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Local modes in LiMg and BeCu alloys
Volume
24A, number
PHYSICS
10
8 May 1967
LETTERS
6. J. Furdyna. Phys.Rev. Letters 16 (1966) 646. 7. R.A.Stradling and A.Anticliffe, Proc.Intern.Conf. on Physics of Semiconductors. Kyoto. 1966, p. 374. 8. S. Groves. private communication. 9. E. N.Adams and T. D. Holstein. J. Phys. Chem.Solids 10 (1959) 254.
10. L , Sniadower . V . I. Ivanov-Omsky and 2. Dziuba . Phys.Stat.Sol. 8 (1965j K 43. 11. W.Szumanska. L.Sniadoaer and W.Giriat, Phys. Stat.Sol. 10 (1965) K II. 12. W.Giriat, Brit.J.App1.Phy.s. 15 (1964) 151.
*****
LOCAL
MODES
IN Li-Mg
AND
Be-Cu
ALLOYS
I. NATKANIEC, K. PARLtiSKI, A. BAJOREK* and M. SUDNIK-I-IRYNKIEWICZ Neutron Physics Laboratory Joint Institute for Nuclear Research, Dubna t USSR and Institute of Nuclear Physics z Cracow, Poland )
Received
23 March
1967
Inelastic neutron scattering experiments with the alloys LiO,O5MgO,g5, LiO.IOMgO.gO, BeO.02CuO.98 and BeO_o5Cuo.95 and the pure metals, magnesium and copper, revealed vibrations of the impurity lithium atoms in the magnesium lattice of an energy of 35 f 2 meV and the beryllium atoms in the copper lattice of an energy of 42 * 2 meV. The energy of these local modes for these systems, calculated with the assumption that the force constants in the vicinity of the impurity atom remain unchanged, equals 43 meV for Li-Mg and about 58 meV for Be-Cu. respectively. It was estimated on the basis of the measured energies of local modes that the average change of the force_constants of the impurity atom is about 45 percent for Li-Mg and 56 percent for Be-Cu.
In crystal, containing impurity atoms of mass m’, smaller than the mass m of the host matrix atoms, there may appear additional modes with frequencies higher than the maximum frequency of the vibrations of the pure crystal, The frequencies of these modes are determined by the difference of the masses, the force constants around the impurity atom, and certain properties of the host matrix. The problem of the change of the force constants around impurity atoms has not yet been solved conclusively [l]. The frequencies of local modes are found by solving the equation ]&02)6X
-
iI =0
(1)
where e(w2) is the Green function for the host crystal, 6E is the perturbation matrix associated with the presence of the impurity atom. According to eq. (1) with an asumption that, the force constants near the impurity atom remain unchanged and the impurity atom is at a lattice site, the frequencies of the local modes are solutions of the equation * Laboratory University,
of Structure Research Cracow, Poland.
