Discussion

The Mössbauer spectral intensity is proportional to the recoilless fraction of the Mössbauer signal[15]. We distinguish two mechanisms that decouple the Mössbauer emitting nucleus from the lattice and decrease the recoilless fraction. One is the nucleus vibrating about a given site in the lattice and the other is the nucleus hopping away from the lattice site.

Vibrations decrease the recoilless fraction[15] by a Debye-Waller factor where, expanding in the anharmonic potential

Here is the Boltzmann constant, q is the Mössbauer gamma ray wavenumber ( for Sn ), and is the mean squared vibration amplitude, . The result is valid in the classical limit , where is the highest frequency of the solid. Terms which contribute proportional to and higher powers are neglected.

The simple one-dimensional potential of Eq. (1) is equivalent to an Einstein model for the vibrational motion of the Mössbauer atom. This crude model is sufficient to parameterize correctly the temperature dependence of in the classical limit as given in Eq. (2). A more exact theory valid in the quantum limit also, though only in the harmonic approximation, has been presented by Mannheim[16] and has been used by Howard and Nussbaum[17] to analyze the behavior of Fe impurities in various hosts.

As seen in greater detail further down, the decrease of with temperature below is almost linear and can be fully accounted for by the anharmonic model. However, this mechanism, even with a T term, cannot account for the rapid drop in intensity above in either Pb or Ag. To explain this behaviour we propose a locally correlated hopping model. The hopping will not contribute to bulk diffusion if the hopping atom remains localized about a particular lattice site. In this sense the hopping is highly correlated. Whereas the onset of bulk diffusion replaces the recoilless line with a broadened one, localized hopping does not entirely eliminate the recoilless fraction.

In our model the Mössbauer atom has two possible states. State 1, with probability P, corresponds to the atom vibrating about the initial site with a mean squared disorder as given by Eq. (2). State 2, with probability , corresponds to the atom hopping from the initial site with a mean squared displacement of . The Mössbauer spectrum for this model is given by

where is a Lorentzian

and / is the decay rate from state 1 (taken as the initial state) to an equilibrium distribution of states 1 and 2. Eq. (3) is derived in the Appendix. The Mössbauer spectral intensity is given by

where the integration is done only over the sharp line, so that any significant broadening contributes to the background.

Making the assumption that the states are in thermal equilibrium at all temperatures, the relative occupation probability is given by

where E is the energy difference between the states and is the degeneracy of the high energy state relative to that of the low energy state. To estimate the degeneracy we assume that, whereas one atom is involved in the low energy state, L atoms are involved in the hopping process and corresponding to typically n states per atom. For a rough estimate we assume n=2. Defining a local configurational entropy

Eq. (6) becomes

We note from Eq. (3) that broadens the Mössbauer line width. For cases in which there is no significant line broadening, either (a) or (b) . Since the high energy state does not contribute to the Mössbauer spectra intensity we know that . For the Sn Mössbauer line with , it is required to have to satisfy this assumption. In case (a) Eqs. (5) and (6) give

while in case (b) they give

Because XAFS has a different measurement time scale than ME, it can distinguish between the two cases (a) and (b) that the Mössbauer measurements allow. The Mössbauer lifetime is s while the XAFS measurement time is s, so that Mössbauer measurements average over many lattice vibrations (s) for each atom, while XAFS measures an essentially instantaneous average distribution of atoms.

In case (a) the nucleus does not change its state during the Mössbauer lifetime. Both ME and XAFS see an essentially instantaneous average of the atomic distribution with a fraction P in the low energy state and in the high energy state, and neither technique sees many transitions between these states. The from XAFS would show the same qualitative behavior as the ME in this case. Although XAFS and Mössbauer measure different (Mössbauer measures the relative to a lattice site and XAFS the relative to its neighbors) the relative size of the should be similar. In order for case (a) to have the Mössbauer spectral intensity decrease dramatically above , the high energy state must have a . Thus, for case (a), as the temperature is varied through the measured by XAFS should also change abruptly from to a much larger .

In case (b) the nucleus makes many transitions between the two states during the Mössbauer lifetime, but does not make transitions between the two states during the XAFS time, and it is possible for the of XAFS and Mössbauer to be quite different above . Since the nucleus is making many hopping transitions during the Mössbauer lifetime, the from ME would be determined by the hopping length . However, the hopping time could be short compared to the time between hops, when the nucleus is vibrating about a particular site with a mean square disorder . The XAFS would still be , and the XAFS signal would be only slightly decreased by the fraction of the time spent hopping, as seen in XAFS measurements on liquid Pb. XAFS measurements on a 2% Sn in Pb sample show no significant change in around besides the ordinary Debye-Waller dependence. This behavior rules out case (a), and case (b) is the one that applies to Pb:Sn alloys.

