57Fe IMPURITY ATOM LATTICE DYNAMICS AND SYSTEMATICS IN GROUP V AND VI HOST METALS

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57Fe IMPURITY ATOM LATTICE DYNAMICS AND

SYSTEMATICS IN GROUP V AND VI HOST

METALS

R. Taylor, D. Erickson, T. Kitchens

To cite this version:

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JOURNAL Dl3 PHYSIQUE Colloque C6, supplbment au no 12, Tome 37, dkcembre 1976, page C6-35

'Fe

IMPURITY ATOM LATTICE DYNAMICS AND SYSTEMATICS

IN GROUP V AND VI HOST METALS

(*)

R. D. TAYLOR, D. J. ERICKSON and T. A. KITCHENS Los Alamos Scientific Laboratory, University of California

Los Alamos, New Mexico 87545, U. S. A.

Resume.

-

La fraction sans-recul f et le deplacement thermique Mossbauer des impuretees trks

diluks de 57Fe ont kt6 mesurks dans les mktaux h6tes cubiques a corps centres V, Nb, Mo, Ta et

W, dans l'intervalle de 4 B 860 K. Ces quantitks expMmentales ont Bte interpretks au moyen d'un modkle de dynamique de rkseau d'un atome impurete de Mannheim. Le paramktre important

yih/yhh est une mesure du couplage de l'atome impuretk relatif au couplage correspondant dans le

rkseau h6te. Nous avons obtenu des valeurs de yihlyhh dans chaque h6te avec les donnks surf et,

indkpendamment, avec les donnCes sur les deplacements et pour chaque h6te, !accord obtenu est bon. La tendance gknkrale des donnkes indique que pour des h6tes voisins d'une m6me ligne de la table periodique la liaison relative de l'impuret6 57Fe est la plus forte dans les h6tes du groupe V

que dans les h6tes du groupe VI. Les resultats soutiennent aussi un postulat enonce auparavant suivant lequel la liaison entre premiers voisins impuretee et h6te soit proportionnelle la moyenne gkometrique du couplage premiers voisins d'un reseau form6 entikrement par des atomes d'impuret6 et de celui d'un reseau des atomes h6tes seuls.

Abstract. - The Mossbauer recoil-free fraction f and thermal shift have been measured for very dilute 57Fe impurities in body-centered cubic V, Nb, Mo, Ta, and W host metals in the range

4-860 K. These experimental quantities have been interpreted in terms of an impurity-atom lattice- dynamical model of Mannheim where the important parameter ~ i h l ~ h h is a measure of the coupling

of the impurity atom to the host lattice relative to the corresponding coupling in the pure bost lattice. We have obtained values of ~ i h l y h h for each host from thef-value data and, independently,

from the shift data, and for each host rather good agreement is obtained. The general trend of the data shows that for neighboring hosts of the same row of the periodic table, the relative 57Fe impurity binding is stronger for the group V host than for the group VI host. The results also support a previous conjecture that the nearest-neighbour binding between the impurity and the host should be proportional to a geometric mean of the nearest-neighbour couplings for a lattice consis- ting entirely of impurity atoms and for a lattice of host atoms only.

1. Introduction.

-

It is well known that dilute atomic impurities can dramatically affect the physical properties of metals. It is clear that the perturbation due t o impurities on the lattice dynamics of a metal can play a major role in determining the mechanical properties, and not too minor a role in the electronic properties, especially in the case of superconductors. Despite the importance of this problem a completely satisfactory lattice dynamical theory has not yet been developed. The realism of existing theories has been severely limited due to the complexities introduced by the broken translational symmetries. Within the last ten years some progress has been made on this theoretical question, spurred t o some extent by the development of a microscopic probe capable of measuring important lattice dynamical parameters specific to impurities

-

the Mossbauer effect. The Mossbauer effect is capable of precisely determining the mean-

(*) Work performed under the auspices of the United States Energy Research and Development Administration.

squared displacement,

<

x2

>,,

from a measurement of the recoil-free fraction

and the mean-squared velocity,

<

v2

>,,

from a measurement of the second-order Doppler (SOD) shift of the recoil-free spectral line,

In these expressions k is the propagation vector of the resonance y ray and c is the velocity of light.

