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Resonant phonon scattering in colored sodium chloride crystals
Solid State Communications, Vol. 5, pp. 899-903, 1967. Pergamon Press Ltd. Printed in Great Britain
RESONANT PHONON SCATTERING IN COLORED SODIUM CHLORIDE CRYSTALS* R.E. Aldrich~, W.J. Burke~,and K.A. McCarthy Department of Physics, Tufts University, Medford, Massachusetts, U.S.A. (Received 7 July 1967 by B.N. Brockhouse) (Revised 30 August 1967) The thermal conductivity of both v-irradiated and additivelycolored sodium chloride crystals containing OW ions has shown two prominent resonances, one at a frequency of 1.6 cm1 and the other at 17 cm’. The 1.6 cm’ resonant frequency is explained by tunneling transitions calculated from a Devonshire model. A model is proposed for the 17 cm’ resonant frequency in which one of the chloride ions in the region of the OW ion is replaced by a negative ion vacancy.
THE RESONANT phonon scattering associated with the depression at 6°K in the thermal conductivity of as-received sodium chloride crystals is now believed to be due to an OW impurity3 the OW dipole substitutes for a Cl ion in the NaC1 lattice. It will be shown here that in both additively- colored and irradiated2 sodium chloride crystals the OH- resonance is enhanced by the coloration, and that the 17 cm~ level associated with the OW depression is similar to that found by Chau, Klein, and Wedding3 for KC1:OW .
The thermal conductivity of several irradiated Optovac sodium chloride crystals is shown in Fig. 1. The data have been analyzed4 using the Callaway model5 with a relaxation time, r’, of the form: -
The coefficients for the crystals given In Fig. 1 are listed in Table 1. The values for N(4. 7 x 10~’°),U(4. 04 x 10~)and F(34. 7) are obtained from fits to pure crystals6, and were used in fitting all crystals. The boundary scattering term, B, is the Casimir7 value; all crystals were sandblasted as the final step before measurement. The thermal conductivity of several additively-colored Optovac sodium chloride crystals is shown in Fig. 2; these crystals were cleaved from a different boule than the crystals used for v -irradiation. These data have also been fitted using a Callaway model; a different relaxation time, ~R3 , is used for the resonance terms and is of the form: A
r’
B+Pw4
A 2 w 2 1 n2)2 —
+
NT2w
+
UT2w2 exp(-F/T) A 2 u?
+~
(~ 2— ~, )2 + 2 2
c
2 (1)
(cu 2~ *Supported in part by the National Science Foundation (Grant GP-4322). ~Now at Itek Corporation, Lexington, Massachusetts. Present address Department of Physics, University of Illinois, Urbana, illinois. 899 +
2w2 A 2w2 3T + 2T (w 2-w2)2+C~n~2 (u~-w2)2+C~w 2 2 2
2
+
+
A
1 f (~, w1 , T) g(w ,w1)
(2)
0 where the third term Is that obtained by Wagner for a quasi-localized mode. TheadditivelyT2 term in the resonant relaxation time orthe colored crystals corresponds to a narrowing of the resonance; the resonance is asymmetric and
900
RESONANT PHONON SCATTERING 10
here do not extend to sufficiently low temperatures to permit observation of the recovery of the thermal conductivity to the Casimir value. However, the machine fit to the available data yields a value for the upper limit of w2 equal to
‘1
5,~
~oo~ 4~
~6~
~ 2.0 p 0’ 1.0
-
/
02 .
~f o8,~f
o.t
-
9’K
~‘ ..!‘~‘
~
~
~
KEY
~
A B
,~‘
/
.05
The resonance at 6°K is not predicted by the and is believed to haveby the Devonshire same originmodel, as the resonances presented Chau et al. The insensitivity of the contribution of th&i~ionanceterms thenot half-widths suggests that the two resonances to may be associated with the same type center. Shore et al.’3 have
./‘~‘
~
~ //
:~ ~
~ ~
0.5
~
1.6 cm’, which in good agreement with a of Devonshire modelisusing a barrier parameter 725Wedding cm~, as of anddetermined Klein. ‘~ from the infrared data
4~’\
~V
~
~ U
~‘ ~
~V’~
Vol. 5, No. 11
°
-
A
~‘
E
• 02’ 10
20
TEMPERATURE
.
