Effect of indium impurities on the de Haas-van Alphen effect in lead

J. Phys. Chem. Solids

Pergamon Press 1971. Vol. 32, pp. 1721-1732.

Printed in Great Britain.

EFFECT OF INDIUM IMPURITIES DE HAAS-VAN ALPHEN EFFECT

ON THE IN LEAD*

P. J. TOBINI", D. J. SELLMYER and B. L. AVERBACH Center for Materials Science and Engineering and Department of Metallurgy and Materials Science, Massachusetts Institute of Technology, Cambridge, Mass. 02139, U.S.A.

(Received 12June 1970; in revised form 10 November 1970) A b s t r a c t - Several aspects of the effect of indium impurities on the de Haas-van Alphen effect in lead were investigated. Studies were made of the nature of several orbits in the third zone Fermi surface of pure lead and the effects of alloying on these orbits. Rigid band theory is able to account for changes in cross-sectional area for one of the orbits. Some data on the scattering temperature obtained from amplitude studies are also reported.

1. INTRODUCTION

THERE are several interesting phenomena which occur in the de Haas-van Alphen (dHvA) effect in a pure metal when impurity atoms are introduced. First, the conduction electron scattering rate increases so that the amplitude of the oscillations decreases. Secondly, the energy levels shift in the alloy and the density of conduction electrons changes, so that the shape and size of the Fermi surface changes. This implies that the d H v A frequency F changes on alloying since F = hcAo/21re,

(1)

where A0 is an extremal area normal to the magnetic field[l]. The decrease of the d H v A amplitude on alloying limits measurements typically to solute concentrations of less than about one atomic per cent. Even at these low concentrations, however, it is possible to study the rate at which various impurities scatter conduction electrons on given orbits *Research supported by the Advanced Research Projects Agency and performed in part at the Francis Bitter National Magneti Laboratory which is supported by the United States Air Force Office of Scientific Research. tPresent Address: Purdue University, Lafayette, Indiana.

in a particular metal, and to compare changes in Fermi surface areas with the simplest theoretical model of alloying, the rigid band model (RBM). Theoretical work on the d H v A effect in alloys has attempted to treat both the geometrical Fermi surface changes and the scattering aspects. Heine was the first to consider changes in d H v A frequencies on alloying[2]. His work was based on the RBM proposed by Friedei[3] and was sufficiently general to handle geometrical effects for a general Fermi surface model. Later, Brailsford [4] considered area changes and also extended the early work of Dingle[5] on the scattering time z. However, the scattering problem was considered only from the viewpoint o f f r e e electrons being scattered by a simple screened Coulomb potential. These concepts have been tested experimentally in a number of dilute alloys [6-12]. The purpose of the present investigation was to study the above-mentioned aspects of the d H v A effect in dilute Pb(In) alloys. This system was chosen because previous investigations by Gold [13] and Anderson and Gold [14] (AG) have produced a rather well defined Fermi surface for pure lead. Also, indium has a wide range of solid solubility

1721

1722

P. J. TOBIN, D. J. SELLMYER and B. L. AVERBACH

in lead and the equilibrium distribution coefficient is ~ 0.96 [15] so that homogeneity problems are minimized. Figure 1 shows the pipe model constructed by Gold for the third zone Fermi surface. Attention has been directed in this work to the orbits around the arms along the (110) directions. In particular, the work of A G sug-

< Ao .q-.(

i i!

Fig. 1. Gold's pipe model of the third zone Fermi surface of lead (Ref. [14l). gested that the single extremum of the pipe model actually consisted of a minimum plus a non-central orbit very much like an inflection point, as shown in Fig. 2. This figure resulted from the parameterized Fermi surface model constructed by A G and produced a convincing explanation for the presence of beats with about 42.5 oscillations per beat in the y oscillations when the field B is parallel to [110]. In the present work we have shown that the frequency of the non-central orbit of the A G model is actually dominant in amplitude over that of the central orbit and have measured the areas of each. In the alloys, we have measured changes in area of the dominant frequency and have compared these with RBM. The orbits in question are actually rather large ( F - 18 MG), so that the corresponding changes in area tended to be rather small with the impurity concentrations that we were able to use. Thus, only a semiquantitative test of the