For an isotopic substitutional impurity, i.e., an impurity with a mass discrepancy but coupled t o the host atoms as is a host atom, Dawber and Elliott [l]

have given an exact solution for the cubic lattice, using double-time Green's functions. They found that if the impurity mass is sufficiently less than the host atomic mass there is a vibrational mode localized on the impurity at a frequency above the continuum associated with the host as discussed by Montroll and Potts [2].

In this theory, which has been the basis of some of our

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C6-36 R. D. TAYLOR, D. J. ERICKSON AND T. A. KITCHENS

earlier work on this problem [3, 41, the localized mode frequency,

<

x2

>,,

and

<

v2

>,

depended only on the mass discrepancy and the phonon density of states of the pure host material.

More recently a significant improvement has been made by Mannheim and coworkers [5]. By using group theoretical techniques they have been able to account for impurities with a mass discrepancy as well as changes in the nearest-neighbour force constant for both bcc and fcc lattices with harmonic interactions. The results are remarkably simple and reduce to the Dawber-Elliott equations if there is no force constant change. Only one parameter is added, the force constant ratio.

The utility of the Mannheim approach was first established by the analysis of the Mossbauer observa- tions on the f-value and the SOD shift for 57Fe in V by Mannheim and Simopoulos [6]. Nussbaum, Howard and coworkers have also applied this approach to the high precision f-values of 57Fe in numerous cubic metals [7, 81, correcting for anharmonic effects [9]. More recently they have studied thef-value of 57Fe in Cu and Pd and obtain nearly temperature-independent force-constant ratios if the host phonon density of states is allowed to be temperature dependent as observed experimentally [lo]. The technique has also been employed by Prince et al. [I 11 to interpret lg7Au f-value data in Cu (2

%

Au) and Ag (5

%

Au). A more

extensive review of previous applications of the Mann- heim model for Mossbauer data is given by Cohen

et al. [lo].

In this work we'present high precision Mossbauer measurements on absolutef-values and SOD shifts for 57Fe in V, Nb, Mo, Ta, and W. These data are analyzed in terms of the Mannheim model by a non-linear least- squares algorithm [12]. The main thrust of the present work is to investigate the systematic variation of the microscopic impurity lattice dynamic quantities in the Group V and VI metals.

2. Impurity lattice dynamics.

-

The mean-squared thermal displacement and velocity of an impurity atom can be written as

ti

"

<

x2

>,

= w-I coth

(3) and

<

v2

>,

=

1:

_{w coth}

(f

loj3) gi(w) d o , (4) where Mi is the mass of the impurity atom,

P

= Ilk, T, and gi(w) is an impurity response function. gi(o), the density of phonon states at the impurity site, describes the dynamical behaviour of the impurity atom coupled to the host lattice.

Mannheim has calculated gi(o) for a substitutional impurity atom in bcc and fcc hosts with harmonic cen-

tral forces allowing for force constant modifications [5]. The Mannheim gi(o) depends on the host response function, g,(w), the impurity-host mass ratio, Mi/Mh, and the force constant ratio yih/yhh and can exhibit localized modes.

For the bcc host metals considered in this study, gh(o) has been determined from inelastic neutron and thermal diffuse X-ray scattering data. Since the mass ratios Mi/Mh are also known, the only adjustable parameter in the Mannheim calculation for

<

x2

>,

and

<

u2

>,

is the force-constant ratio yih/yhh. Because of the large temperature range used in the present measurements, it is necessary to correct for the effects of anharmonic motion. Maradudin and Flynn [9] calculated the effect of such behaviour and found the effective mean-square displacement of the impurity to be approxi'mately

where

<

x2

>,

is the harmonic contribution as described above. The linear term in temperature is controlled by E, a small constant measuring the amount

of anharmonicjty. In the fitting procedures discussed below, thef-value data for 57Fe are therefore described in terms of two parameters, yih/yhh from the impurity model, and 8.

