Displacive transformations in near-equiatomic Nb-Ru alloys—II. Energetics and mechanism

.Acu .blr~aliury~cu.

Vol. 21. pp. 3’4

Pcrgamon

Prcs.

19’6

Pnntsd

m Great Briram

DISPLACIVE TRANSFORMATIONS IN NEAREQUIATOMIC Nb-Ru ALLOYS-II. ENERGETICS AND MECHANISM* BiJOY K. DASt Department

of Metallurgy and Mining Engineering and Materials. Research Laboratory. of Illinois at Urbana-Champaign. Urbana. IL. U.S.A.

University

EDWARD A. STERN: Technion-Israel

Institute of Technology. Haifa. Israel and

DAVID S. LIEBERMAN Department

of Metallurgy and Mining Engineering and Materials Research Laboratory. of Illinois at Urbana-Champaign, Urbana, IL, U.S.A. (Rrceiced 17 April

1972; in rrcisrd~orrn

II

July

University

19753

Abstract-The transformation distortions during the /I+ p’ and /I + 8” transformations in near equiatomic Nb-Ru alloys produce phases that have progressively lower symmetry than the CsCl type j phase and approach the packing of the hexagonal close packed structure. By the use of a simple approximation-the superposition approximation-a model is developed which explains all of the experimental measurements on this alloy system. The driving force for these transformations is electronic in nature. The Fermi energy p of the fl phase alloys is near a sharp peak in the density of status. p(E). The transformations broaden this peak. lowering p(jc) and the energy of the phase. The signiticant experimental measurements accounted for by this model are the variation of magnetic susceptibility with phase and Ru content, the variation of transformation temperatures and transformation distortion with Ru composition, and distortion in the 8” phase with temperature and Ru composition. R&urn&--Les distorsions au tours des transformations /I--p’ et p-j” dans les alliages presque Cquiatomiques Nb-Ru produisent des phases dont la symitrie est progressivement plus basse que celle de la phase /I de type Cscl et tend vers l’empilement hexagonal compact. Une approximation simple-le principe de superposition-pet-met de developper un mod&e expliquant toutes les mesures experimentales effect&s dans ce systeme. La force motrice de ces transformations est de nature electronique. L’inergie de Fermi p des alliages de phase /3 est voisine dun pit etroit de la densiti d’ttat p(E). Les transformations elargissent ce pit. abaissant p(p) et I’inergie de la phase. Les mesures experimentales les plus significatives dont le modtle rend compte sont la variation de la susceptibilite magnetique avec Ia teneur en phase et en Ru, la variation des temperatures de transformation et de la distorsion a la transformation avec la teneur en Ru. et la variation de la distorsion dans les phases 8” avec la temperature et la teneur en Ru. Zusammenfaswng-Die Transformationsverzerrungen, die in nahezu aquiatomaren Nb-Ru-Legierungen wlhrend der /?- p- und F- F-T rans formation auftreten, erzeugen Phasen die fortschreitend geringere Gittersymmetrie als die &Phase vom CsCl-Typ aufweisen und sich der Packung der hexagonal dichtesten Packung annlhern. Mittels einer einfachen Nlherung wird ein Model1 entwickelt. das alle experimentellen Messungen an dieser Legierung erkllrt. Die treibende Kraft dieser Transformationen ist elektronischer Natur. Die Fermi energie p dieser B-Phasen-Legierungen liegt in der NIhe eines scharfen Maximums der Zustandsverteilung p(E). Die Transformationen verbreitem dieses Maximum und emiedrigen p(p) und somit die Energie der Phase. Die wichtigen experimentallen Messungen. sind die dieses Model1 beschreibt. die Anderung der magnetischen Suszeptibilhat mit Phase und Ru-Gehalt, die Anderung der Transformationstemperaturen und -verzerrungen mit der Ru-Zusammensetzung. und die Verzerrungen in der F-Phase mit Temperatur und Ru-Zusammensetzung.

IXFRODUCI’ION

*This research was supported in part by the U.S. Atomic Energy Commission under Contract AT(I l-1)-1 198. Partly based on a thesis submitted by B. K. Das to the University of Illinois at Urbana-Champaign in partial fulfillment of the requirements for the degree of Doctor of Philosophy. t Present address: National Physical Laboratory. New Delhi 12, India. $ Present address: Department of Physics, University of Washington. Seattle, Washington, DC 98105. U.SA.

