Electric field gradient produced by conduction electrons in rare earth metals

Solid State Communications, Vol. 21, pp. 963—965, 1977.

Pergamon Press.

Printed in Great Britain

ELECTRJC FIELD GRADIENT PRODUCED BY CONDUCTION ELECTRONS IN RARE EARTH METALS* L. D’Onofrio and R. Iraldi Departamento de Fisica, Universidad Central de Venezuela, Caracas, Venezuela (Received 21 September 1976; in revised form 22 November 1976 by E.F. Bertaut) The electric field gradient produced by conduction electrons in the heavy rare earth metals has been calculated taking into account non-spherical effects due to the magnetic ordering in these metals. To our knowledge the values obtained are the first to agree with the sign of the experimental results. 1. INTRODUCTION THE RARE earth metals crystallize in the h.c.p. structure charactenstic of metals with s—d conduction bands. The sources of the electric field gradient in these metals have been the object of a number of investigations in 1,2 7 recent years. The particular interest of these studies is the contribution of the conduction electrons to the •

.

•

.

•

.

•

•

effect that can be enhanced in the metal with respect to the single ion case, but his enhancement has not a theoretical base sufficiently solid to explam this big difference (approximately two times the free ion value). It is accepted that the EFG m these metals may be expresseu oy .

.

‘

(EFG) that interacts with the nuclear quadrupole moment. The presence of the conduction electrons in a metal may render the EFG different from the one which results from a point charge calculation. One could envisage strong screening effects which reduce the effective field, however experimental evidence shows that the point charge field may itbeis enhanced even achanged sign. For convenience useful to and coilsider field in gradient in a rare earth metal as arising from three different sources: (1) the EFG produced by the 4f electrons, (2) the EFG produced by the conduction electrons and (3) the EFG produced by the point charges of the lattice. The heavy rare earth metals (Gd to L~)crystallize in h.c.p. structure with a c/a value approximately equal to 1.58, this value leads us to consider the crystalline electi-ic field parameter A~as the most important, then, the electric interaction of the 4f electrons with the crystalline field gives a ground state of the 4f shell that wifi produce an EFG at the nucleus opposed to the EFG produced by the lattice; this is confirmed by the experimental results which give an EFG smaller than that produced by the 4f electrons, nevertheless, this difference is larger than the expected value from point charge calculations taking into account the shielding parameters. Pelzl2 suggested that the origin of this enhancement may be a combined effect of the lattice of point charges and the conduction electrons exterior to the atomic sphere, together with the Sternheimer anti-shielding •

Partially supported by the Consejo de Desarrollo CientIfico y Humanistico. Caracas, Venezuela. 963

=

(1 —RQ)q41— [1 —7(r)]~q~~(int)

+ (1 ‘yo~){qiatt+ q~E(ext)} where ~eu is the EFG at the nucleus, —

2

-3

q~1= 3e(aIIJIIa>{3J~—J(J+ 1)}(.r

.

____________

*

~

)4f,

(with z the axis of the magnetic moment) is the EFG produced by3)the 4f electrons; (JIIafIJ~’ a tabulated 3 iscalculated with constant, (r 41 is the mean value of r 4f wave functions; ~iatt the EFG produced by the point charges of the lattice, (1 —RQ) and (1 ~ are the shielding parameters defined and calculated by Sternheimer;3 q~~(int) the EFG produced by the conduction electrons into the Wigner Seitz sphere of the ion under study. This value must be multiplied by a factor [1 7(r)]av (average of the anti-shielding factor for the conduction electrons). Finally q~~(ext) is the EFG produced by the conduction electrons far from the ion, and we multiply this term by (1 The ~iatt contribution may be determined by a lattice sum calculation; in the case of a h.c.p. structure. Das and Pommerantz4 and Das and Ray,5 &rived an analytic expression to evaluate it. The contribution of the conduction electrons outside the ion cores has been calculated by Das and Ray6 using the APW (augmented plane waves) method. They find that this contribution is proportional to ~iatt and write —

—

—

—

q~~~ext)~ The value they found for Dy is 6 = 0.4, but their computed value for q~~(int) is of the same sign as that of the EFG produced by the 4f electrons, contrary to experimental results. However this has been calculated —

