- Email: [email protected]
Formation and structure of H' centers
J. Phys. Chem. Solids
Pergamon
Press
FORMATION
1970. Vol. 3 1, pp. 70 l-705.
AND
Printed
STRUCTURE
in Great
Britain.
OF H’ CENTERS*
G. J. DIENES Brookhaven
National
Laboratory,
Upton,
N.Y.
11973. U.S.A.
and R. D. HATCHER Queens
College,
Flushing,
New York,
U.S.A.
and R. SMOLUCHOWSKI Princeton (Received
University, 8 August
Princeton,
N. J. 08540,
1969; in revisedform
U.S.A.
30 October
1969)
Abstract-The
theoretically predicted cross-sections for the formation of H’-centers out of H-centers in KC1 and KBr and for the trapping of H-centers at Na impurities are in good agreement with the values obtained experimentally by Itoh. Two H-centers joined along a[1 lo] direction can have nine possible configurations. Their energies have been calculated using a somewhat simplified model with the result that in the lowest state the two H-centers are perpendicular to each other and lie in a (001) plane. In the highest state they are co-linear. 1. INTRODUCTION
of a new band, H’, in proximity of the known H band was first observed in KC1 and KBr crystals, upon prolonged X-ray irradiation at low temperatures, by Faraday and Compton[l]. Further studies by Itoh et af.[2, 31 have shown that initially the concentration of H’-centers is proportional to the square of the concentration of H-centers and that later it saturates. This lead them to suggest that the H’-center is a pair of H-centers. H-centers are anticenters of F-centers and are produced by a crowdionlike motion of the displaced halogen as first proposed by Howard et al. [4] and later used in other mechanism such as Pooley’s[S]. Thus the efficiency of the 2H*H’ reaction during irradiation is proportional not only to the effective cross-section (T of the H-centers for this reaction but also to the mean path of the interstitialcy. The experimentally determined parameter j3 = ha is a volume which can be deduced either from the relative rate of growth of THE
FORMATION
“Work supported Commission.
in part by the U.S. Atomic
Energy
H and H’ bands or from the saturation value. The experimental[3] values of p are 1.2 X IO3 and 2.5 X lo3 for KC1 and 1.4 X lo3 and 2.7 X 103 for KBr expressed in units of the elementary lattice cubes d3. In the first section of this paper the effective cross-section of H-centers is calculated and the product ah compared with experiment under the assumption that the H/centers are stable and that there is no back reaction. In the second section the structure of the various H’ configurations is analyzed. 2. FORMATION
OF H’-CENTERS
The stability and the structure of H-centers and of their immediate neighborhood has been investigated in considerable detail by Dienes et af.[61. At that time, however, no particular attention was paid to more distant neighbors which may be of importance for the present problem. Thus the calculations have been repeated using the same methods and numerical values as previously[6] It was first established that displacements in other than the close-packed cubic directions decrease very rapidly with distance. This per-
702
C. J. DIENES
mitted directing the main effort at calculating displacements in the cubic directions including up to lo-th neighbors. As shown in Fig. 1 in
2 2
I 0
I
2
2
+
2
Fig. 1.Strain pattern around an H-center.
four equivalent directions the ions are placed outward while in the remaining equivalent directions they are displaced ward. The calculated displacements shown in Fig. 2 for KC1 and KBr. The
distwo inare dis-
Fig. 2. Displacements of neighbors of an H-center along cubic directions. Positive (outward) displacements along x and y axes, negative (inward) displacements along zaxis (see Fig. I). Only first neighbors have a significant displacement in non-cubic directions. Displacements expressed as percentage of interionic distance. Note that positively charged neighbors are relatively more displaced while the negatively charged neighbors are less displaced than the mean strain indicated by the curves. Full dots KCI, open dots KBr.
placements do not fall on a smooth the positive ions being displaced more than the negative ones. It pointed out that the inclusion of tional ions alters the total energy
curve with relatively should be these addiof the H-
et al.
