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Slow-neutron scattering study of oxygen and nitrogen interstitial atoms vibrations in vanadium
Physica B 174 (1991) 241-245 North-Holland
Slow-neutron scattering study of oxygen and nitrogen interstitial atoms vibrations in vanadium S . A . D a n i l k i n a, V.P. M i n a e v a a n d V.V. S u m i n b "Institute of Physics and Power Engineering, Obninsk 249020, USSR blnstitute of Physical Chemistry, Obninsk 249020, USSR
The lattice dynamics of the interstitial solid solutions VO x and VN, has been studied. The deformation of the vanadium atom frequency distribution caused by interstitials is observed. The local vibration frequencies of O and N atoms are reported. A computer simulation of the frequency spectra of these compounds were performed.
1. Introduction The local vibrations of the oxygen and nitrogen interstitials in body-centred cubic (BCC) transition metals have frequencies above the boundary of the metal atom spectrum [1-3]. In this case the positions of the local frequencies yield information about the metal-interstitial potential, and the metal-metal interaction can be derived from the acoustic part of the spectrum. This paper reports results of the INS study of V O x and VNx solid solutions with different interstitial atom concentrations. As compared with refs. [1, 2] a frequency distributions of the alloy components is obtained in this paper. In the VN~ alloy we use a different method of sample preparation.
2. Experiment The INS experiment has been carried out on a VO~ (x = 0.03, 0.06, 0.11) and VN0.04 solid solutions. The VO0.~l alloy was prepared from V and V20 5 mixed in an appropriate ratio. This mixture was melted a few times in an electron-beam furnace in an argon gas atmosphere. The samples with x = 0.03 and x = 0.06 were melted from this VOo.i1 alloy with an additional amount of Elsevier Science Publishers B.V. (North-Holland)
vanadium. After melting the samples were annealed in a high-vacuum furnace at 1000°C. The VOo.06 sample was then quenched in liquid nitrogen. The VN004 sample was prepared from a 0.5 mm thick vanadium foil by the addition of nitrogen from the gas phase at 900°C. The samples were then annealed at the same temperature and quenched. X-ray and neutron diffractometry were carried out in order to control a single-phase structure. The content of interstitial atoms was determined from a lattice parameter concentration dependence. The INS experiment was performed with a time-of-flight "direct" geometry DIN-2 spectrometer [4] at the IBR-2 reactor, Dubna. The incident energy was chosen as 10 meV. The measurements were performed with a neutron energy gain. Seven detectors were used to detect the scattered neutrons with scattering angles of 76, 81, 86, 90, 114, 119 and 123°. The scattering data yield a 0(e) function [5]: O(e) = ~2 o', f~ exp( s
m
2Ws)g,(e),
s
where g,(e) is the frequency spectrum of the atoms s and f,, o-s and m, are the fraction of the atoms s, the cross-section and the mass, respectively. W, is the D e b y e - W a l l e r factor.
242
S . A . Danilkin et al. / Vibrations o f 0 and N insterstitial atoms in V
Owing to the mass difference and the strong metal-interstitial interaction the fraction of the interstitial atom vibrations in the acoustic part of the spectrum of the et-VOx alloy (e ~< eB, where e B is the boundary of the metal atom spectrum) is less than 5% of all frequencies [6]. As well as at the energies e ~> e B the main contribution in the spectrum arises from the light-atom vibrations. Therefore we consider the function 0(e) for energies e ~ e B as gMe(e) and as gint(e) for e ~> e a. More details of the treatment procedure have been described elsewhere [3].
