VOLUME DEPENDENCE OF HYPERFINE INTERACTIONS

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VOLUME DEPENDENCE OF HYPERFINE INTERACTIONS

G. Kalvius, U. Klein, G. Wortmann

To cite this version:

G. Kalvius, U. Klein, G. Wortmann. VOLUME DEPENDENCE OF HYPERFINE INTERACTIONS.

Journal de Physique Colloques, 1974, 35 (C6), pp.C6-139-C6-149. �10.1051/jphyscol:1974613�. �jpa-

00215743�

JOURNAL DE PHYSIQUE

Colloque C6, suppliment au no 12, Tome 35, DPcembre 1974, page (26-139

VOLUME DEPENDENCE OF HYPERFINE INTERACTIONS

G. M. KALVIUS, U. F. KLEIN and G. WORTMANN Physik Department der Technischen Universitat Miinchen

D-8046 Garching, James Franck Str., Germany

RBsumB.

L'influence de variations du volume atomique sur les trois types d'interactions hyper- fines (le deplacement isomerique, le couplage quadrupolaire et l'interaction magnktique dipolaire) est passee en revue. La variation d'une interaction hyperfine en fonction du volume est habituelle- ment obtenue par application de pressions elevees ; ces experiences sont discutkes de f a ~ o n assez dktaillke. Caccent est mis sur l'interprktation des coefficients volumiques du dkplacement isomerique dans des mktaux et des composb. Le coefficient volumique d'une interaction hyperfine peut servir a interprkter d'autres phknomknes qui s'accompagnent de changements de volume. Un exemple carac- teristique est la variation avec la tempkrature des interactions hyperfines dans les solides. Afin de donner une idke des problemes relib aces ktudes, une sklection &experiences de ce genre est discutke.

Abstract. - The influence of changes in atomic volume on the three types of hyperfine interac- tions (the isomer shift, the quadrupole coupling and the magnetic dipole interaction) is reviewed.

The volume dependence of a hyperfine interaction is usually obtained from high-pressure experi- ments, which are discussed in some detail. Particular emphasis is given to the interpretation of the volume coefficients of the Mossbauer isomer shift in metals and compounds. The volume coefficient of a hyperfine interaction can be used to interpret other phenomena which are accompanied by volume changes. A typical example is the temperature dependence of hyperfine interactions in solids.

A selection of measurements of this kind is discussed to give some insight into the problems connect- ed with such investigations.

1. Introduction. - In a measurement of the hyper- fine interaction one determines the electromagnetic coupling between a nucleus and its surrounding electronic charges. Jn treating the problem for a solid one commonly differentiates between the contribution arising from the atomic electrons of the nucleus concerned and that of more distant charges, for exam- ple, the ionic charges on the neighbouring lattic sites.

Usually, the first term will be dominant but the latter often also gives a marked contribution and has to be considered. This separation is certainly somewhat artificial and becomes meaningless in cases where non-localized electrons are present ; e. g. in metals and intermetallic compounds. Both contributions, however, will be sensitive to changes in volume of the solid. In condensed matter, the Wigner-Seitz radius is smaller than the radial extent of the outer electrons and the spatial expansion of the atomic electrons is therefore limited by the interatomic separation. A reduction in lattice parameters will result in a compression of the electronic shell around the nucleus. It will also alter the distance to the neighbouring lattice charges. The influence of the latter effect on hyperfine interactions should be calculable in a straight forward manner as long as the simple model of point charges (resp. point dipoles) is applicable. Little is known, theoretically of the influence of the compression of the atomic electron shell on the hyperfine interaction and our

main interest is concentrated on this question. In fact, this problem should be a very good testing ground for more advanced self-consistent field calculations of electronic structure (in particular for band structure calculations) as they have been reviewed, for example, by Freeman at this conference [I].

Volume changes in a solid can be implemented by a variety of means. Most directly, the atomic volume can be varied by the application of external pressure.

Volume changes, however, also occur indirectly by temperature variations (because of thermal expansion), by an exchange of substituents (with different atomic radii) in a compound or an alloy, or by a phase transi- tion of first order, to name just a few possibilities. In order to understand the more complex changes in hyperfine interactions connected with such indirect variations of lattice spacing, it is imperative to know the volume coefficient of the hyperfine interaction from separate high-pressure experiments. Therefore this review will largely deal with static high pressure measurements, typically in the range up to 200 kbar.

We wish to limit our discussion to continuous changes of hyperfine parameters with volume and thus exclude effects like pressure induced phase transitions [2] or changes of ionic charge state [3].

