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PHYSICAL REVIEW B VOLUME 13, NUMBER 9 1 MAY 1976Phonon properties of A-15 superconductors obtained from heat-capacity measurements*

G. S. Knapp and S. D. BaderArgonne National Laboratory, Argonne, Illinois 60439

Z. Fisk~University of California, San Diego, California

(Received 29 October 1975)The heat capacities of Nb3Al, Nb3Sn, and Nb, Sb were measured between 2 and 400 K. Their superconductingtransition temperatures T, are 18.5, 17.9, and 0.2 K, respectively. The higher-temperature entropy wasanalyzed, and it was found that phonon-mode softening occurs on cooling for the two high- T, compounds, butnot for Nb3Sb. Geometric mean phonon frequencies cog are also evaluated for these compounds and for V,X,where X = Si, Ga, Ga05Snoand Sn. As is shown, cog is an appropriate phonon average for evaluatingMcMillan's expression for the electron-phonon mass enhancement. Average phonon properties cannot reliablybe used to calculate T, values for these A-15 materials. It is suggested that, for the higher electronic density-of-states materials, select phonon modes strongly couple to the electronic system and influence the magnitudeof T, to a greater extent than average-phonon correlations would indicate. This view is supported by theobservation that the higher-density-of-states materials exhibit the more pronounced phonon-mode softening oncooling.

I. INTRODUCTIONCompounds that possess the A-15 crystal structure

have the highest known superconducting transitiontemperatures. Accordingly, the A-15 supercon-ductors have been the subject of extensive experi-mental and theoretical investigations. ' Consider-able evidence exists that soft phonons are associat-ed with the A-15 high superconducting transitiontemperatures. Ultrasonic and inelastic neutronscattering experiments clearly indicate that, forthe high-T, members V,Si and Nb, Sn, extensiveacoustic-phonon softening occurs with decreasingtemperature. ' In an earlier paper devoted tovanadium-base A-l. 5 superconductors, we usedheat-capacity measurements to show that the ex-tensive shifting of optical-phonon modes also ap-pears to occur for the higher-T, A-15 compounds. 'The purpose of the present study was to charact-erize certain average phonon properties for someniobium-base, as well as vanadium-base, A-15superconductors, and to relate the phonon physicsto the superconductivity of these interesting mat-erials. To further elucidate the nature of thesuperconductivity of these well-studied A-15 com-pounds, we attempted to determine whether thevariations in the magnitude of the superconductingtransition temperatures were governed primarilyby variations in phonon properties. In principle,the best experimental probes to investigate theproblem under consideration, are superconductivetunneling and inelastic neutron spectroscopy. Un-fortunately, practical limitations often precludetheir use. Although the heat capacity is insensitive

to details of the phonon spectrum, certain aver-aged properties, or moments of the phonon spec-trum, can be assessed by heat-capacity measure-ments. Closely related moments have been identi-fied by McMillan as being the phonon propertiesrelevant to superconductivity theory. 'It is particularly interesting to include the vana-dium-base A-15 superconductors, which we re-ported earlier, ' with the new results for the nio-bium-base compounds for the following reason.The superconducting transition temperature isknown to increase strongly with d-electron densityof states at the Fermi energy N, (E~) for the vana-dium-base compounds, whereas only a weak cor-relation exists between T, and N~(Er) for the nio-bium-base compounds. By a comparison of theproperties of the vanadium- and niobium-basematerials, insight can be gained in understandingthe different roles the d electrons play in modi-fying the superconductive properties of these mat-erials.Concerning purely phononic properties, we findfrom the heat-capacity analysis that temperature-

dependent phonon-mode softening does occur forthe high-T, compounds Nb,A1 and Nb, Sn, as wasobserved for the high-T, vanadium-base A-15superconductors. ' The magnitudes of the averagephonon-mode shifts with temperature, for the high-T,niobium-base A-15 compounds, are smaller thanthose observed for the vanadium-base compounds.This observation is explicable within the frameworkwe used previously in discussing the origin of thephonon-mode shifts in the V,X system. Using theidea that temperature-dependent electronic screen-

