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The role of point defects in melting of solid He
Physica B 280 (2000) 142}145
The role of point defects in melting of solid He Emil Polturak*, Amit Kanigel, Nir Gov, Tuvy Markovich, Joan Adler Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Abstract We have recently observed that the resistance to shear of solid He decreases dramatically near the "rst-order BCC}HCP transition. In our view, the solid shears via a di!usive counter-#ow of atoms and point defects. The mechanism of self di!usion couples point defects with one speci"c phonon which softens as the transition is approached. Since such a scenario can possibly lead to melting, it is important to understand (a) which type of point defect is associated with the reduction of shear resistance, and (b) can the presence of point defects lead to the softening of phonons. We report here the results of numerical simulations and analytic modeling. Our results indicate that split interstitials are much more e!ective than vacancies in lowering the resistance to shear. We suggest that these excitations can be generated as a result of a `local modea excited in the crystal. 2000 Elsevier Science B.V. All rights reserved. Keywords: Helium-4 solid; Melting; Phonon softening; Point defects; Solid helium-4
1. Introduction Understanding the melting process of a 3D solid presents an ongoing challenge to theorists and experimentalists alike. Calculations of the free energy of the liquid and solid are not precise enough to predict the correct melting temperature. Alternately, existing models of melting rely on the presence of crystalline defects or excitations in order to destabilize the lattice. Over the years, melting was attributed to phonons (Lindemann and Born), to vacancies (Frenkel), to interstitials (Granato) and dislocations (Mizushima). Empirically, the melting temperature, ¹ , was found to be proportional + to the activation energy for self di!usion, E , with pro" portionality constant which is approximately the same for solids having the same crystalline structure. Beside that observation, which is of great importance, there is no strong experimental evidence supporting any particular model. In second-order phase transitions, one can learn about the physical mechanism leading to the phase change by looking at the behaviour of the system inside the critical
* Corresponding author. E-mail address: [email protected] (E. Polturak)
region. In 3D melting, the transition is strongly "rst order and the width of the critical region is too small to make experimental work feasible. To make things even harder, the transition is always broadened by the heterogeneity of the samples. Here is precisely where solid He becomes important, since it is relatively easy to grow macroscopic single crystals of very high purity and crystalline quality. Furthermore, the density of point defects in solid He above 1.5 K is an order of magnitude higher than that found in typical solids at ¹ . One could then + expect to see signi"cant in#uence of point defects on the properties of the lattice. To make this region of the phase diagram even more interesting, we discovered that the shear resistance of solid He decreases quite dramatically near a "rst-order structural BCC}HCP phase transition [1]. Loss of shear resistance is characteristic of melting; we thus have an opportunity to study a mechanism which can potentially bring about melting.
2. Experimental Brie#y, the aforementioned experiment involves the motion of a solid object (superconducting wire) through the crystal, under the in#uence of an external force [1]. The wire is situated in a constant magnetic "eld.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 5 2 9 - X
E. Polturak et al. / Physica B 280 (2000) 142}145
Fig. 1. Values of the self-di!usion coe$cient of a solid 1.4% He}He mixture near its BCC}HCP transition.
A Lorentz force is applied by passing a DC current through the wire, causing it to move. At small forces, the velocity of the wire is linear in the force. This type of behaviour of solid He was studied extensively by the Kharkov group [2]. In this regime, the plastic #ow of the solid around the wire takes place via a counter#ow of atoms and point defects (Nabarro}Herring mechanism), and is proportional to the self-di!usion coe$cient [3]. On approaching the HCP}BCC transition while keeping the driving force constant, we observed a dramatic increase of the velocity of the wire, and hence of the di!usion coe$cient. At the transition, the di!usion coe$cient reaches &10\ cm/s, which is typical of a liquid. This data is shown in Fig. 1 for a 1.4% He}He mixture [4] near its BCC}HCP transition; the data for pure He is similar [1]. Self-di!usion is traditionally associated with an exchange process of an atom and a point defect. The di!usion coe$cient is a product of the density of defects times their mobility. Its increase implies that either the density of defects, their mobility, or both increase sharply as one approaches the phase transition. It is instructive to point out the connection between the increase of the di!usion coe$cient and the apparent loss of resistance to shear. It would seem that an ideal crystal containing enough point defects would have zero resistance to shear, as any external shear stress would be relieved by the di!usive #ow of atoms away from the high-stress region to a low-stress region. One has only to wait for a su$cient time (uP0 limit). The analysis of the data in the region near the transition leads to a conclusion that the di!usion process is phonon assisted. This type of di!usion was considered by Schober et al. [5]. The analysis of our data requires only one such phonon, which by analogy with other BCC solids we identify as the T(1 1 0) phonon at the edge of the Brillouin zone. This short wavelength phonon deforms
143
Fig. 2. Softening of the T(1 1 0) phonon near the BCC}HCP transition in He and in Ti (Ref. [6]). Solid lines are guide to the eye.