R B M was possible. However, the sign of the frequency change is certainly in agreement with simple electron concentration ideas, which was n o t the case in some early measurements of Pb(Bi) alloys made by Gold [13]. Scattering temperature measurements as a function of alloying were attempted but, except for the most dilute alloy, were hampered by the introduction of a third frequency into the y oscillation beat pattern. A detailed model of the nature of the non-central orbits in the alloys was constructed to explain this effect. In the following section a brief account is given of experimental procedures. Section 3 contains the experimental results and a comparison of these with theory. The concluding remarks are found in Section 4. 2. EXPERIMENTAL TECHNIQUES The de Haas-van Alphen experiments were performed at the Francis Bitter National Magnet Laboratory using 1 5 0 k G watercooled solenoids. These solenoids were periodically recalibrated, using a coil and integrator, to an accuracy of 0.2 per cent. The coil and integrator were calibrated by nuclear magnetic resonance techniques [16]. In order to improve the field stability of the solenoids, a superconducting shield, acting as a low pass filter, was situated around the sample. This device has been described in detail elsewhere [17]. L o w frequency field modulation techniques were used throughout this work[18,19]. Several articles have been devoted to recent developments in these methods and therefore they will not be described here [20-22]. The alloys were prepared by melting the requisite amounts of material in graphitecoated, pyrex tubing evacuated to 20 microns*. U s e of precision bore tubing and a graphite coating facilitated easy removal of the grown crystals. A point, sufficiently sharp *Starting material of '69' purity was obtained from Cominco, Spokane, Washington.

EFFECT OF INDIUM IMPURITIES I

I

I

1723 I

0.115

&--, 0.112

u.w_l

8Ao -= 0.022

O.IOS

O.lOt 6

r L ii,

U,K

I 0

W

I

I 0.1

--

I O.2

dislance along zone line (2"rr/o)

Fig. 2. Cross-sectional area along the third zone arm in lead according to the parameterized model of Anderson and Gold (Ref. [14]). The point, U, corresponds to the midpoint of the arm.

to initiate single crystal growth by the Bridgman technique, was pulled on one end of the 12 mm. tubing. The alloy components were mixed by agitating the molten metal for 10 min and then allowing it to cool slowly. A furnace moving with a speed of 3 cm/hr in conjunction with a high vacuum system, were used to grow single crystals having indium concentrations of 0.13, 0.27, 0.43 and 0.62 per cent.* These concentrations were determined using standard gravimetric techniques. In order to check for gross inhomogeneities, one crystal was sampled at several points along its length. Except for the last portion to solidify, the variation in indium concentration was found to be within the accuracy of the analytical technique (-0.02at.%). *The solute concentrations here, and in the remainder of the paper, are in atomic per cent.

T w o of the crystals were subjected to spectrographic analysis to check for contamination with ferromagnetic impurities. In both cases, iron was found to be present only to the extent of 1 p.p.m. The single crystals were oriented with an accuracy of 0.5 ~ using standard Laue backreflection techniques. Samples were then spark trepanned to shape using a Metals Research spark cutter. Typically they were cylindrical, having a diameter of 2.5 mm, and a length of 5 mm. 3. RESULTS AND DISCUSSION

(a) T h i r d z o n e a r m orbits in p u r e l e a d As mentioned in the Introduction, the y oscillation beats o f - 42 oscillations/beat were perplexing until the parameterized fit of A G produced the curve of Fig. 2. The presence of the beats limited somewhat the accuracy of