13 3783

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3784 G. S. KNAPP, S. D. BADER, AND Z. FISKing arising from peaked electronic densities ofstates in the vicinity of the Fermi energy causestemperature- dependent phonon- mode frequencies,we expect smaller frequency shifts for the nio-bium-base materials. This is because the nio-bium-base A-15 compounds have electronic densityof states that are less peaked in the vicinity of Ebased on the temperature dependence of their mag-netic susceptibilities and the magnitude of theirnormal-state electronic heat capacities at low tem-peratures.Concerning the relationship of the purely phonon-ic properties to superconductivity, we find thataverage phonon properties cannot reliably be usedto calculate superconductive properties for theseA-15 compounds. The reason, we believe, is thatselect phonon modes can be strongly coupled to theelectronic system and influence the magnitude ofT, to a greater extent than average phonon corre-lations would indicate. Such a situation appears tooccur most dramatically for V,Si, and, in general,plays a larger role in the enhancement of the T,values in the vanadium-base than in the niobium-base super conductors investigated.

the V,X compounds were described previously. 'The heat-capacity measurements on the Nb,X com-pounds were made using the heat-pulse method,employing feedback to regulate the temperature ofthe radiation shields surrounding the sample. ' Theaccuracy of the data above 20 K was of the order of1 jo, based on measurements of American Calori-metry Conference Standard Cu. ' Below 20 K, theaccuracy of the data was of the order of 2% forNb, Sn and Nb,Al. For Nb, Sb, the accuracy was ofthe order of +10%because only a small sample wasavailable and the molar heat capacity of Nb, Sb wassmall at low temperatures.

III. ANALYSIS OF DATA AND RESULTSThe thermodynamics relevant to the analysis ofthe data will be introduced in the first portion ofthis section. We will not restrict ourselves to

the harmonic approximation, since it is clear fromour earlier work' that optic-phonon-mode shiftingcan be an important characteristic of the high T,vanadium-base A-15 superconductors. The latticeentropy S~ can be represented by using the resultsobtained from the harmonic approximation

II. SAMPLES AND EXPERIMENTAL TECHNIQUEThe preparation and characterization of the V,Xcompounds are described in Ref. 2. Each of the

Nb, X compounds were prepared differently be-cause of metallurgical considerations unique toeach system. The Nb, Al was prepared by meltingcompacted powders of Nb and Al in a levitationfurnace, and then heat treating at 1550 and 600 Cfor 5 min and 1 week, respectively. The Nb, Snsample was obtained by hot pressing appropriateamounts of Nb and Sn, in a graphite die -1 cm indiameter, for 15 h at 1175'C and 2 kbar. The re-sultant pill was &90% of its theoretical density.The Nb, Sb sample was prepared at La Jolla by anI,-vapor-transport technique. ' A stoichiometricmixture of 0.25-mm-Nb foil and 99.999% Sb weresealed with 40-mg L, in a 10-cm-long, 1.5-cm-diarn (out-gassed) quartz tube. After an overnightsoakat900'C, the tube was placedinan800-900 Cgradient for 2 weeks, with the charge at the coldend. To increase the transport rate, the chargetemperature was lowered to 750'C and kept at thattemperature for 1 week. All samples were studiedby x-ray and metallographic analyses. The Nb,Aland Nb, Sn samples were at least 90% single phase,and the Nb, Sb sample consisted of a collection oflarge single-phase grains.The T, values were determined from the heat-capacity data for all samples except Nb, Sb, forwhich T, was detected magnetically in a dilutionrefrigerator. The heat- capacity measurements on

S~ = k~ g [Ph u&,n, + ln(n, + 1)],s =&where P=(keT) ', n, =[exp(Pars, )1] ', and allow-ing the ro, normal-mode frequencies of harmonictheory to be temperature dependent. ' The abovestatement has long been known to be correct tolowest order in anharmonicity (phonon-phonon in-teractions) and is now also known to be correct tothe lowest order when the phonon-frequency shiftswith temperature are due to electron-phonon inter-actions. a (In our earlier paper, ' we tacitly use theterm anharmonicity to describe either of the aboveinteractions. We will now use the term nonhar-monicity to refer to either type. )From Eq. (1), it can be shown that S~ approachesits high- temperature limiting form

S~=SNk~ In

where, if X is Avogadro's number, the entropy isexpressed on a per gram-atom basis, the BareBernoulli numbers (B,= , ,B,=,B,=~~,ec.),(&o") is the nth moment of the phonon spectrum de-fined for all n&-3, 10 as

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PHONON PROPERTIES OF A-15 SUPERt ONDUCTORS. . . 3'785TABLE I. Table of smoothed constant-pressure heat-capacity data for the indicated niobium-base A-15 super-conductors. The units are J/ Kg atom.