center only by about one per cent and thus all the earlier conclusions[6] concerning the stability of these centers are valid. If the strain pattern surrounding an H-center were approximately spherical with a radius Y then the effective cross-section o would be given by (2r)S. This is not the case, however, and thus the relationship between the displacements shown in Fig. 2 and the interaction between a moving and a stationary H-center has to be investigated in detail. First of all the mean displacement of an ion in KC1 or KBr from its ideal lattice site near 4°K is given essentially by its zero point amplitude which is about O-3 per cent of the interionic distance d. It is thus reasonable to assume that at this temperature the effective strain pattern surrounding an H-center in KC1 should be cut off at the 7th neighbor in the four equivalent cubic directions and at the 5th neighbor in the two other cubic directions. It follows that an H-center moving, as a crowdion, in a (110) direction presents a cross-like strain pattern, perpendicular to that direction, with one arm (7/d2)d and the other 5d long. A stationary H-center presents two kinds of strain patterns as viewed from various (110) directions: one of them has arms (7lg2)d and 5d long while the other one has arms (71d2)d and 7d long. The second pattern occurs in twice as many (110) directions as the first. The interaction between two such cross-like patterns is illustrated in Fig. 3 for patterns with arms 5d long. The dotted line is the locus of the center of cross A when one of its arms comes within a distance d of the stationary cross B and thus the area enclosed is the required cross-section. A similar pattern is easily obtained when the arms of the crosses are not equal. Taking a properly weighted average of the two kinds of strain patterns of the stationary H-center one obtains w = 208 d2 for KC1 and u = 283 d* for KBr,. The strain pattern illustrated in Fig. 1 suggests that if only an elastic interaction were present then for certain mutual orientations the two H-centers would repel rather than
FORMATION
AND STRUCTURE
---I I
r__--J
I I I L
----1
I
A
:---J
6
L_--q
r---
I
I++
I
L-_-_1
,_--_i
I
!
;
I I__.: Fig. 3. Cross-section patterns
for interaction of identical strain A and E (see text). The direction is
normal to the plane of the figure.
attract. It is reasonable to assume, however, that once two H-centers are close enough they may reorient themselves into other configurations of lower energy. Reorientation of free H-centers [7] does not occur below about 10°K but in proximity of other H-centers, their mobility may be greatly increased. The most direct estimate of the length of the path of the displaced halogen as an interstitialcy in KBr has been obtained by Itoh et al.[fJ] from the observed stabilizing effect of chlorine impurities. The result is A - 10d. No similar measurements in KCl, doped for instance with KF, have been made. The various studies[9, lo] of a-center formation and stability in SrCl, doped KC1 do not permit an unambiguous interpretation in terms of a mean path of the interstitialcy. Behr et al. [ 1 l] expressed a belief that in KC1 this path is about 6d long. Recent studies by Balzer[ 121 of the spontaneous recombination of vacancies and interstitials suggest a value of about 7. Combining these various results one obtains for p = uh the value 1-46 X lo3 d3 for KC1 and 2.70 X 103 d3 for KBr which compare favorably with the experimental values 1.2 - 2.5 x 103d3 and 1.4 - 2.7 x lo3 d3 respectively. An H-center travelling through a KC1 or KBr lattice may encounter not only other H-
OF H’ CENTERS
703
centers to form H’-centers but also other impurities. In particular it may encounter a Na+ substitutional ion. According to Douglas’ calculations [ 131 the nearest Cl- to Brneighbors of the Na+ ion are displaced inward by about 0.04 d while the next nearest, K+, neighbors are displaced outward by about 0.002 d. Using the same criteria as above one obtains for the stationary strain pattern as viewed from a (110) direction a rectangle dV2 long and about one atom wide. The crosssection for interaction with a moving H-center obtained in the same manner as above is about 20dZ. It follows that the parameter p for H’center formation should be 10 times greater than a similar parameter for the capture of Hcenters at Na+ impurities both in KC1 and in KBr. Itoh et a1.[3] data indicate that for KBr this ratio is 9.4 for the lower and 18 for the higher value of p. Thus also here the agreement between theory and experiment is satisfactory within the limits of the available information. It would be interesting to see whether parameters /3 are temperature dependent and whether this dependence could be explained in terms of a progressive cut-off of the calculated strain patterns or of the increasing instability of H’-centers.
3. CONFIGURATION
AND ENERGY OF H’-
CENTERS It is interesting to analyze the various possible configurations which can arise when two H-centers are nearest neighbors along a[ 1 lo] direction. The problem is analogous to that of O,- centers in KC1 and KBr[ 14, 151. The basic method of calculation was the same as that used in the early study[6] of H-centers. In order to compare the energies of the various configurations two quantities were kept constant: The size i.e. the separation between the two chlorine ions in each Cl,- molecule and the separation of the two H-centers. The orientation of the two H-centers was different in each case but it was kept fixed while the surrounding lattice was permitted to relax.