Hyperfine interactions can be obtained by a variety of means and we shall not restrict ourselves to any particular one, although naturally Mossbauer spec-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974613

C6-140 G. M. KALVIUS, U. F. KLEIN AND G. WORTMANN

troscopy will play a central role. This technique is well suited for high pressure experiments because the infor- mation is transmitted by gamma radiation and thus can be extracted from a high pressure cell without extreme difficulties. Its resolution, however, is fairly limited which requires the application of rather high pressures and this offsets the before-mentioned advantage somewhat. Mossbauer spectroscopy has the additional advantage that the contact density of electronic charge and of electron spin imbalance at the nucleus can be measured simultaneously. For the investigation of the pressure dependence of electric quadrupole interactions the technique of time diffe- rential perturbed angular correlations (TDPAC) has shown great promise. The pressure dependence of magnetic hyperfine couplings is also observed by NMR and EPR spectroscopy. The high sensitivity of these methods allows measurements at fairly low pressure. Mossbauer spectroscopy has thus far been only moderately successful in these cases [4, 51.

2. High pressure experiments. - We denote by W the energy of any hyperfine interaction. The pres- sure dependence of W is usually measured at cons- tant temperature and expressed by the differential (6 W/6P),. In the case of a linear relationship between

Wand P one simply has :

where A P is the difference to atmospheric pressure.

From the pressure dependence we then wish to derive the volume coefficient of W which is (6 W/6 In V),.

For this we have to know the compressibility

of the material in question. In a cubic system the compressibility is best measured by the change in lattice constant under pressure using X-ray diffraction techniques. One then approximates for small changes of the unit cell :

Here, isotropic compression has of course been assumed. In a cubic system this is expected to happen if the pressure is applied hydrostatically. In practice this is usually possible only at moderate pressures for experimental reasons (e. g. windows, see ref. [5]). At higher pressures one uses quasi-hydrostatic pressure cells [6] in which the sample is contained in a medium like boron-epoxy which starts to flow upon the appli- cation of high pressures. The pressure cell is located between two anvils pressed together by an external force. More detailed description of pressure cells used in Mossbauer spectroscopy can be found in [6] and 171.

The flow medium (e. g. boron-epoxy, AgCI) will

reduce shearing forces to a minimum and thus in first approximation one may assume isotropic compression forces acting on the sample. Whether this leads to a congruent compression of the unit cell is an open question for anything but a cubic system like the bcc of fcc metals or simple binary compounds with NaCl structure. To answer this question a full structure analysis under pressure has to be carried out which is rarely done. In some hcp transition metals it has been shown that the c/a ratio remains fairly constant up to moderately high pressures (30-50 kbar) and one may assume a congruent compression in first order.

It should be mentioned that compressibilities may vary considerably for different substances. The measu- rement of the pressure dependence of hyperfine interactions alone without the additional knowledge of compressibilities is thus of little value. Compressibi- lities are generally known for the elemental metals but only rarely for compounds. Furthermore, compressi- bilities are generally a function of pressure. In order to substantiate a small non-linear behaviour of the hyperfine interaction with volume change, the compres- sibility has to be known as a function of pressure with high accuracy. Often, however, the initial compressi- bility is applied uniformly throughout the whole pressure range in lieu of better data.

In quasi-hydrostatic pressure cells a pressure gra- dient will usually exist which can easily lead to a pressure distribution of + ¹⁰ % around the mean pressure, causing line broadenings in resonance experiments. Mossbauer measurements are in addition often hampered by intensity problems. Finally, the determination of the actual pressure acting on the sample is not an easy task either and considerable errors may arise. In our own pressure cells we use an internal manometer consisting of a small lead strip whose superconducting transition temperature T, is measured. The pressure dependence of T, is known with good accuracy [8]. The width of the superconduct- ing transition of the lead strip gives information about the pressure gradient within the cell [9]. These limiting factors should be kept in mind when dealing with pressure experiments. The pressure range covered in most experiments discussed in this article corresponds to a relative volume change on the order of 10 % typically.

3. Volume dependence of isomer shifts. - We shall discuss isomer shift data as a typical example of measurements of the volume dependence of hyperfine interactions. The pressure dependence of isomer shifts has recently been reviewed in detail by Williamson [lo]

and we refer the reader to this publication for addi- tional information and references.

The shift of the center of a Mossbauer resonance

pattern arises from two terms, the isomer shift S,, and

the second order Doppler shift S,,,. Both depend on

volume changes. Within the approximation of Debye

model it can be shown [lo] that (dSsoD/d la V ) , is

VOLUME DEPENDENCE O F HYPERFINE INTERACTIONS C6-141

usually small compared to (dS,s/d In V),, at least for the Mossbauer isotopes employed thus far in pressure experiments (that is 57Fe, '19Sn, 1519153E~, 18'Ta and 1 9 7 A ~ ) . Therefore we will denote by S only the isomer shift in the following. We shall further assume that the isomer shift calibration constant a is a known quantity and that electron contact densities can directly be obtained from the measured differences in isomer shifts :

AS

a. Ap(0) . (4) For 57Fe a value of a = - 0.23 mmls. a: [1 1] was used throughout this paper.