In the preceding paper [I]. it was concluded that in near-equiatomic Nb-Ru alloys, the parent CsCl type phase transforms sequentially from CsCI (B)- tetragonal (j’)-+orthorhombic (j”) on cooling and the product phases are slight distortions on the parent cubic phase. In the present paper (II), it will be shown that the driving force for the atomic motions of the 37

38

DAS et al: DISPLACIVE TRANSFORMATIONS IN Nb-Ru ALLOYS

Fig. 1. The atomic arrangement in the (lOl), plane of the parent structure before transformation(s). Arrows indicate relative atomic motions during the fi--+_B’ and /I’-/7 transformations. The interatomic distance between (0) and its nearest (unlike) neighbors (1) remains virtually unchanged during all of the transformations, while next nearest (like) neighbors (2) and next nearest (like) neighbors (3) move ctoser to and farther from (0) respectively. In addition, during the fi --* (r transfor~tion, ail atoms in the nei~~ring plane which were at ‘2 move to the positions shown dashed in one twin: in the other twin this movement is in the opposite direction. These movements permit the planes to become more closely packed and the structure to approach h.c.p.

transformations is related to the band structure changes associated with the changes from a CsCl structure during the transformations, The mechanism proposed is consistent with the observed magnetic susceptibility and structural changes during the transformation. MOVERS OF ATOMS DURING T~S~O~AT~ON~~ As shown in I [ 11, the product of the CsCl (fi)- tetragonal (j’) transformation is a stack of twins, the twin plane being (lil),.. The atomic arrangement in this plane is shown in Fig. 1. If the lattice parameter changes during the /?-_B’ transformation are analyzed, it is Seen that the atoms in the [l 10jp, direction come closer together and atoms in the [112-j,. direction move apart. The interatomic distance along [Oil],. and [lOi& close packed directions, remains the same within 0.2%. These movements produce a closer packing of atoms in the (lil)y plane than in the (lOi), plane from which it came; the packing approaches that of hexagonal close packed arrangement if the ordering is neglected. The angle 6 in Fig. 1 is 70.53’ in the cubic phase and 60” in the hexagonal arrangement. In the tetragonal structure in Nb50R~50 for example, this angle is 68.77”. The atoms in the close packed layer then can move in either [112& or [lE&. direction with respect to the neighboring layer. This movement brings the stacking of the close packed layer closer to the stacking in the hcp structure.

During the p’- 8” transformation. the atomic movements are again such that further closer packing of the atoms results in the above mentioned close packed plane. Atoms along the [liO& direction come closer together and the atoms along the [il2&direction move apart, decreasing the angle f? to 68.31’ for Nb30R~S0. There is also an increase in the atoms per unit area in this plane. The result of all these movements is that during the transformations, the number of nearest (unlike) neighbors and their separation distance remain essentially unchanged while the next nearest (like) neigh~rs move closer and the next nearest (like) neighbors move even further away as the atoms approach the arrangement in the hcp structure. Because they are of such critical importance to the development of the model presented below, two features of the atomic movement during the sequential phase transitions on cooling shown in the preceding paper (I) and described above will be summarized here for etiphasis: (1) the syrnmerry of the phases decreuses progressively as the structure distorts further from the cubic CsCl type of the parent phase and approaches the packing of the hcp, and (-1)the interatomic distance between (unlike) nearest neighbors in the direction of closest atomic packing remains essentially constant and the atomic volume prlr atom is approximately unchanged. MAGNETIC SUSCEPTIBILITY MEASUREMEWS The apparatus and technique for measurement of magnetic susceptibility as a function of temperature has been described elsewhere [Z]. Figure 2 shows the variation of magnetic su~ptibility for the NbSOR~50 alloy with temperature; the start and finish temperatures for /3+ j? and p --) j?” transformations are indicated on the figure. The magnetic susceptibility decreases by nearly 30 percent during the two step sequential transformation. Figure 3(a and b) show the variation of magnetic susceptibility with temperature for NbJZRu4s and Nb,6Ru,, alloys, respectively, dis-

TEMP.t’t-2 Fig. 2. Magnetic susceptibility (;O of Nb5,Ru,o vs temperature. The various transformation temperatures defined in the preceding paper [i] have been indicated.