704
G. J. DIENES et al.
t2IiZ.L**. l
Configurration
.
,__2--, 0
Table 1. Energies (relative to the perfect lattice, in eV) of the various possible H’centers with and without ~~~~at~o~ Molecular orientation
Common plane
No relaxation
With relaxation
[l IO]-[IjO]
(001)
7-54 764 7.75 7.70 7.75 7.83 7.83 7-66 11.87
3.85 399 4.08 4.12 4.14 4.20 4.32 4.96 8.10
.
l c-----a
l
3 l
0
9 2 8 5 6 4
1 3 7
0
0
,_-a-,, .
*.a-----.
*
9
* l
Fig. 4. Possible configuration of two H-centers forming an H’-center. The small cubes are meant to aid in visuatizing the orientation of the H-centers and are not to be confused with the elementary cubes of the lattice.
The various possibilities are illustrated in Fig. 4 using notation introduced in the study of O,- centers[l4]. Depending upon the symmetry of the configuration a number of parameters was necessary to describe the displacements of the neighboring ions. Case 3 required 45 parameters while case 7 only 6. The results are given in Table 1. It is interesting to note that relaxation did not alter much the sequence of energies and that cases 5, 6 and 8 are very close to each other. The relaxation lowers the energies uniformly by about 3.5 to 3.8 eV except for case 3 where it is only 2.7 eV. The lowest energy corresponds to a configuration in which the two H-centers lie in a (001) plane and are pe~en~cul~ to each other. A more detailed calculation in which both the size of each of the constituent H-centers
Iloll-r~lol
[llO]-[llO] [loll-[lO_l] [lol]-[lo_r] [loll-[Ol l] [lOI]-[llO] [lOi]-] 11 [l lo]-[l lo]
(001) (111) (I 11) (111) (hkl)
and also their orientation would be permitted to relax was not made because of the enormous increase in complexity and in the number of variable parameters. Only the high symmetry case 7 was re-calculated permitting the two H-centers to adjust their size but not their orientation. Its final energy turned out to be comparable to that of the unrelaxed case 3 and thus still one of the highest. It is interesting to note that the sequence of configurations here obtained is quite ~fferent from that deduced for 02- centers[ 141 which for KC1 is 1,8,5,6, 9,2,7,3 and 4 while for KBr is 1,9,7,8,.5,6, 2, 3 and 4. In all these calculations only the elastic energy was taken into account and no quantum mechanical corrections included. The latter are known to affect the configuration of Ii-centers themselves[6]. The present case is prohibitively complicated for a complete quantum mechanical treatment and the use of simplified models for configurations of such low symmetry appears too arbitrary to lead to meaningful results. The absolute magnitudes of the energies in Table 1 are undoubte~y too high because of the neglect of the above mentioned relaxations. As far as the sequence of various configurations is concerned, configurations 8, 5 and 6 are so close to each other that their order could be changed if the relaxations were to decrease the energies by, say, a factor of 2. The order of the remaining co~gurations, for
FORMATION
AND
STRUCTURE
which the energy difference may go down to 5 per cent in some cases, probably would not change since the computations themselves are good to O-01 eV. For the same reason, in comparing the energy of the most stable H’center, 345 eV, with the energy of two isolated H-centers in KCl, 2.8 eV [6], it should not be concluded that the H/-center is unstable relative to the H-center. As mentioned above, a more detailed calculation for the Hi-centers is rather prohibitive because of low symmetry. It is also of interest to compare the energy of an H-center with that of a neutral Cl, molecule inserted interstitially in the KC1 lattice. A calculation was carried out for an interstitial Cl, molecule oriented in a [loo] direction with the two nuclei located symmetrically in adjacent cells (dumbbell configuration across a face). The nearest neighbors (those located in the plane perpendicular to the midpoint of the molecular axis) and the next nearest neighbors (at the corners of the cubes) were allowed to relax. The molecular interaction between the two Cl nuclei was represented by a Morse function fitted to the potential energy vs. distance curve for the free molecule. The calculations showed that the nearest neighbors are displaced outward by about 10 per cent and the next nearest neighbors by 5 per cent, and that the interstitial Cl, molecule is slightly expanded in the KC1 lattice (from r = 2.0 A to r = 2.2 A, where r is the internuclear distance in the molecule). The energy of this configuration is 2-l eV relative to the perfect crystal. A similar calculation showed
OF H’ CENTERS
705
that if the interstiti~ Cl, molecule is confined to one cell in a [ lOO] orientation the molecule is compressed slightly (r = 1.9 A) and the energy rises to 2.7 eV. The interstitial Cl, molecule, according to these calculations, has a lower energy (2.1 eV) than two H-centers (2.9 eV) or two isolated neutral chlorine interstitials (3.6 eV). These results support the idea that the final state of the H-center is the interstitial Clz molecule.