of the Jagiellonian
where E = 1 - m '/m , ga(wo) is the frequency spectrum of the pure host crystal, describing the vibrations along the crystal’s principal axis LY, (Y = a, b, c, Wm is the maximum frequency of the vibrations of host crystal. A comparison of the frequencies of the local modes calculated according to eq. (2) with the measured frequencies gives an idea of the behaviour of the force constants of the impurity atom. The method of inelastic scattering of neutrons make possible direct observations of the local modes in metal alloys. In such systems all conditions for the appearance of local modes can be realized. Observations of such vibrations, however, require that the components of the alloy have appropriate neutron cross-sections [ 2,3]. The Lb.05 Mgo.s5 and Lio.ioWo.go a.lloys, at the temperature of the experiments, are homogeneous alloys hexagonal structure, the same as the structure of magnesium [4]. The copper-beryllium alloy is homogeneous only when the beryllium concentration is low [5]. The Beo.o2Cuo.g3 and 517
Volume 24A. number
10
PHYSICS
LETTERS
BeO.O5CuO_g5samples, therefore, were heated to a temperature of 9OOoKand quenched. As a result of this treatment a homogeneous supersaturated solid solution, with a structure of the face-centred cubic lattice similar to the structure of pure copper? is obtained. The neutron diffraction pattern of the examined sample of BeO.O5CuO.g5alloy did not indicate the presence of other phases. Measurements of inelastic scattering of neutrons at an angle of 900 at a sample temperature of 1130K were performed with the time-of-flight spectrometer with the beryllium filter in front of the detector [6]. On the figs. 1 and 2 the energy spectra of inelastic scattered neutrons are compared. The results for the Li-Mg alloys are corrected for lithium absorption. The peaks of the energies E = (35 * 2) meV (fig. 1) and E = (42 * 2) meV (fig. 2) were assigned to the respective local modes of the lithium atoms in the magnesium lattice and the beryllium atoms in the copper lattice. The experimentally determined energy of the local modes is smaller than that calculated from eq. (2) (cf. table 1). This
NEUTRON TRMFER I
20 ,
I
40
ENERGY al , 1
- Idl ,
energy difference can be ascribed to the change of the force constants around the impurity atom. No influence of the concentration of impurity atoms, more than experimental inaccuracy, on the frequency of the peak was observed, even with high concentration of the impurity (10 percent). The change of the force constants of the impurity atom was estimated in the approximation of a very light impurity atom (the frequency of the local mode is high as compared with the mean frequency of the vibrations of the matrix atoms). It was also assumed that the presence of the impurity atom alters the force constants of the interaction between it and the remaining atoms of the. matrix, but does not alter those of the interaction between the atoms of the matrix. With these assumptions the nondiagonal elements of the matrix c(w2;6 I? tend towards zero and from eq. (1) we are able to obtain the following inte20
Table Energy of local Experimental value
518
Li - Mg E = 0.715
35 f 2 meV
Be - Cu E = 0.858
42 + 2 meV
NEUTRON~RANSFER
ENERGY 60
hdl
I
Fig. 2. The energy spectra of inelastic scattered neutrons of Be0 02Cu0 98 and Be0 05CuO.95 alloys divided by the neutron inelastic scatteied spectrum of pure copper.
Fig. 1. The energy spectra of inelastic scattered neutrons of LiO.05MgO.95 and LiO.lOMgO.gO alloys corrected for lithium absorption divided by the neutron inelastic scattered spectrum of pure magnesium.
Alloy
8 May 1967
1 Change of forces constant
modes Calculated from eq. (21
P
&wo)
kY(001 =1-g&@+
43.1 meV
0.45
P.K.Iyengar
54.0 56.1 57.9 61.5
0.43 0.52 0.56 0.60
E . H . Jacobsen [ 8). R.B.Leighton [9]. S.K.Sinha [lo]. Debay spectrum
meV meV meV meV
et al. [7].
Volume 24A. number
10
gral equation determining the frequencies modes
PHYSICS
of local
where
LETTERS
8 May 1967
The value of the parameter pa, calculated by means of eq. (3) for the measured frequencies of the iocal modes and the-published frequency spectra [7-lo]. are given in table 1. The most probable value .of the parameter pa is 45 percent and 56 percent for the Li impurity in Mg and Be impurity in Cu, respectively. In both cases the frequency spectra, used in calculations, we_re obtained by fitting the dispersion curves of the crystal to the experimental points. The authors thank Prof. F. L. Shapiro, Prof. J. A. Janik and Prof. B. Buras for their valuable comments, Prof. P. K. Iyengar for providing us the frequency spectra of magnesium, Dr. A. Galanty and Mr. S. Bednarski for preparing the alloy samples, and Mr. W. Olejarczyk for assistance during measurements.