Nb3A1 Nb3Sn Nb38b2025303540455060708090100120140160180200220240260280300320340360380400

0.771.442.273.254.365.466.508.4710.3711.9613.4514.6816.6618.4119.7120.7121.5222.1322.6323.0423.3923.7224.0224.3024.5624.7025.02

1.262.103.154.405.716.958.1710.4512.3914.0715.3416.4218.3119.7421.0621.9022.5122.9823.2723.6423.8724.1824.3624.5524.7224.8524.95

0.350.771.392.163.224.295.467.659.9011.7513.3514.7317.0418.9820.3321.4522.3522.9923.4623.8524.1924.4524.6724.8525.0025.1325.21and co, is the geometric mean frequency

(4)We will show in Sec. IV that co is closely relatedto the average over the phonon spectrum that Mc-Millan' indicated to be important for superconduc-tivity. It is also useful to assign n different Debyetemperatures 8n(n), each of which corresponds toa Debye phonon spectrum that has the same nthmoment as the actual spectrum

n+38n(n) =& (&u"), n & -2,WO.BWithin this formalism, 8n(-2) is defined as theDebye temperature obtained from the elastic con-stants. For n =0,

8n(0) =e' ~'k~, /ks,i.e. , the expressions for (&u") and 8n(n) are wellbehaved in the limit as n-0. Representing themoments by their associated Debye temperaturesfacilitates comparisons in that the 8n(n) all havethe same dimensionality, whereas the (~") do not.The 8n(n) are expected to vary smoothly with n,and, in general, vary slowly for n greater than

approximately -1, especially for solids that aremetallically bonded. 9The high-temperature limiting form of the lat-tice heat capacity C~ can be obtained from Eq. (2),since C =T(SS/ST)~,1 )f'((u')C1, =3NkB 1+, ~ ~ +AT,12 k,2Z (7)where only the first term in the power series ofEq. (2) was retained, and

A din(a), 1 ~ 1 d~,3NkB dT 3N ~ (o dT 'In harmonic theory, the (d, are independent ofT and A =0, hence AT can be identified as an ex-plicit nonharmonic contribution to the heat cap-acity in the high-temperature limit.Table I contains smoothed constant-pressureheat-capacity data C~ above 20 K. The C~ was notconverted to constant volume C. The dilation con-tribution to the heat capacity C~C(typically -1%of C~ for these materials at room temperatures)is an anharmonic contribution, and it is absorbedin the A coefficient at high temperatures. Also,to facilitate a comparison of our results with in-elastic neutron scattering studies, when they be-come available, the total frequency shifts are therelevant quantities of interest.To analyze our data for phonon properties, wefirst need to characterize the electronic contribu-tions to C and S. To lowest order, the normal-state electronic heat capacity C~ and entropy SEare linear in temperature, In the low-temperaturelimit, CB=S~ =ypT Table II contains y, values ob-tained in the usual manner. ' To improve upon thelinear approximation, we have used the resultsof model calculations to describe the additionaltemperature dependences to CE and S~ caused byboth electron-phonon renormalization" and band-structure effects. ' For the latter effect, the temp-erature dependences of additional electronic prop-erties (i.e. , the magnetic susceptibility or "Vnuclear spin-lattice relaxation time T,) were util-ized as constraints. ' For the Nb, X compounds,C~ and S~ tend to be smaller in magnitude and havesmaller temperature-dependent band- structurecontributions than in the V,X compounds. Hence,although T, measurements are currently lackingfor the Nb, X system, both systems can be ade-quately characterized electronically. Using theprocedure described in detail in Ref. 2, room- tem-perature y values of 3.0 +0.3, 3.6 +0.4, and 0.6 +0.1mJ/'K'-g-atom are estimated for Nb,A1, Nb, Sn,and Nb, Sb, respectively.Thus, we can now proceed with the thermody-namic evaluation of the phonon properties. Equa-tion (7), as written, is convergent in the vicinity

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G. S. KNAPP, S. D. BADER, AND Z. FISK 13TABLE II. Calorimetrically determined properties of A-15 compounds.