3. Results and discussion
The frequency spectra of the vanadium atom vibrations in VO x (x = 0.03, 0.06, 0.11) and the vanadium metal spectrum are plotted in fig. 1. With increasing oxygen content the phonon spectrum gv(e) becomes harder: e B and the lowenergy maximum (19 meV) of the spectrum shifts to higher values. In the VO0.11-tetragonal phase spectrum, in addition to the two peaks, a third peak is obtained at 34 meV. For x = 0.03 and x = 0.06 a "shoulder" is seen in this energy range. Similar deformation of the vanadium atom spectrum have been found in ot-VN~ solid solutions (x = 0.04, 0.05) [2], The moments of the frequency
Table 1 Moments (e 2) of the vanadium atom frequency spectrum in
VO~ and VN~ solid solutions• VO x
VNx
V
0.03
0.06
0.11
0.04
0.05
Experimental
475
536
569
633
530
534
(s:) (meV2) Calculated
468
488
513
550.)
492
510
(2) (meV2)
") Calculations have been made for the BCC structure.
spectra (e 2) a r e listed in table 1. It is seen that ( e 2) increases with increasing oxygen content. The frequency spectrum of the oxygen atoms in vanadium is shown in fig. 2 for VO0.06. The spectrum contains two peaks corresponding to interstitial atom vibrations (57 and 86 meV). The degeneracy of the vibrations is related to the point symmetry of the oxygen octahedral interstitial position in the BCC structure. The excitation with higher frequency (w3) has a polarization vector along the tetragonality axis. The twofold degenerate modes oJ1,2 polarize in the perpendicular plane. Similar spectra were obtained for all the VO x and VN x solid solutions and the frequencies of the local modes are listed in table 2. The low-energy peak (44meV) in fig. 2 as shown by the computer-simulated results corresponds the closest to oxygen-vanadium atom vibrations. It is seen (table 2) that the w3 values decrease
6O q0 u,0
J
20
0
"
20 1o
I0
, 20
°I
0 30
,
HO Fig. 1. Vanadium atom frequency spectra in VO x solid solutions: O, vanadium, x, VO003; ©, VO0.06; A, VO0 , .
.
"" .I. "~
&,M~
60
8
I0
IZO ~,,Mab
Fig. 2. Frequency spectrum of oxygen atoms in VOo.o6.
S . A . Danilkin et al. / Vibrations o f 0 and N insterstitial atoms in V
243
Table 2 Frequencies of the oxygen and nitrogen atom local vibrations in V O , and VN x (meV), Experiment
Model calculations
Phase
to~,z
to3
a-VO0.03 a-VO0 06 et'-VO0 . 13'-VO0 2
55.1 -+ 0.9 56.7 -+ 0.9 57.9 - 1.1 58.4 +--2.0 "~ 56.9 -+ 0.8 b) 70.0 -+ 1.4 80 -+ 1
91 86 84 79 80 97 -
~x-VN0.04 13-V3N [7]
%,2
t°3
70 (84.5) c) 74.6 (86.5)
97.4 (97.4) -
- 2 -+ 2 --+7 -+ 2
") Incident energy of neutrons is E 0 = 8.2 meV [1]. b~ E0 = 33.5 meV. c~ Values in brackets correspond to the V - N potential of J o h n s o n et al. [8].
with increasing oxygen content, at same time the value of to1.z has a weak tendency to increase. The decrease in to3 correlates with the concentration dependence of the lattice parameter (a in the ot and et' phases and c in the 13' phase): it changes from 3.03/~ and 3.06,~, in the BCC VO0.03 and VOo.o6 to 3.11 A and 3.32 A, respectively, in BCT ot'-VO0.11 and 13'-VO0.2. This increase of the lattice parameter probably results from a softening of the metal-interstitial interaction. To calculate the deformation of the frequency spectrum caused by the interstitials and the local vibrations we used two different models. For a description of the lattice distortions near the interstitials and the local vibrations of the interstitial atoms a computer simulation method was used [8, 9]. A short-range pairwise interatomic potential was commonly used in this method. However, as was shown for hydrogen in refractory metals the local modes are determined by the short-range part of this interaction [10]. Because it is difficult in the computer simulation method to take into consideration the perturbation of metal-metal interaction by interstitials, we used a virtual-crystal approximation to calculate the deformation of the metal atom Spectrum. Following the paper [11] we assume that oxygen atoms in VO x occupy each octahedral site and have mass x m o and force constant x k . This is qualitatively the same as adding an additional interaction between the metal atoms
[12]. The calculated spectra for VO003 and VO0.06 are shown in fig. 3, the (e 2) values are listed in table 1. As seen in the figures the deformation of the vanadium atom spectrum corresponds qualitatively to the experimental results. However, the calculated ( e ) values grow slower with increasing oxygen content than do the experimental results. Nevertheless, this result shows that the deformation of the spectrum is caused by a collective process that changes the metal-metal interaction throughout the entire volume of the metal. This can be connected with the variation of the electron properties of the metals leading to the strengthening of the metalmetal interaction. It should be noted that there is a correlation between the density of the electron
80 60 40 20 O0
lu
20
30
0
fO
20
50 ~,,,,,~,
Fig. 3. Frequency spectra of vanadium atoms in VOo.o3 (left, shaded) and VOo.o6 (right, shaded). For comparison the v a n a d i u m metal s p e c t r u m is shown.