DAS et al:

DISPLACIVE

TRANSFORMATIONS

39

IN Nb-Ru ALLOYS

06-

TEMP. IFI (a)

I

I

1

I

I

I

Oh 40

45 At 96 Ru

50

Fig. 4. Magnetic susceptibility (x) of Nb-Ru alloys vs composition.

w0

T2 200

ODoaunP-94

TI 400 600 TEMP l’ct

800

IO00

04 Fig. 3. The magnetic susceptibility (x) as a function of temperature for two non-stoichiometric alloys. The electric resistivity data from (I) are also shown; the significance of the temperature T1 - T3 is discussed in (I).

cussed in (I) [l]. The resistivity data from Figs. 7(a and b) of that paper has been included here for completeness. The total magnetic susceptibility can be written as: % = %C+ XL+

i&b

+

%OT

mated to be N&/E, where N is the numkr of electrons per mole, pLgis the Bohr magneton. and E is the mean of the separation between the energy levels which is estimated to be of the order of the width of the d-band in the case of transition metals. Hence xOrbis not expected to change very much with change in crystal structure or temperature since the width of the d-band does not change much with such a change. The change in magnetic susceptibility of the Nb-Ru alloys on cooling through the transformations then is concluded to be d;e to the change in the spin paramagnetic susceptibility, x0, which is directly proportional to the density of states at the Fermi surface. Figure 4 summarizes the variation z with composition for the Nt+Ru alloy structure.

(1)

48-

where x %C %L

%orb %O

= total magnetic susceptibility = diamagnetic contribution of the core electrons = diamagnetic contribution of the s and d electrons (Landau diamagnetism) = orbital paramagnetism = spin paramagnetic susceptibility due to the s and d electrons. (Pauli paramagnetism).

The contribution of the core, xr. is usually small and independent of crystal structure and temperature. The change in the diamagnetic contribution of the d electrons, the main contribution to zL. is expected to be small because the volume per atom remains approximately the same and the average overlap with other d-states, and thus its spatial form, does not change much during the transformations. The contribution of the orbital paramagnetism [3], lorb is esti-

0a . ;4.65 .

ri 0 tt!

. . -

444-

9?T-e-e At

X Ru

Fig. 5. Lattice constants of the product phase at WC and that of the cubic phase at 1000°C for Nb-Ru alloys vs composition. Note the increase in the transformation strains with increasing Ru content.

DAS er at:

40

40

DISPLACIVE T~~SFO~.~ATIONS

%?u

At Fig. 6. Pseudo phase diagram showing the ‘stable’ phases at different temperatures and compositions for the near equiatomic Nb-Ru alloys. CO~ffOS~ION DEPE~ENCE OF THE T~SFO~~ATION D~ORTIO~S AND T~SFO~~ATION T~ERA~R~ Since the e/u ratio for the alloy increases with increasing Ru content and since the electronic energies and driving force for the transformation would be expected to also depend on the number of electrons, it is constructive to examine the composition dependence of the transformation strains and temperatures to facilitate comparison with variation of x with composition. The X-ray results described in I are summarized in Fig. 5 which shows that the distortion of the parent cubic phase on transfo~tion increases with increasing Ru content: The pseudo phase diagram of Fig. 6 was constructed from the results described in I and II. It summarizes the variation of transformation temperature(s) with composition and hence shows the variation of phase stability with temperature and composition. The dependence of transformation temperature with Ru content or e/a ratio is rather large, being _ lW’C/at.% Ru. RAND STRC’CTURE OF ALLOYS Aii S~E~S~ION ~PROX~ATION

THE

The magnetic su~ptibility data of Figs. 2 and 3 indicate that the structures stable at lower temperatures have a lower density of states at the Fermi energy ,o&) than the higher temperature phase. One thing that must be explained therefore is why it is that a smaller p(p) implies a structure with lower energy. It is of interest to point out that this behavior is opposite from that expected ‘using the argument given in the past to explain the Hume-Rothery rules [4]. Although today this argument is not generally accepted [SJ, it does emphasize the necessity of understanding the correlation between pCu) and the

IN Nb-Ru ALLOYS

energy of the phase. Further, the change of p@) with the change in crystal structure must be correlated and explained. These are treated below. Band calculations [6] and general considerations of transition metal alloys [7] indicate that the density of states curve p(E) for an ordered or disordered alloy is approximately equal to that of a pure metal with the same crystal structure whose potential is the average potential of the constituents (virtual crystal approx~ation~ if the valence difference between the two constituents is a small fraction of the valence of each constituent. In Nb-Ru alloys, the valance difference is almost as large as the valence of one of its components (Nb). In this case, and where neither constituent has a nearly filled d-band, another approximation is used. For this approximation, called the superposition approximation, p(E) of the alloy is given by p(E) = &;b(-@+ PI;,(E),