From the conditions of invariance of the force constants with respect to macroscopic displacements of the crystal it follows that the change of the force constants of the zero coordination sphere is equal to (taken with the opposite sign) the sum of changes of all the remaining force constants of the impurity atom
References where &gA(Ol), Cp,,(OZ) are the force constants
1, A.A.Maradudin.
of the host matrix atom and impurity atom, respectively. The considered atom is found at the lattice site I = 0. The parameter pN defines the magnitude of the change of the force constants of the impurity atom along the direction of the (Y crystal’s principal axis. For the magnesium alloys (a = b # c) the frequency spectra g&wo) and g,(wO) are different. The energy of local modes calculated from eq. (2) for these two frequency spectra is equal 43.1 meV and 41.6 meV, respectively. The resolution of the spectrometer did not allow to separate of the peaks corresponding to these energies. In this case the parameter PO! was estimated using the full frequency spectrum of magnesium [7]. For Be-Cu alloys Pa = pb = PC.
2. 3. 4. 5.
6.
7.
8. 9. 10.
Solid State Physics, Vol. 18. (Academic Press, New York and London, 1966). Yu. Kagan and Ya.Iosilevekii, Zh. Eksp. i Teor. Fiz. 44 (1963) 1375. K. Parlitiski. Postepy Fizyki 16 (1965) 667. F.H.Herbstein and B. L.Averbach, A&a Mel. 4 (1956) 407. M.Hansen and K.Anderko. Constitution of binaq alloys. Vol. II (McGraa-Hill Book Comoanv. . .. . Nea York. Toronto, London, 1958). A. Bajorek, T. A. Machekhina. K. Parliriski and F. L.Shapiro, Inelastic scattering of neutrons. Vol. II (International Atomic Energy Agency. Vienna, 1965) p. 519. P.K.Iyengar. G. Venkataraman, P.R.Vijayaraghavan and A.P.Roy. Inelastic scattering of neutrons Vol. I (International Atomic Energy Agency, Vienna, 1965) p. 153. E.H.Jacobsen, Phys.Rev. 97 (1955) 654. R.B. Leighton, Rev.Mcd. Phys. 20 (1948) 165. S.K.Sinha, Phys.Rev. 143 (1966) 422.
*****
519
24A, number
PHYSICS
10
8 May 1967
LETTERS
6. J. Furdyna. Phys.Rev. Letters 16 (1966) 646. 7. R.A.Stradling and A.Anticliffe, Proc.Intern.Conf. on Physics of Semiconductors. Kyoto. 1966, p. 374. 8. S. Groves. private communication. 9. E. N.Adams and T. D. Holstein. J. Phys. Chem.Solids 10 (1959) 254.
10. L , Sniadower . V . I. Ivanov-Omsky and 2. Dziuba . Phys.Stat.Sol. 8 (1965j K 43. 11. W.Szumanska. L.Sniadoaer and W.Giriat, Phys. Stat.Sol. 10 (1965) K II. 12. W.Giriat, Brit.J.App1.Phy.s. 15 (1964) 151.
*****
LOCAL
MODES
IN Li-Mg
AND
Be-Cu
ALLOYS
I. NATKANIEC, K. PARLtiSKI, A. BAJOREK* and M. SUDNIK-I-IRYNKIEWICZ Neutron Physics Laboratory Joint Institute for Nuclear Research, Dubna t USSR and Institute of Nuclear Physics z Cracow, Poland )
Received
23 March
1967
Inelastic neutron scattering experiments with the alloys LiO,O5MgO,g5, LiO.IOMgO.gO, BeO.02CuO.98 and BeO_o5Cuo.95 and the pure metals, magnesium and copper, revealed vibrations of the impurity lithium atoms in the magnesium lattice of an energy of 35 f 2 meV and the beryllium atoms in the copper lattice of an energy of 42 * 2 meV. The energy of these local modes for these systems, calculated with the assumption that the force constants in the vicinity of the impurity atom remain unchanged, equals 43 meV for Li-Mg and about 58 meV for Be-Cu. respectively. It was estimated on the basis of the measured energies of local modes that the average change of the force_constants of the impurity atom is about 45 percent for Li-Mg and 56 percent for Be-Cu.