244
S.A. Danilkin et al. / Vibrations of 0 and N insterstitial atoms in V
Table 3 Nitrogen-vanadium and oxygen-vanadium interatomic potentials ~(R) (eV) = A3 R3 4A2 R2 + A I R + A o, (R in angstroms). Range(A)
m 3
A2
A1
A0
Nitrogen-vanadium potential ~<1.95 -5.968 1.95-2.007 - 14.4 2.007- 2.733 - 3.565
41.269 90.596 25.347
-93.454 - 189.642 - 58.669
68.897 131.420 43.786
Oxygen-vanadium potential ~<1.95 -5.968 1.95-2.015 -19.0 2.015-2.896 - 1.999
41.269 117.506 14.727
-93.454 -242.117 -34.999
68.897 165.528 26.402
states at the Fermi level N~ and the value of ( e 2) of the metal atom spectrum. As was shown in refs. [13, 14] the increase of N F corresponds to a decrease of the value of ( e 2) in the substitutional alloys of transitions metals. The addition of oxygen and nitrogen to vanadium reduces the electronic specific heat and the magnetic susceptibility, which are determined essentially by the total density of states at e F. This is in the case of the V O x and VN x solid solution (and in ot-NbO x and ot-TaN~ [2, 3]) we see the same correlation between N F and ( e 2) as was established in ref. [14]. In the computer simulation of the lattice dynamics of the interstitial compounds one of the main problems is the choice of the appropriate interatomic potential. There are only a few potentials for the metal-interstitial interaction. We used the V - N potential of Johnson et al. [8]. At first, using the variation method, we determined the positions of the metal atoms in a BCC vanadium crystallite containing 637 atoms and a nitrogen atom. Then for 225 inner atoms a frequency spectrum was calculated by B o r n Karman method. In this approximation the deformation of the vanadium atom spectrum reduced on the formation of the frequency levels in the energy region 34-41 meV (with increasing interstitial atom content a band is formed in this energy interval for both nitrogen and oxygen). In experiments the additional peaks were observed at 45-50 meV for nitrogen and 45 meV for oxygen (fig. 2). As was mentioned earlier this excitation corresponds to vibrations of the vanadium
atoms that are neighbours of the interstitial atom. The nitrogen atom local-mode frequencies were obtained as 84 and 97 meV for the V - N potential from [8] (table 2). The agreement for to1,2 is not good enough, therefore this potential was corrected to reproduce the experimental values of %,2 and to3 (table 3). With these two potentials the nitrogen atom frequencies in H C P [3oV3N were calculated. It is seen (table 2) that the deviation from the experimental value is about 10% in both cases. This means that the Johnson metal-interstitial potential gives good estimates of the local-mode frequencies but some corrections must be made to reproduce the exact frequency values. The V - N potential (table 3) is not appropriate in the case of the O interstitial in vanadium. Only the o.)3 frequency calculated with this potential agrees with the experimental value. Therefore we corrected this potential to reproduce the experimental frequencies of oxygen in vanadium. The results are listed in table 3. As in the V - N potential the short-range part of this potential corresponds to the data of Johnson et al.