FIG. 1 Thermal conductivity of v -irradiated Optovac socium chloride crystals. Curve A, an as-received crystal; Curve B, an as-received annealed at 600°C; Curve C, acrystal v irradiated crystal containing 3. 8 x 10’s F centers; Curve D, a v-irradiated crystal containing 5. 6 x 1017 F centers; Curve E, a v-irradiated crystal containing 7.6 x io~’F centers. The dotted lines are the machine fits. The color-center concentrations are determined from op.tical measurements. The concentration of OW is estimated to be —~3ppm. results in a lowering of the frequency,
analyzed the properties of an OW the alkali halides in the presence of anTii~rityin electric field, and have shown that the energy of the groundstate singlet (A 11) is greatly reduced with the tric field applied along each of the three principal crystallographic directions. In their models the impurityfield is assumed to be in an octahedral crystalline with potential minima along the six [100J directions, as suggested in the experiments of Luty.’4 It may be assumed that the tunneling thatand takes is at 90° to original orientation, thatplace the probability of the tunneling by 180° is small. To explain the w, resonance, we propose that one of the next-nearest-neighbor C1 ions in the region of some of the OW ions is replaced by a negative ion vacancy changing the symmetry group from °h to C~, Thus the ion is effectively in a large static electric5 and fieldlowers along the [l10J two direction, which splits These off’ two levels markedly of the six levels. would have a splitting A’, as compared to the splitting 2A in the °h symmetry; A’ and A are the respective tunneling rates between adjacent sites, for which A’ > A The OH- ion can be visualized as oscillating between the [100J and E010 directions. The reduced barrier potential can be related to A by introducing a one-dimensional tunneling mode1’~for which A’ /A is proportional to exp 1 6 k~2, where k is the dimensionless barrier parameter Introduced by Devonshire and 8 is the fractional amount by which the barrier is reduced. 6 has been approximated from the results of the thermal conductivity data for NaCl:OW for which A’ /3A equals 10. Using .
.
-
The resonance frequency ui 2 for both the irradiated samples and the additively-colored samples can be explained by considering the tunneling transitions fromand the suggesDevon9 model and the calculated Shore’°model ted shireIn the works of Chau etal. and of Seward and Narayanamurtl.” The thermal data presented
this value of A’ /3A for KC1, the model yields tunneling transitions of 22 cm” for KC1:OH’ with A equal to 0. 8 cm’, and 26 cm~ for RbC1:OW estimatedwith to beChau’s 1 cm’~ values are with to beAcompared and these Klein’s’6 values of 32 cm’ and 30 cm~, respectively. In the irradiated crystals the frequency
Vol. 5, No. 11
RESONANT PHONON SCATTERING
901
TABLE 1 Coefficients of the relaxation times for several v-irradiated NaC1 crystals
Crystal
5_C
B 5 x10
P W 1 A1 x].0”~ em’~ x1032
W2
A2 x1029
em”
A
5.0
Ii..6
17
0.33
1.6
0.63
B
6.7
I~.1i
.17
0.914.
1.6
2.25
C
5.0
Ls..L 1.8
17
1.23
1.6
6.614.
D
5.0
8.37
17
14..014.
1.6
E
5.0
9.73
16
14..59
1.6
i.,,.,..
21.1 9.52
‘.‘‘‘
-
‘..~...
2•o
r
A
A -
?~“ A
I0
o z0.~ o C.)
)1~ ~‘
x
:~
o
,~
A.P
/
x
~
~5OlO.
~0.