(2)

&(4 = c&J&) d”(E) = (1 - C)&(E)

(3)

where

for E near the fermi energy pt: here as well as subsequent discussions it will not be necessary to know the properties of p(E) far from /I. ps,(E) and p,&) are the density of states of pure Nb and Ru, respectively, (but with the same lattice structure as the alloys), while c and (1 - c) are respectively the fractional per cent of Nb and Ru in the alloy. The energy origins of P&E) and p,,(E) are fixed by the requirement that their Fermi energies must be equal. On physical grounds, it is expected that there is approximate charge neutrality around each Nb and each Ru atom and hence approx. 5 electrons around the former and 8 electrons around the latter. However, it is important to note here that in this approximation the long range order in this alloy system and its consequences, viz. that the unit cell size of the CsCl structure is twice that of bee, and that (hence) the Brillouin zone of the former is half that of the latter, can be neglected. Thus, for example, the /I phase of this alloy is treated as a b.c.c. structure in this approximation. Band structure calculations by Deegan [S]. Peetifor [93, and DaIton and Deegan [IO], have shown that the crystal structure of a pure transition metal depends upon the number of valence electrons, z: the hexagonal close packed structure is favoured for z = 3, 4, 7 and 8: the b.c.c. structure for I = 5 and 6. Figure 7 shows the density of states for b.c.c., and h.c.p. structures for a transition metal with the same atomic volume in all three structures. To a first approximation p(E) vs E is found to be independent of z. i.e. the rigid band approximation is valid for the prre transition metals, even though it is not for Nb-Ru. The density of states curve for the b.c.c. structure has essentially three peaks; the small peak at low energy is due to the s-d hybridization and the other two peaks are due to the bonding and antibonding states of the d-electrons, the bonding peak

DAS er at:

DISPLACIVE TRANSFORMATIONS

IN Nb-Ru ALLOYS

-Ii

The orthorhombic B” phase is distorted more from b.c.c. than the tetragonal p-phase and thus the broadening of the peaks will be expected to be greater for the p” phase. COMPARISON BETWXEN EXPERIMEX AND THEORE-IXXL PREDICTfOXS

Ekydl

(a) -

hCP

I

(b) Fig. 7. Density of states, p(E), for the d-band in b.c.c. (a). and h.c.p. (b) structures. The dotted curve represents the integrated density of states. The Fermi level for the elec-

tron per atom ratio appropriate for Nb,,R+, according to the rigid band model is also shown in (a). being broader and at a lower energy. The density of states curves for the two close packed structures show no such large peaks however, being rather ‘flat’ as can be seen by comparing Fig. 7(b) with Fig. 7(a). For subsequent discussions, it is important to understand why the b.c.c. structure has such large peaks in p(E) compared to the close packed structure. Its first brillouin zone, is a rhombic dodecahedron. All of its twelve zone faces are the same distance from the origin. A peak of p(E) occurs whenever a constant energy surface just touches a zone boundary. In the b.c.c. structure, this peak reaches a high value because, by s~etry, all twelve faces of the zone contribute s~ul~neously. The two main peaks in Fig. 7(a) correspond to the constant energy surfaces of the bonding and antibonding d-states touching the zone faces. The brillouin zone of the h.c.p. have several sets of faces at different distances from the center. Thus the peaks in p(E) in this structure is not expected to be as large as in the b.c.c. structure since not as many faces are contributing simultaneously. If the structure is distorted from the b.c.c. toward h.c.p. with a decrease in symmetry, as occurs in the /3- p and /$‘+ @” transfo~ations, all of the faces of the first b~llouin zone will no longer be at the same distance from the center and the peaks in &E) will be expected to broaden and decrease in height.