References
[1] V.V. Sumin, S.A. Danilkin and S.I. Morozov, Soviet Phys. Solid State 20 (1978) 1001. [2] S.A. Danilkin, M.G. Zemlyanov et al., Fiz. Tverd. Tela 29 (1987) 2112. [3] S.A. Danilkin, M.G. Zemlyanov et al., Fiz. Tverd. Tela 31 (1989) 8.
S.A. Danilkin et al. / Vibrations of 0 and N insterstitial atoms in V
[4] A.V. Abramov, N.M. Blagoveschenski et al., Atomnaya Energiya 66 (1989) 316. [5] M.G. Zemlyanov, E.G. Brovman et al., in: Inelastic Scattering of Neutrons, Proc. Symp. Bombay, India Vol. 2 (IAEA, Vienna, 1965) p. 431. [6] S.A. Danilkin and T.V. Alenicheva, Preprint PhEI-1718 (Obninsk, 1985). [7] S.I. Morozov and V.V. Sumin, Preprint PhEI-1922 (Obninsk, 1988). [8] R.A. Johnson, G.J. Dienes and A.C. Damask, Acta Metall. 12 (1964) 1215.
245
[9] V.P. Zhukov and V.G. Kapinos, Fiz. Tverd. Tela 19 (1977) 3126. [10] Y. Fukai, Japan. J. Appl. Phys. 22 (1983) 207. [11] H. Rafizadeh, Phys. Rev. B 23 (1981) 1628. [12] W.A. Kamitakahara and D. Khatamian et al., Phys. Rev. B 21 (1980) 4500. [13] I.R. Commersal and B.L. Gyorfi, Phys. Rev. Lett. B 3 (1974) 1286. [14] G.F. Syrykh, M.G. Zemlyanov et al., Zh. Eksp. Teor. Fiz. 81 (1981) 308.
Slow-neutron scattering study of oxygen and nitrogen interstitial atoms vibrations in vanadium S . A . D a n i l k i n a, V.P. M i n a e v a a n d V.V. S u m i n b "Institute of Physics and Power Engineering, Obninsk 249020, USSR blnstitute of Physical Chemistry, Obninsk 249020, USSR
The lattice dynamics of the interstitial solid solutions VO x and VN, has been studied. The deformation of the vanadium atom frequency distribution caused by interstitials is observed. The local vibration frequencies of O and N atoms are reported. A computer simulation of the frequency spectra of these compounds were performed.
1. Introduction The local vibrations of the oxygen and nitrogen interstitials in body-centred cubic (BCC) transition metals have frequencies above the boundary of the metal atom spectrum [1-3]. In this case the positions of the local frequencies yield information about the metal-interstitial potential, and the metal-metal interaction can be derived from the acoustic part of the spectrum. This paper reports results of the INS study of V O x and VNx solid solutions with different interstitial atom concentrations. As compared with refs. [1, 2] a frequency distributions of the alloy components is obtained in this paper. In the VN~ alloy we use a different method of sample preparation.