We consider another possible interpretation of the dramatic decrease in Mössbauer intensity, suggested by J. Mullen[8], that the impurity does not remain dissolved at all temperatures but partially precipitates out as the temperature is lowered or raised. The decrease in Mössbauer spectral intensity is caused by the precipitated particles having a smaller force constant k and consequent larger as given by Eq. (2). The phase diagram[18] does show a decreasing solubility limit with temperature near room temperature which, if extrapolated, may cross the 2% and even 1% limits at low enough temperatures. This possibility can be ruled improbable by the low temperatures of 150K where the putative precipitation would occur. At such low temperatures the possibility of any coalescence of the impurities into a precipitate would be improbable because of the absence of bulk diffusion or grain boundary movement. The large decrease would require a sudden precipitation of most of the impurities which means that the solubility must decrease to a value well below one percent. Any reasonable extrapolation of the solubility limits would not be consistent with such a low value.

Three measurements we performed show clearly that precipitation does not occur. The first is to measure any hysteresis in the temperature dependence of the rapid decrease near . As shown in Fig. (4) there is no hysteresis found for the Mössbauer spectral intensity for the 2% Sn in Pb sample. A hysteresis was found for the 3% sample in Pb which is above the solubility limit ( 2%) at room temperature and may be expected to have some further precipitation at lower temperatures.

For comparison, in an investigation of the precipitation of Sn in Pb, the spectral intensity for an 8 at.% Sn in Pb sample was measured by Arriola and Cranshaw[19] up to 520K. They found that the spectral intensity had a hysteresis effect in the temperature range of 320 - 360K, as precipitation occurred. Above 360K the Sn is completely dissolved, and a of 104K was found. This value is consistent with our value of 107K, determined at low temperatures for the 2 and 3% samples. Extrapolating the results of Arriola and Cranshaw above 320K to 80K, assuming the = 104K, we find that their spectral intensity is about 8 times that of ours, as shown in Figure (2), in agreement with the ratio of concentrations. Our 2% Sn-Pb sample did not show the hysteresis expected if precipitation had occurred. Moreover, the Debye temperature for this sample is consistent with other measurements for fully dissolved Sn-Pb alloys.

The second measurement was the value of the line shift. The isomer shift of tin in tin is 2.5 mm/sec at room temperature[20] while the measured value is 2.0 mm/s for the AgSn alloys, and 3.1 mm/s for the PbSn alloys, all relative to calcium stannate. The temperature dependence of the isomer shift in our samples is as expected from relativistic effects. This isomer shift is in disagreement with tin precipitates of appreciable size.

The third measurement was the XAFS of the Sn K-edge in dilute alloy samples. The XAFS at T = 80K showed that the Sn was surrounded by only Pb near neighbor atoms for the 2% sample as expected by the phase diagram. Figure (7) shows the predicted XAFS, using FEFF theory, of the isolated first neighbors if they are all Pb, all Sn, and 3 Sn and 9 Pb neighbors. It is estimated that the XAFS could detect Sn neighbors if they are 10% or more of the total. These three experiments rule out the possibility of precipitation as an explanation for the observed rapid drop at in the 1% Sn in Pb sample.

Although the Sn in Ag alloys are always in a range well below the solubility limits, a check of the reversibility of the temperature dependence of the Mössbauer spectra intensities was made for both the 2 and 4% Sn in Ag samples. No significant hysteresis was detected as shown in Fig. (3). Thus, we conclude that both for the 1% and 2% Sn in Pb samples and all of the Sn alloys in Ag, no precipitation is occurring in the temperature range where the deviates from a linear T behavior.

Another suggested interpretation for our observations is that impurities at the grain boundaries dominate the measurements, and so what we see is not a bulk phenomenon but a ``premelting" of the grain boundary. There is indeed an enhanced impurity concentration at the grain boundaries. For binary alloys with an impurity solubility limit on the order of 10%, the concentration of impurities at the grain boundaries is 10 times greater than in the bulk[22]. The grain size of our samples is typically 1. Since grain boundary widths are of the order of , the fraction of the volume enclosed by grain boundaries is 0.001. Thus, only 1% of the impurities are in the grain boundaries, and the effect we observe must be a bulk phenomenon. In addition, the amount by which the grain boundary concentration is greater than the bulk concentration decreases with increasing temperature[22]. This too is contrary to the suggested interpretation that, as the sample is heated, the impurities move to the grain boundaries where ``premelting" is more likely.