3.1 METALLIC

- We turn our attention first towards metallic systems. Figure 1 shows the change in isomer shift with atomic volume for the

2, ^0.05

(r

(-13b kbar)

t; 0.10

W i

FIG. 1. -Volume dependence of the isomer shift of s7Fe in the bcc and hcp phases of iron (Drickamer et al., ref. [21]).

14 keV resonance of 57Fe in metallic iron. One finds a nearly linear increase in contact density up to pressures near 130 kbar. Here a phase transformation from the bcc into the hcp lattice structure takes place and the shift changes abruptly. The hcp phase again shows a nearly linear decrease of contact density but with a much reduced slope. As we have stated above, only the continuous changes in p(0) are of interest to us, thus the phase transition will not further be discussed.

The variation of contact density can be fitted tothe simple relation

Ap(0) cc const. V-P ( 5 ) with p close to one [2]. This means that within the accuracy of all measurements there is a linear variation of p(0) with the inverse of the volume. Quite similar behaviour is found for the 77 keV resonance of 197Au in gold metal [12], for the 6 keV resonance of " ' ~ a i n tantalum metal [13] and for the 22 keV resonance of '"Eu in europium metal [14]. In contrast, the 24 keV transition of '19Sn in tin metal shows a practically linear decrease of contact density with reduced volume [15] as shown in figure 2. In this figure we also note that all intermetallic compounds of Sn exhibit the

FIG. 2.

Pressure dependence of the isomer shift of l19Sn in metallic systems of tin ( ~ o l l e r , ref. [IS]).

same properties except for Mg,Sn which shows a strongly non-linear dependence of isomer shift with pressure and even changes the sign of slope. This behaviour is due to a rather broad semiconductor-metal transition in the region between 30 and 50 kbar [15].

At high pressures (> 60 kbar) the sample shows indeed the same behaviour as the other metallic tin compounds. Again, the low pressure data of Mg,Sn is considered anomalous within the present concept and will not be further treated. In figure 3 we give a

1 ^fcc,

FIG. 3. - Schematic representation of the volume dependence of charge density at 57Fe in different transition metal hosts (after

Williamson, ref. [lo]).

schematic representation of the change in contact density p(0) with the reduction in relative volume for 57Fe as a dilute impurity in various other transition metals [16]. From all these data we may draw the following basic conclusions :

a) The electron density at the nucleus increases with

reduced atomic volume for all d and f transition metals

measured thus far. This means that the pressure acts

primarily on the outermost conduction electrons which

C6-142 G . M. KALVIUS, U. F. KLEIN AND G. WORTMANN

have s character. It should be noted that a direct compression of d or f electron shells would lead to a decrease of p(0).

b) The increase in electron density at the nucleus for Fe as an impurity in other d metals (as well as for iron metal itself) increases faster with reduced volume for bcc lattices than for the closepacked fcc or hcp structures. Ingalls et al. [16] explained this behaviour with an enhanced s

d transfer in the closepacked structures which can also be extracted from the syste- matic of the isomer shift of 57Fe and other Mossbauer isotopes in the transition metals [17]. It is quite possible that the behaviour is caused by two mecha- nisms : (i) substantial differences in band structure for fcc (resp. hcp) and bcc metals ; (ii) the marked decrease in s-conduction electron density when going to the right within a series of transition metal hosts [IS].

Unfortunately, too little pressure data are available up to now for the other Mossbauer resonances in d-transition elements to substantiate the behaviour seen in figure 3. The few results indicate the same behaviour for lg7Au [lo], but not for l8'Ta [ 5 ] .

c) All isomer shift data presently available for d-transition metal systems can be fitted to eq. (5) with p

1. This supports a simple picture of the compres- sion of a free electron gas (the s-like part of the conduction electrons) neglecting any further pressure induced mechanism, as it has been done in the inter- pretation of the first high pressure (3 kbar) MBssbauer measurements on iron [4]. It appears that the actual behaviour over a much extended pressure range is astonishingly close to such a primitive model.

d) In metallic Sn compounds with their (5s 5p) valence electron structure the pressure data indicate that the outer electrons which are compressed must have largely p character. The increased p shielding overwhelms obviously the s-electron compression.

More systematic data for these series of elements are clearly needed.

An attempt to discuss the volume dependence of electron contact density p(0) on the basis of band structure calculation has first been undertaken by Ingalls [19] for bcc iron metal. The variation of the total contact density is divided into two terms

where pc(0) is the contribution from the Is, 2s and 3s core electrons and p,(O) arises from the 4s volume electrons in the 3d-4s conduction band. The valence dependence of the term p,(O) is calculated on the basis of a simple band structure to be

respectively

with p = 1.25.

It should be kept in mind, however, that the pro- cedure used is not a true self-consistent field calculation.

It was shown that the observed difference in contact density arises from a compression of the 4s band wave function taken at the r1 point (bottom of the band).