the lattice locally, thus allowing one atom to squeeze readily through the potential barrier presented by its neighbours. The enhancement of the di!usion near the transition is a consequence of the softening of this phonon. This e!ect was observed previously in several BCC metals undergoing a BCC}HCP transformation [6]. In Fig. 2 we compare the softening of the relevant phonon in BCC He calculated from our data with that observed in Ti using inelastic neutron scattering [6]. The data is plotted versus reduced temperature, so that the two sets of data can be shown on the same scale. One can see that relative degree of softening is similar in both materials. The appearance of soft modes near a second order or a weak "rst-order phase transition is well known [7]. It was not considered previously in the present context, as a way for the solid to lose its shear resistance and eventually melt. Beside enhancing the process of di!usion, a soft phonon mode can lead directly to destabilization of the lattice and melting via either the Born or Lindemann mechanism. There is a signi"cant di!erence between such a scenario and the original models; unlike in the original predictions, not the whole phonon spectrum is responsible for melting, but only one speci"c mode which softens. A direct observation of a phonon mode which softens near the melting transition would be of course very important; experiments aiming to test this hypothesis are in the planning stage.
3. Molecular dynamics simulations In parallel, one would like to examine the question as to what is it that makes phonons soften. In general, any instability that may arise in the solid must feed o! some positive feedback mechanism. This feedback mechanism
144
E. Polturak et al. / Physica B 280 (2000) 142}145
Fig. 3. Dependence of C on the concentration of point defects in Cu. Data points are a result of molecular dynamics simulation. Open symbols * vacancies, closed symbols * split interstitials. The temperature is 1400 K (very close to the temperature of mechanical melting). Solid lines are "ts to the data.
is one ingredient which is missing in most of the models of melting mentioned above. In our case, we wish to test the viability of a very speci"c mechanism, in which point defects lead to phonon softening, and softer phonons then enhance the di!usion coe$cient and at the same time reduce the shear moduli. In the following, we report some recent results of molecular dynamics simulations testing this scenario [8]. The simulations were done on samples up to 2000 atoms in size held under constant stress, using the method developed by Parrinello and Rahman [9]. Since we are aiming at understanding melting in general, the simulations were done for copper, for which there exists an interatomic potential yielding accurate results for a large body of experimental properties [10]. The "rst question which we addressed regards the type of point defect which has a signi"cant e!ect on the shear moduli. Two types of defects were considered, the vacancy and the (1 0 0) split interstitial. These two were chosen for having the lowest formation energies. From the simulation, we obtained values of the elastic coe$cients associated with shear, C and C "(C !C )/2. In Fig. 3, we show the dependence of C on the density of point defects. A similar dependence is found for C . Quite surprisingly, vacancies do not a!ect these elastic coe$cients at all. In contrast, there is a pronounced decrease resulting from the split interstitials. The in#uence of the split interstitials on the elastic coe!"cients increases with temperature, up to the point where one of these coe!"cients vanishes, and the crystal melts. This mechanism, usually called mechanical melting, essentially generalizes the Born model for the case where the crystal contains defects. The phase diagram resulting
Fig. 4. In#uence of split interstitials on the phase diagram of Cu. The points are the results of a molecular dynamics simulation, while the dotted line separating the solid and the melt is a guide to the eye. The scatter results from the "nite time span of the simulation.
