TEMPERATURE DEPENDENCE OF THE ELECTRIC FIELD GRADIENT AT TRANSITION IMPURITIES IN CLOSE-PACKED HEXAGONAL METALS

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TEMPERATURE DEPENDENCE OF THE ELECTRIC FIELD GRADIENT AT TRANSITION

IMPURITIES IN CLOSE-PACKED HEXAGONAL METALS

M. Piecuch, Ch. Janot

To cite this version:

M. Piecuch, Ch. Janot. TEMPERATURE DEPENDENCE OF THE ELECTRIC FIELD GRADI- ENT AT TRANSITION IMPURITIES IN CLOSE-PACKED HEXAGONAL METALS. Journal de Physique Colloques, 1974, 35 (C6), pp.C6-291-C6-294. �10.1051/jphyscol:1974644�. �jpa-00215801�

JOURNAL DE PHYSIQUE Colloque C6, supple'ment au no 12, Tome 35, DCcembre 1974, page C6-291

TEMPERATURE DEPENDENCE

OF THE ELECTRIC FIELD GRADIENT AT TRANSITION IMPURITIES IN CLOSE-PACKED HEXAGONAL METALS

M. PIECUCH and Ch. JANOT

Laboratoire de Physique du Solide (L. A. no 155) C. 0 . 140, 54037 Nancy-Cedex, France

Rhsumk. - La variation du gradient de champ Clectrique avec la temperature mesuree sur une impuretk de transition dans un metal hexagonal normal peut Ctre expliquk par la presence &&tats lies virtuels 3d. En Btudiant la levee de dCg6nerescence de ces niveaux liBs virtuels par le champ cristallin, on Btablit une relation lineaire entre le gradient de champ klectrique local et la densite Bectronique des Ctats 3d au niveau de Fermi. Ainsi, les donnCes experimentales obtenues par spectro- mCtrie Mossbauer pour l'alliage diluCBeFe semblent etre expliquBes par des effets de fluctuations de spin localis6s.

Abstract. - The temperature dependence of the electric field gradient at transition impurities in normal close-packed hexagonal metals is shown to be consistent with an appreciable contribution from virtual bound 3d states. By studing the crystal field influence on this virtual level in a Friedel- Anderson model, the localized electronic contribution to the EFG appears to be linearly related to the density of 3d states at the Fermi level. So, the experimental data from Mossbauer spectroscopy for the temperature dependence of the EFG in the &-Fe dilute alloy can be ascribed to the effects of localized spin fluctuations.

1. Introduction. - There has recently been increas- ing interest in the observation of electric-quadrupole hyperfine interactions in hexagonal metals, with a view to elucidating the relative contributions of local and distant lattice and conduction electrons to the electric field gradient (EFG) [I]-[lo].

In an earlier paper we have reported [6] the experi- mental results obtained from Mossbauer spectroscopy regarding the temperature dependence of the electric- quadrupole hyperfine interaction at dilute iron impu- rities in polycrystalline and single crystal beryllium

t

GOO 900

Tet>,,>,.yat,~r,: '1 f"51

,>Oh

. a l l I-- . _ _ .., +. +.-

FIG. 1. - Quadrupole splitting versus temperature measured by

Mossbauer spectroscopy on iron dilute impurity in beryllium host metal [6].

metal in the temperature range from 77 to 1 200 K.

The curve in figure 1 summarizes the Mossbauer data : (i) The sign of the EFG on iron is found to be negative whereas precise methods of calculation and NMR experiments [I 1, 1121, [13] give just the opposite sign for the EFG on beryllium atoms.

(ii) While heating the specimen the magnitude of the quadrupole splitting clearly decreases and the slope of AEh versus T is larger at high temperature than at low temperature. This temperature dependence differs from what could be expected from the anisotropy of thermal expansion [6], [14] between the c axis and the basal plane of the beryllium lattice. This anisotropy would lead to changes of only a few percent over the whole temperature range and the magnitude would slightly increase instead of decreasing strongly.

So, we have proposed [6] that the temperature dependence of the EFG is mainly ascribed to the effects of local conduction electrons of virtual bound 3d- states at the iron impurities.

In this work, we give a theoretical investigation of this localized electronic contribution eq'OC to the EFG on iron in beryllium, by studying the crystal field influence on a virtual level in the Friedel-Anderson model.