A possible re-arrangement of the electronic structure, like the amount of s-d transfer is not calculated. Eq. (7) suggests that Apc(0) = const. under volume compres- sion. This is not correct since a change in Apc(0) might be indicated by a modification of the shielding of 3s core electrons by either a variation of radial extent of 3d band electrons or by an increase of 3d band electron number by electron transfer. The magnitude of the first contribution was coarsely estimated by Ingalls from free atom wave functions and added as a correction to eq. (7). It was also shown that the second contribution is significant should s-d transfer occur. The original purpose of Ingalls was to obtain from pressure measu- rements a value for the isomer shift calibration cons- tant a. The strong influence of Apc(0) on the total change in contact density renders this task impossible as long as reliable values for the electron transfer under pressure are not available from other sources. This may be demonstrated that in earlier days (when had about twice the now accepted value) a pressure induced cherge transfer from s to d could be concluded, where today improved band calculations and the present value of a suggest that no charge transfer occurs [lo, 1 I].

In view of the many other uncertainties and simplifi- cations present in the band calculation of p,(O), the latter conclusion can neither be guaranteed at this time. The relative volume dependence of contact density, however, is rather satisfactorily described byeq. (7) despite the simple model used.

The volume dependence of contact density in gold metal has been calculated by Roberts et al. [12] using a free atom Dirac-Fock-Slater self-consistent field calculations for finite nuclear size. Their program allows the introduction of appropriate boundary conditions on the major and minor part of the electron wave functions at the Wigner-Seitz radius R,,. The volume dependence is thus simply expressed as the appropriate change in R,, and its influence on p(0) is calculated. The result can be expressed in the form

with y - 0.86 and (p6,(O)),, = 145 a i 3 . This result explains the experimental data with sufficient accuracy.

The latter factor leads to an isomer shift calibration constant which is, within the limits of error, in good agreement with values derived from chemical argu- ments [ZO].

In summary we have to stress the statement that one

will need rather elaborate calculations of electronic

wave functions in metals to explain the details of the

volume dependence of electronic contact density. The

rough overall behaviour is obviously already well

VOLUME DEPENDENCE O F HYPERFINE INTERACTIONS C6-143

represented by fairly primitive and coarse model calculations.

3.2 COMPOUNDS. - Data for non-metallic com- pounds of d-transition elements are available for Fe and Au. We shall first discuss briefly compounds of iron. In figure 4 the pressure dependence of contact

Fe- Ionic Compounds

(Drickamer e t al )

pressure

FIG. 4. - Schematic plot of the pressure dependence of the isomer shift at 57Fe in ionic compo~mds (Drickamer et al.,

ref.

density in ionic salts of Fez+ and Fe3' are shown schematically [3, 211. Exact compressibility data are not available for most of these materials and little can thus be said with respect to the apparent non-linearity.

Compared to metallic materials the curvature is certainly more pronounced. We see that the charge density increases with pressure like in the metal. This increase is more pronounced in ~ e salts, but the ~ + order of magnitude is the same as in metallic iron samples. Indeed, this can be demonstrated by one of the rare volume coefficients of S for an ionic ~ e , ' salt.

In FeF2 one can derive (dS/d In V), = 1.3 mm/s (considering the assumptions made in ref. [22]), which is exactly the same as for metallic iron 12, 4, 161. These findings are on first sight rather astonishing. The ionic iron salts possess no filled 4s electron shell which could be compressed. A simple compression of 3d electrons should lead to a decrease of p(0) with pressure.

Obviously the situation is more complex than that.

It has been argued that the reduced inter-nuclear separation under pressure would lead to an overlap of the 3d electrons of the iron with the valence electron shells of the neighbouring atoms [23]. This then would lead to an expansion of the spatial distribution of 3d electrons and thus to a reduced shielding of the 3s core electrons at the iron. In keeping with this picture is the more pronounced pressure dependence for ~e~

ions.

One then should expect an even larger increase of Ap(0) with pressure in covalent compounds. Some typical data 1211 are shown in figure 5. The behaviour

I I I I ,

50 100 150 200 k b a r Pressure

FIG. 5. - Schematic plot of the pressure dependence of the isomer shift at 57Fe in two covalent compounds.

is even more complex and we see that both a positive and negative slope of the contact density are possible.

The slope may even change sign. Some insight in the origin of this behaviour can be drawn from the calculations of Walch and Ellis [24] who investigated the iron contact density as a function of bond length in an FeAr, , cluster using Hartree-Fock self-consistent field calculations. Earlier work has always used a frozen orbital approach. That is to say the electronic structure was not allowed to re-arrange itself under reduced volume which led to an over-estimate of the influence of overlap distortion on Ap(0). The varia- tional molecular orbital calculation of [ l l ] show that the net change in Ap(0) arises from two sources. The overlap distortion tends to increase Ap(0) with reduced bond length while the potential distortion leads to a decrease of Ap(0). The latter effect arises from the change in distortions of atomic wave functions due to the changed crystalline potential under reduced volume. Both contributions are roughly of the same magnitude, but at small bond length the overlap distortion should win and Ap(0) thus increase. Details clearly will depend on the exact nature of the bond and on characteristics of the ligand. The interplay of at least two competing mechanisms, however, is well substantiated by the experimental data on different iron compounds. A most dramatic example is a-Fe,O, which shows practically no change in isomer shift up to pressures of 200 kbar in contrast to pyrite (FeS,) although there is little difference in compressibility between these two materials [53, 541.