Since the spectral line shift data do not determine (AEIE),,, directly, it is also necessary to consider another parameter to describe the shift measurements. In addition to the SOD shift, there is an additive shift (AEIE), due to the chemical properties of the impurity atom in its environment. We will assume in the follow- ing that (AEIE), is temperature independent. The total line shift AE/E as a function of temperature is then written as

where

<

v2

>=

is from eq. (4). In fitting the shift data for 5 7 ~ e , two parameters are again used, the impurity force constant ratio yih/yhh and the chemical shift (AE/E)c.

3. Experimental.

-

The temperature dependence of the recoil-free fraction and of the line shift of the 14.4-keV y rays of dilute 5 7 ~ e impurities in the several host metals was determined using 57Co-doped sources and room temperature absorbers. The wide-black absorber technique 1131 was employed to obtain absolute f-values, and a narrow line absorber was used to obtain precision shift data.

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57Fe IMPURITY ATOM LATTICE DYNAMICS AND SYSTEMATICS I N GROUP V AND VI HOST METALS C6-37

thermocoupIe and controller to obtain preset tempera- tures stable to about f 2 OC. Below 290 K LHe, LN, and dry ice-acetone cryogens were used. The temperature of the drive and of the absorber was maintained constant to within f 1 OC.

57Co sources were prepared by standard electro- plating techniques on polished surfaces of the high purity metals. Except for Nb, all were randomly oriented single crystals, 99.99

%

pure. Diffusion of the 57Co was carried out in an atmosphere of He contain- ing a few percent of H,, followed by a high vacuum

anneal to remove any dissolved hydrogen. A11 sources exhibited narrow, nearly temperature-independent Iine widths.

A narrow-line enriched K,Fe(CN), . 3 H,O resonant absorber was used to measure the line shift with tem- perature and to confirm the relativef-values. The wide-

black absorber used previously [13] was used to

determine absolutef-values. The ratio of good gammas to bad gammas [14] in the single channel analyzer

window was overdetermined using a thin Cu absorber and two thicknesses of A1 absorbers.

All Mossbauer spectra were fitted by computer. Relativef-values given by the relative area of the ferro- cyanide spectra confirmed the higher precision black absorber data.

Shift and f-value data for 57Fe in W are given in figure 1. These and similar data for the other sources were analyzed in terms of the Mannheim model discussed above. The quality of the fit shown for 57Fe in W is typical.

Kagan [15] has proposed for a harmonic system a relationship between the absolute f-value at T = 0 and the temperature dependent SOD shift. We have computed the Kagan integral from the shift data by a scheme outlined earlier [16] for comparison with the absolute f-vaiues measured at 4 K. The agreement is very good for all cases except Nb, as shown in table I. The correction for the bad y's is less dependable for

Nb and Mo because of their x rays near the 14.4-keV y-ray window. Thef-values for Nb and Mo have been renormalized in terms of the Kagan integral values in the analysis which follows.

FIG. 1. - The recoil-free fraction and shift data for 57Fe in W.

The solid lines are calculated from the Mannheim model. See table I.

4. Determination of model parameters. - A non- linear least-squares computer algorithm has been designed to fit recoil-free fraction and shift data to the forms expressed in eq. (5) and (6). To determine the effective force constant ratio yih/yhh from either type of data, it is necessary to have well defined phonon density-of-states functions gh(o) for the pure hosts. Cohe- rent inelastic neutron scattering studies have been made on Nb 1171, Mo [18], Ta [19] and W [20] at 296 K. In each of these studies, a Born-von KArmAn analysis was made providing force constant tensors that adequa- tely described the measured phonon dispersion curves. Using these force constant tensors, which often involve a relatively large number of neighbors, we have calculated gh(o) for each of the hosts using the method of Gilat and Raubenheimer [21].

The phonon-dispersion relations of V cannot be

f data

--

AE/E data Temp.

Range - Yih E ~ 1 0 4

f

(4) (")

f

(0) ( b )

Host (K) Yhh (K-l) Yhh Ref, No.