1724

P . J . T O B I N , D. J. S E L L M Y E R and B. L. A V E R B A C H

the determination of the central cross-sectional area by A G . In a beat pattern the frequency of the carrier will be that of the oscillatory term with dominant amplitude [23]. Therefore, unless the sign of the frequency difference can be determined, it is not known whether to assign the carrier frequency, measured over an integral number of beats, to the smaller or larger orbit. In cases, where the number of cycles per beat is small, it is possible to determine the sign of the frequency difference from a comparison of the peak spacing (in B -1) at a minimum to that at a beat maximum. This method still works in principle for a large number of cycles per beat, but in that case the accuracy required is prohibitive. A G apparently avoided this problem by assuming the component terms to have equal amplitudes in which case the measured carrier frequency is the average of the two components. In the present work we have been able to infer from the alloy data that the amplitude dominant frequency in pure lead, for our field range of 20-150 kG, is that of the non-central orbit. Presentation of the evidence for this conclusion will be deferred until the next section. For the present, we will use this information to deduce the cross-sectional areas of the central (C) and noncentral (NC) orbits expressed in units of (2rr/a) 2:

Ac = 0.1031 ___0-0005 ANC= 0" 1055 +--_0.0005. As mentioned before these values are free of any ambiguity due to the presence of beats. It should be emphasized that deduction of these values depends upon our acceptance of the A G model (Fig. 2) and upon our observation of a simple two-frequency beat pattern in pure lead. (b) Extremal area changes on alloying The problem of predicting the shape of the Fermi surface in a Concentrated or even dilute solid solution alloy is an extremely difficult

one. Aside from the facts that the Fermi surface is smeared out and k is no longer a good quantum number, the problem is one of selfconsistently determining the charge deposited in the vicinity of each solvent and solute atom and solving the Schroedinger equation for the true potential seen, in some statistical sense, by a conduction electron in the alloy. Recent work of Stern[24,25] and Soven[26], for example, has been directed to this problem but most of their results are either qualitative in nature or refer to a simple, soluble band model in one or more dimensions. Thus outside of a recent virtual crystal approximation calculation[27] for a brass, which was later criticized by Stern[28], there appear to have been no determinations of the energy dispersion relations for a real alloy. Under these circumstances one is left with the predictions based on the RBM. In the case of dilute alloys Heine[2] showed that the fractional change in d H v A frequency is given by

8F= Z mc g . F 2tZBFN (EF)

(2)

In this equation F is the frequency for the pure metal, ~F the change in frequency for an X atomic fraction solute concentration, Z the valence difference between the solute (3 for In) and solvent (4 for Pb),/XB the Bohr magneton, mc the cyclotron mass in units of the free electron mass, and N(EF) the density of states at the Fermi level. The lower portion of Fig. 3 shows the results of measurement of the third zone T frequency orbit for BII(110). The open circles represent measurement of the carrier frequency. Except in the case of the 0.13 at. % alloy, the frequency variation appears to be small but regular. In seeking an explanation for this apparent discontinuity at 0.13 per cent, we must consider the possibility that the amplitude dominant orbit changes as a function of alloy concentration. The data point for the 0.13at.% alloy lies approximately 2.4 per

EFFECT i

Z L,d (...) ,,m Ir 0,.

=

i

i

OF INDIUM

,

=

J

)" OSCILLATIONS AT <110> VS CONC.

8F/F

20

,,-.~ I.C r I

O,

F VS CONC.

1.830

1.800 r 1.790 r 1.780~1.770 0

I

I

I

I

f

I

I

.I

.2

.3

.4

.5

.6

.7

INDIUM CONCENTRATION (ATOMIC PERCENT) Fig. 3. d H v A f r e q u e n c y as a f u n c t i o n o f indium c o n c e n tration. T h e solid line in the u p p e r plot is the R B M prediction.

cent below the value expected on the basis of the other points (dotted line). Since the beat frequency is 2.2 per cent of the carrier frequency in this alloy (and all the others), this strongly suggests that the frequencies have interchanged in amplitude. In Fig. 3, therefore, the carrier frequency plus the beat frequency is plotted (open square). This point and the four referring to pure lead and the more concentrated alloys now describe the frequency variation of the same o r b i t - t h e one which is larger and therefore non-central (Fig. 2). This conclusion was used in part (a) of this section in the discussion of pure lead. F o r comparison with alloy theory the data is re-plotted in a different form in the upper portion of Fig. 3. Here, the quantity 8F/F for the non-central orbit is shown as a function of indium concentration. The heavy line was drawn with the aid of equation (2) and several assumptions to be explained in what follows. mc and N(EF) values were experimentally determined values from cyclotron mass measurements and electronic specific heat measurements. The mass value used was mc = 0.55 at