In gold, both Au(1) and Au(II1) compounds have

been investigated [25, 261. The main purpose of these

studies was to test the interpretation of apparent

correlations between isomer shifts and quadrupole

splittings exhibited by both types of compounds. The

quadrupole splitting is found to increase with increasing

isomer shift in Au(1II) compounds while it decreases

(and has negative values) in Au(1) compounds. These

correlations are interpreted as arising from changes in

C6-144 G . M. KALVIUS, U. F. KLEIN AND G . WORTMANN

the overlap between the ligand-gold orbitals with different ligands. The pressure results for the Au(II1) are in keeping with the aforementioned correlation whereas the Au(1) compounds deviate markedly from the predicted behaviour. The results for KAu(CN), and AuCN are interpreted as being due to the de-locali- zation of d electrons to antibonding ^n:* orbitals thereby increasing p(0) (via reduced shielding) and decreasing the electric field gradient at the gold nucleus [26].

3.3 EU,+

Eu3+

; Eu-METAL. -

Finally, we wish to discuss some recent results of our own group on Eu, a 4f transition metal. This element is the most favourable candidate in the rare earth region, since it possesses two resonances with a high sensitivity for isomer shift and occurs in contrast to most other rare earths in two stable charge states (Eu2+ and Eu3+) both in conducting and non- conducting compounds. Investigations were carried out on EuO and EuS as examples of covalent Eu2+

compounds, on EuSO, as an ionic Eu2+ salt, on the trivalent compounds Eu,O, and Eu2Ti,0, [27] and on

EU metal [28]. In all cases a linear dependence of isomer shift as a function of pressure was found within experimental accuracy. The slopes of the pressure induced change in contact density vary greatly for the different materials as seen in figure 6, but all show an

0 5 0 100

P r e s s u r e [ k b a r ]

FIG. 6. - Pressure dependence of the charge density at 15l31S3Eu in Eu metal and Eu compounds (Klein et al., ref. [14, 271).

increase of p(0) with pressure. The slope is steeper for the divalent than for the trivalent compounds. This difference, however, largely vanishes if the compressi- bilities are taken into account. As we see from figure 7,

FIG. 7. - Volume dependence of the isomer shift of

in Eu metal and Eu compounds.

"153

[z]

all materials exhibit the same volume coefficient (dp(O)/d ln V), within 30 %. Possibly, the volume coefficient becomes somewhat larger for conducting and covalent materials. Unfortunately we have no compressibility data for EuSO, and thus cannot draw any conclusion for fully ionic divalent compounds.

The apparent differences in slope of figure 6 thus mostly represent the different compressibilities. Of course, the compressibility is also a quantity depending on the electronic structure (i. e. on the chemical bond) of the sample. One finds that the compressibility is strongly dependent on the type of ligand, while the volume dependence of contact density is not. Within a series of compounds like the divalent chalcogenides (EuO, EuS, EuSe, EuTe) the compressibilities change by more than a factor of two (even when normalized to the same bond length). The increase in contact density with volume reduction found for Eu compounds can be more easily understood than the similar beha- viour seen in d-transition elements. The radial extent of the 4f electrons is in any case smaller than that of the 5s (and 5p) electrons of the Xe-core. Thus we may postulate that we have mostly a compression of these 5s electrons which lead to the derived increase of p(0).

Volume dependent, free atom self-consistent field calculations are unfortunately not yet available to test whether the magnitude of changein contact density can be explained with this picture. The 4f electrons must be little affected since the volume coefficient of p(0) is nearly independent of 4f configuration.

It is most astonishing that even for Eu metal the volume dependence of p(0) is roughly the same as for non-conducting materials, a behaviour which is very similar to iron. As mentioned above one expects the

Apt01

QEUS /

0 EuO /

VO1,UME DEPENDENCE OF HYPERFINE INTERACTIONS C6-145 change in p(0) t o arise mainly from a uniform compres-

sion of 6s conduction electrons. Indeed, the 6s-electron density of Eu metal (calculated by Kobayasi et al. [28]) yields a volume coefficient of

which is in good agreement with our experimental value of figure 7. Since the high-pressure investigations on metallic systems of Eu are in an initial state, we want to draw no further conclusions on the contri- butions of conduction and inner shell electrons to the observed volume coefficient in Eu metal. It should be mentioned, however, that the magnitude of the observed volume coefficient of the electron density in Eu metal can well explain the isomer shifts of Eu2+

dilute in Sm and Gd metal [31] and in the alloy systems with Yb, Ca, Sr and Ba [30].

4. Indirect volume dependences of isomer shift. -

We shall discuss in this section a few typical experi- ments, which show the application of volume coeffi- cients of isomer shifts to other measurements.

