School of Physics, University of Warwick, Coventry, U. K.

Received May 5, 1971

An account is given of a variational calculation to estimate the amplitudes of resonance factors e in a recent theory to describe the enhancement of crystalline field potentials by conduction electrons in heavy rare earth metals. I t is demonstrated that the values of e obtained by minimising the energy of interaction between conduction electrons and rare earth ions are consistent with those previously used to form a comparison with experiment. These latter values were obtained by maximising the A ~ crystal field coefficient with respect to e. Consistency is exhibited in both the sign and order of magnitude of the resonance amplitudes and renders the theory parameterless. The values of e show an approximate linear dependence with the number of electrons in the incomplete ] shell of the rare earth ions.

I . Introduction

I t has been shown recen t ly (Dixon and Dupree , 1971, I , I I ) t h a t c rys ta l l ine field pa r ame te r s ca lcu la ted using a l a t t i ce sum m e t h o d (de W e t t e and Ni jboer , 1958) for ra re ea r th meta ls , are cons iderab ly enhanced when the in te rac t ion be tween conduct ion electrons and local ised 4 / e l e c t r o n s is t a k e n into considerat ion. This t h e o r y res ts on the form t a k e n for the conduct ion e lec t ron wave- func t ion ~ which is no t free electron-l ike b u t is or thogonal i sed to some of the occupied orhi ta ls of a rare ea r th ion and to those of i ts twelve neares t neighbours . ~ is a ssumed to be or thogonal to all the r a re -ea r th orb i ta l s up to 5/9 and 5s except one of the 4f orbi ta ls . I t was assumed in a n y pa r t i cu l a r case, t h a t th is l a t t e r o rb i t a l (denoted b y ~e) was t h a t 4~ orb i ta l in to which a n e x t r a e lec t ron would go when the ful ly occupied orb i ta l s are those of the g round configurat ion. The a m p l i t u d e of this s ta te in ~g was t a k e n to be eas where e was a resonance amp l i t ude and ae was the over lap of the orb i ta l ~p~ wi th a p lane wave having the F e r m i wave vector . I n pr inciple , therefore, i f we assume t h a t the in te rac t ion be tween conduc- t ion electrons can be neglec ted (as in I ) , we could look upon ~0g as a t r ia l funct ion for a va r i a t iona l ea lcula t ion in which the t o t a l energy of the rare ea r th ion inter- ac t ing wi th a conduct ion e lec t ron could be mhfimised wi th respect to e t he r e by giving i ts value. The above t heo ry would therefore become paramete r less and a d i rec t compar ison wi th expe r imen t eould be made i f the values of e deduced were comparab le wi th those t a k e n in I I . I n I I i t was assumed t h a t the enhancement resul t ing f rom or thogona l i sa t ion of ~k to the orb i ta l s of the centra l ra re ear th , could be max imi sed wi th respect to e.

This p a p e r will e s t ima te the va lue of e for t he rare e a r t h meta l s and a l loys s tud ied in I using a va r i a t iona l technique. These values will be shown to be

1 Phys. kondens, Materie, Bd. 14

2 J.M. Dixon:

consistent with enhancement of the crystal field parameters in both sign and magnitude.

II. A Derivation of e

We set up a system of spherical polar co-ordinates with the origin at a rare earth ion and the Z-axis along the c-axis as defined in I. Only those terms of the total energy E which involve the parameter s Will be retained since on differen- tiation of E with respect to e other terms will vanish. We define our trial func- tion qJk by

~ = ~ { e i " ' - - ~ a ~ ( l + ~ ) ~ f l ~ } (1) (I

N is a normalisation constant given by

and V is the volume enclosed by the nearest neighbours of the rare earth under consideration, k will be taken to be the Fermi wave vector throughout and d is uni ty for the state with amplitude eae and zero otherwise. Bearing in mind our

first remark we may divide the appropriate energy E (retaining only terms which contain e) into three distinct parts. We study kinetic energy and central potential energy of the conduction electron together, and secondly the Coulomb interaction between the conduction electron and the electrons of the rare earth. The third par t will contain the exchange energy between the itinerant and loealised electrons.

(i) The Kinetic and Central Potential Energy o[ the Conduction Electron

Firstly we examine P~ z~ _

2m re

where Pc is the momentum of the electron, m its mass, re its radius vector and Z the valence of the rare earth. The pseudo-expectation value of this energy for the non-loealised state ~ is

eiV~ f - i k . r ~ - __ V _ _ / ~ ~ ~ ~ e + i k . ~ d v

(2) N2 F +

We expand e ~k'~ in terms of spherical harmonics so tha t

e ik'r = 4 ~ i z Y~(O, q~) YFm(~, fl) j~(kr). (3) l,m

Inserting (3) in Eq. (2) gives

J~l= /.~ ~9))$(Yrm(~

V ,

Resonance Amplitudes for a Theory of CrystMtine Field Parameters 3

We now average over all directions (~, fl) of k and note t h a t

a~ = 4~ i3 g~-,(~, fl) I ]'I(r) j3(# r) r2 d 0

where ~o 8 has been wr i t ten [4i(r)Y+~(O, of) with + s the az i thmal q u a n t u m number for the s ta te ~0e. Since the opera tor ~ only acts on the conduct ion electron co-ordinates in (4) the harmonic eontMning (0~, fl) mus t be the same on bo th sides of ~ to obta in a non-vanishing result.