Compounds T. ('K) 'Yo (m~/ K g atom) e&(2) ( K) h(d~/k~ ( K)V3 SiV)GaV3Gao. 5Sn(. 5V3Sn

16.514.3

3.8

16.721.610.27.22

498399371364

326254

259Nb3A1Nb3SnNb3sb

18.517.90.2

12.2370308335

239201246

of -0.7&8~(2)/1.3. Since Cs and the nonhar-monic contribution to C~ are approximately linearin temperature, the slope of a (C3R)/T vs T 'plot yields a (&u') value, and hence a 8c(2) value,for each material. " The values of 8~(2) appearin Table II. The experimental entropy for eachmaterial as a function of temperature was obtainedby evaluating the area under appropriate C/T vsT plots. The high-temperature values of ~orthe equivalent 6n(0), for each material, could beobtained from Eq. (2). To follow the temperaturedependence of 8~(0) for temperatures above ,8the lowest temperature for which Eq. (2} israpidly convergent, the terms in Eq. (2) throughorder (w') were retained and two assumptionswere made. First, we assumed 8~(6}=8~(4)=6~(2). For a Debye spectrum, the (&o') and (&o')terms in Eq. (2) are small compared with the totallattice entropy above -M3~, and, for any spectrumwith a reasonably well-defined cutoff frequency,the 8 (n) values vary slowly and smoothly for n &2.Hence, this assumption is not unreasonable andcannot cause significant errors in the determina-tion of 8~(0). The second assumption is that thetemperature dependence of 6~(2) is one-half thatof 8n(0). This required that Eq. (2) be iterated toobtain consistent 6~(0) vs T plots. The precisevalue of one-half was chosen somewhat arbitrari-ly. However, if the one-half value is varied by+100%, 6~(0) values shift a maximum of only a fewpercent and only below a temperature of,6~(0).It is expected that 6~(2) is less temperature sensi-tive than 6D(0}, since the present work taken to-gether with V,Si and Nb, Sn ultrasonic and inelasticneutron scattering studies indicate that the lowestmoments exhibit the largest shifts with tempera-ture. In Fig. 1, the temperature dependence of8~(0} (and hence &o,) is displayed for the materialsof interest. Phonon-mode softening with decreas-ing temperature occurs most dramatically forV,Si and V,Ga. Only the slope of 8~(0) vs T plotfor Nb, Sb has the expected sign, if dilation governs

500480460440 4Q380360

0

360340340340300"

IIOO

Nb~ SnI200T( K)

y~ Gaog Sn~v~ sn

&by Sb

I300

38036Q

=6040404 P=340~ 30080

FIG. 1. Temperature dependence of the Debye tem-perature associated with the n th moment of the phononspectrum in the limit n 0. This Debye temperature isproportional to the geometric-mean frequency of thephonon spectrum and is closely related to the appropriatephonon properties for calculating A. Note the trend ofphonon softening with decreasing temperature, which ismost pronounced for the high-T, compounds V3Si, V&Ga,and Nb3Sn.

the phonon-mode shifts (i.e., since C~C&0,d&o /dT &0, and mode softening occurs as the melt-ing temperature is approached).From the slopes of the straight lines drawnthrough the data in Fig. 1, the values of the fre-quency-shift parameters A/3R are obtained [Eqs.(6} and (8)]. These -A/3R values are listed in thesecond column of Table III. It should be emphasiz-

ed that ambiguity exists in interpreting the magni-tudes of these A/3R values. For example, forV,Si at room temperature, a value of -0.07 isfound for A/3R times temperature, but is is not

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PHONON PROPERTIES OF A-15 SUPERCONDUCTORS. . . 3787TABLE III. Additional properties of A-15 superconductors.

Compounds -A/3R =d inn~ /d T {10 'K ) &(E~) (states/eV atom) A, hg(dg (eV/A, ) A.MGdg (eV/A. ~)V3SiV,GaV3Gao 5Sno 5V3SnNb3AINb3SnNb3Sb

2.42.30 40.10.20.81.0

3.84.82.72.0

2.40.4

0 860.910.620.561.071.170.3

8.616.437.308.147.827.1810.85

7.4

4.68.48.4

known whether, for instance, all the modes shift7% on cooling from room temperature, or half themodes shift 14'fg on cooling, etc. We will again re-fer to this important point in Sec. IV. The errorsassociated with the A/3R values are +4 x 10 "K '.For the V,X compounds, A values were obtainedearlier' from a high-temperature heat-capacityanalysis, which are the same as those listed inTable III, to within the stated error limits.