IMPURITIES

1725

(110) [29]. This value of Mina and Khaikin agreed with that of Phillips and Gold[30] (0.56___0.01) and our result (0.53__+0-03) at (110). The masses for all of the alloys at (110) also fell within the range of 0-53----0-03. N(EF) was determined to be 17.6 states atom -~ Ry -1 from the experimentally measured value of electronic specific heat coefficient, ~/= 3-05 mJ mole -1 deg-2[31]. N o w mc and N(E~), as measured experimentally, are known to be enhanced by manybody effects, especially electron-phonon interactions [32, 33]. Thus, the use of empirical values for mc and N(EF) in equation (2) is justified only if the single-particle values of each of these quantities are enhanced equally. The work of A G appears to bear this out since the measured masses are - 2.2 ___0.2 times the band structure values. The other assumption used in applying equation (2) to the data was that the difference between the masses for the two orbits is so small as to have no significant influence on the conclusions to be drawn from Fig. 3. That is, a difference of masses of, for example, 5 per cent would affect the slope of the R B M line by the same amount and the accuracy of the 8F[F data is not sufficient to resolve this in any case. N o w the cyclotron masses were evaluated at (110) by measuring the temperature dependence of the amplitude at the beat maxima. This assumes that the mass is the same for both orbits. Since the resulting plots were linear to within several per cent, this assumption appears to be justified to an accuracy of approximately 5 per cent. There has been no mention in the above of possible corrections to the R B M due to lattice parameter changes on alloying. It is easily shown, using the measured lattice parameter changes [34] and the treatment of Shepherd and Gordon[9], that one can neglect these corrections to an accuracy of approximately 5 per cent in a 1 at.% alloy. It is concluded therefore that, within the accuracy of the data, the R B M is able to explain the results reasonably well for BII < ] 10).

1726

P. J. TOBIN, D. J. SELLMYER and B. L. A V E R B A C H

(c) Introduction o f a new frequency on alloying The exact nature of the non-central orbit was found to be a sensitive function of alloying. Results on the most dilute alloy, Fig. 4, indicate that the amplitude ratio of the component frequencies is closer to unity than in pure lead. This is apparent from the fact that the minima in the beat pattern are much narrower in the alloy. Moreover, the amplitude at the beat waists is not decreasing with

BI[(ll0)) for the 0.62 per cent In sample, which is typical of the three more concentrated alloys. The amplitude at the beat waist goes through a maximum at high field, where we see the occurrence of a 'long beat' (~ 62 cycles between points A and A'), and then begins to decrease. It will now be shown that this behavior can be accounted for by the presence of three frequencies. For simplicity, we consider the addition of three terms having equal, field-independent.

J/" j''" . ./--" " "'" " ./-"

"~ I

z n.I.e~ Z 3

40

50

60

70

80 MAGNETIC

90

I00

I10

120

150

140

FIELD IN KILOGAUSS

Fig. 4. dHvA effect in Pb + 0.13 per cent In showing beat pattern characteristics of nearly equal amplitude components.

increasing field as we observe in pure lead. but instead appears to be scaling with field in a manner similar to the beat maxima. This implies that the scattering temperature-mass products for the two orbits are nearly equal in this alloy. By measuring the amplitude at the beat maxima and making the usual plot, the scattering temperature may be determined from Fig. 5. The resulting value is To -----2.95 _ 0.2~ In the more concentrated alloys the observed beat pattern has a form which is qualitatively different from that found in pure lead and the 0.13 per cent In alloy. Figure 6(a) shows a recorder tracing (still with

amplitudes and frequencies co, t o - Ato~, to -- Ato2. The signal S is then given by S - cos tox+ cos ( t o - Atol)x + cos (to-- Ato2)x.