We now turn our discussion to the Pb host alloys. The 1% and 2% concentration samples clearly have no precipitates and the Sn is dissolved throughout the temperature range of the measurements. They show no irreversibility and are thus in thermal equilibrium. The very rapid decrease of the Mössbauer spectral intensity above is much too rapid to be explained by anharmonicity alone, and localized hopping of the atom off the lattice site must occur as discussed above. Also, as discussed in the experimental section, the 1% sample represents an isolated impurity. The temperature dependence of the Mössbauer spectral intensity below for the 1% sample is first fit to a as in Eq. (2), giving the value of , the temperature slope of , listed in Table ( i). Above , the ratio between an extrapolation of this normal vibrational behavior and the observed anomalous drop in intensity gives from Eqs. (8) and (9b) the values of E and S presented in Table ( i). With the crude model of n=2 for the hopping state, this entropy translates to L = 43 atoms involved in the hopping process. As another interpretation of this entropy, we compare it to the entropy of fusion for Pb ( e.u. per atom)[23] which allows for 30 atoms to have the same entropy as that in a liquid. To reiterate, the implication of is that there are 30 atoms around the Sn that are involved in a liquid-like hopping process with an entropy much higher than that of solid state diffusion. We call this region the high entropy or ``premelted" bubble. It is much larger than the region over which the impurity itself moves.

A striking feature of Fig. (4) is the strong concentration dependence of the temperature dependence of the Mössbauer spectral intensity. The 2% sample has a smaller , a higher and a more gradual temperature variation than the 1% sample. Fitting the 2% data with Eqs. (8) and (9b) gives the values for , E, S, and L in Table ( i).

The results of this analysis give an understanding of the concentration and temperature dependence found for the dilute Sn in Pb alloys. At low enough concentrations, the impurities have a premelted bubble of 30 atoms surrounding them. This premelted bubble does not have the full disorder of a liquid since it is surrounded by crystalline Pb and the short range order this induces orders the bubble substantially. It should be noted that a bubble of 30 atoms contains about 8 unit cubes or a radius of about one unit cube, i.e., about 1 1/2 nearest neighbor distances. Short range order in a liquid occurs over this same dimension and, therefore, will impose a strong constraint on the amount of disorder within the bubble. Yet, our experiments indicate that the Sn atom can still hop about within the bubble with the surrounding atoms participating in the motion.

When the concentration of the impurities becomes large enough to have substantial overlap of the bubbles surrounding the impurities, they interact and, as the experiment indicates, the bubbles harden, i.e., the value of E increases, and their size increases, i.e., S increases. The hardening of the bubbles explains why the solid does not lose its rigidity as the concentration increases. The bubble surrounding the impurity explains why the interaction between impurities occurs at such low concentrations as 2%.

The behavior of the Ag based alloys is qualitatively somewhat different. The decrease below a linear behavior in Fig. (3) is initially much more gradual and at 900K a more rapid drop occurs. Some concentration dependence is found. The most striking concentration dependence is exhibited by the line shift plots (Fig. (5). The 2% sample shows a linear T-dependence and reversibility on heating and cooling. The 4% and 8% samples show a shift from the linear dependence at 500K, and an irreversibility on an initial cycling. The concentration dependence suggests some interaction effect between the impurities is at the basis of the shift. We conclude that only the 2% Sn in Ag sample is an isolated impurity case.

The gradual decrease of in Fig. (3) from a linear behavior is not anomalous and can be fit by anharmonicity as given in Eq. (2). The coefficients of the terms linear and quadratic in T in are listed in Table ( i). The quadratic term is the anharmonic term and its magnitude is reasonable compared to the linear harmonic term, e.g., at 1000K, near the melting temperature, the two terms are comparable. However, the more rapid drop at requires the hopping mechanism. A fit to this drop using Eqs. (9b) and (8) gives the values listed in Table ( i). A fit to the temperature behavior of the 4% Sn in Ag sample results in the values listed in Table ( i). The values of L are obtained from for Ag[23]. The concentration dependence in E and S is similar to that in the Pb host indicating a similar hardening and growth in bubble size with increasing concentration.