“~
~t
‘,~
‘~
-
‘
8’ 4..
0
5’O.’
0R
20
TEMPERATURE
50
10.
“~
20.
~.•
.1
•K
FIG. 2 Thermal conductivity of additively-colored Optovac sodium chloride crystals. Curve M, an as-received crystal; Curve N, an as-received crystal annealed at 600°C; Curve P. crystal M after additive coloration, containing 3.2 x 10’4 F centers; Curve R, crystal N after additive coloration, containing 3. 1 x 1O’4 F centers.
The dotted lines are the machine fits.
902
RESONANT PHONON SCATTERING
Vol. 5, No. 11
TABLE 2 Coefficients of the relaxation times for several additively-colored NaC1 crystals
B Crys
P
W
A1 28 cm”~- xlO 1
x].0-’
x10’~’
M
6.58
14.32
N
6.0
P
6.37
1.67
15
R
6.58
14.18
13
10.0
13 13
16.0 0.614. 15.2 5.33
w~is unchanged by first-stage and early secondstage irradiation; the scattering from this mode Increases monotonically with irradiation. Similar results are found for the 53 resonance, cxcept for the heaviest irradiation. If the OH- ion were present only as a free ion in a substitutional position, one would expect that with irradiation the scattering from this impurity would decrease with irradiation; as shown by Etzel and Patter8, and Walker~, the effect of irrason”~, Ltlty’ diation is to break up the substitutional OW to form U centers and oxygen defects. A comparison of the coefficients for crystals M and N shows that with annealing, the scattering of the quasi-localized mode at 40°Kas well as that of both the w, and W2 modes is reduced, and the point defect scattering is increased. It is believed that the w 1 resonance is associated with the presence of a divalent impurity ~ and that M(OH)2 is formed during annealing. An optical absorption band observed at 2. ‘7~iis consistent with the formation of this complex. ~
~“2 1
om’~ 1.6 1.6 1.6 1.6
A2 28 xlO
A cm~
12.0
5.144 18.2
110
14.59
110
2.02
110 110
9.63
17
xlO”
10.6 14.89
The results of Fritz et al. 23 on the effects of OW ions on the electrical conductivity of KC1 crystals also show that OH- is present as a reaction product involving divalent impurities. The model recently proposed by Baur and Salzman~suggests that the center of mass of the OW ion is displaced, changing the symmetry group to C Theshould polarization expertment which they 4,. suggest distinguish between the two symmetry groups, and the two proposed models. More extensive work on both irradiated crystals and additively-colored crystals contaming colloids produced by thermal treatment will be discussed in subsequent papers. Acknowledgment We thank Professor L. M. Roth for her valuable theoretical discussion. -
References 1.
KLEIN M.V.,
2.
Irradiation performed using the Co~source of the U. S. Army Quartermaster Research and Development Laboratory, Natick, Massachusetts. CHAU C.K., KLEIN M.V. and WEDDING B., Phys. Rev. Lett. 17, 521 (1966); KLEIN M.V., WEDDING B. and CHAU C.K., Bull. Am. Phys. Soc. 12, 78 (1967).
3.
4.
Phys. Rev.
122, 1393 (1961).
These calculations were made at the Massachusetts Institute of Technology Computation Center.
RESONANT PHONON SCATTERING
Vol. 5, No. 11 5.
CALLAWAY J.,
Phys. Rev. 113, 1046 (1959).
6.
Data were taken from Klein’s measurements, Phys. Rev.
7.
CASIMIR H.B.G.,
8.
WAGNER M.,
9.
DEVONSHIRE A. F., Proc. Roy. Soc. (London) A153, 601 (1936).
Physica 5,
1393 (1961).
495 (1938).
Phys. Rev. 131, 1433 (1963).
Recalculated by SAUER P., 10. SHORE H. B.,
122,
Z. Physik 194, 360 (1966).
Phys. Rev. 151, 570 (1966).
