Short-range versus long-range order in a model binary alloy

We calculate the free energy of a linear binary metallic alloy using an exact transfer-matrix formalism. We obtain for the first time the electronic band structure when there is nonvanishing short-range order and calculate the temperature dependence of the short-range order parameter. Following Foo and Amar, we also introduce long-range order via two sublattices, but we find this necessarily raises the free energy. We conclude that the first-order phase transition of Foo and Amar is spurious, being an artifact of their assumption of long-range order and neglect of short-range order.

VOLUME 27, NUMBER 1 PHYSICAL REVIEW LETTERS 5 JULY 1971 Short-Range Versus Long-Range Order in a Model Binary Alloy* Michael Plischke and Daniel Mattis Belfer Graduate School of Science, Yeshiva University, New York, New York 10033 (Received 26 March 1971) We calculate the fr~e energy of a linear binary metallic alloy using an exact transfer­matrix formalism. We obtain for the first time the electronic band structure when there is nonvanishing short-range order and calculate the temperature dependence of the short­range order parameter. Following Foo and Amar, we also introduce long-range order via two sublattices, but we find this necessarily raises the free energy. We conclude that the first-order phase transition of Foo and Amar is spurious, being an artifact of their assumption of long-range order and neglect of short-range order. In this Letter we present what we believe to be the first rigorous and self-consistent treatment of the effects of short-range order (SRO) on the electronic density of states of a disordered medi­um. Recently Foo and AmarI reported a surpris­ing first-order phase transition in a simple mod­el of a one-dimensional binary alloy. In this model the atoms do not interact with one another, but contribute to the free energy of the system directly only through their disorder entropy and, indirectly, through the effect of disorder on the electronic density of states. After assuming a variable long-range ordering of the atoms, Foo and Amar calculated the total free energy in the coherent-potential approximation and indeed found a discontinuous disappearance of the sublat­tice ordering parameter at a finite temperature. The existence of their phase transition would have far-reaching implications for the theory of amorphous materials. For example, it could be the mechanism for a metal-insulator phase tran­sition if a gap in the electronic density of states depended on long-range order. We disagree with both their procedure and their conclusion. We shall recall Landau's prooe that no matter what long-range order (LRO) we might assume in a linear array, it would "melt" at infinitesimal temperature above absolute zero. Assume an ordered configuration with total free energy F LRO, then break L of the N electronic bonds of the chain at random. This results, on the one hand, in an increased electronic free en­ergy L lEI, where E is a temperature-dependent quantity which can be rigorously bounded: 0 < IE I < 1. This increase is more than compensated by a configurational disorder-related decrease -kTN[[lnt + (l-f) In(l-f)], where t= LIN. Min­imizing with respect to t, we find that the total free energy is now lower than F LRO, thus proving that LRO is thermodynamically unstable. In equi- 42 librium we find that f=(l+eBIElrl, and the average length f of the independent dis­connected segments is which is finite at any finite temperature. More quantitative arguments are given following Eq. (7). On the other hand, we compute the short-range order parameter, and find it to be a smooth func­tion of the temperature (see Fig. 1), vanishing slowly with increasing temperature. This also implies the lack of any Significant low-order thermodynamic phase transition within the accu­racy of our calculation, as well as the absence of LRO in this model. Finally, we obtain the elec- 1.0 0.9 0.8 0.7 0.6 o.s+------+------+------+------~----~ o 0.1 0.2 0.3 0.4 0.5 T FIG. 1. Plot of SRO parameter PAB(T) as