4 . 1 TEMPERATURE EFFECTS. - The knowledge of both volume and temperature coefficient of a hyperfine interaction enables the extraction of explicit tem- perature effects. A careful analysis of temperature and pressure dependence of the isomer shift in metallic iron yielded an explicit temperature dependence 1321.

Similar results, but of much higher accuracy were obtained recently by analogous investigations using the 6.2 keV resonance of 18'Ta in various transition metal hosts. We shall not discuss these measurements in detail here since they are covered in a contributed paper by Binder et al. following this review. The comparison of temperature and pressure dependencies of isomer shifts leads to the conclusion that a substan- tial thermal re-arrangement effect must be present in the conduction band of Ta. Possible explanations for the observed d

s transfer are based on a theory developed by Kasovski and Falicov [33]. A similar model has recently been applied by Shrivastava [34]

to the temperature dependence of the 57Fe isomer shift in FeF, as observed by Perkins and Hazony [22].

In both cases the observed transfer originates from the electron-phonon interaction.

In the rare earths, the only systems studied so far are 1 % Eu2+ in CaF, and EuF, by Nitsche et al. [35]

The experiments were performed in the range between 300 K and 900 K ; the results are shown in figure 8.

In EuF, no temperature shift was observed beyond the contribution expected by the second order Doppler shift. The thermal expansion coefficient of EuF, is not known. Assuming an average expansion of Aa/AT = A/K which is roughly the value for CaF, above room temperature [35] one expects a change in isomer shift of A S

0.15 mm/s for a tem- perature difference of 500 K from the pressure data of Eu compounds as discussed above. This variation is

3 ( T )

/

14 5 ^{S O D}

FIG. 8. - Temperature dependence of the isomer shift of 151Eu in EuF2 and Euz+ : CaF2. The change of the isomer shift by volume expansion is denoted by S(P) (Nitsche et nl., ref. [35]).

of the same order of magnitude as the second order Doppler effect. Thus the observed result is in keeping with the pressure data. Nitsche et al. [35] originally had derived a different conclusion. The change of isomer shift with temperature is much larger in

EU"

: CaF, (see Fig. 8). It had been assumed that this represents the volume expansion effect and the result in EuF, was explained as arising from a cancella- tion effect due to an additional contribution of opposite sign. It is our belief, based on the now available pres- sure data, that the dilute system EuZ+ : CaF, shows the untypical behaviour. From concentration effects as discussed below one may draw additional evidence that p(0) for EuZ+ in dilute systems show a volume dependence that is considerably different from the simple compression behaviour seen by the application of pressure to Eu compounds. I t probably arises from changes in chemical bond.

4 . 2 CONCENTRATION EFFECTS. - A change in ato- mic volume can also be produced by varying the concentration of one constituent in a continuously miscible system. Examples we wish to discuss are the EuF, : CaF, and EuS : CaS systems. The variation of the lattice parameter has been determined by X-rays and was found to follow Vegard's rule [37]. For example, in EUF, : CaF, the lattice constant varies linearly with Eu concentration between the values for EuF, (5 840 A) and CaF, (5 463 A).

Maletta et al. [38] find a very steep increase of p(0) at Eu2+ for the concentration range between 0 and 3 % Eu2+. These data were later verified by Wickman et al. [37] together with the change in the

11

C6-146 ^G. M. KALVIUS, U . F. KLEIN AND G. WORTMANN lattice constant already mentioned, this leads to a

volume coefficient of p(0) which is nearly two orders of magnitude larger than the coefficients derived from pressure experiments on Eu compounds. Although Vegard's rule may not properly describe the change of volume available to the Eu2+ ion in the CaF, lattice, it is difficult to imagine that this can explain the large difference to the pressure data. For concentrations larger than 3.5 mol. % Eu no further change in isomer shift is found for the EU'+ line. Based on the volume coefficient of p(0) as derived from pressure data for the undiluted compounds, one would expect only a small variation of the isomer shift (as indicated in figure 9)

FIG. 9. -Concentration dependence of the isomer shift of 151Eu in C ~ I - ~ E U ~ S and C ~ I - ~ E U ~ F ~ (Wickman et al., ref. [37]).

The variation of the isomer shifts with lattice constant is taken from the pressure measurements on undilutesystems (see Fig. 7).

which is beyond the accuracy of the experimental data [37, 381. Thus we feel that on the high-dilution side an effect other than the compression of outer electron shells must be dominating. This conclusion is further supported by isomer shift data of Wickman et al. [37] on the EuS/CaS system also shown in figure 9. These authors find that, in contrast to the EuF,/CaF, system, the contact density p(0) decreases in the range of small Eu concentrations (up to 10 % Eu), but increases in the high concentra- tion (80-100 % Eu) range. The increase of p(0) in the high concentration range appears to be somewhat, but not significantly smaller than the increase derived from pressure experiments on EuS. It appears that the concentrated samples show the usual variation of p(0)

with volume. At low concentration a different mecha- nism must lead to opposite slopes of the p(0) versus concentration curves for CaF, and CaS as host crystals. Wickman et al. [37] explain this behaviour qualitatively by the change in the p - 5dt, type dona- tion, which leads to opposite trends in the isomer shift for the 6-fold octahedral coordination of CaS and the 8-fold cubic coordination in CaF,.