However, the behavior of the two alloy based systems is different in some respects. The rapid decrease in the spectral intensity occurs at a much lower temperature for the Pb based alloys than the Ag based ones. This is true even if the data are normalized to the different melting temperatures of the host materials. One possible explanation of this difference is the difference of the relative sizes of the host atoms to the Sn atoms. The metallic radii of Ag, Pb and Sn are 1.44, 1.75, and 1.40, respectively. Tin impurities contract[24] the lattice constant of Pb but expand[25] that of Ag. Thus Sn has much more ``rattling" space in a Pb host than it does in a Ag host. This large excess free volume available for Sn in Pb should greatly expediate a liquid-like hopping behavior. However, it should be noted that one should not treat the atoms as hard spheres with a fixed atomic radius. The XAFS measurement of 2% Sn in Pb, shown in Fig. (7), indicates that the Sn-Pb distance is 3.479, only slightly smaller than the average interatomic distance of 3.498 found by x-ray diffraction, and considerably larger than the sum of the metallic radii of Sn and Pb (which gives 3.15).

One can obtain an appreciation of how enhanced the local motion of the impurity is by comparing to the solid bulk diffusion rate. The extrapolated value of the solid bulk diffusion rate[26] of Sn in Pb at 150K is and it would take seconds to diffuse 3.5, an atomic spacing. The local hopping is occurring in less than the Mössbauer lifetime ( seconds), about 26 orders of magnitude more rapid! For the Ag host, the corresponding numbers at 900K are[27]: , and seconds to diffuse 2.9. Here the bulk diffusion is comparable to the local hopping rate, but the sudden drop in the spectral intensity cannot be explained by bulk diffusion because there is no increase in line width. A more careful estimate from Singwi and Sjölander taking into account the discrete hopping motion shows that the line broadening is given by , where 2.9 . This gives a contribution of only 2% to the measured line width. Thus, bulk diffusion only slightly broadens the line width within experimental uncertainties, but does not decrease its intensity.

In an interesting paper Martin and Singer (MS)[9] investigated the behavior of point defects in a model crystal near melting by computer simulation. They misinterpreted our concept of a ``premelted" bubble and assumed an actual bulk-like liquid within the bubble without any short range correlation with the surrounding solid. Their simulations could not sustain any such liquid bubble below and wrongly concluded that their simulations were in disagreement with our model. It is clear why a bulk-like liquid cannot be maintained below in a bubble of diameter only a few atom spacings. The bubble, in contact with the surrounding solid, will have imposed on it short range order as discussed above. This short range order, even in a bulk-like liquid, is not too different from a solid within an atom spacing or so. Thus, such a small bubble will not appear much more disordered than a solid. Its main feature is its ability to have correlated movements of the enclosed atoms to allow the impurity to move about its original site by more than 0.2. This motion is different than hopping in a solid because it is a higher entropy process, more like a flow than a hopping.

Martin and Singer discovered the formation of interstitial-vacancy (I-V) pairs for an impurity with a substantially smaller radius than the host at temperatures well below melting. Their I-V pairs cannot explain our results because the interstitial in their case unbinds easily and contributes to the bulk diffusion rate, causing it to increase noticeably--an effect not seen. They speculate that there may be some lower temperature where the I-V formation is frequent, but the interstitial and vacancy remain closely bound to each other and eventually recombine.

Our excited state is a bound I-V pair in the MS terminology. The motion of the atom off the original lattice site produces a vacancy there and the atom, being off a lattice site, is in an interstitial position. However it is different in two important respects. One, the motion is not a hopping process as in a solid but a higher entropy one where the surrounding atoms participate more by flowing as in a liquid. Two, it is unlikely that the displacement of the pair is as large as in a classical interstitial of the FCC structure of Ag and Pb. It is hard to understand how such a well separated I-V pair (2.47 for octahedral and 2.14 for tetrahedral interstitial sites in Pb) can remain so tightly bound as required by our experiment. For example, such I-V pairs have been suggested[26] as a mechanism for the diffusion of Hg in Pb. In that case the various correlation coefficients are calculated to be greater than 0.25 above 500K, e.g., the I-V pair will move in less than four hops of the impurity. If this were the mechanism for the rapid local motion of Hg impurities in Pb seen by XAFS, the bulk diffusion rate would be close to the liquid value, i.e., many orders of magnitude larger than the true value. Our experiment requires that the I-V separation be greater than 0.2, but the strong localization of the I-V pairs that corresponds to localized motion in our model argues that the separation be significantly less than 2. It should be emphasized that in addition to the motion of the impurity we find a high entropy bubble that contains 30 surrounding atoms, and is thus about 5 in radius.

