Low-temperature entropy in Cu-based shape-memory alloys and the boson peak

We have measured the energy and entropy differences between b and martensitic phases in Cu-Zn and Cu-Zn-Al shape-memory alloy systems which display martensitic transitions in a range of temperatures between 30 K and 130 K. We show that the entropy difference originates in the low-energy TA2 vibrational modes and displays the predicted features for a system showing the so-called boson peak, characteristic of structurally disordered solids. In the studied systems the boson peak is conveniently revealed by the ratio between the entropy difference and the Debye entropy.


I. INTRODUCTION
Cu-based shape-memory alloys display an ͑open͒ bcc phase or ␤ phase at high temperature, and transform martensitically towards a close-packed structure at a lower temperature.For a considerable time, 1 it has been recognized that the stability of the high-temperature open phase is a consequence of its excess of vibrational entropy compared with its close-packed counterpart, which is more favorable from purely energy considerations. 2This excess of entropy originates from low restoring forces in specific lattice directions related to the path that leads the system from the hightemperature parent phase to the martensite ͑combination of shears and shuffles on ͕110͖ planes along ͗11 ¯0͘ directions͒.Such low restoring forces manifest themselves by a lowenergy TA 2 ͓110͔ phonon branch with a correspondingly small elastic constant CЈ, which slightly softens as the martensitic transition is approached.In contrast, in the martensitic phase, the boundary frequency of the TA 1 ͓100͔ branch 3 is three times larger than the corresponding boundary frequency of the TA 2 branch in the ␤ phase.At the same time the elastic anisotropy C 44 /CЈ decreases from ϳ13 to ϳ2. 4,5 This peculiar vibrational behavior shares some similarities with that of a number of structurally disordered solids.Actually, these systems also show an excess of low-frequency states not present in their crystalline counterpart. 6,7This fact suggests that the open ␤ phase can be considered, from the point of view of its vibrational properties, as a vibrationally disordered phase, and the martensitic transition as a condensation, with a first-order character, towards a vibrationally ordered phase.In disordered systems the excess of lowfrequency states is usually seen by a higher contribution compared with the Debye behavior which, in principle, is supposed to apply in the low-frequency regime.This excess of low-frequency states gives rise to the so-called boson peak ͑BP͒, which is shown by a maximum in the vibrational density of states ͑DOS͒ divided by 2 at frequencies far below the Debye frequency ͑without a peak in the DOS itself͒.The BP is also revealed by thermodynamic quantities.For instance, the heat capacity C shows a peak in the scaled C/T 3 vs T curve.
In the present paper we report measurements of the difference of enthalpy and entropy between ␤ and martensitic phases in two shape-memory alloy systems: Cu-Zn and Cu-Zn-Al.In the ternary alloy the compositions are selected with a constant valence electron to atom ratio e/a.The martensitic structures of these two alloy families are 9R and 18R, respectively, which are monoclinic structures, and are conveniently described by a large unit cell ͑approximately orthorhombic͒ containing, 9 and 18 close-packed atomic planes, 8 respectively.The distinct packing sequence is a consequence of the different ordering configurations in the two systems.Cu-Zn displays a B2 (Im3m) ordered structure ͑nearest-neighbor ordering͒, while Cu-Zn-Al is L2 1 (Fm3m) ͑next nearest-neighbor ordering͒.It is worth mentioning that measurements of enthalpy and entropy changes have been previously reported for a number of Cu-based shape-memory alloys, including those studied in this work.However, all previous data correspond to alloy systems transforming at relatively high temperatures.For this temperature range, ⌬S has reached, within experimental error, the classical high-temperature value, and thus it does not show any appreciable temperature dependence.In the present paper measurements have been extended to systems transforming at relatively low temperatures.A simple vibrational model based on the ideas described above has been used in order to show that the entropy difference accounts fairly well for the excess of vibrational entropy of the high-temperature ␤ phase.Moreover, these results corroborate the existence of the BP in the studied shape-memory alloys.