(a) Absolute f-value determined from black-wide absorber with statistical error in last digit. ( b ) $value at T = 0 from Kagan integration.

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C6-38 R. D. TAYLOR, D. J. ERICKSON AND T. A. KITCHENS

determined by conventional coherent neutron scattering techniques since the neutron scattering cross-section is almost totally incoherent, but another approach involv- ing the measurement of phonon dispersion curves via thermal diffuse scattering of X rays has been used [22] to generate a gh(w) for V at room temperature.

For each of the five hosts, we have used the corresponding g,(w) to least-squares fit the Mannheim model independently to thex-value data and to the AEIE data, The results of these fits are summarized in table I, which tabulates the two nearest-neighbour force constant ratios. Also listed are the anharmonicity parameters and (AEIE), relative to pure iron at 296 K.

5. Discussion and conclusions.

-

From table I, we note the rather good agreement between the force constant ratios obtained from independent fits to the impurity f-value and shift data for 5 7 ~ e in each of the host metals, except perhaps for V. Such agreement supports the validity of the Mannheim approach to the impurity problem. The agreement is especially significant since the two lattice dynamical quantities of interest are determined by quite different phonon frequency weightings of gi(w). The poorer agreement for 5 7 ~ e in V may be due to deficiencies in the g,(w) used. The discrepancies in the yih/yhh values suggest in part the need for further refinements of the impurity model such as consideration of impurity-host interactions more distant than nearest-neighbour and thermal expan- sion of the lattice. The latter factor is only partially offset by the anharmonicity correction factor 8. This

assumption has been necessary since the (w) has been assumed to be temperature independent. The temperature dependence of g,(w) has not yet been measured for the hosts used here. The yi,/y,, values from our f-value data should be considered as upper limits [lo]. Several previous results on 57Fe jn these same hosts obtained from the same impurity model are relevant to the work presented here. Howard and Nussbaum [8] have also made precision f-value measurements for 57Fe in Nb between 290 K and 800 K. Using the same gh(m), they determined yih/yhh = 0.85 f 0.02, with a somewhat larger E than ours. Agreement with our

results is reasonable considering the absence of low temperature measurements. Raj and Puri [23] using our previously published but uncorrected f-value data for 5 7 ~ e in Mo have obtained a yih/yhh = 0.45 omitting the anharmonic correction. We obtained a similar value before correcting the f-value data as discussed in section 3.

In figure 2 we have plotted as ordinates the yih/yhh values listed in table I. We have also included a value for Cr, due to Brace et al. [7], from a Mannheim analy- sis of precision 57Fe in Crf-value data between 80 K

Ci f r o m f d a t a (Broce,Howard and N u s s b o u m )

/ /

'

I

FIO. 2. -The correlation of 'yihlyhh with y ~ ; ' 2 . See the text.

and 600 K. The general trend of the yih/yhh data shows that for neighbouring hosts of the same row of the periodic table, the 57Fe impurity binding is stronger for the group V host than for the group VI host (i. e., V vs Cr, Nb vs Mo, and Ta vs W).

Previously we have found that the frequency moment ratio

<

w

>/<

w-'

>

for dilute 57Fe impurities in cubic transition-metal hosts, as obtained in f-value and shift studies, is correlated in a simple and unexpected way with the host Debye temperature and the impurity- host mass ratio [3, 41. The correlation was consistent with the hypothesis that the effective force constant associated with the impurity was proportional to the geometric mean of the force constants for the impurity in bulk and for the host in bulk, that is yih

4 K .

Such a hypothesis further implies that yih/yhh7 the parameter determined in this study, should be linearly proportional to y;.12 for a given impurity. Such a dependence is suggested in figure 2 where yih/yhh values

obtained from the Mossbauer data are plotted vs

yil'z.

yhh is a nearest-neighbour bond-stretching force cons-

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57Fe IMPURITY ATOM LATTICE DYNAMICS AND SYSTEMATICS IN GROUP V AND VI HOST METALS C6-39

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