2. Experiment The INS experiment has been carried out on a VO~ (x = 0.03, 0.06, 0.11) and VN0.04 solid solutions. The VO0.~l alloy was prepared from V and V20 5 mixed in an appropriate ratio. This mixture was melted a few times in an electron-beam furnace in an argon gas atmosphere. The samples with x = 0.03 and x = 0.06 were melted from this VOo.i1 alloy with an additional amount of Elsevier Science Publishers B.V. (North-Holland)
vanadium. After melting the samples were annealed in a high-vacuum furnace at 1000°C. The VOo.06 sample was then quenched in liquid nitrogen. The VN004 sample was prepared from a 0.5 mm thick vanadium foil by the addition of nitrogen from the gas phase at 900°C. The samples were then annealed at the same temperature and quenched. X-ray and neutron diffractometry were carried out in order to control a single-phase structure. The content of interstitial atoms was determined from a lattice parameter concentration dependence. The INS experiment was performed with a time-of-flight "direct" geometry DIN-2 spectrometer [4] at the IBR-2 reactor, Dubna. The incident energy was chosen as 10 meV. The measurements were performed with a neutron energy gain. Seven detectors were used to detect the scattered neutrons with scattering angles of 76, 81, 86, 90, 114, 119 and 123°. The scattering data yield a 0(e) function [5]: O(e) = ~2 o', f~ exp( s
m
2Ws)g,(e),
s
where g,(e) is the frequency spectrum of the atoms s and f,, o-s and m, are the fraction of the atoms s, the cross-section and the mass, respectively. W, is the D e b y e - W a l l e r factor.
242
S . A . Danilkin et al. / Vibrations o f 0 and N insterstitial atoms in V
Owing to the mass difference and the strong metal-interstitial interaction the fraction of the interstitial atom vibrations in the acoustic part of the spectrum of the et-VOx alloy (e ~< eB, where e B is the boundary of the metal atom spectrum) is less than 5% of all frequencies [6]. As well as at the energies e ~> e B the main contribution in the spectrum arises from the light-atom vibrations. Therefore we consider the function 0(e) for energies e ~ e B as gMe(e) and as gint(e) for e ~> e a. More details of the treatment procedure have been described elsewhere [3].
3. Results and discussion
The frequency spectra of the vanadium atom vibrations in VO x (x = 0.03, 0.06, 0.11) and the vanadium metal spectrum are plotted in fig. 1. With increasing oxygen content the phonon spectrum gv(e) becomes harder: e B and the lowenergy maximum (19 meV) of the spectrum shifts to higher values. In the VO0.11-tetragonal phase spectrum, in addition to the two peaks, a third peak is obtained at 34 meV. For x = 0.03 and x = 0.06 a "shoulder" is seen in this energy range. Similar deformation of the vanadium atom spectrum have been found in ot-VN~ solid solutions (x = 0.04, 0.05) [2], The moments of the frequency
Table 1 Moments (e 2) of the vanadium atom frequency spectrum in
VO~ and VN~ solid solutions• VO x
VNx
V
0.03
0.06
0.11
0.04
0.05
Experimental
475
536
569
633
530
534
(s:) (meV2) Calculated
468
488
513
550.)
492
510
(2) (meV2)
") Calculations have been made for the BCC structure.
spectra (e 2) a r e listed in table 1. It is seen that ( e 2) increases with increasing oxygen content. The frequency spectrum of the oxygen atoms in vanadium is shown in fig. 2 for VO0.06. The spectrum contains two peaks corresponding to interstitial atom vibrations (57 and 86 meV). The degeneracy of the vibrations is related to the point symmetry of the oxygen octahedral interstitial position in the BCC structure. The excitation with higher frequency (w3) has a polarization vector along the tetragonality axis. The twofold degenerate modes oJ1,2 polarize in the perpendicular plane. Similar spectra were obtained for all the VO x and VN x solid solutions and the frequencies of the local modes are listed in table 2. The low-energy peak (44meV) in fig. 2 as shown by the computer-simulated results corresponds the closest to oxygen-vanadium atom vibrations. It is seen (table 2) that the w3 values decrease
6O q0 u,0
J
20
0
"
20 1o
I0
, 20
°I
0 30
,
HO Fig. 1. Vanadium atom frequency spectra in VO x solid solutions: O, vanadium, x, VO003; ©, VO0.06; A, VO0 , .
.
"" .I. "~
&,M~
60
8
I0
IZO ~,,Mab
Fig. 2. Frequency spectrum of oxygen atoms in VOo.o6.