The problem we intend to solve here is to calculate the expression of [I51 :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974644

C6-292 M. PIECUCH AND Ch. JANOT

where cp is the electronic wave function in the metal, V is the electronic potential on the nuclei, r and 6 the usual radial and angular coordinates for the electrons ; z is related to the beryllium c-axis.

2. General expression for the localized EFG. -

To obtain e q 1 O C through the relation (1) one must know exactly the electronic wave functions and even, to be more precise, modifications of these wave func- tions around the iron impurity. In the Friedel-Anderson model [16], [17] the problem is approached by mixing impurity atomic states cpdm(r) and lattice Bloch func- tions B,(r) so that :

So, using the expression (I), the localized EFG is given by :

kk'

(m,, m, labeling the various d orbitals and k the conduction electron wave vector).

In the following sections, we will neglect the two last terms in the eq'OC expression. Consequently, the above relation reduces to :

which is exact if Bk(r) are s-like electron wave functions, a good approximation in the case of p-like conduction electrons but probably quite erroneous for a transition host having a d-like conduction band. Then, the follow- ing approach is valid for normal host metal in which we will consider that the main contribution to ^eqloc comes from the deformation of the Bk(r) functions near the impurities ; this approach can be accepted in the Be-Fe dilute alloy.

So, if cpdm(r) is a 3d atomic function, the expres- sion (2) can be rewritten :

where C f and C are the usual creation and destruction operators. Now, the problem is to obtain the value of

< C: Cm > which implies the choice of a good Hamil- tonian for the localized iron atom d orbitals and the host-metal conduction electrons (denoted s in the following sections).

3. Expression of the electronic Hamiltonian. -

Assuming that the crystal field influence can be considered as a perturbation, the Hamiltonian will be written :

with

where Vmlm, are the matrix elements corresponding to the interaction of the electrons with the crystal field ; X I is a diagonal matrix in the case of an hexagonal host which exhibits a main c-axis, and becomes

where nm is the number of electrons in each m state.

Let us note that in a lower crystallographic symmetry the Hamiltonian would have to be rewritten on the basis of a irreductible representation for the spatial- symmetry group of the lattice.

An important point is to choose the main term X, of the Hamiltonian so that it remains strictly invariant by spin or spatial rotation, to be consistent with our assumption of a complete degeneracy for the 3d states which will splitt only through the crystal field termX,.

This implies the use of a degenerate Anderson Hamil- tonian modified by two extra terms in the d-d mixing interaction 1181, [19] so that :

with :

Xsd =

z

k m a

U ^{A A} U - J ^{A A}

X d d = y m l m z a C ~ r n 1 a ~ r n 2 - a + - ~ E n m i a n m 2 - a - o,ml f m2

X, and Xd are the Hamiltonians for the host conduction electrons and the d electrons respectively, X,, repre- sents the s-d mixing interaction, a denotes the spin, E:, and E:, are the host-metal conduction electron state energies and the d-state energies without interactions,

U and J are the Coulomb and exchange integrals between d localized states in the metallic environment,

TEMPERATURE DEPENDENCE O F THE ELECTRIC FIELD GRADIENT AT TRANSITION IMPURITIES C6-293 V,, is the admixture matrix element between d states trons for each m state whatever the spin, is then given and host conduction-electron states. by :

The third term in Ed, [18] restores the possibility for

X o to be invariant by spin rotation and the fourth term An, = - - Vmm x does the same regarding the spatial rotation [19]. 5

- 7

So, we have really ^eq'OC = 0 if XI = 0, that is to say P ~ ( E F ) - ^L 5 ^U P ~ , ( E F ) -,(&F) when the crystal field influence is neglected. _{x -}

P d U ( u - 2 4

4. Solutions in the Hartree-Fock approximation. - - (U - J , - 5 (&F) + ₂₅ ~du(&F) ^{~d -a(%)} The solutions of the above Hamiltonian are readily

where ~ ~ ( 8 ~ ) = P ~ , ( E ~ ) + pd-,,(eF) is the electronic obtained using the Green's function technic as shown

by Anderson [17]. In the degenerate case the Hartree- density of 3d states at the Fermi level whatever the spin.