function of T. A, E=1.0; B, E=2.0; andC, E=3.0. Atlowtem­peratures curve B is almost indistinguishable from curve A. VOLUME 27, NUMBER 1 PHYSICAL REVIEW LETTERS 5 JULY 1971 tronic density of states and study the energy gap as a function of temperature. Let us consider the same infinite one-dimen­sional chain as in Ref. 1, with ''hopping'' parame­ter set equal to unity: H=L)!i)Ej(i! -L) !i)(j!. (1) i i,J;: j ~ 1 Here E j = EA or EB• We assume an equal number of A and B atoms and without loss of generality adjust the zero of energy such that EA = -EB = E. In the case of SRO one introduces the parameter P AB, defined to be the probability that an A atom on the nth site has a B atom as its right-hand neighbor. By left-right symmetry P AB =PBA. Al­so P AA= 1-P AB =PBB• To compute the electronic density of states we use the exact formalism of Schmidt.3 The integrated density of states, M(w) = L~dw' p(w'), may be obtained from the following equations: M(w) = -Mw A(W, -IT) + W B(W, -IT)], where W A and W B obey the equations W A(W, cp) =P AAW A(W, T ACP) +P ABW B(W, T BCP) -P AAW A(W, -IT)-P ABW B(W, -IT), (2) W B(W, cp) =PBAW A(W, T ACP)+PBBW B(W, T BCP) -PBAW A(W, -IT)-PBBW B(W, -IT), (3) with the conditions Wj (w,cp+2lT)=W j(w,cp)+1, Wj(w,O)=O, j=A,B, (4) and where Wj(cp) is monotically increasing; also, (5) We obtain W A(W, cp), W B(W, cp) by dividing the in­terval 0<Scp<S2lT into N subintervals. By linear in­terpolation over a subinterval we obtain N-1 lin­ear equations. It has been shown by Agacy3 that while this procedure does not converge very well to give the density of states, p(w) =dM(w )/dw, it nevertheless does yield the integrated denSity of states accurately even for choices of N as low as 20. The electronic free energy in the case of a half-filled band, including the entropy of the elec­trons, can be expressed in terms of M(w) by a partial integration: F el(T,P AB) = -kT l:dwp(w)ln(e-BW + 1) = i:dwM(w)(eBW + 1)-1. (6) We estimate that F el obtained from M(w) calcu­lated for N = 60 is correct to within 0.1 % at all temperatures by comparison with spot checks us­ing N = 120 and N = 240. In one dimension, the configurational atomic entropy associated with SRO parameter P AB has the simple expression S(P AB) = -k[P AB InP AB + (1-P AB) In(1-P AB)]. (7) We now wish to buttress the general arguments against LRO in one dimension by a quantitative calculation of the extra energy required to intro­duce LRO, the results of which are given in Fig. 2. For definiteness, we consider precisely the two-sublattice model of LRO defined in Ref. 1. As it happens, the identical formula (7) also ob­tains for LRO if one suitably redefines P AB to represent the probability of finding a B atom on an A-sublattice site (or an A atom on the B sub­lattice) and allows P AA =PBB to stand for the probability of finding an atom on its proper sub­lattice. Once we have the equation suitable for determining W in this case, we can find M(w) ap­propriate to LRO and obtain an exact4 expression for F el using Eq. (6). The appropriate equations for WLRO are W A(W, cp) =P AAW B(W, T BCP) + (1-P AA)W B(W, T ACP)-W B(W, -n), W B(W, cp) =P AAW A(W, T ACP) + (l-P AA)W A(W, TBCP)-WA(w, -IT), (8) where W A and W B obey the same conditions (4) and T j is again defined by (5). We numerically calculate F LRO (not shown here), and compare it 0.02 6F 0.0 1 c B A o 2 4 5 T FIG. 2. Plot of AF=FeILRO-FeISRO as function of T for some typical fixed values of P Ab at E = 1. A. P Ab =0.6; B, PAB=0.7; andC, PAB =0.8. AtPAB=l (per­fect LRO) and PAB = 0.5 (disorder) AF= O. 43 VOLUME 27, NUMBER 1 PHYSICAL REVIEW LETTERS 5 JULY 1971 with F SRO. In Fig. 2 the differences in electronic free energies F el SRO and F el LRO are shown as func­tions of temperature for various P AA' As seen in this figure, we find F el SRO <F el LRO at the same val­ue of P AA' It immediately follows that at any tem­perature T the total free energies obey the rela­tionFSRO( T)<FLRO(T), whereF",Fe1-TS. For, suppose that we have obtained the stationary LRO state by minimizing F LRO with respect to P AA' At the value