In crystal, containing impurity atoms of mass m’, smaller than the mass m of the host matrix atoms, there may appear additional modes with frequencies higher than the maximum frequency of the vibrations of the pure crystal, The frequencies of these modes are determined by the difference of the masses, the force constants around the impurity atom, and certain properties of the host matrix. The problem of the change of the force constants around impurity atoms has not yet been solved conclusively [l]. The frequencies of local modes are found by solving the equation ]&02)6X
-
iI =0
(1)
where e(w2) is the Green function for the host crystal, 6E is the perturbation matrix associated with the presence of the impurity atom. According to eq. (1) with an asumption that, the force constants near the impurity atom remain unchanged and the impurity atom is at a lattice site, the frequencies of the local modes are solutions of the equation * Laboratory University,
of Structure Research Cracow, Poland.
of the Jagiellonian
where E = 1 - m '/m , ga(wo) is the frequency spectrum of the pure host crystal, describing the vibrations along the crystal’s principal axis LY, (Y = a, b, c, Wm is the maximum frequency of the vibrations of host crystal. A comparison of the frequencies of the local modes calculated according to eq. (2) with the measured frequencies gives an idea of the behaviour of the force constants of the impurity atom. The method of inelastic scattering of neutrons make possible direct observations of the local modes in metal alloys. In such systems all conditions for the appearance of local modes can be realized. Observations of such vibrations, however, require that the components of the alloy have appropriate neutron cross-sections [ 2,3]. The Lb.05 Mgo.s5 and Lio.ioWo.go a.lloys, at the temperature of the experiments, are homogeneous alloys hexagonal structure, the same as the structure of magnesium [4]. The copper-beryllium alloy is homogeneous only when the beryllium concentration is low [5]. The Beo.o2Cuo.g3 and 517
Volume 24A. number
10
PHYSICS
LETTERS
BeO.O5CuO_g5samples, therefore, were heated to a temperature of 9OOoKand quenched. As a result of this treatment a homogeneous supersaturated solid solution, with a structure of the face-centred cubic lattice similar to the structure of pure copper? is obtained. The neutron diffraction pattern of the examined sample of BeO.O5CuO.g5alloy did not indicate the presence of other phases. Measurements of inelastic scattering of neutrons at an angle of 900 at a sample temperature of 1130K were performed with the time-of-flight spectrometer with the beryllium filter in front of the detector [6]. On the figs. 1 and 2 the energy spectra of inelastic scattered neutrons are compared. The results for the Li-Mg alloys are corrected for lithium absorption. The peaks of the energies E = (35 * 2) meV (fig. 1) and E = (42 * 2) meV (fig. 2) were assigned to the respective local modes of the lithium atoms in the magnesium lattice and the beryllium atoms in the copper lattice. The experimentally determined energy of the local modes is smaller than that calculated from eq. (2) (cf. table 1). This
NEUTRON TRMFER I
20 ,
I
40
ENERGY al , 1
- Idl ,
energy difference can be ascribed to the change of the force constants around the impurity atom. No influence of the concentration of impurity atoms, more than experimental inaccuracy, on the frequency of the peak was observed, even with high concentration of the impurity (10 percent). The change of the force constants of the impurity atom was estimated in the approximation of a very light impurity atom (the frequency of the local mode is high as compared with the mean frequency of the vibrations of the matrix atoms). It was also assumed that the presence of the impurity atom alters the force constants of the interaction between it and the remaining atoms of the. matrix, but does not alter those of the interaction between the atoms of the matrix. With these assumptions the nondiagonal elements of the matrix c(w2;6 I? tend towards zero and from eq. (1) we are able to obtain the following inte20
Table Energy of local Experimental value
518
Li - Mg E = 0.715
35 f 2 meV
Be - Cu E = 0.858
42 + 2 meV
NEUTRON~RANSFER
ENERGY 60
hdl
I
Fig. 2. The energy spectra of inelastic scattered neutrons of Be0 02Cu0 98 and Be0 05CuO.95 alloys divided by the neutron inelastic scatteied spectrum of pure copper.