S . A . Danilkin et al. / Vibrations o f 0 and N insterstitial atoms in V
243
Table 2 Frequencies of the oxygen and nitrogen atom local vibrations in V O , and VN x (meV), Experiment
Model calculations
Phase
to~,z
to3
a-VO0.03 a-VO0 06 et'-VO0 . 13'-VO0 2
55.1 -+ 0.9 56.7 -+ 0.9 57.9 - 1.1 58.4 +--2.0 "~ 56.9 -+ 0.8 b) 70.0 -+ 1.4 80 -+ 1
91 86 84 79 80 97 -
~x-VN0.04 13-V3N [7]
%,2
t°3
70 (84.5) c) 74.6 (86.5)
97.4 (97.4) -
- 2 -+ 2 --+7 -+ 2
") Incident energy of neutrons is E 0 = 8.2 meV [1]. b~ E0 = 33.5 meV. c~ Values in brackets correspond to the V - N potential of J o h n s o n et al. [8].
with increasing oxygen content, at same time the value of to1.z has a weak tendency to increase. The decrease in to3 correlates with the concentration dependence of the lattice parameter (a in the ot and et' phases and c in the 13' phase): it changes from 3.03/~ and 3.06,~, in the BCC VO0.03 and VOo.o6 to 3.11 A and 3.32 A, respectively, in BCT ot'-VO0.11 and 13'-VO0.2. This increase of the lattice parameter probably results from a softening of the metal-interstitial interaction. To calculate the deformation of the frequency spectrum caused by the interstitials and the local vibrations we used two different models. For a description of the lattice distortions near the interstitials and the local vibrations of the interstitial atoms a computer simulation method was used [8, 9]. A short-range pairwise interatomic potential was commonly used in this method. However, as was shown for hydrogen in refractory metals the local modes are determined by the short-range part of this interaction [10]. Because it is difficult in the computer simulation method to take into consideration the perturbation of metal-metal interaction by interstitials, we used a virtual-crystal approximation to calculate the deformation of the metal atom Spectrum. Following the paper [11] we assume that oxygen atoms in VO x occupy each octahedral site and have mass x m o and force constant x k . This is qualitatively the same as adding an additional interaction between the metal atoms
[12]. The calculated spectra for VO003 and VO0.06 are shown in fig. 3, the (e 2) values are listed in table 1. As seen in the figures the deformation of the vanadium atom spectrum corresponds qualitatively to the experimental results. However, the calculated ( e ) values grow slower with increasing oxygen content than do the experimental results. Nevertheless, this result shows that the deformation of the spectrum is caused by a collective process that changes the metal-metal interaction throughout the entire volume of the metal. This can be connected with the variation of the electron properties of the metals leading to the strengthening of the metalmetal interaction. It should be noted that there is a correlation between the density of the electron
80 60 40 20 O0
lu
20
30
0
fO
20
50 ~,,,,,~,
Fig. 3. Frequency spectra of vanadium atoms in VOo.o3 (left, shaded) and VOo.o6 (right, shaded). For comparison the v a n a d i u m metal s p e c t r u m is shown.