964

ELECTRIC FIELD GRADIENT IN RARE EARTH METALS

Vol. 21, No. 10

with a muffin tin potential without considering nonspherical effect, and this approximation is certainly not

denoting the fraction of states at Fermi level with character i byf1, the change in population of these states

valid for magnetically ordered metals. 7 calculated the EFG taking into Devine and Dixon, account non-spherical effects using the OPW approxi. mation, taking the magnetic moment of the rare earth ion in the direction of the c-axis of the h.c.p. structure. This is not the case for Tb, Dy and Ho, for which they found agreement with experimental results. If their calculation is repeated with the magnetic moment in the perpendicular direction, the result would differ in sign, which is contrary to experimental results. For Tm, where the magnetic moment is parallel to c-direction, Devine and Dixon changed the sign of the experimental result quoted by Pelzl. 8 calculated the EFG produced by the Watson et of al.the conduction electron states near the redistribution Fermi level when the perturbing potential of the lattice is introduced. This effect gives a contribution which

is given by: i~nj= 2’q(E~)f 6EF} 1{(V)1 for an orbital with where is the expectation value spatial character i, o(EF) is the density of state at Fermi level defined for electrons of a single spin, 6EF is the shift in the Fermi energy required by the conservation of particles, which is expressed by —

(.

.

—

.)~

~

=

o.

The i-th component contribution to the EFG is: q = 2 ~n 3P 1(r 2(cos O))~ and the total contribution is q

1ICE =

1. differs in sign from the lattice contribution, and therefore it cannot explain the experimental results in the rare earth metals. We have calculated the electric field gradient produced by the redistribution of the conduction electron states at Fermi level due to the interaction with the ordered 4f shell assuming that the conduction electrons form predominant a band, with d-character. a density of states theintroduction Fermi level with Beforeatthe of the perturbation we have supposed that the number ofd-states is equal for all components and therefore the charge density is spherically symmetric, after the perturbation some orbitals will be more occupied than the others and the charge density will produce a non zero electric field gradient at the nucleus.

Let us assume that individual conduction electrons within the sphere can be expressed as a product of a radial function times an angular part. We will also suppose that of the three electrons of the conduction band of the rare earth metals, one has a d-character and the other two have 10 s-character and do not contribute to our calculation.The conduction electrons orbitals with d-character may be written V’kF

~(kT)Y~(Or,

=

~r))’~(0kF~~kF),

where 0,. and ~r are the direction angles of r, and and the direction angles of kF. Using the expansion ~ =

2. CALCULATIONS In the ordered region the 4f shell will have a Coulomb interaction with non spherical symmetry with

V(r),

duction electrons are subject to the potential =

~~

$ Ip~,—ri \(Y~(o~, if4f(P)i _____

~~)j2

3p~

——

1

)d

?

ym*(Oct,)yjm(0Ø)

p—ri l=o m=-j > the expected value of V(r), can be written as

the conduction electrons. We may consider that the con-

V(r)

~

~

f

=

where

J

=

m1—i

1oç~’A7’

2/ + 1

~

r>

(1)

drdp

where f 4~(p)is the radial function of the 4f electrons and Y~are the spherical harmonics, and and Ø, are the direction estimates have taken theangles valuesofp~.For from the numerical table of Herman andwe Skilman,9 the sum over p is extended over all the OCCU~ pied electron states of the second half of the 4f shell. We are interested only in the direct Coulomb interaction, since the inclusion of the indirect one would give a spin dependent term. We normalize our electron state functions to a single atomic sphere; doing this, and Q,,

and r> and r< are greater and smaller between r and p respectively, and 2 Yr(0r, ~r) d~2r

or

=

A~ =

f

iY~(0r,~r)i

f

tiY~(0p,Ø~)I2

Y[~l*(O~,Ø~) ~

Due to well known selection rules the only elements that will differ from zero are those with I = 0, 2, 4. and A~are integrals of products of three Legendre

or

Vol. 21, No. 10

ELECTRIC FIELD GRADIENT IN RARE EARTH METALS

965

Table 1. Experimental results of the electricfield gradient of the heavy rare earth metals and calculated value of the different contributions. Along the c-axis (c) and the magnetic axis (J). All value in Vcm2