of P AA for which F LRO has a minimum, FSRO(P AA, T) < FLRO(p AA, T). But F SRO is not yet at its own minimum and can be further mini­mized. Thus SRO is, by a finite margin, more stable than LRO. (The possibility that yet other forms of LRO, not considered by Foo and Amar or by us, such as AABB· .. , etc., might be more stable than SRO has already been discounted by the Landau-type arguments offered at the begin­ning of this paper.) From the minimization of F SRo(T) we obtain the equilibrium free energy and the SRO parameter P AB(T) which is plotted in Fig. 1 for € = 1, 2, 3. In all cases, P AB(T), together with the free ener­gy and its various derivatives, varies smoothly with temperature, supporting the conclusion that there is no phase transition in this system. The first-order phase transition of Foo and Amar is an artifact of their assumption of LRO. One of our calculated results is the differential density of states, p(w,P AB)' As has been shown by Agacy3 it is an extremely "noisy" function. The following features, however, are clear: At P AB = 1 (perfect LRO) p(w) has a gap of width 2€ centered about w = O. For P AB < 1, states appear in the gap and begin to fill it up. We arbitrarily define a temperature-dependent "gap" tl(P AB) as twice the distance between w = 0 and the point € c at which the density of states exceeds the "round­off error noise," p(€c):~ 10-3• In Fig. 3 we plot tl/2€ as function of P AB for € = 1,2,3. With the use of Fig. 1 for P AB as a function of T, this yields tl(T). For € = 3 some gap remains even at total disorder, in accord with the Saxon-Rutner theorem, whereas at € = 2 and 1 the gap disap­pears before total disorder (P AB = O. 5 or T = 00) is reached. We find the slope dtl(T)/dT to be dis­continuous at the point where the gap vanishes (see curves A and B). On the other hand, we see no discontinuity in any other thermodynamic vari­ables we have calculated, and in one dimension we do not expect any. Therefore, we can only conclude that, within the accuracy of our calcula­tion, and for reasons we do not yet fully grasp, the gap may not be chosen as a proper thermody- 44 1.0 , 0.8 'I " I" , \ I, 0.6 I \ \ Ii ~2< II \ , 0.4 I I c 0.2 A o 1.0 0.9 0.8 0.7 0.6 0.5 FIG. 3. Plot of the gap parameter tl(PAB)/2E as func­tionofPAB( T). A, E=1; B, E=2; andC, E=3. Those parts of the curves which we have extrapolated, but not computed, are indicated by dashed lines. Insert shows a typical density of states, for E = 2 and P AB = 0.9, for w>O, p(w)=p(-w). namic variable. The question of a metal-insula­tor transition is left open as it is possible that some, or all, of the states are localized, regard­less of whether or not a gap exists. In conclusion we would like to draw attention to the crossing of the curves in Fig. 1. This seems to indicate that this model cannot be approximat­ed by an Ising model with a temperature-indepen­dent coupling constant, and thus casts doubts on the traditional Bragg- Williams approach to the structure of binary alloys. We thank Dr. J. E. Gubernatis and Professor Philip L. Taylor for a very useful commentS on the computation of solutions to Eq. (2)-(5). *Research sponsored by the U. S. Air Force Office of Scientific Research, Air Force Systems Command, un­der Grant No. AFOSR-69-1642. 1E _N. Foo and H. Amar, Phys. Rev. Lett. 25, 1748 (1970). - 2L. Landau and E. Lifschitz, in Statistical Physics (Oxford U. Press, Oxford, 1938), p. 232. 3H. Schmidt, Phys. Rev. 105, 425 (1957). This, as well as R. L. Agacy, Proc. Phys. Soc., London 83, 591 (1964), is reprinted in Mathematical Physics in One Di­mension: Exactly Soluble Models of interacting Parti­cles, edited by E. H. Lieb and D. C. Mattis (Academic, New York, 1966). See also the more recent work of B. Y. Tong, Phys. Rev. 175, 710 (1968). In the appli­cation of these methods,7relaxation method as sug­gested to us by Gubernatis and Taylor (private commu- VOLUME 27, NUMBER 1 PHYSICAL REVIEW LETTERS 5 JULY 1971 nication) converges very fast without requiring vast computer storage. This allows one to set N as large as 103 without much effort, and