Wi th a l imited a m o u n t of algebra, which i t is no t necessary to give, one can now show t h a t

2r~ rr

S S ( Y~ (0, ~))* V 2 ( Y~ (0, ~) ja (/c r)) sin 0 dO dq~ 0 0

- i~ ' (~ , ) + ~ - / ~ ( ~ , ) - 2 2 . - ~ - ~3 (~ r) (5) 2r~

and f ~(Y~(O, qD))*V2(y~(o,q))f4f(r))sinOdOdq~ 0 0

,, 2 , 12 =/4f(r) ~- r / 4 f ( r ) - - ~Y/41 (r) (6)

for all values of m appropr ia te for a 4] wave function. The single and double dashes on the spherical Bessel funct ion and 4] radial wave funct ion denote differentiat ion once and twice respect ively wi th respect to r. To be consistent with I I we use a simple Slater Orbi ta l for 4], name ly

/4f(r) ~-- Ur ~~ exp ( - - A r)

where U is a normMisat ion constant and p and A are defined b y Griffith (1964) for any par t icular ion. Eq. (4) therefore becomes

EI~- + ~ X 16~ 2 • 14f(r)j3(lcr)r2dr • 2m

• {p~ + p - 1 2 } / 4 f ( ~ ) j 3 ( k r ) d r - - 2A I (P + 1)/4~(r)js(~r) rdr 0

4- j" A 214;,(r) j~ (k r) r~ dr 4- k ~ .~ l*~.~(r) j l (k r) r~ dr 0 0

5 # * -- I I ~ f ( r ) j ~ ( k r ) rd r - -4 I I * f ( r ) j~ (k r )dr 0 0

x { P ~ - F p - - 1 2 } ] I a ] ( r ) p d r - - 2 A ( p § I la~( r )prdr 0

- F ~ X 1 6 ~ x Z e ~ x (r)ja(kr)r~dr X 4i(r)ja(kr)rdr

N~Ze~ (e ~ + 2 e ) X 16~r ~ X 1~I* (r)j3(kr)r2dr x /4i[2r dr (7) V

4 J.M. Dixon:

(ii) The Coulomb and Exchange Energies o/the Conduction Electron The crystal field energies are very much smaller than the Coulomb, exchange

or central potential and kinetic energies so we shall omit them for simplicity. However the importance of the interaction of the conduction electron with the / electrons of the rare earth will become apparent and we consider these next, dividing them into Coulomb and exchange interactions as in I. Using the notation of I, the first order Coulomb energy may be written

n

=i 1 ff :(1) u (2) vo (8)

where Ui is one of the / orbitals with spin included, n is over all electrons in the / shell and Ve = e2/r12. Ve may be expanded as a sum of products of Legendre Polynomials for the conduction electron and an / electron. Hence the integral over d~2 will be zero for a total angular momentum l > 6 and for polynomials with odd l or with azithmal quantum numbers not equal to zero. Legendre polynomials involved are therefore

P0 ~ 02), p0(cos 02), p0(cos 02) and P08(cos 02).

Using Eqs. (1) and (3), (8) becomes

V-- f l ~ a* a~ e ~o,* U~ (2) Ve yJZ U, (2) dT1 dr2 i = 1 fl

-{- f l Z a*~ aee V*~ U* (2) Ve ~0e UI (2) dT1 d~2

-~- I f e2 I ae [ 2 ~o~ V/* (2) Ve ~oe Ui (2) dT1 dT2 (9)

- ~ 4 ~ IS a* ~ W* U* (2) V~ U, (2) i~ Yr (0, ~) rz - ~ (~, ~) i~ (k r~) d ~ d~2 1,m

- - ~ 47~ I I ~ ae (-- i) t Y~- m (0, ~v) Y+ ra (~,/3) ]z (k rl) u~ (2) Ve ~p~ Ui (2) dT1 dr21 . J

The component expressions in (9) can be simplified by observing that when an average is taken over the orientation of k the first two terms will vanish unless

= / 3 which is in turn equal to the azithmal quantum number of ~0~ and for a similar reason the last two terms will vanish unless Y~nis the spherical harmonic expressing the angular dependence of yJ~. Following standard works of Condon and Shortley (1964) and Ballhausen (1962) for Coulomb repulsion energies in free ions, E2 reduces to a sum of products of tabulated angular integrals with radial integrals defined by

In:2/'/ar~(r-2)[2{!'r]+2'/4/(rl)[2drl}

co

Kn---- 2 I~o r~+ /*4$(rl)]3(t~rl)drl dr2

o o

o r~-I / ~(rl)lur~+2drl dr2.

