HAL Id: jpa-00221180
https://hal.archives-ouvertes.fr/jpa-00221180
Submitted on 1 Jan 1981
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
THE ULTRASOUND-INDUCED ELECTRIC FIELD
GRADIENT (EFG) IN METALS
B. Ströbel, K. Läuger, H. Bömmel, V. Müller
To cite this version:
JOURNAL DE PHYSIQUE
CoZloque C6, suppZ6ment au n o 12, Tome 42, dicembre 1981
B. Strabel, K. Lzuger, H.E. ~ 6 m m e l and V. ~iiller*
Universittit Konstanz, FakuZtBt fiir Physik, Postfach 5560, 0-7750 Konstanz
,
F. R. G.
"Freie Universittit Berlin, Fachbereich Physik, Kijnigin-Luise-Str. 28-30, 0-1000
Berlin-33, Germany
Abstract.- The method of quadrupolar nuclear acoustic resonance (NAR) allows to determine the tensor S relating the ultrasound-induced dynamic EFG (DEFG) to the ultrasonic strain. The observed temperature dependence of S in cubic Ta metal supports the "phonon-model" for the empirical ~'&-law of the nuclear quadrupolar interaction in metals.
1 . Introduction.- The study of the static EFG in non-cubic metals and around defects in metals has been of considerable interest over the past decade (for a review see Ref. 1). One of the motives for this interest is the empirical temperature dependence of the EFG as (1
-
B*T~.') which is observed in nearly all metals. In order to explain this ~I"-law, two models were propounded: a "phonon- model"2 attributing the temperature dependence essentially to a Debpe-
Waller factor (DWF),
and a "quadron-mode11'3 based on the assumption of quadrupolar elementary interactions in non-cubic metals, in analogy to Bloch's ~'"-1aw in ferromagnets which is explained by the excitation of magnons.A
crucial test for any form of "quadron-model" would be the absence of a temperature dependence in an undisturbed cubic metal (where the static EFG vanishes, and thus no quadrupolar elementary excitations should be present).2. The ultrasound-induced EFG.- Although the static EFG vanishes for cubic symmetry, one is still able to observe the DEFG accompanying an ultrasonic wave. When interacting with quadrupolar nuclei in a magnetic field, the DEFG can induce transitions between Zeeman
-
levels, giving rise to quadrupolar NAR'. From the intensity of the NAR signal one can determine the fourth-rank tensor S relating the DEFG Vij at the site of a nuclens to the applied strain ekl:Vij
= z
Sijkl Ekl.k, 1
The analysis of the S-tensor components allows one also to infer on the electron- phonon interaction in metals s. The present contribution is particularly concerned with the temperatur dependence of S which is studied both theoretically (on the basis of the "phonon-model") an experimentally. (A detailed account can be found in Ref. 6).
3. Screened-potential approach.- In the approach of Nishiyama et al.' the EFG at a certain nucleus is given by a superposition of screened-potential contributions from all other nuclei, with the screening charges following adiabatically the thermal motions of the nuclei. For our needs we have expanded6 this approach to a lattice with both thermal und ultrasonic vibrations. We obtain eventually the
+
S-tensor components as a sum over all reciprocal lattice vector points G:
where a and (I-ym) are temperature-independent prefactors, Deff (T) is an effective
..
DWF, V denotes the spatial derivative in the Fourier space, 6 is the Kronecker
*
-+symbol, and V (G) is the Fourier transform of the screened-potential EFG at the
-+
site of the reciprocal lattice vector point G. It can be shown6 that within this "phonon-model" the effective DWF (and thus the temperature dependence) for the S-tensor in a cubic metal is essentially the same as one would expect for the static EFG if the metal were non-cubic.
4.
Experiment.- The temperature dependence of S has been investigated in cubic tantalum metal which due to its high nuclear quadrupole moment is especially well suited for a NAR experiment. In order to increase the experimental accuracy a novel calibration method was applied, using the magnetic-field-dependent ultra- sonic attennation known as Alpher-
Rubin effect for an internal calibration standard. In Ta there are two independent linear combinations of S-tensor components. For these the following experimental results were obtained at 300 K:(Note that with a conventional NAR experiment only the sign of the ratio of the S-tensor components is accessible.)
temperature ( K )
Figure: Temperature dependence of the ratio (Sll-S12)/2Sb4 (ab0v.e) and of the absolute value of S4 '(below) in tantalum using a TI*' scale. The fit for I S 4 4 1 corres-
to a dependence as b~(0)
(
* ( I - B - T " ~ ) withJOURNAL DE PHYSIQUE
The Figure shows the measured temperature dependence of the S-tensor in Ta. The ration of the components is found to be temperature-independent within the experimental error. In contrast to this, the S-tensor as a whole shows a decrease with temperature which can be fitted to a
law,
in analogy to the static EFG. From this fit the parameter B is obtained as +7(3) ~ o - ~ K - ' " .5. Conclusion.- Whereas the "qnadron-model" results in a zero temperatur
dependence for cubic metals, our observations are in complete agreement with the "phonon-model" which predicts a temperature dependence of the S-tensor as a whole. An estimation7 of the parameter B from the tantalum atomic mass and Debye
temperature yields +6*
lo-'
K-~". The observed temperature dependence in tantalum thus supports the idea that the empirical ~''~-1aw of the nuclear quadrupolar interaction in metals can be essentially explained by the effect of lattice vibrations.E.N. Kaufmann and R.J. Vianden, Rev. Mod. Phys.
51,
161 (1979)K. Nishiyama, F. Dimmling, Th. Kornrumpf, an D. Riegel, Phys. Rev. Lett.
31,
357 (1976)3. Christiansen, P. Heubes, R. Keitel, W. Klinger, W. Loeffler,
W.
Sandner, and W. Witthuhn, Z. Physik B2,
177 (1976)'
D.I. Bolef, in Physical Aconstics, ed. by W.P. Mason (~cademic, New York, 19691, Vol. IV A, Chap. 3V. Miller, G. Schanz, and