from these calculations is shown in Fig. 4. One can see that the temperature at which mechanical melting takes place decreases by about 60 K with 0.5% of split interstitials. In contrast, vacancies have no e!ect at all. Common wisdom has it that if anything, vacancies are the better candidate to induce melting, simply because their density is much larger than that of interstitials. However, if we accept the premise that the density of vacancies, typically 10\ of the atomic density even at ¹ , is not large enough to a!ect the crystal directly, one + should keep an open mind to the possibility that the density of defects is not the only relevant parameter, and there may be more than one type of defect involved. A model of melting based on split interstitials was proposed by Granato [14]. In order to understand the surprising `detachmenta of vacancies from the elastic properties of the crystal shown in Fig. 3, it is useful to view the vacancies as a dispersionless excitation mode of the crystal, at an energy which is much higher than that of any phonon. Consequently, this excitation branch does not hybridize with the elastic modes, namely phonons. On the other hand, the split interstitial, composed of two atoms, has internal degrees of freedom which are low enough in energy so that they do couple to the phonons in a resonant way (so called resonance modes) and hence a!ect the phonon dispersion relation directly [11}13]. Indeed, our simulations con"rm this point of view [8]. How does all this relate to solid He? We have recently developed a model [15] in which the inherently large anharmonic behaviour of the crystal is addressed by treating the solid as a perfectly harmonic solid which also supports `local modesa. Due to symmetry considerations, the two kinds of excitations, i.e. phonons and
E. Polturak et al. / Physica B 280 (2000) 142}145
local modes, interact rather weakly. In the BCC solid, the coupling between these two types of excitations leads to softening of one particular phonon branch, the T(1 1 0) phonon, while the other branches remain described by the self-consistent harmonic approximation. The resulting phonon spectrum is in good overall agreement with the neutron scattering data [16]. At the same time, coupling to the phonons raises the energy of the `local modesa to 14 K. The spatial extent of the 14 K mode is over two unit cells [15], hence it constitutes a point defect, which however resembles a split interstitial more than a vacancy because it involves two atoms. The value of the energy, 14 K, is very close to the experimental value of E determined for the BCC phase [1,2]. Thus, both " ingredients of our picture, a soft phonon mode and a point defect resembling a split interstitial can be modelled in the BCC solid phase of He. This mode may have been observed in neutron scattering experiments [16]; we plan to extend these experiments in the near future to test our model. In conclusion, we combined experimental data and molecular dynamics simulations into a detailed scenario of 3D melting. In our picture, there is a feedback between the phonons and the point defects which together bring about a reduction of the shear resistance of the crystal. We found, by looking at data obtained for BCC metals, that in the present context solid He is very similar to usual solids. We feel that this similarity may eventually present us with yet another case in which studying He contributes to understanding a rather general problem in solid state physics.
145
Acknowledgements This work was supported in part by the Israel Science Foundation and by the Technion VPR Fund for the Promotion of Research.
References [1] I. Berent, E. Polturak, Phys. Rev. Lett. 81 (1998) 846. [2] N.E. Dyumin, N.V. Zuev, S.V. Svatko, V.N. Grigor'ev, J. Low-Temp. Phys. 17 (1991) 468. [3] F.R.N. Nabarro, Theory of Crystal Dislocations, Oxford Univ. Press, New York, 1967. [4] I. Berent, E.P. Polturak, J. Low-Temp. Phys. 112 (1998) 337. [5] H.R. Schober, W. Petry, J. Tampenau, J. Phys.: Condens. Matter 4 (1992) 9321. [6] W. Petry et al., Phys. Rev. B 43 (1991) 10933. [7] R.A. Cowley, Adv. Phys. 29 (1980) 1. [8] A. Kanigel, Thesis, Technion, Israel, 1999. [9] M. Parrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182. [10] F. Cleri, V. Rosato, Phys. Rev. B 48 (1993) 22. [11] P.H. Dederichs, C. Lehmann, A. Scholz, Z. Phys. B 20 (1975) 155. [12] R.F. Wood, M. Mostoller, Phys. Rev. Lett. 35 (1975) 45. [13] H.R. Schober, V.K. Tevary, P.H. Dederichs, Z. Phys. B 21 (1975) 255. [14] A.V. Granato, Phys. Rev. Lett. 68 (1992) 974. [15] N. Gov, E. Polturak, Phys. Rev. B 60 (1999) 1019. [16] E.B. Osgood, V.N. Minkiewicz, T.A. Kitchens, G. Shirane, Phys. Rev. A 5 (1972) 1537.