(3)

In this expression, the independent variable x is proportional to B -~ and the three terms are taken to be in phase at x = 0. The second and third terms may be expanded and combined with the first term giving, after some manipulation, S - [3 + 2 cos Atolx + 2 cos Ato2x + 2 cos (Atol--Ato~)x] 1t2 sin (tox+ r

EFFECT OF INDIUM I

I

I

I

I

I

IMPURITIES

1727

with

,

200 \ I 0 0 ---- x ~ x

-

-

tO = tan -1 [1 2 sin f~l x cos f~2x ] + 2 cos 121x cos f~2xJ"

(7)

50

\

20

,,r ILl r% I-._1 12.

I0 u

•

-~ -

5

2

-

I

-~ _

0.5

0.2 001

\ I 08

I I0

RECIPROCAL

I 12

I 14

I 1.6

I 18

I 2.0 FIELD IN UNITS O F I0-5 (GAUSS) -~

Fig. 5. Amplitude plot for determining the scattering temperature in P b + 0 . 1 3 per cent In. Each point corresponds to a beat maximum. T = 3.77~ IIII(110).

where sin Ao~lx+ sin Ato2x ] to = tan-I 1 -~ co-s ~ ~ A---~zxJ"

(4)

The cartier frequency is amplitude modulated by a rather complicated but slowly varying envelope function. The phase to is also a slowly varying function. If the experimental trace can indeed be explained by the presence of three frequencies, apparently two of the frequencies are considerably closer in value than either is from the third. To quantify this, we define:

f~1 = 89

+ AoJ2)

~-~2= 89

"-1-

AO,)2)

(5)

and take l~l/f12 >> 1. In terms of these new variables, (4) may be written: S - [5 + 4 cos fll x c o s l'~2x-- 4 sin 2 ~'~2X] ll2 x sin (tox+ r

(6)

In Fig. 6(b) is shown a plot of the envelope function for f~l/lI2 = 40. It is found that the essential features of the trace in Fig. 6(a) are reproduced. The 'long beat' feature is apparent between points B and B' and is due to the change in sign of cos f~2x in the above envelope function. In seeking an explanation for the appearance of a third frequency, we were led by previous experimental results and a theoretical prediction to consider several mechanisms. These included: (1) the presence of microstructure in the crystals with indium concentrations greater than 0.13 at.%; and (2) a beating introduced by localized levels predicted by Ermolaev and Kaganov[35]. To these we add (3), a geometrical effect peculiar to the nature of the non-central third zone orbits. With regard to [1], Condon[36] claimed that a new frequency was introduced in Be by - 0 . 5 ~ of microstructure. Similar effects were seen in Be and Be(Cu) alloys be Goldstein e t a/.[12] in samples with - 1 ~ microstructure. On the other hand, Gold [ 13] claims a rod-like 'bundle' structure (- 89176 misorientation) in lead cannot explain the beats in the y oscillations at (110) because the carder frequency is stationary with respect to angle there. In the present case, it is difficult to see how the amount of microstructure would depend so critically en indium concentration such that the third frequency would be absent in the 0.13 per cent alloy and present in the others. It also should be pointed out that the beat frequency produced by the non-central orbit is also stationary with respect to angle. This mechanism is therefore thought to be unlikely in producing the observed behavior. Ermolaev and Kaganov[35] have shown how beats may appear as a result of the creation of quasi-local levels by impurities. Their conclusions apply to the perturbation of an

1728

P . J . TOBIN, D. J. SELLMYER and B. L. AVERBACH

9-|-~~:i

:i:~i...i -~-:

~o ~'r::

0

,,t=

r

Or

r

I~ ,'~

~~ r

.~_

0 |

P4

~,

l)

c~

X

,..,

m~ SJ.INN AYV~IJ.I~]8V NI 3Qn/I-IcllNV I .

:.,

;,.-Lf

' i

~r

_.-L-"?""