11. SEWARD W.D. and NARAYANAMURTI V., 12. WEDDING B. and KLEIN M.V.,
Phys. Rev. 147, 463 (1966).
Bull. Am. Phys~Soc. 11, 228 (1966).
13. SAUER P., SCHJLRMER 0. and SCHNEIDER J., 14. KuHN V. and LTJTY F.,
Phys. Stat. Sol. 16, 79 (1966).
Solid State Comm. 2, 281 (1964);
HARTEL H. and LUTY F., Phys. Stat. Sol. 12, 347 (1965). The possibility that this is not the proper configuration has been pointed out by HANDLER P. and ASPNES D, E., Phys. Rev. Left. 17, 1095 (1966). 15. See, for example, BOHM D., New Jersey. 16. KLEIN M.V.,
Quantum Theory, Chap. 12. Prentice-Hall, Englewood Cliffs,
Bull. Am. Phys. Soc. 12, 340 (1967).
17. ETZEL H.W. and PATTERSON D.A., Phys. Rev. 112, 18. LiJTY F., J. Phys. Chem. Solids 23, 19. WALKER C.T., Phys. Rev. 20. STOEBE T, G.,
1112 (1958).
677 (1962).
132, 1963 (1963).
Bull. Am. Phys. Soc. 11,
21. FRITZ B., LUTY F. and ANGER J., 22. BAUR M.E. and SALZMAN W.R.,
886 (1966).
Z. Physik 174, 240 (1963).
Phys. Rev. Left.
18, 590 (1967).
La conductibilité thermique des cristaux de chlorure de sodium a la lois irradiês par rayons gamma et colords additivement, contennant ions OH-, a montré deux résonances prêdomthantes, l’une a la frequence de 1.6 cm’ et l’autre de 17 cm’. On explique In frequence de résonance de 1.6 cm” par des transitions de tunnel calculées selon un modèle Devonshire. On propose un modèle pour in fréquence de resonance de 17 cm~”dana lequel un des ions de chiorure dana la region du ion OW est remplacé par une lacune d’un ion négatif.
903
RESONANT PHONON SCATTERING IN COLORED SODIUM CHLORIDE CRYSTALS* R.E. Aldrich~, W.J. Burke~,and K.A. McCarthy Department of Physics, Tufts University, Medford, Massachusetts, U.S.A. (Received 7 July 1967 by B.N. Brockhouse) (Revised 30 August 1967) The thermal conductivity of both v-irradiated and additivelycolored sodium chloride crystals containing OW ions has shown two prominent resonances, one at a frequency of 1.6 cm1 and the other at 17 cm’. The 1.6 cm’ resonant frequency is explained by tunneling transitions calculated from a Devonshire model. A model is proposed for the 17 cm’ resonant frequency in which one of the chloride ions in the region of the OW ion is replaced by a negative ion vacancy.
THE RESONANT phonon scattering associated with the depression at 6°K in the thermal conductivity of as-received sodium chloride crystals is now believed to be due to an OW impurity3 the OW dipole substitutes for a Cl ion in the NaC1 lattice. It will be shown here that in both additively- colored and irradiated2 sodium chloride crystals the OH- resonance is enhanced by the coloration, and that the 17 cm~ level associated with the OW depression is similar to that found by Chau, Klein, and Wedding3 for KC1:OW .
The thermal conductivity of several irradiated Optovac sodium chloride crystals is shown in Fig. 1. The data have been analyzed4 using the Callaway model5 with a relaxation time, r’, of the form: -
The coefficients for the crystals given In Fig. 1 are listed in Table 1. The values for N(4. 7 x 10~’°),U(4. 04 x 10~)and F(34. 7) are obtained from fits to pure crystals6, and were used in fitting all crystals. The boundary scattering term, B, is the Casimir7 value; all crystals were sandblasted as the final step before measurement. The thermal conductivity of several additively-colored Optovac sodium chloride crystals is shown in Fig. 2; these crystals were cleaved from a different boule than the crystals used for v -irradiation. These data have also been fitted using a Callaway model; a different relaxation time, ~R3 , is used for the resonance terms and is of the form: A
r’
B+Pw4
A 2 w 2 1 n2)2 —
+
NT2w
+
UT2w2 exp(-F/T) A 2 u?