REFERENCES 1. FARADAY 3. .I. and COMPTON W. D., Phys. Rev. 138, A893 (1965). 2. ITOH N., KAWAMATA T., HIRAO T. and KANZAKI H., J. phys. Sot. Jnpan 23,453 (1967). 3. ITOH N. and SAIDOH M., Phys. Status Solidi 33,
No. 2 (1969). 4. HOWARD R. E., VOSKO S. and SMOLUCHOWSKI R.,Phys. Rev. 122,1406(1961). 5. POOLEY D., Proc. Phys. Sot. 87, 245,257 (1966). 6. DIENES G. J., HATCHER R. D. and SMOLUCHOWSKI R., Phys. Rev. 157,692 (1967). 7. BACHMANN K., Thesis. EidgenSssische Technische Hochschde, Ziirich (1968). 8. ITOH N., ROYCE B. S. H. and SMOLUCHOWSKI R.. Phvs. Rev. 138. Al766 (196.51. 9. BEHR’ A.: PEISL ti. and WAIdELICH W., J.
Phys. 28, C4-163 (1967). 10. GIULIANI
G. and REGUZZONI
E., Phys. Status
Solidi 25,437 (1968). 11. BEHR A., PEISL H. and WAIDELICH W., Phys. Lat. 24A, 379 (1967). 12. BALZER R., Thesis, Technische Hochschule, Darmstadt (1969). 13. DOUGLAS T. B.,J. &em. Phys. 45,457 l(l966). 14. SHUEY R. T. and BEYELER H. U.,J. uppl. Math. Phys. 19,278 (1968). is. VON WALDKIRCH T., ZELLER H. R. and KANZIG W., H&v. Phys. Actri 40,823 (1967).
Pergamon
Press
FORMATION
1970. Vol. 3 1, pp. 70 l-705.
AND
Printed
STRUCTURE
in Great
Britain.
OF H’ CENTERS*
G. J. DIENES Brookhaven
National
Laboratory,
Upton,
N.Y.
11973. U.S.A.
and R. D. HATCHER Queens
College,
Flushing,
New York,
U.S.A.
and R. SMOLUCHOWSKI Princeton (Received
University, 8 August
Princeton,
N. J. 08540,
1969; in revisedform
U.S.A.
30 October
1969)
Abstract-The
theoretically predicted cross-sections for the formation of H’-centers out of H-centers in KC1 and KBr and for the trapping of H-centers at Na impurities are in good agreement with the values obtained experimentally by Itoh. Two H-centers joined along a[1 lo] direction can have nine possible configurations. Their energies have been calculated using a somewhat simplified model with the result that in the lowest state the two H-centers are perpendicular to each other and lie in a (001) plane. In the highest state they are co-linear. 1. INTRODUCTION
of a new band, H’, in proximity of the known H band was first observed in KC1 and KBr crystals, upon prolonged X-ray irradiation at low temperatures, by Faraday and Compton[l]. Further studies by Itoh et af.[2, 31 have shown that initially the concentration of H’-centers is proportional to the square of the concentration of H-centers and that later it saturates. This lead them to suggest that the H’-center is a pair of H-centers. H-centers are anticenters of F-centers and are produced by a crowdionlike motion of the displaced halogen as first proposed by Howard et al. [4] and later used in other mechanism such as Pooley’s[S]. Thus the efficiency of the 2H*H’ reaction during irradiation is proportional not only to the effective cross-section (T of the H-centers for this reaction but also to the mean path of the interstitialcy. The experimentally determined parameter j3 = ha is a volume which can be deduced either from the relative rate of growth of THE
FORMATION
“Work supported Commission.