Fig. 1. The energy spectra of inelastic scattered neutrons of LiO.05MgO.95 and LiO.lOMgO.gO alloys corrected for lithium absorption divided by the neutron inelastic scattered spectrum of pure magnesium.
Alloy
8 May 1967
1 Change of forces constant
modes Calculated from eq. (21
P
&wo)
kY(001 =1-g&@+
43.1 meV
0.45
P.K.Iyengar
54.0 56.1 57.9 61.5
0.43 0.52 0.56 0.60
E . H . Jacobsen [ 8). R.B.Leighton [9]. S.K.Sinha [lo]. Debay spectrum
meV meV meV meV
et al. [7].
Volume 24A. number
10
gral equation determining the frequencies modes
PHYSICS
of local
where
LETTERS
8 May 1967
The value of the parameter pa, calculated by means of eq. (3) for the measured frequencies of the iocal modes and the-published frequency spectra [7-lo]. are given in table 1. The most probable value .of the parameter pa is 45 percent and 56 percent for the Li impurity in Mg and Be impurity in Cu, respectively. In both cases the frequency spectra, used in calculations, we_re obtained by fitting the dispersion curves of the crystal to the experimental points. The authors thank Prof. F. L. Shapiro, Prof. J. A. Janik and Prof. B. Buras for their valuable comments, Prof. P. K. Iyengar for providing us the frequency spectra of magnesium, Dr. A. Galanty and Mr. S. Bednarski for preparing the alloy samples, and Mr. W. Olejarczyk for assistance during measurements.
From the conditions of invariance of the force constants with respect to macroscopic displacements of the crystal it follows that the change of the force constants of the zero coordination sphere is equal to (taken with the opposite sign) the sum of changes of all the remaining force constants of the impurity atom
References where &gA(Ol), Cp,,(OZ) are the force constants
1, A.A.Maradudin.
of the host matrix atom and impurity atom, respectively. The considered atom is found at the lattice site I = 0. The parameter pN defines the magnitude of the change of the force constants of the impurity atom along the direction of the (Y crystal’s principal axis. For the magnesium alloys (a = b # c) the frequency spectra g&wo) and g,(wO) are different. The energy of local modes calculated from eq. (2) for these two frequency spectra is equal 43.1 meV and 41.6 meV, respectively. The resolution of the spectrometer did not allow to separate of the peaks corresponding to these energies. In this case the parameter PO! was estimated using the full frequency spectrum of magnesium [7]. For Be-Cu alloys Pa = pb = PC.
2. 3. 4. 5.
6.
7.
8. 9. 10.
Solid State Physics, Vol. 18. (Academic Press, New York and London, 1966). Yu. Kagan and Ya.Iosilevekii, Zh. Eksp. i Teor. Fiz. 44 (1963) 1375. K. Parlitiski. Postepy Fizyki 16 (1965) 667. F.H.Herbstein and B. L.Averbach, A&a Mel. 4 (1956) 407. M.Hansen and K.Anderko. Constitution of binaq alloys. Vol. II (McGraa-Hill Book Comoanv. . .. . Nea York. Toronto, London, 1958). A. Bajorek, T. A. Machekhina. K. Parliriski and F. L.Shapiro, Inelastic scattering of neutrons. Vol. II (International Atomic Energy Agency. Vienna, 1965) p. 519. P.K.Iyengar. G. Venkataraman, P.R.Vijayaraghavan and A.P.Roy. Inelastic scattering of neutrons Vol. I (International Atomic Energy Agency, Vienna, 1965) p. 153. E.H.Jacobsen, Phys.Rev. 97 (1955) 654. R.B. Leighton, Rev.Mcd. Phys. 20 (1948) 165. S.K.Sinha, Phys.Rev. 143 (1966) 422.
*****
519