If we suppose the spin density at the Fermi level Fock approximation for the 0 K limit < im, > is given

pda - pd-a ⁼ 0 (non magnetic states of the impurity), by :

we obtain :

A

J E F 1m G;.(E) d~ = f ( ~ , , ) (4)

< n m a > = - - _An = - - vmm P ~ ( E F )

7-c 0 5 U ^- 2 J

where the Green's functions for the virtual levels are : 1 -- l o P ~ ( & F )

- 1

and then the 3d contribution to the EFG is easily given

- Emu) - C

kkLJ

tion of self-consistency : eqloC = 1 e 1 < r - 3 > 3 d C ^m2

m

The crystal field splitting can be expressed by writting Emu = Eda + AEm,

where Ed, and Ed, have respectively the same values for all m. This splitting involves a number n,, of electrons in each (mo) state so that :

< A nrna > ^{= nda} + ^Anma

with

An,, ⁼ 0

m

and n,, the same for all m.

If the An,,, 's are not zero, there will appear a localiz- ed contribution to the E. F. G.

A

By putting Em, and < nm, > in the self-consistency relation, we obtain both the shift and splitting level for the d-states :

Ed, ^{= E:,} + (5 U + J ) n d - , + 4(U - J ) n,, AE,, ⁼ J Anm-, - (U - J ) An,, + 1/,, . (6) The An,, can be expressed through the relation (4) in the usual way :

df 1

An,, ⁼ ---- AE,, = - P~,(E~).AE,, dEm,

In a close-packed hexagonal metal as beryllium, the matrix elements Vm, of the crystal field interaction are expressed with only two parameters [15] :

V o o = - 2 A 2 + 6 A 4 V l l = V-1-1 = - A2 - 4 A, V Z 2 = V-2-2 = 2 A 2 + A S . In this case :

where A,, evaluated by a lattice-sum method [I I], is given by :

eqIa" is the lattice contribution to the electric field gradient. Finally, the expression of e q ' O C becomes :

where Pda(EF) is the electronic density of 3d states for If we are far from the Hartree-Fock condition for the spin o at the Fermi level. Using the AE,, value from existence of magnetism :

relation (6), An,, is deduced for each (mo) state and the

(U + ^{4 J )}

%

change An, ⁼ An,, + Anm-, in the number of elec-

M. PIECUCH AND Ch. JANOT C6-294

involving :

which leads to an approximated value of the localized E F G :

(7) 5. Discussion. - So, the localized contribution to the EFG is linearly related to the density nd(&,) of 3d states at the Fermi level. This result explains the linear correlation of the quadrupole splitting with the impu- rity resistivity as experimentally noticed by Window [3].

I t can even be concluded that this linear correlation will exist with all the physical properties depending linearly on the amplitude of the charge oscillation around the impurities.

The sign of the experimental quadrupole splitting on iron in beryllium can also be explained through the relation (7) which implies an opposite sign of the localized contribution with respect to the lattice contri- bution and a magnitude of the 3d contribution much larger than the lattice contribution. Indeed, the coeffi- cient in relation (7) can be roughly evaluated using the results from Freeman et al. [20] giving

for iron in various electronic configurations (3d4, 3d5 3d6, 3d7) ; so :

where pd(cF) is always a few electron-volts-' per atom.

On the other hand, even if we must be suspicious about the applicability of the Friedel-Anderson model and its solution in the Hartree-Fock limit when the temperature is not 0 K, it can be shown that the rela-

tion (7) may be used again as a first approximation as long as kT is smaller than 6,. So, the temperature dependence of the EFG will be strongly related to the changes of the 3d density at the Fermi level as a func- tion of the temperature. The experimental changes of the quadrupole splitting on iron impurity in beryllium metal [6] as shown on figure 1 from Mossbauer data shows :

- a low temperature behaviour which can be described by a T 2 law as :

with 0 = (850 f 50) K.

- A (( high ⁾⁾ temperature behaviour which can be described by a T law as :

with 6' = (2 200 f 70) K leading to

Such a change of the temperature dependence of pd(cF, T) from T 2 to T behaviour is usually explained on the basis of the localized spin fluctuation model [21], [22], with a theoretical value of O f / @

-

The model we have developed here, although admittedly schematic, contains the essential physics of the problem of the temperature dependence of a virtual bound state contribution to the EFG on a transition impurity when the host metal is not a transition metal, in the Hartree-Fock approximation. We intend to reexamine the problem by taking many-body effects into account and for transition metal hosts where the band electrons are d-like.

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