The variation of p(p) in the afioys can be understood by using (2 and 3). Figure 7(a) indicates the location of &p) and ~&4 for the pure metals in a b.c.c. structure. They are both near peaks of p(E) and thus from (2 and 3). p(p) of the /?-phase is also expected to be near a peak. When the distortion from the CsCl type lattice occurs in the fi- J? and p -+ p” transformatio&. p&) and p&) of the pure metals for these structures will decrease-the greater the distortion, the greater the decrease-since the peaks will broaden and decrease in height when the structure distorts. Again from (2 and 3). the same behaviour is expected to occur in the alloy. The variation in &t) inferred from the magnetic su~eptibilit~ data of Figs. 2 and 3 is in agreement with this explanation and is associated with the first experimental feature emphasized above, viz. that the symmetry of the phases decreases with the sequential transitions. The question remains as to why the smaller value of ~01) implies a lower energy for the phase. The first point to make in understanding this is that the average energy of a given band is not shifted by overlap interactions which produce the band width. The average energy is given by the diagonal matrix element of the Hamiltonian between the Wannier wave functions for the band; for transition metals. they are approximated by atomic &states. The wave functions change very little in the /?--+ /Y’- /Y’transformations because the overlap of the d-states does not change much since the interatomic distance between nearest neighbors is essentially c~nsrunt and the average volume per atom remains approximately the same during the transitions, as emphasized earlier. Thus the average energy of each alloy d-band is essentially the same in all phases. In particular, the average energy of the states that correspond to the antibonding d-states will be essentially the same in all phases. In Fig. 8 is sketched the behavior of dE) expected, using the superposition approx~ation, for the #Xfl and p phases in the vicinity of p (set equal to zero) for an approximately equiatomic NbRu alloy. p(E) for the phase was calculated using Fig. 7(a) and has contributions from &, and pk, indicated in the figure. p(E) for the p’ and p” phases were schematically drawn according to the prescription delineated earlier; the figure illustrates the progressive broadening and depression of a peak and the constancy of its average value during the sequential transformations. This broadening, depression, or -smearing* of the peaks in p(E) has a net effect on the average energy only for the peak which occurs at p. For any peak below g which is completely filled with electrons. the

DAS et al:

42

DISPLACIVE TRANSFORMATIONS IN Nb-Ru ALLOYS formations are first order transitions since both the parent and product phases can coexist and one grows at the expense of the other as shown in the previous paper [l]. From thermodynamics, at a first order transformation, T,AS = AU,

0

/ ’

-0-l

-005

0

005

1

04

E kyd) Fig. 8. The p(E) of the alloy phases in the vicinity of fl (set equal to zero) determined and schematically sketched as described in the text. In the b-/3’ phase transformation, the states in the cross hatched region effectively shift to the (equivalent area) dotted region. Since both sets of states lie below the Fermi energy, the net effect of the transformation is to depress the energy of the 8’ phase relative to that of the /3phase. Note that both the vertical and horizontal scales differ from those in Fig. 7 by a factor of two. smearing to first approximation does not change the average value of the energy since it increases the energy of about the same number occupied states as it decreases. Only for the peak at p does the smearing have the net effect of lowering the energy. The states whose energy is increased by the smearing of this peak are not occupied, while those whose energy is decreased are occupied. For example, during the j3 to 8’ transitions, smearing shifts the states shown cross hatched to the region shown dotted in Fig. 8, and consequently lowers the energy of the electrons in these states represented by the area shifted. The net effect of the smearing is thus to lower the energy and the correlation between P(P) and the energy of the phase is thus understood. A smaller p(J) indicates more smearing of the peak which means more occupied electron states are lowered and a lower total energy. The model can quali~tively describe the variation of transformation temperatures and strains with composition. The sharp peak at the Fermi energy conks from p;(, and hence as Ru content increases, equation (3) indicates that peak at E = fi becomes more prominent and po1) for B-phase will increase. There is a definite trend of increasing x and hence p(p) with increasing Ru content (except for the equiatomic composition) as shown in Fig. 4. It can be seen from Fig. 6 that the transforknation temperatures increase with increasing Ru content. This increase of transfo~ation temperature with increasing Ru content is perhaps not surprising since the stable crystat structures of Nb and Ru are b.c.c. and h.c.p., respectively. The p- 8’ and j?‘-+ j$” trans-

(4)

where AS is the difference in entropy and AU is the difference in energy between the two phases, and To is the transfo~ation temperature. As the Ru content increases, AU increases, and thus T,, or AS or both will increase. The b.c.c. type of structure of the fi phase has a higher entropy at a given temperature than the closer packed p and j?” structures because of the soft shear type phonon modes as first pointed out by Zener [I I]. As the j?’ and j?” phases increase their distortions from the b.c.c. type towards the h.c.p. structures, [ ASI is expected to increase; increased distortion indicated in Fig. 5, suggests that /ASi increases with increasing Ru content. However, To also could increase with Ru content. Our discussion explains the driving force for the transformations and subsequent distortions but it is worthwhile to also consider what stops the distortion at a given value in the p-phase. At any finite temperature the equilibrium distortion is that which minimizes the free energy F, F = I/ - TS.