IV. DISCUSSIONIn Sec. III, certain phononic properties, such asthe geometric-mean frequency co, and its tempera-ture coefficient, -d luau, /dT =A/3R, were obtainedfrom the heat-capacity measurements. In this sec-tion, we will first discuss these purely phononicproperties and then their relationship to supercon-

ductivity.Referring again to Fig. 1, it is clear that, ex-cept for Nb, Sb (for which dilation effects play adominant role), phonon-mode softening with de-creasing temperature is a characteristic of all theA-15 compounds investigated. Although the effectis not pronounced for V,Sn and Nb, A1, it should benoted that even in these cases the mode softeningwith decreasing temperature is sufficiently strongto totally mask the usual dilation effect. In ourearlier paper, we argued that the phonon-modesoftening was due to conduction-electronic screen-ing of the bare-ion potentials. ' Temperature-de-pendent softening occurs because the sharp struc-ture in the d-band density of states in the vicinityof the Fermi energy, which causes electronicproperties such as the magnetic susceptibility to betemperature dependent, produces temperature-dependent electronic screening of the phonon-modefrequencies. The present study supports that view-point. Nb, Sb does not soften on cooling because dcharacter at E~ is barely present, based on themagnitude of its low-temperature electronic heat-capacity coefficient in the normal state yo (TableII). The compounds V,Sn and Nb,A1 exhibit less

e -1.04(1+z)1.45 Xw (1+0.62KI) ' (9)where p, * is a Coulomb repulsion parameter, whichis approximately equal to 0.13 for transition met-als, and 8 is a characteristic phonon parameter.We use 8~(0) extrapolated to its value at T, forthe e in Eq. (9}. McMillan indicated that

N(Ez) &I'&M(& &/& '&)'

where I is the electronic matrix element of thechange in crystal potential when one atom is mov-ed, and for M we use the gram-atomic weight.Using Eqs. (5} and (5) and since e~(n) varies slow-ly and smoothly as a function of n for n &-1, onefinds

&~&/&~ '& = 2 (}te/E)'[eg(+1)]leg(-1)]= ~,, (ll)hence, X=N(E~)&P&/M&u', .McMillan noted that, for a number of bcc transi-tion metals, the numerator in Eq. (10) is empiric-ally found to be approximately constant. ' Hopfield"has since provided a theoretical basis for under-standing McMillan's empirical observation. How-ever, Hopfield's arguments might not be valid inthe presence of gross lattice instabilities. Hesuggested that both the product N(Ez)&I'& is inde-pendent of N(E~} for metals with large d electroncontributions to N(Ez), and the magnitude of

softening on cooling than the remainder of the com-pounds under consideration (except Nb, Sb) becauseless peaking of their d-band density of states oc-curs at E~. Again, this is based on the magnitudesand temperature dependences of their electronicproperties, such as y, and y.We now attempt to relate the phonon propertiesto the magnitudes of the observed T, values. Mc-Millan's strong-coupling formulation of the BCStheory is used, whereby T, is related to the elec-tron-phonon mass enhancement X via the semi-empirical expression'