4 . 3 ALLOY

- Systematic trends of the isomer shift of d-metal impurities in tran- sition metal host have been studied by various authors [17, 39, 401. These trends should have their origin in two sources ; (1) the change in volume available to the impurity in the various host metals ; (2) the effective charge transfer arising from the different number of d valence electrons present in the various hosts. The first contribution can be estimated (as discussed in Chap. 3.1) to be roughly proportional to Y - ' , where Y is given by the Wigner-Seitz cell of the host lattice. A plot of the observed isomer shift against 1/Y gives, however, no apparent correlation.

This indicates that charge transfer effects must be dominating, which can also explain in part the varia- tion of the volume coefficients of the isomer shift in these host lattices [16]. Ingalls 1181 gives a simple treatment for the case of ''Fe based on OPW pseudo- potential calculations. For a review of pressure work on other alloy systems see reference [lo].

As mentioned above, the volume coefficient of S in Eu metal can roughly explain the observed isomer shifts of Eu2' in other metallic matrices. For any detailed discussion, e. g. the influence of the electronic structure of the host lattice on the properties of the EuZf impurity, pressure experiments on the system in question are needed.

5. Volume dependence of electric and magnetic hyperfine interactions. - We shall give a brief outlook on the situation concerning the other hyperfine interactions in discussing a few selected examples. A review on this subject is not intended.

5.1 ELECTRIC

- Of central interest is the behaviour of metallic systems.

The investigations are best carried out in simple hexagonal lattices since for this system the electric field gradient (efg) is rotationally symmetric. For the evaluation of pressure data it is necessary to know the linear compressibilities and thermal expansion coeffi- cients of the lattice (i. e. the variation of both lattice constants c and a under pressure or temperature), Some of the hexagonal metals show a nearly cons- tant c/a ratio, both as a function of pressure and tem- perature. They are the most favourable candidates.

The sources of the efg at the nucleus in a non-cubic

metal are commonly separated into a contribution

arising from the lattice charges and into a contribution

having its origin in the non-spherical arrangement of

valence electrons. In the latter contribution the efg

VOLUME DEPENDENCE OF HYPERFINE INTERACTIONS C6-147

generated by conduction electrons of non-s character is included. Little is known a priori on the relative magnitude of these two contributions to the efg, in particular since the Sternheimer shielding factor is difficult to evaluate for conduction electrons. It has been shown for pure Cd metal that the relative magni- tude of the lattice and the electronic efg can be obtained by a combined study of the temperature and the pres- sure dependence of the quadrupole interaction without the use of a specific theoretical model on electronic structure [41]. The measurements were carried out using the technique of time differential perturbed angular correlations. It is found that the lattice efg cannot describe either the pressure or the temperature dependence of the quadrupole interaction. The dominating role is played by the spatial resp. thermal re-arrangement of the conduction electrons. The pressure variation of the efg in Cd metal is shown in figure 10. The lattice efg is calculated using a point

0

5l

0 10 2 0 3 0 4 0 5 0 60 7 0 80 Pressure

[

1

FIG. 10. - Pressure variation of the electric field gradient at

11

1Cd in Cd metal (Data points from Raghavan

at., ref.

charge plus uniform background model [42]. The addition of the electronic efg as obtained from a recent band structure calculation [43] leads to a theore- tical pressure dependence of the total efg which is not too seriously different from the experimental curve (see Fig. 10). Such experiments have been extended to impurity systems like dilute Ta in Zr [49]. The results on the efg at the Ta nucleus are in keeping with the findings described for pure Cd metal. No theoretical calculations, however, exist for the electronic efg in impurity systems.

A pure point charge model is nearly always incapable of describing the volume dependence of the electric field gradient even in rather ionic non-conducting compounds. In Eu2Ti207, for example, we have

observed a variation of the efg with V W 4 (under the assumption of a congruent compression of the unit cell), whereas the point charge model predicts a V-' dependence [27]. A fair amount of Mossbauer pressure data exists for iron compounds [45]. A careful study of the temperature dependence of the quadrupole interac- tion and charge density in FeF,, KFeF,, FeC1, and FeF, has been carried out by Perkins and Hazony 1221.

Following their interpretation, the temperature as well as the pressure variation of the quadrupole splitting in FeF, is caused by a radial expansion respectively contraction of the t,, and e, orbitals. This model can also explain the temperature dependence of the isomer shifts in the investigated iron fluorides and settles their explicit temperature variation.