in part by the U.S. Atomic
Energy
H and H’ bands or from the saturation value. The experimental[3] values of p are 1.2 X IO3 and 2.5 X lo3 for KC1 and 1.4 X lo3 and 2.7 X 103 for KBr expressed in units of the elementary lattice cubes d3. In the first section of this paper the effective cross-section of H-centers is calculated and the product ah compared with experiment under the assumption that the H/centers are stable and that there is no back reaction. In the second section the structure of the various H’ configurations is analyzed. 2. FORMATION
OF H’-CENTERS
The stability and the structure of H-centers and of their immediate neighborhood has been investigated in considerable detail by Dienes et af.[61. At that time, however, no particular attention was paid to more distant neighbors which may be of importance for the present problem. Thus the calculations have been repeated using the same methods and numerical values as previously[6] It was first established that displacements in other than the close-packed cubic directions decrease very rapidly with distance. This per-
702
C. J. DIENES
mitted directing the main effort at calculating displacements in the cubic directions including up to lo-th neighbors. As shown in Fig. 1 in
2 2
I 0
I
2
2
+
2
Fig. 1.Strain pattern around an H-center.
four equivalent directions the ions are placed outward while in the remaining equivalent directions they are displaced ward. The calculated displacements shown in Fig. 2 for KC1 and KBr. The
distwo inare dis-
Fig. 2. Displacements of neighbors of an H-center along cubic directions. Positive (outward) displacements along x and y axes, negative (inward) displacements along zaxis (see Fig. I). Only first neighbors have a significant displacement in non-cubic directions. Displacements expressed as percentage of interionic distance. Note that positively charged neighbors are relatively more displaced while the negatively charged neighbors are less displaced than the mean strain indicated by the curves. Full dots KCI, open dots KBr.
placements do not fall on a smooth the positive ions being displaced more than the negative ones. It pointed out that the inclusion of tional ions alters the total energy
curve with relatively should be these addiof the H-
et al.
center only by about one per cent and thus all the earlier conclusions[6] concerning the stability of these centers are valid. If the strain pattern surrounding an H-center were approximately spherical with a radius Y then the effective cross-section o would be given by (2r)S. This is not the case, however, and thus the relationship between the displacements shown in Fig. 2 and the interaction between a moving and a stationary H-center has to be investigated in detail. First of all the mean displacement of an ion in KC1 or KBr from its ideal lattice site near 4°K is given essentially by its zero point amplitude which is about O-3 per cent of the interionic distance d. It is thus reasonable to assume that at this temperature the effective strain pattern surrounding an H-center in KC1 should be cut off at the 7th neighbor in the four equivalent cubic directions and at the 5th neighbor in the two other cubic directions. It follows that an H-center moving, as a crowdion, in a (110) direction presents a cross-like strain pattern, perpendicular to that direction, with one arm (7/d2)d and the other 5d long. A stationary H-center presents two kinds of strain patterns as viewed from various (110) directions: one of them has arms (7lg2)d and 5d long while the other one has arms (71d2)d and 7d long. The second pattern occurs in twice as many (110) directions as the first. The interaction between two such cross-like patterns is illustrated in Fig. 3 for patterns with arms 5d long. The dotted line is the locus of the center of cross A when one of its arms comes within a distance d of the stationary cross B and thus the area enclosed is the required cross-section. A similar pattern is easily obtained when the arms of the crosses are not equal. Taking a properly weighted average of the two kinds of strain patterns of the stationary H-center one obtains w = 208 d2 for KC1 and u = 283 d* for KBr,. The strain pattern illustrated in Fig. 1 suggests that if only an elastic interaction were present then for certain mutual orientations the two H-centers would repel rather than
FORMATION
AND STRUCTURE
---I I
r__--J
I I I L
----1
I
A
:---J
6
L_--q
r---
I
I++
I
L-_-_1
,_--_i
I
!
;
I I__.: Fig. 3. Cross-section patterns
for interaction of identical strain A and E (see text). The
normal to the plane of the figure.