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3788 G ~ S ~ KNAPP, S. D. BADER, AND Z. FISK 13N(Er}(I') is largely determined by the identity ofthe transition-metal element for binary alloyscomposed of transition and nontransition metalssuch as Nb, X and V,X. An important consequenceof this finding is that X should be inversely pro-portional to average force-constant values such asMco' within the V,X and Nb, X systems. The Xvalues obtained from Eq. (9) are shown in column4 of Table III. In columns 5 and 6, the M+' andXM~' values, respectively, are listed. The non-constancy of the XM(d', values in column 6 and thelack of any simple trend in their values indicatethat A-15 superconductors under consideration areexceptions to the McMillan-Hopfield expectations.[This statement remains true even if Nb, Sb is ex-cluded from consideration on the basis that it doesnot meet Hopfield's criterion of substantial d-elec-tron contributions to N(Er) ]For.instance, we notefirst that V,Ga and V,Si have X values that differby only -6/0, but their Mco', values differ by 30%0.Second, V,Sn and V,Si have X values that differ by-40%, but their M&, values are quite similarhelower-T, compound V,Sn is actually -6% softer onthe average. Hence, average phonon propertiescannot be reliably used to calculate superconduc-tjoe properties, such as T, or x, for these interesting A -15compounds.We now attempt to explain our observation thatvariations in M(d, do not quantitatively correlatewith variations in the T, values of the A-15 super-conductors of interest. We focus primarily on theV3X compounds be cause our obse rvation above ismost firmly established for the vanadium-basecompounds. We recall that an important correla-tion has been found between T, and density of statesat E~ for the V,X compounds. NMR studies in-dicate that, for the pseudobinaries V,Ga, SnandV,Ga, Pi, T, (normalized by a Debye tempera-ture) increases strongly and monotonically withthe bare or "band-structure" N(E}value." Thisis a particularly striking observation since the T,vsx curves are different for the two pseudobinarysystems and nonmonotonic for V,Ga, Pi. We re-call also that a strong correlation between the bareelectronic density of states at E~ and the frequen-cy-shift parameter A has been found (Table III).Hence, N(E), an electronic property, stronglycorrelates with both Ta superconductive prop-erty, and A, a phonon property. To understandthe lack of quantitative correlation between T, andthe phonon property M(d', we return to the ambig-uity in the interpretation of the magnitude of theA parameter. For a given A value, for instance,it is not clear whether essentially all the phononmodes are uniformly shifting a certain percentagewith temperature, or whether a small number ofmodes are shifting much more dramatically with

temperature. Ultrasonic and inelastic neutronscattering experiments on V,Si and Nb, Sn, whichhave probed acoustic modes to approximately one-half the Brillouin-zone boundary small fractionof the total number of phonon modesndicate thatanomalous softening only occurs for certainbranches and over certain ranges of wave num-bers. ' Adopting the viewpoint that the calorimet-rically determined frequency-shift parameter isbeing influenced by a small fraction of the totalnumber of modes, which are selectively softening,we can understand the lack of quantitative correla-tion between Mv' and X. Select phonon modescouple strongly to the electronic system and in-fluence the magnitude of T, to a greater extent thanaverage phonon property correlations zoould in-dicate. Extensive theoretical work has provided abasis for the above statement. '4The recent work of Sinha nad Harmon is particu-larly relevant to our discussion in that it presentsa clear theoretical basis for selective electronic-

ally driven phonon softening that enhances T, inhigh-density-of- states materials. " Sinha andHarmon address their work to the phonon anomal-ies in Nb and NbC. The anomalies were detectedby neutron inelastic scattering measurements, andconsist of dips in the frequency versus wave-num-ber curves for certain acoustic-phonon branchesnear the zone boundary. According to Sinha, theI values [cf. Eq. (10)]for these select branchesand wave numbers will be enhanced by roughly(I+V,It, ) ', where V, is a q-dependent potential-energy term, which is negative at high-q values,and X, is the generalized susceptibility, which isapproximately equal to the d-band density of statesat E~." Therefore, if V,X,~1 over certain rangesof q, these modes will dominate over the otherelectron-phonon couplings and strongly enhance X.These same soft modes, because they are relative-ly few in number, will have a much smaller effecton (d, and on the frequency-shift parameter obtain-ed from heat-capacity measurements.Based on the rather erratic magnitudes of theM~' values in column 6 of Table III, we believethat the extremely strong coupling of select phononmodes to the conduction electrons, which, for ex-ample is addressed by the Sinha-Harmon theory, "occurs most dramatically for V,Si. The high-T,compounds Nb,Al and Nb, Sn have considerablysmaller values of N(Er) than the high-T, compoundsV,Ga and V,Si. The frequency-shift parametersof these niobium-base A-15 compounds are accord-ingly smaller than those of the high-T, vanadium-base compounds. Also, although a correlationbetween TgB (or a} and N(Er) is not well estab-lished for the niobium-base A-15 superconductors,it is safe to conclude that X is a weaker function of