The role of point defects in melting of solid He Emil Polturak*, Amit Kanigel, Nir Gov, Tuvy Markovich, Joan Adler Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Abstract We have recently observed that the resistance to shear of solid He decreases dramatically near the "rst-order BCC}HCP transition. In our view, the solid shears via a di!usive counter-#ow of atoms and point defects. The mechanism of self di!usion couples point defects with one speci"c phonon which softens as the transition is approached. Since such a scenario can possibly lead to melting, it is important to understand (a) which type of point defect is associated with the reduction of shear resistance, and (b) can the presence of point defects lead to the softening of phonons. We report here the results of numerical simulations and analytic modeling. Our results indicate that split interstitials are much more e!ective than vacancies in lowering the resistance to shear. We suggest that these excitations can be generated as a result of a `local modea excited in the crystal. 2000 Elsevier Science B.V. All rights reserved. Keywords: Helium-4 solid; Melting; Phonon softening; Point defects; Solid helium-4
1. Introduction Understanding the melting process of a 3D solid presents an ongoing challenge to theorists and experimentalists alike. Calculations of the free energy of the liquid and solid are not precise enough to predict the correct melting temperature. Alternately, existing models of melting rely on the presence of crystalline defects or excitations in order to destabilize the lattice. Over the years, melting was attributed to phonons (Lindemann and Born), to vacancies (Frenkel), to interstitials (Granato) and dislocations (Mizushima). Empirically, the melting temperature, ¹ , was found to be proportional + to the activation energy for self di!usion, E , with pro" portionality constant which is approximately the same for solids having the same crystalline structure. Beside that observation, which is of great importance, there is no strong experimental evidence supporting any particular model. In second-order phase transitions, one can learn about the physical mechanism leading to the phase change by looking at the behaviour of the system inside the critical
* Corresponding author. E-mail address: [email protected] (E. Polturak)
region. In 3D melting, the transition is strongly "rst order and the width of the critical region is too small to make experimental work feasible. To make things even harder, the transition is always broadened by the heterogeneity of the samples. Here is precisely where solid He becomes important, since it is relatively easy to grow macroscopic single crystals of very high purity and crystalline quality. Furthermore, the density of point defects in solid He above 1.5 K is an order of magnitude higher than that found in typical solids at ¹ . One could then + expect to see signi"cant in#uence of point defects on the properties of the lattice. To make this region of the phase diagram even more interesting, we discovered that the shear resistance of solid He decreases quite dramatically near a "rst-order structural BCC}HCP phase transition [1]. Loss of shear resistance is characteristic of melting; we thus have an opportunity to study a mechanism which can potentially bring about melting.
2. Experimental Brie#y, the aforementioned experiment involves the motion of a solid object (superconducting wire) through the crystal, under the in#uence of an external force [1]. The wire is situated in a constant magnetic "eld.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 5 2 9 - X
E. Polturak et al. / Physica B 280 (2000) 142}145
Fig. 1. Values of the self-di!usion coe$cient of a solid 1.4% He}He mixture near its BCC}HCP transition.
A Lorentz force is applied by passing a DC current through the wire, causing it to move. At small forces, the velocity of the wire is linear in the force. This type of behaviour of solid He was studied extensively by the Kharkov group [2]. In this regime, the plastic #ow of the solid around the wire takes place via a counter#ow of atoms and point defects (Nabarro}Herring mechanism), and is proportional to the self-di!usion coe$cient [3]. On approaching the HCP}BCC transition while keeping the driving force constant, we observed a dramatic increase of the velocity of the wire, and hence of the di!usion coe$cient. At the transition, the di!usion coe$cient reaches &10\ cm/s, which is typical of a liquid. This data is shown in Fig. 1 for a 1.4% He}He mixture [4] near its BCC}HCP transition; the data for pure He is similar [1]. Self-di!usion is traditionally associated with an exchange process of an atom and a point defect. The di!usion coe$cient is a product of the density of defects times their mobility. Its increase implies that either the density of defects, their mobility, or both increase sharply as one approaches the phase transition. It is instructive to point out the connection between the increase of the di!usion coe$cient and the apparent loss of resistance to shear. It would seem that an ideal crystal containing enough point defects would have zero resistance to shear, as any external shear stress would be relieved by the di!usive #ow of atoms away from the high-stress region to a low-stress region. One has only to wait for a su$cient time (uP0 limit). The analysis of the data in the region near the transition leads to a conclusion that the di!usion process is phonon assisted. This type of di!usion was considered by Schober et al. [5]. The analysis of our data requires only one such phonon, which by analogy with other BCC solids we identify as the T(1 1 0) phonon at the edge of the Brillouin zone. This short wavelength phonon deforms
143
Fig. 2. Softening of the T(1 1 0) phonon near the BCC}HCP transition in He and in Ti (Ref. [6]). Solid lines are guide to the eye.