Tb D~ H 0 Tm

viatt

viatt

0.0200 0.0203

—0.0100 —0.0101

77.8 50

—2.28 —2.8

4.56 5.6

230 280

0.0229 0.0277

—0.0114 0.0277

17.4 —119

—4.5

9

5.0

5

440 180

ij4f

i,exp ~

—

j~4f ~.,

function; their values were calculated by Gaunt.~~ In order to make numerical estimates we have taken —‘ Fr,~

~—

TJexp

—

~

rj4f

1

—

•

ucaic

LPCaiC F1

—0.24 —0.24

—2.4 —2.4

—0.1 0.24

—l 2.4

‘.1.,

~‘

—

~(

~i

value of this factor is (1 ‘y,,.,) = 801. In the seventh column we quote the value of the component of the —

EFG the direction the magnetic moment calculated in thisinpaper, without of considering antishielding effect.

J~ Fr,,

/2(kFr) is a second order spherical Bessel function, and we have neglected the shift in energy produced by the terms with I = 0 and I = 4 of equation (1); the / = 0 term produces a shift in energy which is independent of the spatial character i; the term with 1 = 4 is very small compared with the 1 = 2 term. We have taken f, = ~ for all d-state components and ?7(EF) = 2 state per electron volt.

The value of the EFG produced by the conduction electrons that we have calculated has the opposite sign of that produced by the 4f contribution, in agreement with experimental results. We have presented here only a rough estimation of the conduction electron contribution to the EFG in rare earth metals in order to find a contribution of different sign from that of the 4f electrons. The obtained values must be multiplied by [1 ‘y(r)]av. The value of this factor lies between 0.8 and 80. The first value of the 3. DISCUSSION antishielding factor correspond to the case in6which shell, the the In Table 1 we quote the measurements and principal charges are localized in the interior of the 5p results of the calculation of the EFG in rare earth metals. second one correspond to the case in which the charges In the first column we give the value of the component are localized at the exterior of the 5p6 shell. Since the of the EFG produced by the lattice of point charges in charge distributiori density of the conduction electrons the direction of the c-axis of the hexagonal structure. In lies predominantly at the exterior of the 5p6 shell, a the second column the value of this gradient is given in value of [1 y(r)]au of ten for all rare earth metal may the direction of the magnetization, in the third one the indicate a good approximation for the order of magnitude component of the EFG produced by the 4f shell in the of our calculation. The calculated EFG multiplied by direction of magnetization. In the fourth and fifth this antishielding factor is reported in the last column. columns the experimental values of the EFG are shown, The effects of the orthogonalization of ~/1kF relative in the sixth co(umn we give the value of(l ~y~)which to the core electrons, as well as the shielding effects may fits the experimental results, assuming that there is no be analyzed when more precise measurement of the contribution from conduction electrons [the free ion hyperfine parameters will be made. —

—

—

REFERENCES 1.

LOH E., CHIEN C.L. & WALKER J.C., Phys. Lett. 49A, 5,357 (1974).

2.

PELZLJ.,J. Phys. 251, 13(1972).

3.

STERNHEIMER R.M.,Phys. Rev. 146, 140 (1966).

4.

DAS T.P. & POMERANTZ M.,Phys. Rev. 123,6,2070(1961).

5.

DAS K.C. & RAY D.K., Phys. Rev. 187, 3, 777 (1969).

6.

DAS K.C. & RAY D.K., Solid State Commun. 8, 2025, (1970).

7. 8.

DEVINE R.A.B. & DIXON J.M., Phys. Rev. B7, 11, 4902 (1973). WATSON RE., GOSSARD A.C. & YAFET Y., Phys. Rev. 140, A375 (1965).

9.

HERMAN H. & SKILLMAN S., Atomic Structure Calculation. Prentice Hall, Englewood Cliffs, NJ (1963).

10. 11.

EAGLES D.M., Solid State Commun. 12, 291 (1973). GAUNT J.A., Trans. R. Soc. (London) A228, 195 (1928). A more accesible table of results may be found in Condon and Shortley The Theory ofAtomic Spectra. Cambridge Ijnivervitv Press, New York (1951).