"

: -'.--9 ~r - -

9

S.LINfl A~VH.LIgHV NI 3 O N i l ' l d n V

";

3

E F F E C T OF I N D I U M I M P U R I T I E S

electron piece of Fermi surface by the addition of donor impurities or a hole piece by acceptor impurities. In the case of the third zone Fermi surface in Pb(In) alloys, we have acceptor impurities and an electron surface. The theory therefore does not apply and it appears that no simple modification can be made to make it more general. It now remains to consider the third of the possibilities suggested above. The appearance of a third frequency seems to be consistent with the description of the non-central orbit in pure lead as essentially an inflection point. When indium is added to lead, the third zone Fermi surface tends to shrink according to simple electron concentration ideas. It is hypothesized that the cyclotron masses (most of which are non-extremal) along the arm do not have the same value, resulting in a nonuniform decrease in area. This is shown schematically in Fig. 7 which illustrates how a maximum and minimum area could develop in place of an inflection point. Curves A

1729

through E correspond to decreasing electron concentration with curve B representing the situation in pure lead. The simplest assumption for the mass values at points 1 and 2 in Fig. 7, which will produce the indicated behavior is (me)2 > (mc)l. Since the cyclotron mass is simply a measure of the rate of change of area with energy, the crosssection at point 2 will therefore decrease faster than that at point 1. In this interpretation the absence of a three-frequency beat pattern in the 0.13 per cent alloy is assumed to indicate that the minimum has n o , yet deepened enough for a maximum and a minimum to be separately resolved. It should be added that the discussion of Section 2(b) is not contradicted by this explanation. The two frequencies at points 1 and 2 in Fig. 7 are so close together in frequency (f~2/to ~ 0.05 per cent) that any amplitude switching between them as a function of alloy concentration is beyond the accuracy of our measurements (0.4 per cent). (d) Effect o f impurities on scattering rate The scattering or Dingle temperature TD, which appears in the amplitude of the d H v A effect [5], is a measure of the total scattering rate for electrons on a given cyclotron orbit. TD is related to a relaxation time by [4]

I>n,-

TD =

h

2wknz"

(8)

n,"

Brailsford has shown how r is related to a resistivity relaxation time ~-pdefined by

Z tad it,"

_

rp

0 DISTANCE

.05

.I

ALONG ZONE

.15 LINE (~) U

Fig. 7. Schematic illustration of the area along the third zone arm for decreasing electron concentration (,4 through E).

m

(9)

Ne2p,

where m is the free electron mass, N the conduction electron density, and p the resistivity. Brailsford assumed scattering of free electrons by a screened Coulomb potential and obtained [4]

r-T-= 9 ( 1 + q2/2kF 2) Tp

( 1O)

1730

P . J . TOBIN, D. J. S E L L M Y E R and B. L. A V E R B A C H

compare with equation (10), we define To =

where

h/27rk~'p so that the effect of the dislocation

[,k+l~

1], J

q-1 is the screening length, and kp is the Fermi wavevector. Free electron values for q and k~. appropriate to Pb give rh'p = 0.58. As mentioned in Section 3(c), the introduction of the third frequency for concentrations above 0.13 per cent In made TD measurements in the more concentrated alloys impossible. Thus, c o m p a i o n with the theory is possible only for the 0.13 per cent alloy. N o w the values obtained for To in pure lead (1.62~ and 1.73~ are considerably larger than those obtained by Phillips and Gold[30]. Their work showed that unless special precautions are taken with respect to sample perfection, the measured values of To in pure lead are due to broadening of the Landau levels caused by dislocations and not to impurity scattering. The lowest value of To obtained by Phillips and Gold for the y oscillations at (110) is 0.31~ indicating that the mechanism they suggest is probably the explanation of the higher values measured here. In the 0.13 per cent alloy To was 2-95~ If we assume that Mathiessen's rule is approximately obeyed (as it should be for two elastic scattering mechanisms) and that the scattering rate due to dislocations in the alloy is roughly the same as in pure Pb, we may obtain an experimental ratio (~'h'p), due to impurity scattering, to compare with the theoretical value. In determining rp an enhanced electron mass (2.2m) is used[33] and a tabulated value of the resistivity (22 micro--ohm-cm) is employed in conjunction with the measured residual resistance ratio ( R R R ----p(295 K)/p(4.2 K)). Since these alloys are superconducting at 4.2~ the R R R for each alloy was determined be extrapolating the transverse magnetoresistance to zero field. The resulting R R R for the 0.13 per cent alloy was 200. To obtain the experimental quantity to