the lattice locally, thus allowing one atom to squeeze readily through the potential barrier presented by its neighbours. The enhancement of the di!usion near the transition is a consequence of the softening of this phonon. This e!ect was observed previously in several BCC metals undergoing a BCC}HCP transformation [6]. In Fig. 2 we compare the softening of the relevant phonon in BCC He calculated from our data with that observed in Ti using inelastic neutron scattering [6]. The data is plotted versus reduced temperature, so that the two sets of data can be shown on the same scale. One can see that relative degree of softening is similar in both materials. The appearance of soft modes near a second order or a weak "rst-order phase transition is well known [7]. It was not considered previously in the present context, as a way for the solid to lose its shear resistance and eventually melt. Beside enhancing the process of di!usion, a soft phonon mode can lead directly to destabilization of the lattice and melting via either the Born or Lindemann mechanism. There is a signi"cant di!erence between such a scenario and the original models; unlike in the original predictions, not the whole phonon spectrum is responsible for melting, but only one speci"c mode which softens. A direct observation of a phonon mode which softens near the melting transition would be of course very important; experiments aiming to test this hypothesis are in the planning stage.
3. Molecular dynamics simulations In parallel, one would like to examine the question as to what is it that makes phonons soften. In general, any instability that may arise in the solid must feed o! some positive feedback mechanism. This feedback mechanism
144
E. Polturak et al. / Physica B 280 (2000) 142}145
Fig. 3. Dependence of C on the concentration of point defects in Cu. Data points are a result of molecular dynamics simulation. Open symbols * vacancies, closed symbols * split interstitials. The temperature is 1400 K (very close to the temperature of mechanical melting). Solid lines are "ts to the data.
is one ingredient which is missing in most of the models of melting mentioned above. In our case, we wish to test the viability of a very speci"c mechanism, in which point defects lead to phonon softening, and softer phonons then enhance the di!usion coe$cient and at the same time reduce the shear moduli. In the following, we report some recent results of molecular dynamics simulations testing this scenario [8]. The simulations were done on samples up to 2000 atoms in size held under constant stress, using the method developed by Parrinello and Rahman [9]. Since we are aiming at understanding melting in general, the simulations were done for copper, for which there exists an interatomic potential yielding accurate results for a large body of experimental properties [10]. The "rst question which we addressed regards the type of point defect which has a signi"cant e!ect on the shear moduli. Two types of defects were considered, the vacancy and the (1 0 0) split interstitial. These two were chosen for having the lowest formation energies. From the simulation, we obtained values of the elastic coe$cients associated with shear, C and C "(C !C )/2. In Fig. 3, we show the dependence of C on the density of point defects. A similar dependence is found for C . Quite surprisingly, vacancies do not a!ect these elastic coe$cients at all. In contrast, there is a pronounced decrease resulting from the split interstitials. The in#uence of the split interstitials on the elastic coe!"cients increases with temperature, up to the point where one of these coe!"cients vanishes, and the crystal melts. This mechanism, usually called mechanical melting, essentially generalizes the Born model for the case where the crystal contains defects. The phase diagram resulting
Fig. 4. In#uence of split interstitials on the phase diagram of Cu. The points are the results of a molecular dynamics simulation, while the dotted line separating the solid and the melt is a guide to the eye. The scatter results from the "nite time span of the simulation.