+~
(~ 2— ~, )2 + 2 2
c
2 (1)
(cu 2~ *Supported in part by the National Science Foundation (Grant GP-4322). ~Now at Itek Corporation, Lexington, Massachusetts. Present address Department of Physics, University of Illinois, Urbana, illinois. 899 +
2w2 A 2w2 3T + 2T (w 2-w2)2+C~n~2 (u~-w2)2+C~w 2 2 2
2
+
+
A
1 f (~, w1 , T) g(w ,w1)
(2)
0 where the third term Is that obtained by Wagner for a quasi-localized mode. TheadditivelyT2 term in the resonant relaxation time orthe colored crystals corresponds to a narrowing of the resonance; the resonance is asymmetric and
900
RESONANT PHONON SCATTERING 10
here do not extend to sufficiently low temperatures to permit observation of the recovery of the thermal conductivity to the Casimir value. However, the machine fit to the available data yields a value for the upper limit of w2 equal to
‘1
5,~
~oo~ 4~
~6~
~ 2.0 p 0’ 1.0
-
/
02 .
~f o8,~f
o.t
-
9’K
~‘ ..!‘~‘
~
~
~
KEY
~
A B
,~‘
/
.05
The resonance at 6°K is not predicted by the and is believed to haveby the Devonshire same originmodel, as the resonances presented Chau et al. The insensitivity of the contribution of th&i~ionanceterms thenot half-widths suggests that the two resonances to may be associated with the same type center. Shore et al.’3 have
./‘~‘
~
~ //
:~ ~
~ ~
0.5
~
1.6 cm’, which in good agreement with a of Devonshire modelisusing a barrier parameter 725Wedding cm~, as of anddetermined Klein. ‘~ from the infrared data
4~’\
~V
~
~ U
~‘ ~
~V’~
Vol. 5, No. 11
°
-
A
~‘
E
• 02’ 10
20
TEMPERATURE
.
FIG. 1 Thermal conductivity of v -irradiated Optovac socium chloride crystals. Curve A, an as-received crystal; Curve B, an as-received annealed at 600°C; Curve C, acrystal v irradiated crystal containing 3. 8 x 10’s F centers; Curve D, a v-irradiated crystal containing 5. 6 x 1017 F centers; Curve E, a v-irradiated crystal containing 7.6 x io~’F centers. The dotted lines are the machine fits. The color-center concentrations are determined from op.tical measurements. The concentration of OW is estimated to be —~3ppm. results in a lowering of the frequency,
analyzed the properties of an OW the alkali halides in the presence of anTii~rityin electric field, and have shown that the energy of the groundstate singlet (A 11) is greatly reduced with the tric field applied along each of the three principal crystallographic directions. In their models the impurityfield is assumed to be in an octahedral crystalline with potential minima along the six [100J directions, as suggested in the experiments of Luty.’4 It may be assumed that the tunneling thatand takes is at 90° to original orientation, thatplace the probability of the tunneling by 180° is small. To explain the w, resonance, we propose that one of the next-nearest-neighbor C1 ions in the region of some of the OW ions is replaced by a negative ion vacancy changing the symmetry group from °h to C~, Thus the ion is effectively in a large static electric5 and fieldlowers along the [l10J two direction, which splits These off’ two levels markedly of the six levels. would have a splitting A’, as compared to the splitting 2A in the °h symmetry; A’ and A are the respective tunneling rates between adjacent sites, for which A’ > A The OH- ion can be visualized as oscillating between the [100J and E010 directions. The reduced barrier potential can be related to A by introducing a one-dimensional tunneling mode1’~for which A’ /A is proportional to exp 1 6 k~2, where k is the dimensionless barrier parameter Introduced by Devonshire and 8 is the fractional amount by which the barrier is reduced. 6 has been approximated from the results of the thermal conductivity data for NaCl:OW for which A’ /3A equals 10. Using .