allowed us to obtain quite satisfactory accuracy in the final results, as discussed in the text. (The methods used in Ref. 1 might be criticized on account of the several approximations (such as the use of the coherent-potential approximation, and approxi­mating F el by the electronic internal energy with ne­glect of the electronic entropy). The free energy we calculate, on the other hand (denoted F L R 0), is exact within the postulates of the model and free of any ap­proximations except those which arise in numerical computation. Symmetric Fission Observed in Thermal-Neutron-Induced and Spontaneous Fission of 257 Fm t W. John, E. K. Hulet, R. W. Lougheed, and J. J. Wesolowski Lawrence Radiatirm Laboratory, University of California, Livermore, California 94550 (Received 14 May 1971) The observed mass distribution for thermal-neutron-induced fission of 257Fm is strong­ly symmetric, whereas that for spontaneous fission of 257Fm is found to be predominant­ly asymmetric with, however, some symmetric fission present. The mass distributions were derived from energy measurements on coincident fission fragments. The onset of symmetric fission at 257Fm supports theories relating asymmetric fission to the sec­ond hump of the fission barrier. The understanding of the mass distribution in nuclear fission is an outstanding problem. The Simple liquid-drop model predicts symmetric fission, yet mass distributions observed for low­energy fission have always been asymmetric. Our measurements on the thermal-neutron-in­duced fission of 257Fm show a strong mode of symmetric fission. We also observe the sym­metric- fission mode occurring weakly in the spontaneous fission of 257Fm, in agreement with the recent data of Balagna et ai. 1 These unusual results afford an additional test of theories of the mass distribution in fission and, indeed, support a recent suggestion by Moller and Nils­son2 relating the mass distribution to the double­humped fission barrier. Measurements were made on the kinetic ener­gies of the coincident fission fragments from a thin sample placed between two Si detectors. The sample, containing 4x 108 atoms of 257Fm, was obtained from the Hutch underground nuclear explosion. 3 Ci emission is the predominant decay mode of this 100-day isotope although a small (0.2%) spontaneous-fission branching also occurs. After final purification, a few microliters of solution containing 257Fm was evaporated on a 200- fJ.g/cm2 Pt foil. The foil was then mounted on a four-position sample wheel located between the two Si detectors with collimators to limit the maximum fragment entry angle to 50° from the normal to the detectors. Other positions of the wheel contained a 252Cf spontaneous-fission source for energy calibration, a 2S'U source (2 ng) for neutron-flux determination and a check on the energy calibration, and a blank Pt foil for background measurement. All samples were covered by 200-flg/cm2 Pt foils to prevent de­tector contamination. The assembly was placed in the thermal column of the Livermore reactor in a flux of 2 x 1011 neutrons/cm2 sec (cadmium ratio 600). The techniques for counting in the high neutron flux included cooling the detectors and using fast linear electronics as previously described.4 Data were accumulated alternately with the reactor on and then off. Two separate runs of about three weeks each were made. The fragment masses for each event were calculated from the kinetic energies, assuming conserva­tion of momentum and mass. A correction was made for the pulse-height defect but not for neu­tron emission. Thus the masses correspond to provisional (approximate pre-neutron) masses of Schmitt, Neiler, and Walter. 5 In Fig. 1 the results for spontaneous fission of 257Fm are displayed as a contour diagram of counts versus fragment mass and total kinetic energy. The symmetric part of the distribution is evident as a ridge along the line of mass sym­metry extending up to 250 MeV. The greatest number of events occurred on the asymmetric peaks at masses 115 and 142 with a total post­neutron kinetic energy of 188 MeV. A graph of 45