(5)

Consider temperatures below T’w $) so that we remain in the p phase. increasing the distortions will lower U and S. As the temperature is lowered, S becomes less important and the min~um F will thus occur at a greater distortion from the &phase. From this we can understand the data illustrated in Figs. 2 and 3 which indicate that as the temperature is lowered in the p-phase, the distortions do indeed increase while x and thus p@) decreases. As the temperature is lowered to near T = 0, the equilibrium is determined by V alone and it is useful to estimate that distortion which minimizes U. Our discussion focused on the smearing around the sharp peak of pkU because we assumed that pk,(p) dominated the behaviour of &). This assumption is valid as long as pA,(&)% ph&). When &b) z &,(j~) then the effect of the distortion on P;&jf) must be taken into account. From Fig. 7(a) we note that as the distortions increase the unoccupied sharp peak in &,(E) above p&(l() becomes smeared for exactly the same reason as for the peak in pk,(E). If this initially sharp peak in &,(E) above j&,(p) becomes smeared enough it could start increasing p&@) with further distortions. We can estimate from Fig. 7(a) that such a situation would occur when the sharp peak has been smeared so much that its peak value has been lowered to about the initial value of &,(JL). However, at these distortions A,(p) will also be decreased to the values where ~“~) z ,&&) and we must include the variation of ,L&&) to estimate U. The increase of &,(p) with increasing distortions in

DAS er al:

DISPLACIVE

TRANSFORMATIONS

this region will increase L’ in opposition to &J/l). Thus we expect that the distortions would cease to increase at the temperature where &,(@) = f&l&). Using (6) and assuming that z is proportional dp) we estimate from Figs. 7(a) and 2 that ;ca - ;la(T = 0)

&&T = 0)

(6) to

I 0.3 at w< Ru z 0% at 50?; Ru.

(7)

The corresponding experimental values when ~425°C) is used in place of E(@ = 0) are in surprisingly good agreement with the values in (7). namely the same to one significant figure. Such good agreement is clearly fortuitous since the lowest experimental temperature is 25% not OK. Yet the trend with electron per atom ratio is clearly correct giving further support to the model. The electrical resistivity data in I [ 1J. and included here in Figs. 3(a and b), were used p~mar~iy to indicate the transformation temperatures and the volume fraction of the phases during the transformations to compare with the X-ray intensity measurements. It can be seen that the resistivity of the product is considerably higher than that of the parent cubic phase and that the slopes of the resistivity vs temperature curves are appreciable. If the same electrons were responsible for the magnetic susceptibility and electrical conductivity, as is assumed in simple metals, the resistivity data would be consistent with the model herein presented. However, since the effect of distortion on s electron behaviour and s-d scattering is not known, no correlation is possible in this treatment; an understanding of the resistivity behaviour would require a more detailed examination.

SUiMMARY AND DISCC’SSION

The model of the Nt+Ru alloy and its transformations depends on the use of the superposition approximation and it is essential that its validity and applicability be critic&y examined. One basic assumption is that a11 atoms are approxi~teIy neutral. Since the alloy is metallic such an a~umption should be quite valid. Estimates [7] indicate that expected deviations from charge neutrality are usually less than cu. 0.3 of electronic charge. A deviation of 0.3, from charge neutrality can have significant effects if p(p) is in a region where it changes rapidly with changes in the number of electrons per atom. For instance consider the TiNi alloy. In the D-phase. Ni, according to Fig. 7(a), would have a low value of p&(p) if it were filled with 10 electrons as required by charge neutrality. Permitting a deviation from charge neutrality of 0*3e which lowers the number of electrons around nickel to 9-7, the value of p&t doubles from the previous value. For Ru its &,(~O is not so sensitive to charge neutrality deviations and the assump-