.
-
The resonance frequency ui 2 for both the irradiated samples and the additively-colored samples can be explained by considering the tunneling transitions fromand the suggesDevon9 model and the calculated Shore’°model ted shireIn the works of Chau etal. and of Seward and Narayanamurtl.” The thermal data presented
this value of A’ /3A for KC1, the model yields tunneling transitions of 22 cm” for KC1:OH’ with A equal to 0. 8 cm’, and 26 cm~ for RbC1:OW estimatedwith to beChau’s 1 cm’~ values are with to beAcompared and these Klein’s’6 values of 32 cm’ and 30 cm~, respectively. In the irradiated crystals the frequency
Vol. 5, No. 11
RESONANT PHONON SCATTERING
901
TABLE 1 Coefficients of the relaxation times for several v-irradiated NaC1 crystals
Crystal
5_C
B 5 x10
P W 1 A1 x].0”~ em’~ x1032
W2
A2 x1029
em”
A
5.0
Ii..6
17
0.33
1.6
0.63
B
6.7
I~.1i
.17
0.914.
1.6
2.25
C
5.0
Ls..L 1.8
17
1.23
1.6
6.614.
D
5.0
8.37
17
14..014.
1.6
E
5.0
9.73
16
14..59
1.6
i.,,.,..
21.1 9.52
‘.‘‘‘
-
‘..~...
2•o
r
A
A -
?~“ A
I0
o z0.~ o C.)
)1~ ~‘
x
:~
o
,~
A.P
/
x
~
~5OlO.
~0.
“~
~t
‘,~
‘~
-
‘
8’ 4..
0
5’O.’
0R
20
TEMPERATURE
50
10.
“~
20.
~.•
.1
•K
FIG. 2 Thermal conductivity of additively-colored Optovac sodium chloride crystals. Curve M, an as-received crystal; Curve N, an as-received crystal annealed at 600°C; Curve P. crystal M after additive coloration, containing 3.2 x 10’4 F centers; Curve R, crystal N after additive coloration, containing 3. 1 x 1O’4 F centers.
The dotted lines are the machine fits.
902
RESONANT PHONON SCATTERING
Vol. 5, No. 11
TABLE 2 Coefficients of the relaxation times for several additively-colored NaC1 crystals
B Crys
P
W
A1 28 cm”~- xlO 1
x].0-’
x10’~’
M
6.58
14.32
N
6.0
P
6.37
1.67
15
R
6.58
14.18
13
10.0
13 13
16.0 0.614. 15.2 5.33
w~is unchanged by first-stage and early secondstage irradiation; the scattering from this mode Increases monotonically with irradiation. Similar results are found for the 53 resonance, cxcept for the heaviest irradiation. If the OH- ion were present only as a free ion in a substitutional position, one would expect that with irradiation the scattering from this impurity would decrease with irradiation; as shown by Etzel and Patter8, and Walker~, the effect of irrason”~, Ltlty’ diation is to break up the substitutional OW to form U centers and oxygen defects. A comparison of the coefficients for crystals M and N shows that with annealing, the scattering of the quasi-localized mode at 40°Kas well as that of both the w, and W2 modes is reduced, and the point defect scattering is increased. It is believed that the w 1 resonance is associated with the presence of a divalent impurity ~ and that M(OH)2 is formed during annealing. An optical absorption band observed at 2. ‘7~iis consistent with the formation of this complex. ~
~“2 1
om’~ 1.6 1.6 1.6 1.6
A2 28 xlO
A cm~
12.0
5.144 18.2
110
14.59
110
2.02
110 110
9.63
17
xlO”
10.6 14.89
The results of Fritz et al. 23 on the effects of OW ions on the electrical conductivity of KC1 crystals also show that OH- is present as a reaction product involving divalent impurities. The model recently proposed by Baur and Salzman~suggests that the center of mass of the OW ion is displaced, changing the symmetry group to C Theshould polarization expertment which they 4,. suggest distinguish between the two symmetry groups, and the two proposed models. More extensive work on both irradiated crystals and additively-colored crystals contaming colloids produced by thermal treatment will be discussed in subsequent papers. Acknowledgment We thank Professor L. M. Roth for her valuable theoretical discussion. -
References 1.