5.2 MAGNETIC

INTERACTIONS. - Studies of the pressure dependence of magnetic hyperfine fields have not been carried out in a large and systema- tic manner. Most data exist for the magnetic 3d metals (Fe, Co, Ni) either in the pure metals or as impurity in the other 3d magnetic hosts. These measurements are partially obtained by NMR and partially by Mossbauer spectroscopy. The latter technique, how- ever, suffers in this case from limitations in resolution.

The hyperfine field of 57Fe in Fe decreases by about 2 % per 100 kbar [46, 21. It increases by about the same amount for 57Fe in Co [46], at least initially. The situation for 57Fe in Co and Ni appears to be complex.

The data suggest a sharp increase for low pressure (20-50 kbar) and then a slow decrease with higher pressures up to 200 kbar [46, 471. The data, however, are not of truly convincing accuracy. In table I we

TABLE I

Pressure dependence of magnetic hyperfine fields from NMR measurements

Resonant Host d ln B/dP nucleus material kbar-'

- -

61Ni Ni + 8.1

57Fe Fe - 1.69 References in

5 9 ~ 0

Ni + 13.8 Raimondi and

59Co Co + 6.1 Jura [47]

59Co Fe + ^1.6

have compiled additional NMR data on the pressure dependence of the magnetic hyperfine field B,, of Fe, C o and Ni in 3d metals. Thus far only Fe metal shows at low pressures a decrease of B,, with reduced volume.

Especially for impurity systems it would be worthwhile to compare the pressure variation of the local hyperfine field with the change in magnetic moment of the matrix. Systematic studies of this type are unfortunately at present not available.

The investigation of the volume dependence of

transferred hyperfine fields have been carried out with

C6-148 G. M. KALVIUS. U. F. KLEIN AND G. WORTMANN

diamagnetic Sn as an impurity in Fe [48]. A linear increase (d In B/dP ⁼ + 20 x kbar-l) of B,, with pressure was found, as expected. Recently, this measurement has been used to extract the explicit temperature dependence of the '19Sn hyperfine field in the iron matrix [49]. A contributed paper following this review deals with the pressure variation of the trans- ferred hyperfine field on Sn nuclei in ferromagnetic Co2MnSn. This investigation also finds an increase of B,, with reduced volume.

In our own group we have studied the pressure dependence of the europium hyperfine field in the Eu2*-chalcogenides and Eu metal. A typical result is shown in figure 11. While B,, increases in the non-

I ^{87 kbar}

FIG. 11. - Volume dependence of the magnetic hyperfine fields in Eu metal and EuS (Klein et al., ref. [14, 271).

conducting chalcogenides EuS and EuO, it decreases in Eu metal. We conclude from these data, that the core polarization contact field which is largely responsible for B,, in the chalcogenides, increases in magnitude under reduced volume

e. since B,,, is negative, it

[l] FREEMAN, A. J., J. Physique Collq.

(1974) C6-3.

121 WILLIAMSON, D. L., BUKSPAN, S. and INGALLS, R., Phys.

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PANJUSKIN, V. N., SOY. Phys., Solid State 10 (1968) 1515.

becomes more negative with pressure). In the metal however, the resultant hyperfine field arises from three terms [50] :

I) Core polarization (- 340 kOe) ;

2) Conduction electron polarization by its own 4f electrons (+ 190 kOe) ;

3) The field produced by the neighbouring Eu atoms (- 115 kOe).

Our pressure data suggest that the second term has the overriding pressure dependence ; that is to say, the compression of the conduction electrons at the Eu is responsible for the reduction in magnitude of the total hf field. In order to illuminate this pro- blem further we plan pressure measurements on inter- metallic compounds of Eu where the relative magni- tude of the conduction electron field is found to be smaller [29]. Dilute paramagnetic impurities of Eu2+

in Cap,, SrF, and BaF, have also been investigated under pressure using EPR techniques [51]. In cont~ast to the Eu chalcogenides, no remarkable change in the magnetic hyperfine coupling constant was found. The results can be used, however, to settle a strong explicit temperature dependence of the Eu2" hyperfine coupling constant [52] in these systems.

6. Conclusion. - We hope to have shown with this review that pressure data and other investigations of volume dependent hyperfine interaction will give useful information on the electronic structure in solids.

Besides good band structure calculations one clearly needs more precise systematic data. The technical problems of high pressure measurements have largely been solved and nearly routine measurements are pos- sible now. It should be emphasized, however, again, that in addition to hyperfine data one needs fairly precise X-ray measurements on the change of lattice structure with pressure. This is not an easy task in cases of multi-atom unit cells and the lack of such data has largely hindered the evaluation of hyperfine measure- ments in non-metallic systems.

Acknowledgments. - The authors wish to thank A. Freeman, T. P. Das, W. B. Holzapfel and T. Butz for helpful discussions. We also wish to thank J. D. Cas- hion for his help in preparing the manuscript. The financial support of the Deutsche Forschungsge- meinschaft in part of this work is gratefully acknow- ledged.

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