IN Nb-Ru ALLOYS

43

tion of charge neutrality is nof very critical in the NbRu alloy system. Near the Fermi energy the bonding states in Nb atoms and the antibonding states in Ru atoms will be involved in the alloy wave functions. The bonding states of the ?4~ atoms will have an appreciable amplitude where they overlap the Ru nearest neighbors and there could be an appreciable change in their potential energy values compared with that of b.c.c. pure Nb. However. the anti~nd~g states of the Ru atoms do not have much amplitude where they overlap the nearest neighbor &ib atoms and their potential energy (and thus total energy) is not greatly changed from that of pure b.c.c. Ru. Since the main features of the Nb-Ru alloy system came from the sharp peak in p(E) contributed by Ru, the superposition approximation should give this peak with good accuracy. Although the superposition approximation may distort the contribution to dE) given by the Nb atoms, this contribution is not critical to the alloy behavior. It is worthwhile to consider the possible effects of a deviation from charge neutrality further. especially in those cases where the resultant change can be large as in TiNi. The deviation from charge neutrality depends on several interacting factors[7]. In order to approach charge neutrality in an alloy like TiNi it is necessary that a given electron state deposit on the average two and a half times more charge on a Ni atom than it does on a Ti atom. The xii atom attracts more charge around itself because it has a more attractive potential, the more attractive, the more charge attracted. At the same time the potential around a given atom depends on the charge around it because the electronic charge contribute to the potential. The final charge distribution has to be solved for self-consistentfy by accounting for this interaction between electron charge density and the potential. The experimentally similar behavior [ 121 of TiNi and NbRu can be explained by assuming that each Ni atom has CII.9.7 electrons around it instead of 10 electrons to give the large peak in &,(J& Theoretical estimates of the deviation from charge neutrality required to support an electron charge ratio of 2: 1 in a CsCI type lattice (75 gave a value of 03 electrons, already sufficient to move p&t) near enough to the peak to explain why TiNi could cause even a larger charge deviation from neutrality which makes it even easier to exptain the behavior of T&i by the superposition approximation. In &%Ru the charge ratio is 1.6 and charge deviation from neutrality is expected to be smaller than for TiNi. Coupling this with the fact that pk&) is not so sensitive to deviations from charge neutrality justifies the neglect of this possibility when discussing the NbRu alloy system. The other assumption made in the superposition approxi~tion is that there is no charging near ,tt so that the ampIitude of the wave-unctions on both the Ru and Nb atoms are the same. i.e. the amount of Wannier function centered about the Ru atom that

4-t

DAS et al:

DISPLACIVE

TRANSFORMATIONS

contributes to the alloy state is the same as that of the Wannier function centered about the MJ atom. This charging is to be distinguished from the deviation from charge neutrality discussed just above. Such an assumption is not rigorously true. It is known only that at the Fermi energy, both amplitudes should be the same order of magnitude. But a rough estimate of the validity of this assumption can be ascertained from the experimental values of l vs Ru content of the b-phase in Fig. 4. If the amplitudes are not the same then equation (3) must be modified by multiplying pNCby the amplitude squared of the alloy wave function on the Nb atom and correspondingly for the Ru atoms. Using (3) and the values of P&) and P&) given by Fig. 7(a) the necessary weighting factor introduced by charging can be estimated to agree with the composition variation of x of the b-phase (Fig. 4). It is found that the weighting factor for Nb is I.1 while that for Ru is 0.9 and hence the assumption of equal weighting is approximately valid. As mentioned earlier, long range order has been neglected in this discussion and the CsCl lattice has been treated as a b.c.c. lattice. This should be valid because what enters into the argument is the bonding and antibonding d-states. The bonding states have a higher density at the midpoint along the line between nearest neighbors and this is determined by the b.c.c. structure. The antibonding state has a minimum in electron density at this same point-again determined by the b.c.c. symmetry. In summary, the superposition approximation should accurately describe the peak in p(E) contributed by the Ru atoms and how this peak width varies with crystal symmetry. Since this is the critical feature that drives the transformations, the main ex-

IN Nb-Ru ALLOYS

perimental features of the phase transformations are expected to be accounted for by the model based on this approximation, and this appears to be the case for near equiatomic NbRu alloys. Acknowledgements-The authors gratefully acknowledge the help of Dr. J. Peters for the magnetic susceptibility measurements. They wish to express their appreciation for partial support of the research presented herein, BKD and DSL to the United States Atomic Energy Commission and EAS to the National Science Foundation for a Senior Postdoctoral Fellowship. EAS and DSL also would like to express their thanks to the Physics and Materials Science Departments, respectively, of the Technion, Israel Institute of Technology. Haifa, Israel, for their hospitality during sabbatical leaves.

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