scattering may be eliminated approximately by putting

imp = TD(0"13) -- To(0"0)' where T o ( 0 . 1 3 ) = 2.95~ and To(O'O) = 1.67 ~ This gives (z/~'p)lmp = 1"7. This value is quite obviously very much larger than the theoretically predicted one. It is not clear whether this large discrepancy, which is in the same sense as for Be alloys[12] and A1 alloys [9], is a result of a failure of the theory or of band structure effects on the effective Tp.

4. SUMMARY AND CONCLUSIONS

Several aspects of the effect of indium impurities on the de Haas-van Alphen effect in lead have been investigated. Attention was focused on the third zone electron arm orbits with BII(ll0). It was shown from the dilute alloy data that the dominant frequency of the beat pattern in pure lead is that of the noncentral orbit. Values for the extremal crosssectional areas of the central and non-central areas were obtained. These values are free of any ambiguity due to the presence of beats in the oscillations. The changes on alloying of the extremal area of the non-central orbit were compared with the rigid band model. The area decreased on alloying as expected because the arms are electron pieces of Fermi surface and adding indium to lead decreases the electron concentration. Because of the limited accuracy of the data, it is possible to conclude only that deviations from rigid band behavior are less than about twenty-five per cent in Pb(ln) alloys. It is perhaps not surprising that deviations from the R B M are not large in this alloy system. This follows since the perturbation theory treatment of Friedel, upon which the R B M result (equation (2)) is based, would be expected to be valid in the Pb(In) alloy

EFFECT OF INDIUM IMPURITIES

system. This statement, in turn, is based upon the validity of pseudopotentials and freeelectron-like pseudowavefunctions in Pb[14] and upon the conclusion of Stern[25], who states that perturbation theory should be correct in alloys for which the valence difference is a small fraction of the valence of either the solvent (4 for Pb) or solute (3 for In). In alloys with greater than 0.13at.% indium a third d H v A frequency was observed in the beat pattern. Several possible origins for this effect were considered but the most likely of these is that the apparent inflection point in the third zone changes into a resolvable maxima and a minima on alloying. This suggestion could be checked in two ways. One is that the cyclotron masses could be evaluated from a parameterized pseudopotential fit in the vicinity of the orbits in question. Secondly, increasing the electron concentration, in an alloy like Pb(Bi), would not only fail to produce a third frequency, but might also completely eliminate the extremum at the inflection point. Preliminary results[37] in a Pb (0-5 per cent Bi) alloy showed in fact that one of the two beating 3, frequencies had essentially disappeared although it was possible to follow the y oscillations only over about 200 cycles. Finally, an attempt was made to investigate the iiafluence of indium impurities on the scattering rate appearing in the d H v A amplitude. It was impossible to study concentrations higher than 0.13 at.% indium because of the third frequency appearing in the beat pattern. The limited data gave a z/~p which was considerably higher than the free electron predictions, as is the case in other alloys of AI and Be. However this phenomenon should be studied carefully as a function of solute concentration, preferably in crystals of low dislocation content. Therefore it would be well to attempt such measurements in a system like Pb(Bi) where, as discussed above, the introduction of the third frequency is not likely.

1731

Acknowledgements-We wish to thank Drs. I. S. Goldstein and J. Ahn for helpful discussions and assistance in the experiments. We are also indebted to the staff of the Francis Bitter National Magnet Laboratory for the use of its facilities. Support of this research by the Advanced Research Projects Agency is gratefully acknowledged.

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