Resonance Amplitudes for a Theory of Crystalline Field Parameters 5

Hence, 8 3 N 2 e2

[~2= ~ ~ L(s, 2m) i ( t , 2m) mt V t = 0 m = 0

• {la i ( 2 + e )z2 - +

The elements of the matr ix L are defined by 2 ~

L(i, ]) = ~ ](Y~)* Y~ P~ 0) sin 0 dO dq9 0o

and mt is defined to be the number of occupied states in the ] shell with azithmal number • s is the component of angular momentum along the axis of quantisa- tion for the resonant state ~p, and is given as in I , by In - - 10 I. 5~ is defined by

oo * 5e = ~ /4/(r) ]8 (k r) r 2 dr.

0

Appropriate matr ix elements of L are displayed in Table 1 and the polynomials po are those defined by Hutchings (i964) as in paper I.

T a b l e 1. Matrix Elements o / L

] 0 2 4 6

=L 3 1 -- (li3) + (1/11) -- (5/13 • 11 • 3) =~ 2 1 0 -- (7/33) + (10/13 • 11) • 1 1 + (1/5) + (1/33) -- (25/13 x 11)

0 1 + (4/15) + (2/11) + (100/13 • 11 x 3)

We now examine the exchange interaction for the determinantal states ]A> defined in I, i.e. the top row of this determinant are the elements ~ (n) and the remaining rows formed from n/-orbi ta ls with spin, ~h (h = 1, 2 . . . . , n). The form of this interaction can easily be deduced from (9) by interchanging scripts 1 and 2 to the right of Ve multiplying by - -1 and changing the summations to those appropriate in all the terms. Again for reasons outlined above :r = fl = e. Suppose Ui has an angular dependence Y~ and any term of Ve (when expanded in spherical harmonics), a dependence Y~' then from the integration over the script 1 we obtain (since the integral must be non-vanishing)

- - s + n ' + m = O and over the script 2,

- - m + n ' + s .

Hence adding we obtain n ' = 0. This argument holds for all the terms. Whereas for state ]A> (in which the conduction electron spin quantum number is Ms ---- + �89 we obtain a finite pseudo-exchange energy, for state [B> we obtain zero (i.e. for the par t dependent on e). Thus for state I A> we obtain for the exchange energy,

3 N 2 e 2 , Ea = ~ (L (s, 2m)) 2 ~ { + 16~ 2 e ae (Kum + K2m) -- l a, ]2 (e2 + 2 ,) Ism}.

m = 0

6 J. i~I. Dixon: Resonance Amplitudes for a Theory of Crystalline Field Parameters

The var ia t ional technique results in the values of s displayed in Table 2, where for information, values of e (used in I) to maximise R1 are also exhibi ted together with the case when state I/~} is lowest in energy, i.e. excluding the exchange

energy E3. Table 2. Resonance Amplitudes s

Er : Zr Tm Tb Dy Er : Mg

a -- 0.73 -- 0.73 -- 0.78 -- 0.77 -- 0.74 b -- 0.73 -- -- -- 0.73 -- 0.73 c -- 0.72 -- 0.72 -- 0.76 -- 0.75 -- 0.73

a Variational calculation; -- b To maximise R1; -- c Without exchange E3.

Conclusion

The var ia t ional approach above has shown tha t a m i n i m u m in the in terac t ion energy of a conduct ion electron and a rare earth ion is consistent with making the enhancement of the A ~ component of the crystall ine field potent ial , R1, a m a x i m u m with respect to e. This la t ter cont r ibut ion to A ~ was shown in I to be the dominan t par t dependent on s. To obta in agreement with exper iment in I I s was chosen to make R1 a m a x i m u m so t ha t this result confirms the results of I I . We note also t h a t there appears to be an approximate l inear dependence between s and the n u m b e r of electrons in the f shell of the rare earth. I n Table 2 the l ine excluding exchange energy was included to invest igate the possibili ty of there being a significant change in s between the case when state ] B> is lowest in energy and tha t when [A} is lowest. Clearly the method is too insensi t ive to detect a ny meaningful change in s. The var ia t ional technique is p robab ly inadequate to detect such differences because when exchange is excluded, values of s are li t t le changed, suggesting t ha t perhaps components of the in terac t ion energy have min ima stretching over a range of s and tha t the to ta l energy is no t appreciably changed over this region. The fact t ha t values of the resonance factor which maximises R1 differs from the var ia t ional est imate is to be expected from the very approximate na tu re of the model. The Slater wave funct ions used together with the neglect of interact ions between electrons would also lend suppor t to this view. We conclude therefore t h a t the assumpt ion t h a t dR1/d~ be zero in I I is consis tent with the requh-ement t ha t the in te rac t ion energy between conduct ion electrons and rare earth ions is a min imum.

References

Ballhausen, C. J. : Introduction to Ligand Field Theory. New York: McGraw-Hill 1962. Condon, E. U., Shortley, G. H. : The Theory of Atomic Spectra, C.U.P. (1964). Dixon, J. M., Dupree, R.: I, II, in press, I.P.P.S. (J. Phys. F) (1971). De Wette, F. W., Nijboer, B. R. A. : Physica 24, 1105 (1958). Griffith, J. S. : The Theory of Transition Metal Ions, C.U.P. (1964). Hutchings, M. T. : Solid State Phys. 16 (1964).

J. M. Dixon School of Physics University of Warwick Coventry, U.K.