244
S.A. Danilkin et al. / Vibrations of 0 and N insterstitial atoms in V
Table 3 Nitrogen-vanadium and oxygen-vanadium interatomic potentials ~(R) (eV) = A3 R3 4A2 R2 + A I R + A o, (R in angstroms). Range(A)
m 3
A2
A1
A0
Nitrogen-vanadium potential ~<1.95 -5.968 1.95-2.007 - 14.4 2.007- 2.733 - 3.565
41.269 90.596 25.347
-93.454 - 189.642 - 58.669
68.897 131.420 43.786
Oxygen-vanadium potential ~<1.95 -5.968 1.95-2.015 -19.0 2.015-2.896 - 1.999
41.269 117.506 14.727
-93.454 -242.117 -34.999
68.897 165.528 26.402
states at the Fermi level N~ and the value of ( e 2) of the metal atom spectrum. As was shown in refs. [13, 14] the increase of N F corresponds to a decrease of the value of ( e 2) in the substitutional alloys of transitions metals. The addition of oxygen and nitrogen to vanadium reduces the electronic specific heat and the magnetic susceptibility, which are determined essentially by the total density of states at e F. This is in the case of the V O x and VN x solid solution (and in ot-NbO x and ot-TaN~ [2, 3]) we see the same correlation between N F and ( e 2) as was established in ref. [14]. In the computer simulation of the lattice dynamics of the interstitial compounds one of the main problems is the choice of the appropriate interatomic potential. There are only a few potentials for the metal-interstitial interaction. We used the V - N potential of Johnson et al. [8]. At first, using the variation method, we determined the positions of the metal atoms in a BCC vanadium crystallite containing 637 atoms and a nitrogen atom. Then for 225 inner atoms a frequency spectrum was calculated by B o r n Karman method. In this approximation the deformation of the vanadium atom spectrum reduced on the formation of the frequency levels in the energy region 34-41 meV (with increasing interstitial atom content a band is formed in this energy interval for both nitrogen and oxygen). In experiments the additional peaks were observed at 45-50 meV for nitrogen and 45 meV for oxygen (fig. 2). As was mentioned earlier this excitation corresponds to vibrations of the vanadium
atoms that are neighbours of the interstitial atom. The nitrogen atom local-mode frequencies were obtained as 84 and 97 meV for the V - N potential from [8] (table 2). The agreement for to1,2 is not good enough, therefore this potential was corrected to reproduce the experimental values of %,2 and to3 (table 3). With these two potentials the nitrogen atom frequencies in H C P [3oV3N were calculated. It is seen (table 2) that the deviation from the experimental value is about 10% in both cases. This means that the Johnson metal-interstitial potential gives good estimates of the local-mode frequencies but some corrections must be made to reproduce the exact frequency values. The V - N potential (table 3) is not appropriate in the case of the O interstitial in vanadium. Only the o.)3 frequency calculated with this potential agrees with the experimental value. Therefore we corrected this potential to reproduce the experimental frequencies of oxygen in vanadium. The results are listed in table 3. As in the V - N potential the short-range part of this potential corresponds to the data of Johnson et al.
References
[1] V.V. Sumin, S.A. Danilkin and S.I. Morozov, Soviet Phys. Solid State 20 (1978) 1001. [2] S.A. Danilkin, M.G. Zemlyanov et al., Fiz. Tverd. Tela 29 (1987) 2112. [3] S.A. Danilkin, M.G. Zemlyanov et al., Fiz. Tverd. Tela 31 (1989) 8.
S.A. Danilkin et al. / Vibrations of 0 and N insterstitial atoms in V
[4] A.V. Abramov, N.M. Blagoveschenski et al., Atomnaya Energiya 66 (1989) 316. [5] M.G. Zemlyanov, E.G. Brovman et al., in: Inelastic Scattering of Neutrons, Proc. Symp. Bombay, India Vol. 2 (IAEA, Vienna, 1965) p. 431. [6] S.A. Danilkin and T.V. Alenicheva, Preprint PhEI-1718 (Obninsk, 1985). [7] S.I. Morozov and V.V. Sumin, Preprint PhEI-1922 (Obninsk, 1988). [8] R.A. Johnson, G.J. Dienes and A.C. Damask, Acta Metall. 12 (1964) 1215.
245
[9] V.P. Zhukov and V.G. Kapinos, Fiz. Tverd. Tela 19 (1977) 3126. [10] Y. Fukai, Japan. J. Appl. Phys. 22 (1983) 207. [11] H. Rafizadeh, Phys. Rev. B 23 (1981) 1628. [12] W.A. Kamitakahara and D. Khatamian et al., Phys. Rev. B 21 (1980) 4500. [13] I.R. Commersal and B.L. Gyorfi, Phys. Rev. Lett. B 3 (1974) 1286. [14] G.F. Syrykh, M.G. Zemlyanov et al., Zh. Eksp. Teor. Fiz. 81 (1981) 308.