Tin is not an anomalous diffusor in Pb[26] nor in Ag[27]. Its diffusivity in those hosts is similar in value to that of the self diffusion of the hosts themselves. The mechanism usually postulated for such normal diffusion is a vacancy one[26]. With such a mechanism the diffusion rates of the host and impurity are within an order of magnitude of one another. Anomalous diffusors are impurities which have diffusion rates many orders of magnitude more rapid than the host. The mechanism for this anomalous diffusion is not known but it is usually associated with those impurities that dissolve as interstitials[26] and not substitutionally as Sn does in Pb and Ag.

An unusual local hopping phenomena was found[28] for Co impurities in Al. The Al interstitials were produced by low-temperature irradiation with high-energy electrons. Mössbauer spectroscopy on the Co impurities indicated a rapid decrease in the recoilless fraction by more than a factor of 4 in the temperature interval between 13 and 20 K. This unusual behavior was attributed to the Co atom being itself in an interstitial site as it formed a dumbbell defect with an Al interstitial across the face of the unit cell cube. It was postulated that the Co interstitial hopped between the six equivalent sites corresponding to forming dumbbell defects across each face of the unit itself. This hopping was thermally activated in a temperature interval 13-20 K to cause the rapid drop in the recoilless fraction. Such a model cannot fit our situation as our samples are well annealed and we do not expect any significant number of self-interstitials to exist. The phase diagrams for Pb-Sn and Ag-Sn as verified by diffraction and by XAFS for Pb-Sn indicate that Sn is a substitutional impurity, not an interstitial one.

There have been attempts to detect ``premelting" phenomena in solids with the motive to detect an instability in the solid which causes the melting transition. The classical thermodynamical picture of melting is that it is not caused by an instability, but it is a first order phase transition wherein the liquid phase becomes thermodynamically more stable. The solid is still mechanically stable and the transition to the liquid phase is a nucleation and growth process. Molecular simulations[1] confirm the thermodynamic picture. Our ``premelting bubbles" are not an instability of the solid but an equilibrium state. There is no reason to expect the solid to be completely homogeneous on an atomic scale when it has impurities to destroy the periodicity.

The mechanism of melting therefore requires specifying the nucleation site where the growth initiates. For small samples where the surface or interface is not negligible, nucleation can initiate on these surfaces[1,2,3,4]. These interfaces show ``premelting" disorder below the melting temperature and they subsequently act as the nucleation sites at melting. However, in large single crystals (rigorously, when the volume goes to infinity) where the surface and grain boundary areas can be neglected, the initiation sites must scale with the volume. The regions surrounding those impurities which exhibit the ``premelted" bubbles do scale correctly and will have a smaller barrier to a fluctuation to a nucleation site and could serve as the initiation sites for melting in 3-d in the thermodynamic limit of very large single crystals, analogous to the ``premelted" interfaces for smaller crystals. Impurities are only one type of a possible whole class of point defects which could act as initiation sites for 3-d melting. Any point defect that increases the free volume for hopping such as vacancies may also show ``premelting" in its vicinity and thus act as an initiation site. Since vacancies are thermally excited in all solids, this could be a mechanism for melting even in pure materials. It is pertinent to mention here that very recent molecular dynamics calculations of R. Car[29] near the melting point of Si show such ``premelting" characteristics around vacancies. Whereas at low temperatures the diffusion in the vicinity of the vacancy consists of a single atom hopping, near the melting point a significant fraction of the diffusion processes consists of several atoms moving coherently, i.e., a high entropy type flow as in a liquid.

The temperature dependence of the spectral intensities that we measure are consistent with other measurements. In unpublished thesis work[30], R. C. Knauer, Jr. found a temperature dependence similar to ours for the spectral intensity of Sn in Ag, but measured up to only 743K, and so did not see the temperature range with the rapid drop. Room temperature values for the recoilless fraction of dilute Sn impurities in Pb have been measured[31] to be 0.016 (0.015).

Finally, it should be noted that possible premelting of the Ag ionic sublattice surrounding impurity Sn atoms has been reported[32] in a Mössbauer experiment on superionic AgSe with tin impurities.