A. General features
We have collected reported DOS for Cu-Zn, 9 Cu-Al-Be, 10 and Cu-Al-Mn ͑Ref.11͒ ␤ Cu-based alloys.From these data the ratio g()/g D () has been calculated, where g() is the actual DOS and g D () is the Debye DOS ͓g D ()ϳ 2 ͔.The ratio is shown in Fig. 1 as a function of / D ( D is the Debye frequency given in the corresponding references͒ for the three systems.A bump is clearly observed in the region of small frequencies (ϳ0.25-0.35D ).It is worth noting that this bump is consistent with the flat TA 2 dispersion relation over a broad region of the Brillouin zone, and with corresponding low zone-boundary frequency which is close to the frequency of the maximum of the bump.This behavior is common to all Cu-based shape-memory alloys.These TA 2 anomalous modes correspond to displacements of ͕110͖ planes along ͗11 ¯0͘ direction, which are involved in the transformation from the ␤ phase to the close-packed martensitic structure.The behavior illustrated in Fig. 1, suggests that at low frequencies the DOS for shape-memory alloys can be expressed as where the term 2 describes 3d Debye phonons, and the term f (Ϫ E ) corresponds to a narrow distribution of phonons centered about E and related to the anomalous phonon modes.The effect of the excess of low-frequency modes on the thermodynamic properties can be accounted for by simply assuming that f (Ϫ E ) is a ␦ function.This means that the low-frequency transverse acoustic modes are essentially described in terms of an Einstein model with characteristic frequency E .While this is certainly too crude an approximation for the DOS, it is expected to provide reasonable results in terms of integral quantities.Thus, the following vibrational spectrum is assumed: where p is the fraction of anomalous or Einstein modes.The heat capacity of the ␤ phase is thus given by where D(T/ D ) and E(T/ E ) are Debye and Einstein heat capacities, respectively ( D ϭប D /k is the Debye temperature and E ϭប E /k is the Einstein temperature͒.The BP is revealed by a peak on the ratio Within the present model this ratio is given by

͑4͒
It is straightforward to show that for T→0, R C →1Ϫp; for T→ϱ, R C →1, and that R C displays a maximum at a temperature T max ϭ E /5.Therefore, the proposed approach is adequate to model systems displaying the BP.

B. The martensitic transition
The martensitic phase is close packed and its vibrational properties can be adequately accounted for by a Debye model.Moreover, we will assume that the corresponding Debye temperature is the same as the Debye temperature of the ␤ phase.This means that the heat capacity of the martensitic phase is C M (T)ϭD(T/ D ).Within this framework, the modes responsible for the BP provide the excess of entropy of the ␤ phase with respect to the martensitic phase.Therefore, this excess of entropy for a system that transforms at a temperature T M is given by where ⌬SϭS ␤ ϪS M .Note that the BP also shows up in the ratio ⌬S/S D given by: The classical high-temperature limit of ⌬S is given by From this equation it is clear that the stability of the hightemperature ␤ phase is, for a given fraction p, favored by a low value of E .On the other hand, the high temperature mean-square displacement of TA 2 (110)͓11 ¯0͔ Einstein modes is given by FIG. 1. g()/g D () vs / D for Cu; 47 at.% Zn, Cu; 22.72 at.% Al; 3.55 at.% Be and Cu; 25 at.% Al; 7 at % Mn.The peak reveals the existence of a BP in these systems.
where M is the average atomic mass.Indeed, large-amplitude TA 2 (110)͓11 ¯0͔ displacements enhances the instability towards the martensitic phase.Therefore, low E also favors higher transition temperatures.From this point of view T M is the temperature at which both, the entropy stabilizing effect ͑thermodynamical͒ and the large-amplitude displacements destabilizing effect ͑mechanical͒ balance each other.This balance leads to the following relationship, . ͑9͒ Note that this expression only applies at temperatures T ӷ E .Within the preceding approach, the difference in vibrational energy between ␤ and martensitic phases is given by Therefore, the difference in configurational energy between both ␤ and martensite is given by where T M ⌬S is the total-energy difference between the two phases.

III. EXPERIMENT
We have studied two shape-memory alloy families, Cu 1Ϫx Zn x and Cu 1Ϫ(xϩy) Zn x Al y , with constant valence electron concentration e/aϭ1ϩxϩ2yϭ1.48͑Cu contributes 1 electron, Zn 2 electrons and Al 3 electrons͒, which corresponds to the eutectoid value.Single crystals have been grown in sealed quartz tubes by the Bridgman method.The purity of the metals used was 99.999% in all cases.In the case of Cu-Zn samples, after the growth process, the encapsulated ingots were kept in contact with a piece of steel at 1100 K while being taken out of the furnace, and rapidly quenched into water at room temperature by breaking the quartz tube.This method ensures that homogeneous ␤-phase Cu-Zn single crystals are obtained.For this reason the quenched Cu-Zn single crystals were not heat treated further except for long-term annealing at room temperature.From the ingots of both Cu-Zn and Cu-Zn-Al, small 5-mmdiameter disks of 2 mm thickness were cut using a low-speed diamond saw.In order to eliminate any surface damage, samples were polished with fine grinding paper.The actual composition of the single crystals was determined by means of electron dispersion analysis of x rays.Cu-Zn-Al samples were annealed for 900 s at 1073 K, air cooled down to room temperature and annealed at room temperature for a very long time ͑several days͒.
Transition temperatures T M , entropy changes ⌬S, and enthalpy changes ⌬H were obtained by means of calorimetric measurements.A high-sensitivity differential scanning microcalorimeter, specifically designed to investigate solidsolid transitions was used.The working temperature range was from 5 K to 350 K, which is adequate for the study of martensitic transitions with T M down to ϳ20 K.An example of a calorimetric curve, represented as the ratio of the ther- mal power dQ/dt to the ͑instantaneous͒ temperature rate dT/dt vs T is shown in Fig. 2 for a Cu-Zn-Al crystal.The calorimetric curves show the jerky structure that is common to martensitic transitions in shape-memory alloys.After a proper correction of the base line, the enthalpy, and entropy changes can be obtained as:

dT, ͑13͒
where T i and T f are temperatures located above ͑below͒ and below ͑above͒ the starting and finishing temperatures on cooling ͑heating͒ calorimetric runs, respectively.

IV. RESULTS
Figure 3 shows the transition temperature T M as a function of composition.For data reduction a weighted composition xϩ␣y was used.It is seen that the value giving the best linear representation is ␣ϭ1.357.Actually, this value is lower than but close to, the value proposed in Ref. 12 where low-temperature transition temperatures were estimated from the resolved stress needed to induce the martensite at 80 K.This estimation assumes a constant value of d/dT which, from the Clausius-Clapeyron equation, implies a constant value of ⌬S.As we will see, this is not a good assumption at low temperature and, therefore, the method supposes an underestimation of T M .This is in agreement with the larger value of ␣ reported in that work compared with the present value.
Figure 4 shows the obtained values of ⌬S as a function of T M for Cu-Zn and Cu-Al-Zn alloys.To our knowledge present calorimetric measurements are the only direct measurements of entropy ͑and enthalpy changes͒ for shapememory alloys which transform below liquid nitrogen.A number of data corresponding to alloys with higher T M have been collected from the literature and were included in the plots ͑open symbols͒.Since it is known that in Cu-based alloys the electronic contribution to the entropy change is negligibly small, 13 it is thus reasonable to assume that the measured entropy difference has a purely vibrational origin.At high temperatures the entropy change is almost independent of temperature, and it decreases as the transition tem- perature lowers.Such a decrease is in agreement with the third law of thermodynamics.
In order to compare the measured ⌬S values with the predictions of the above-presented model, we need estimations of the parameters D , p, and E .We started by taking the values of p and D reported in previous works.For Cu-Zn the Debye temperature was obtained by Veal and Rayne 14 from very low-temperature heat-capacity measurements.An average value for the compositions studied in the present paper is D ϭ265 K.For Cu-Zn-Al alloys, D was estimated in Ref. 15.For the composition range of interest here we take an average value of D ϭ290 K.
On the other hand, p was also estimated in Ref. 15 by fitting ⌬S values of alloys transforming at high temperatures to expression ͑7͒ taking E ϭប ZB /kӍ58 K, where ZB is the zone-boundary frequency of the TA 2 branch measured at a temperature close to the martensitic transition temperature.These fits yield a value of pӍ0.047, which was found to be nearly independent of alloy composition.We thus fixed this value of p for both Cu-Zn and Cu-Zn-Al alloys.As regards E , the estimation based on the TA 2 boundary frequency ZB certainly represents an overestimation.Therefore, this parameter will be left as a free fitting parameter.We fitted expression ͑6͒ to the ratio between the measured ⌬S and S D .S D and S ␤ were determined by numerical integration.Best fits were obtained with E ϭ53 K for both Cu-Zn and Cu-Zn-Al.As expected, these values are slightly lower than previous estimations based on the zone-boundary frequency.The fits are depicted in Fig. 5, where R S Ϫ1 (ϭ⌬S/S D ) vs T M / D is plotted for Cu-Zn and Cu-Zn-Al alloys, respectively.The experimental results clearly show the tendency expected for a system displaying the BP.The continuous lines in Fig. 4 are obtained from Eq. ͑5͒ with the values estimated for the parameters D , E , and p.The goodness of the fit gives further support to the existence of the BP.
Since the volume change at the martensitic transition in Cu-based shape-memory alloys is negligibly small, the calorimetrically measured ⌬H is an excellent estimation of the energy difference between the two phases.In Fig. 6 the measured values of ⌬E are compared with T M ⌬S, where ⌬S is obtained with the values of p, D , and E previously determined.The agreement is very satisfactory.⌬E vib calculated from Eq. ͑10͒ is also shown in the figure.This quantity is small, but plays an important role in the low-temperature transition region.For alloys transforming at temperatures T M Ͼ E , the energy difference is essentially determined by the configurational energy difference and shows a linear dependence on the transition temperature.