KLEIN M.V.,
2.
Irradiation performed using the Co~source of the U. S. Army Quartermaster Research and Development Laboratory, Natick, Massachusetts. CHAU C.K., KLEIN M.V. and WEDDING B., Phys. Rev. Lett. 17, 521 (1966); KLEIN M.V., WEDDING B. and CHAU C.K., Bull. Am. Phys. Soc. 12, 78 (1967).
3.
4.
Phys. Rev.
122, 1393 (1961).
These calculations were made at the Massachusetts Institute of Technology Computation Center.
RESONANT PHONON SCATTERING
Vol. 5, No. 11 5.
CALLAWAY J.,
Phys. Rev. 113, 1046 (1959).
6.
Data were taken from Klein’s measurements, Phys. Rev.
7.
CASIMIR H.B.G.,
8.
WAGNER M.,
9.
DEVONSHIRE A. F., Proc. Roy. Soc. (London) A153, 601 (1936).
Physica 5,
1393 (1961).
495 (1938).
Phys. Rev. 131, 1433 (1963).
Recalculated by SAUER P., 10. SHORE H. B.,
122,
Z. Physik 194, 360 (1966).
Phys. Rev. 151, 570 (1966).
11. SEWARD W.D. and NARAYANAMURTI V., 12. WEDDING B. and KLEIN M.V.,
Phys. Rev. 147, 463 (1966).
Bull. Am. Phys~Soc. 11, 228 (1966).
13. SAUER P., SCHJLRMER 0. and SCHNEIDER J., 14. KuHN V. and LTJTY F.,
Phys. Stat. Sol. 16, 79 (1966).
Solid State Comm. 2, 281 (1964);
HARTEL H. and LUTY F., Phys. Stat. Sol. 12, 347 (1965). The possibility that this is not the proper configuration has been pointed out by HANDLER P. and ASPNES D, E., Phys. Rev. Left. 17, 1095 (1966). 15. See, for example, BOHM D., New Jersey. 16. KLEIN M.V.,
Quantum Theory, Chap. 12. Prentice-Hall, Englewood Cliffs,
Bull. Am. Phys. Soc. 12, 340 (1967).
17. ETZEL H.W. and PATTERSON D.A., Phys. Rev. 112, 18. LiJTY F., J. Phys. Chem. Solids 23, 19. WALKER C.T., Phys. Rev. 20. STOEBE T, G.,
1112 (1958).
677 (1962).
132, 1963 (1963).
Bull. Am. Phys. Soc. 11,
21. FRITZ B., LUTY F. and ANGER J., 22. BAUR M.E. and SALZMAN W.R.,
886 (1966).
Z. Physik 174, 240 (1963).
Phys. Rev. Left.
18, 590 (1967).
La conductibilité thermique des cristaux de chlorure de sodium a la lois irradiês par rayons gamma et colords additivement, contennant ions OH-, a montré deux résonances prêdomthantes, l’une a la frequence de 1.6 cm’ et l’autre de 17 cm’. On explique In frequence de résonance de 1.6 cm” par des transitions de tunnel calculées selon un modèle Devonshire. On propose un modèle pour in fréquence de resonance de 17 cm~”dana lequel un des ions de chiorure dana la region du ion OW est remplacé par une lacune d’un ion négatif.
903