attract. It is reasonable to assume, however, that once two H-centers are close enough they may reorient themselves into other configurations of lower energy. Reorientation of free H-centers [7] does not occur below about 10°K but in proximity of other H-centers, their mobility may be greatly increased. The most direct estimate of the length of the path of the displaced halogen as an interstitialcy in KBr has been obtained by Itoh et al.[fJ] from the observed stabilizing effect of chlorine impurities. The result is A - 10d. No similar measurements in KCl, doped for instance with KF, have been made. The various studies[9, lo] of a-center formation and stability in SrCl, doped KC1 do not permit an unambiguous interpretation in terms of a mean path of the interstitialcy. Behr et al. [ 1 l] expressed a belief that in KC1 this path is about 6d long. Recent studies by Balzer[ 121 of the spontaneous recombination of vacancies and interstitials suggest a value of about 7. Combining these various results one obtains for p = uh the value 1-46 X lo3 d3 for KC1 and 2.70 X 103 d3 for KBr which compare favorably with the experimental values 1.2 - 2.5 x 103d3 and 1.4 - 2.7 x lo3 d3 respectively. An H-center travelling through a KC1 or KBr lattice may encounter not only other H-
OF H’ CENTERS
703
centers to form H’-centers but also other impurities. In particular it may encounter a Na+ substitutional ion. According to Douglas’ calculations [ 131 the nearest Cl- to Brneighbors of the Na+ ion are displaced inward by about 0.04 d while the next nearest, K+, neighbors are displaced outward by about 0.002 d. Using the same criteria as above one obtains for the stationary strain pattern as viewed from a (110) direction a rectangle dV2 long and about one atom wide. The crosssection for interaction with a moving H-center obtained in the same manner as above is about 20dZ. It follows that the parameter p for H’center formation should be 10 times greater than a similar parameter for the capture of Hcenters at Na+ impurities both in KC1 and in KBr. Itoh et a1.[3] data indicate that for KBr this ratio is 9.4 for the lower and 18 for the higher value of p. Thus also here the agreement between theory and experiment is satisfactory within the limits of the available information. It would be interesting to see whether parameters /3 are temperature dependent and whether this dependence could be explained in terms of a progressive cut-off of the calculated strain patterns or of the increasing instability of H’-centers.
3. CONFIGURATION
AND ENERGY OF H’-
CENTERS It is interesting to analyze the various possible configurations which can arise when two H-centers are nearest neighbors along a[ 1 lo] direction. The problem is analogous to that of O,- centers in KC1 and KBr[ 14, 151. The basic method of calculation was the same as that used in the early study[6] of H-centers. In order to compare the energies of the various configurations two quantities were kept constant: The size i.e. the separation between the two chlorine ions in each Cl,- molecule and the separation of the two H-centers. The orientation of the two H-centers was different in each case but it was kept fixed while the surrounding lattice was permitted to relax.
704
G. J. DIENES et al.
t2IiZ.L**. l
Configurration
.
,__2--, 0
Table 1. Energies (relative to the perfect lattice, in eV) of the various possible H’centers with and without ~~~~at~o~ Molecular orientation
Common plane
No relaxation
With relaxation
[l IO]-[IjO]
(001)
7-54 764 7.75 7.70 7.75 7.83 7.83 7-66 11.87
3.85 399 4.08 4.12 4.14 4.20 4.32 4.96 8.10
.
l c-----a
l
3 l
0
9 2 8 5 6 4
1 3 7
0
0
,_-a-,, .
*.a-----.
*
9
* l
Fig. 4. Possible configuration of two H-centers forming an H’-center. The small cubes are meant to aid in visuatizing the orientation of the H-centers and are not to be confused with the elementary cubes of the lattice.