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PHONON PROPERTIES OF A-15 SUPERCONDUCTORS. . . 3789N(E~) for the niobium-base than for the vanadium-base A-15 superconductors of interest. The largevalues of ) for Nb, A1 and Nb, Sn appear to be large-ly associated with inherently strong electron-phonon coupling at the niobium sites, as suggested,for instance, by Hopfield. " The V,X compoundsappear to have an inherently weaker electron-phonon coupling at the vanadium sites comparedwith that at the niobium sites in the Nb,X com-pounds. High-T, values occur for some V,X com-pounds because X can be significantly enhanced byselective electron-phonon couplings in the higher-d electronic density-of-state V,X compounds. Ingeneral, since selective coupling seems to play a

larger role in producing high-T, values in the van-adium-base than in the niobium-base A-15 super-conductors investigated, we predict that neutroninelastic scattering experiments should reveal thatthe phonon anomalies of the type observed in NbCwill be much more pronounced in V,Si than inNb, Sn.

ACKNOWLEDGMENTSWe would like to thank S.K. Sinha for an illumin-

ating discussion, F. Y. Fradin for his constantsupport, and R. A. Conner, J.W. Downey, andT. E.Klippert for technical assistance.

*Work supported by the U. S. Energy Research andDevelopment Administration.)Work supported by the Air Force Office of ScientificResearch Contract No. AFOSR/F44620-C/0017.'See, for instance, L. R. Testardi, in Physical Acous-tics, edited by W. P. Mason and R. N. Thurston (Aca-demic, New York, 1973), Vol. 10, p. 193; L. R. Tes-tardi, Rev. Mod. Phys. 47, 637 (1975);M. Weger andI. B.Goldberg, in Solid State Physics, edited byH. Ehrenreich, F. Seitz, and D. Turnbull (Academic,New York, 1973), Vol. 28, p. 1.2G. S. Knapp, S. D. Bader, H. V. Culbert, F. Y. Fradin,and T. E ~ Klippert, Phys. Rev. B 11, 4331 (1975).W. L. McMillan, Phys. Rev. 167, 331 (1968).4H. Schafer and W. Fuhr, J. Less-Common Metals 8,375 (1965).5R. J. Trainor, G. S. Knapp, M. B. Brodsky, G. J.Pokorny, andR. B. Snyder, Rev. Sci, Instrum. 46, 95(1975); see, in particular, Ref. 3 therein.D. W. Osborne, H. E. Flowtow, and F. Schreiner, Rev.Sci. Instrum. 38, 159 (1967); also see G, T. Furukawa,W. G. Saba, and M. L. Reilly, Critical Analysis ofHeat-Capacity Data of the Literature and Evaluation ofThermodynamic Properties of Copper, Silver, andGold from 0 to 300 'K, Report No. NSTDS-NBS-18(U. S. GPO, Washington, D. C. , 1968).

'T. H. K. Barron, in Lattice Dynamics, edited by R. F.Wallis (Pergamon, London, 1965), p. 247; or A. P.Miller and B.N. Brockhouse, Can. J. Phys. 49, 704(1971).J, C. K. Hui and P. B.Allen, J. Phys. C 8, 2923 (1975);J. C. K. Hui, Ph.D. thesis (State University of NewYork at Stony Brook, 1975) (unpublished).9See, for instance, B.Yates, Thermal Expansion(Plenum, New York, 1972), pp. 73-104;or T. H. K.Barron, W. T. Berg, and J. A. Morrison, Proc. R.Soc. A.242, 478 (1957).' G. Grimvall, J. Phys. Chem. Solids 29, 1221 (1968);F. Y. Fradin, Solid State Commun. 16, 1193 (1975).' Some of these rather unfamiliar plots appear in Ref.2.~2J. J. Hopfield, Phys. Rev. 186, 443 (1969).'3F. Y. Fradin and D. Zamir, Phys. Rev. B 7, 4861(1973);F. Y. Fradin and J. D. Williamson, Phys ~ Rev.B 10, 2803 (1974).4See, for instance, P. B.Allen and R. C. Dynes, Phys.Rev. B 11, 1895 (1975);P. B. All, en and M. L. Cohen,Phys. Rev. Lett. 29, 1593 (1972),' S. K. Sinha and B. N. Harmon, Phys. Rev. Lett. (un-published) .S. K. Sinha (private communication).