from these calculations is shown in Fig. 4. One can see that the temperature at which mechanical melting takes place decreases by about 60 K with 0.5% of split interstitials. In contrast, vacancies have no e!ect at all. Common wisdom has it that if anything, vacancies are the better candidate to induce melting, simply because their density is much larger than that of interstitials. However, if we accept the premise that the density of vacancies, typically 10\ of the atomic density even at ¹ , is not large enough to a!ect the crystal directly, one + should keep an open mind to the possibility that the density of defects is not the only relevant parameter, and there may be more than one type of defect involved. A model of melting based on split interstitials was proposed by Granato [14]. In order to understand the surprising `detachmenta of vacancies from the elastic properties of the crystal shown in Fig. 3, it is useful to view the vacancies as a dispersionless excitation mode of the crystal, at an energy which is much higher than that of any phonon. Consequently, this excitation branch does not hybridize with the elastic modes, namely phonons. On the other hand, the split interstitial, composed of two atoms, has internal degrees of freedom which are low enough in energy so that they do couple to the phonons in a resonant way (so called resonance modes) and hence a!ect the phonon dispersion relation directly [11}13]. Indeed, our simulations con"rm this point of view [8]. How does all this relate to solid He? We have recently developed a model [15] in which the inherently large anharmonic behaviour of the crystal is addressed by treating the solid as a perfectly harmonic solid which also supports `local modesa. Due to symmetry considerations, the two kinds of excitations, i.e. phonons and
E. Polturak et al. / Physica B 280 (2000) 142}145
local modes, interact rather weakly. In the BCC solid, the coupling between these two types of excitations leads to softening of one particular phonon branch, the T(1 1 0) phonon, while the other branches remain described by the self-consistent harmonic approximation. The resulting phonon spectrum is in good overall agreement with the neutron scattering data [16]. At the same time, coupling to the phonons raises the energy of the `local modesa to 14 K. The spatial extent of the 14 K mode is over two unit cells [15], hence it constitutes a point defect, which however resembles a split interstitial more than a vacancy because it involves two atoms. The value of the energy, 14 K, is very close to the experimental value of E determined for the BCC phase [1,2]. Thus, both " ingredients of our picture, a soft phonon mode and a point defect resembling a split interstitial can be modelled in the BCC solid phase of He. This mode may have been observed in neutron scattering experiments [16]; we plan to extend these experiments in the near future to test our model. In conclusion, we combined experimental data and molecular dynamics simulations into a detailed scenario of 3D melting. In our picture, there is a feedback between the phonons and the point defects which together bring about a reduction of the shear resistance of the crystal. We found, by looking at data obtained for BCC metals, that in the present context solid He is very similar to usual solids. We feel that this similarity may eventually present us with yet another case in which studying He contributes to understanding a rather general problem in solid state physics.
145
Acknowledgements This work was supported in part by the Israel Science Foundation and by the Technion VPR Fund for the Promotion of Research.
References [1] I. Berent, E. Polturak, Phys. Rev. Lett. 81 (1998) 846. [2] N.E. Dyumin, N.V. Zuev, S.V. Svatko, V.N. Grigor'ev, J. Low-Temp. Phys. 17 (1991) 468. [3] F.R.N. Nabarro, Theory of Crystal Dislocations, Oxford Univ. Press, New York, 1967. [4] I. Berent, E.P. Polturak, J. Low-Temp. Phys. 112 (1998) 337. [5] H.R. Schober, W. Petry, J. Tampenau, J. Phys.: Condens. Matter 4 (1992) 9321. [6] W. Petry et al., Phys. Rev. B 43 (1991) 10933. [7] R.A. Cowley, Adv. Phys. 29 (1980) 1. [8] A. Kanigel, Thesis, Technion, Israel, 1999. [9] M. Parrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182. [10] F. Cleri, V. Rosato, Phys. Rev. B 48 (1993) 22. [11] P.H. Dederichs, C. Lehmann, A. Scholz, Z. Phys. B 20 (1975) 155. [12] R.F. Wood, M. Mostoller, Phys. Rev. Lett. 35 (1975) 45. [13] H.R. Schober, V.K. Tevary, P.H. Dederichs, Z. Phys. B 21 (1975) 255. [14] A.V. Granato, Phys. Rev. Lett. 68 (1992) 974. [15] N. Gov, E. Polturak, Phys. Rev. B 60 (1999) 1019. [16] E.B. Osgood, V.N. Minkiewicz, T.A. Kitchens, G. Shirane, Phys. Rev. A 5 (1972) 1537.