V. DISCUSSION
In this paper we have reported low-temperature measurements of the energy and entropy differences between ␤ and martensitic phases in Cu-Zn and Cu-Zn-Al with an eutectoid composition.We have confirmed that the excess of entropy of the ␤ phase with respect to the martensitic phases arises from the low-energy TA 2 phonon modes.These low-energy modes show up in the density of states as a BP with similar features as the BP reported for a number of disordered solids.This anomaly is observed about frequencies close to the zone-boundary frequency of the TA 2 branch.This behavior has been accounted for by assuming a DOS with a certain fraction p of localized modes described by an Einstein model and the remaining modes described by a Debye model.The DOS of the close-packed martensite is assumed to be Debye type with the same Debye temperature as in the corresponding ␤ phase.We have found that this simple model provides excellent results regarding thermodynamic quantities such as the entropy difference between the two phases.In particular, the model enables understanding of why the entropy change takes a rather constant value over a broad temperature range ͑see Fig. 4͒: from high temperature down to temperatures slightly above the Einstein temperature which characterizes the anomalous low-energy modes.Moreover, the small contribution of the vibrational energy to the energy difference between the two phases is also explained by the model taking into account the small value of the fraction p of these anomalous modes.
The three relevant parameters of the proposed model are the Einstein E and the Debye D temperatures and the fraction p of anomalous modes.E and p are seen to be, to a good approximation, composition independent, while D depends on composition essentially through the average atomic mass.It is interesting to analyze Eq. ͑9͒ with the estimated values of these parameters, together with the measured value of the entropy change at high temperatures.Assuming that T M Ӎ300 K, the mean amplitude displacements of (110) planes along ͓11 ¯0͔ direction that enable the instability of the ␤ phase to yield the martensitic structure are of the order of 10% of the interplanar distance between (110) planes.Actually, this is a reasonable value for Cu-based shape-memory alloys taking into account the fact that the transformation strain is of the order of 15%.
At a first glance, the existence of the BP in shape-memory alloys could be considered as an unexpected result since this feature is commonly associated with structural disorder.Actually, in disordered solids its appearance is still a subject of speculation and different explanations have been proposed. 16t seems, however, that disorder-induced localization of crystal vibrations is at the origin of such an effect.Since disorder is not the only possible localization effect, this point of view enables us to propose that the BP should be present in any system with localized vibrational modes.This is, for instance, supported by recent inelastic neutron scattering measurements in Tl-filled skutterudites. 17Results show a peak in the DOS, absent in the unfilled system, which is unambigu-ously attributed to localized modes associated with the Tl atoms.The results of the present paper are consistent with this viewpoint.In shape-memory systems, the strong anisotropy of the open ␤ phase, characteristic of these systems, 5 causes the localization of the soft anomalous (110)͓11 ¯0͔ modes.This is in agreement with very general arguments claiming that strong anisotropy and/or noncentral forces are factors that favor the existence of low-energy modes giving rise to the BP. 18In the close-packed martensitic phase with low elastic anisotropy the anomaly disappears and the material displays, to a large extent, the usual Debye behavior.It is foreseeable that the observed behavior could also occur in other crystalline solids with low-lying soft branches such as certain ferroelectrics.

FIG. 4 .
FIG. 4. Entropy change ⌬S vs reduced transition temperature T/ D for Cu-Zn and Cu-Zn-Al alloys.Solid symbols: data from the present paper.Open symbols: data collected from the literature.Continuous lines are obtained from Eq. ͑5͒ with D ϭ265 K, E ϭ53 K, and pϭ0.047 for Cu-Zn and D ϭ290 K, E ϭ53 K, and pϭ0.047 for Cu-Zn-Al.

FIG. 6 .
FIG. 6.Energy difference ⌬E as a function of reduced transition temperature T/ D .Solid symbols: this paper.Empty symbols: data from the literature.The continuous line is T M ⌬S and the dashed line the difference in vibrational energy (⌬E vib ) obtained from the model with D ϭ265 K and pϭ0.047 for Cu-Zn, and D ϭ290 K and pϭ0.047 for Cu-Zn-Al.For both alloys the fit renders E ϭ53 K.