A Method for Determining Impurity-Host Force-Constant Ratios
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A METHOD FOR DETERMINING IMPURITY-HOST FORCE-CONSTANT RATIOS
BEREND KOLK Physics Department,
Boston University,
I11 Cummington Street, Boston MA 02215
ABSTRACT One of the parameters of interest in studies on dilute alloys is the ratio of the effective host-host to impurity-host interaction. A method is presented which allows the evaluation of the effective impurity-host force-constant ratio, A'/A, for substitutional, isolated, impurity atoms in a cubic lattice by combining recoilless-fraction data (Nossbauer effect) of the impurity atoms with x-ray- or neutron-diffraction data of the host lattice. With this method, values of A'/A 7 are obtained for dilute 5 Fe impurities in various hosts.
INTRODUCTION Mtssbauer discovered in 1958 that there exists a non--zero probability f for recoilless emission or absorption of a y ray by a nucleus embedded in a solid [17. In these recoilless processes, no energy transfer occurs between the Y ray and the lattice via creation or annihilation of phonons. Hence, the state of the lattice before and after the nuclear transition is the same. The prohability f is often referred to as the recoilless fraction, and its temperature dependence is determined by f(T)
= exp[-2W'(T)1
(1a)
where
2W'(')
T
(Ib)
is called the Debye-oaller factor. Here ýx2/T represents the mean-square displacement (mad) of they-ray emitting absorbing atom along the direction of the y-ray wave vector k. Hence, recoilless fraction measurements permit the determination of the msd Tx*/. Iý is shown in the following section that from the temperature depen2 dence of 'X /T, certain moments and corresponding anharmonic constants of the phonon-frequency distribution can be derived. The msd of atoms in solids can be evaluated from x-ray or neutron diffraction measurements as well. The temperature dependence of the intensity of the diffracted radiation is given by a formula similar to eqn. (la) with a Debye-Waller factor, 2W(T)
=
('
-
k')2x2>T
(2)
where k and k'/are the wave vectors of the incident and diffracted x-rays or 2 neutrons, and tx 'T the msd of the atoms along the direction of k - k'. Note that - k'I = 4tsin0/l 4k where e is the angle of incidence. The latter relation inserted in eqn. (2) yields the more familiar expression in x-ray literature,
542
= 16•r2EIŽ
2W(2)
(2)
(3)
The recoilless fraction and diffraction measurement:; s'vu each their specific advantages and disadvantages. Recoilless fraction measurements permit the study of the msd's of very dilute impurities in solids. However, the number of isotopes suitable for the Mdssbauer effect (ME) is limited. Diffraction measurements are not very appropriate for determining msd's of very dilute impurities but are not restricted to a limited number of isotopes as is the ME. Thus, the ME lends itself very well to the study of msd's of impurities with such dilute concentrations that the single-impurity approximation is valid, while the msd's of the host atoms can be determined from diffraction measurements. It is the purpose of this paper t
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