The various possibilities are illustrated in Fig. 4 using notation introduced in the study of O,- centers[l4]. Depending upon the symmetry of the configuration a number of parameters was necessary to describe the displacements of the neighboring ions. Case 3 required 45 parameters while case 7 only 6. The results are given in Table 1. It is interesting to note that relaxation did not alter much the sequence of energies and that cases 5, 6 and 8 are very close to each other. The relaxation lowers the energies uniformly by about 3.5 to 3.8 eV except for case 3 where it is only 2.7 eV. The lowest energy corresponds to a configuration in which the two H-centers lie in a (001) plane and are pe~en~cul~ to each other. A more detailed calculation in which both the size of each of the constituent H-centers
Iloll-r~lol
[llO]-[llO] [loll-[lO_l] [lol]-[lo_r] [loll-[Ol l] [lOI]-[llO] [lOi]-] 11 [l lo]-[l lo]
(001) (111) (I 11) (111) (hkl)
and also their orientation would be permitted to relax was not made because of the enormous increase in complexity and in the number of variable parameters. Only the high symmetry case 7 was re-calculated permitting the two H-centers to adjust their size but not their orientation. Its final energy turned out to be comparable to that of the unrelaxed case 3 and thus still one of the highest. It is interesting to note that the sequence of configurations here obtained is quite ~fferent from that deduced for 02- centers[ 141 which for KC1 is 1,8,5,6, 9,2,7,3 and 4 while for KBr is 1,9,7,8,.5,6, 2, 3 and 4. In all these calculations only the elastic energy was taken into account and no quantum mechanical corrections included. The latter are known to affect the configuration of Ii-centers themselves[6]. The present case is prohibitively complicated for a complete quantum mechanical treatment and the use of simplified models for configurations of such low symmetry appears too arbitrary to lead to meaningful results. The absolute magnitudes of the energies in Table 1 are undoubte~y too high because of the neglect of the above mentioned relaxations. As far as the sequence of various configurations is concerned, configurations 8, 5 and 6 are so close to each other that their order could be changed if the relaxations were to decrease the energies by, say, a factor of 2. The order of the remaining co~gurations, for
FORMATION
AND
STRUCTURE
which the energy difference may go down to 5 per cent in some cases, probably would not change since the computations themselves are good to O-01 eV. For the same reason, in comparing the energy of the most stable H’center, 345 eV, with the energy of two isolated H-centers in KCl, 2.8 eV [6], it should not be concluded that the H/-center is unstable relative to the H-center. As mentioned above, a more detailed calculation for the Hi-centers is rather prohibitive because of low symmetry. It is also of interest to compare the energy of an H-center with that of a neutral Cl, molecule inserted interstitially in the KC1 lattice. A calculation was carried out for an interstitial Cl, molecule oriented in a [loo] direction with the two nuclei located symmetrically in adjacent cells (dumbbell configuration across a face). The nearest neighbors (those located in the plane perpendicular to the midpoint of the molecular axis) and the next nearest neighbors (at the corners of the cubes) were allowed to relax. The molecular interaction between the two Cl nuclei was represented by a Morse function fitted to the potential energy vs. distance curve for the free molecule. The calculations showed that the nearest neighbors are displaced outward by about 10 per cent and the next nearest neighbors by 5 per cent, and that the interstitial Cl, molecule is slightly expanded in the KC1 lattice (from r = 2.0 A to r = 2.2 A, where r is the internuclear distance in the molecule). The energy of this configuration is 2-l eV relative to the perfect crystal. A similar calculation showed
OF H’ CENTERS
705
that if the interstiti~ Cl, molecule is confined to one cell in a [ lOO] orientation the molecule is compressed slightly (r = 1.9 A) and the energy rises to 2.7 eV. The interstitial Cl, molecule, according to these calculations, has a lower energy (2.1 eV) than two H-centers (2.9 eV) or two isolated neutral chlorine interstitials (3.6 eV). These results support the idea that the final state of the H-center is the interstitial Clz molecule.
REFERENCES 1. FARADAY 3. .I. and COMPTON W. D., Phys. Rev. 138, A893 (1965). 2. ITOH N., KAWAMATA T., HIRAO T. and KANZAKI H., J. phys. Sot. Jnpan 23,453 (1967). 3. ITOH N. and SAIDOH M., Phys. Status Solidi 33,
No. 2 (1969). 4. HOWARD R. E., VOSKO S. and SMOLUCHOWSKI R.,Phys. Rev. 122,1406(1961). 5. POOLEY D., Proc. Phys. Sot. 87, 245,257 (1966). 6. DIENES G. J., HATCHER R. D. and SMOLUCHOWSKI R., Phys. Rev. 157,692 (1967). 7. BACHMANN K., Thesis. EidgenSssische Technische Hochschde, Ziirich (1968). 8. ITOH N., ROYCE B. S. H. and SMOLUCHOWSKI R.. Phvs. Rev. 138. Al766 (196.51. 9. BEHR’ A.: PEISL ti. and WAIdELICH W., J.
Phys. 28, C4-163 (1967). 10. GIULIANI
G. and REGUZZONI
E., Phys. Status
Solidi 25,437 (1968). 11. BEHR A., PEISL H. and WAIDELICH W., Phys. Lat. 24A, 379 (1967). 12. BALZER R., Thesis, Technische Hochschule, Darmstadt (1969). 13. DOUGLAS T. B.,J. &em. Phys. 45,457 l(l966). 14. SHUEY R. T. and BEYELER H. U.,J. uppl. Math. Phys. 19,278 (1968). is. VON WALDKIRCH T., ZELLER H. R. and KANZIG W., H&v. Phys. Actri 40,823 (1967).