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Temperature effects on the hyperfine coupling of a surface centre
SURFACE
SCIENCE 25 (1971) 625-632 0 North-Holland
TEMPERATURE
EFFECTS
Publishing Co.
ON THE HYPERFINE
SURFACE
COUPLING
OF A
CENTRE
A. J. TENCH Chemistry
Received 2 November
Division,
A.E.R.E.,
Harwell,
Berks.,
England
1970; revised manuscript received 20 November 1970
The ESR spectra of an electron trapped at a surface anion vacancy (F+ centre) show temperature dependent hypertine splitting constants. This temperature dependence is proportional to the absolute temperature and is much larger for surface centres than the equivalent bulk centres. The effect of lattice vibrations at the surface is discussed and the change in the hfs-constants is explained in terms of the variation of the mean square displacements of surface ions with temperature.
1. Introduction The temperature dependence of hyperfine splitting (hfs) constants can give information on the coupling of lattice vibrations to point defects in crystals. Two kinds of defects have been reported, the first where lattice vibrations couple to excited states of an impurity ion1v2) and the second where both lattice and to a lesser extent molecular vibrations are important3,4). A number of measurements of the temperature dependence for Mn2+ in alkaline earth oxides have been reported and the temperature effect has been ascribed to the mixing of higher configurational states into the 3d5 ground state of the Mn2+ ion by non-cubic lattice vibrations. Another type of temperature dependence has been measured for the hfs constants of the diatomic halogen centres where it has been shown that both lattice and molecular vibrations contribute to the temperature dependence. In such cases, the hfs is found to decrease with increasing temperature. The temperature dependence of the hfs constant for a point defect on the surface would be expected to give information about the lattice dynamics at the surface of the solid. No previous work of this kind on surfaces has been found in the literature. In this paper, the temperature dependence of the hfs for the F: centre in the surface of MgO is investigated and found to increase with increasing temperature. 2. Experimental High surface
area samples
of MgO were prepared 625
by a thermal
decom-
626
A. J. TENCH
position of the hydroxide or carbonate in vacuo5) and the samples sealed off in silica ampoules under 10 Torr of hydrogen and y-irradiated to form surface centres6). The centres will be referred to as Fz following the nomenclature of Henderson and Wertz’), the subscript referring to the surface. Similar samples were n-irradiated in oxygen to form F+ centres in the bulk, in this case the oxygen reacted with the F,f centres leaving only the bulk centres *). The samples were measured on a Varian 4502 spectrometer, modulated at 100 kHz, at a frequency of 9.3 GHz. The magnetic field was controlled to 1 part in lo6 by a Mark II Field-dial. Low microwave powers were used to prevent saturation; the sample temperature was varied using a system of flowing nitrogen gas and the actual temperature measured using a copperconstantan thermocouple. The temperature gradient across the sample was found not to exceed 3 “K and the maximum errors in the measured temperatures are given as f 1.5”K. Additional measurements were carried out at 77 “K and 4°K using a dewar with a cold finger inserted into the microwave cavity. 3. Results 3.1. ANALYSIS The investigation of the properties of defects at the surface of a solid is more difficult than in the bulk since in general single crystals cannot be used because the number of defects formed is too small. When details of hyperfine interactions are studied it is necessary to use high surface area materials; these powders give an ESR spectrum which is an envelope of all the single crystal orientations. The hyperfine energy for a trapped electron interacting with one nucleus in the nearest neighbour position is given by,
for IAl $ ]B], where A and B are the isotropic and anisotropic components. Turning points should be observed in the powder spectrums) corresponding (A- B)m, and (A +2B)m, which allow the magnitudes of A and B to be determined. 3.2. F,+ CENTRE The ESR spectrum of the F,+ centre [described previously as the S, centrev)] shows 6 hyperfine lines arising from the interaction of a single 25M g nucleus (I=+). The Mg hyperfine splitting constant is found to increase with increasing temperature, fig. 1, but the small splitting of 2 gauss which has been attributed to a nearly proton remained constant. The average “Mg splitting constant measured from peak to peak for the +$ to -3 and + 5
TEMPERATURE
EFFECTS
627
to -_5 lines was found to increase almost linearly with temperature (fig. 2) from 100 to 450°K. The spectra were analysed using the treatment outlined above and values of A = 8.9 and B=0.4gauss were obtained at 104”K, while at 470 “K analysis gave A = 11.0 and B=0.5 gauss. These are maximum values for B since no clear structure was visible and the line extremities were taken as the turning points.
100 GAUSS , 1
Fig, 1. The ESR spectrum of the Fs+ centre at (a) 100°K and (b) 42O’K. Only the four outer hyperfine lines are shown because of overlap with 24Mg line in the centre of the spectrum.
A similar series of measurements were carried out on the bulk in a similar sample. In this sample the hype&e splitting constant to decrease by - 1% as the temperature was raised from < 100°K This variation was too small to measure accurately on the powder
F” centre was found to 300°K. spectrum.
4. Discussion The origin of this large increase in hfs with increasing temperature is not obvious and it is clear that none of the mechanisms discussed previouslyr-4) are able to account for this effect. Comparison with the results obtained for the bulk F+ centre suggests that in order to explain the behaviour of the FS+ centre we must look in detail at the surface properties of the solid. Any theoretical explanation must account for both the magnitude of the increase in hfs and its dependence on temperature.
628
A. 1. TENCH
The F,+ centre is an electron trapped at a surface anion vacancy (fig. 3) and the observed hfs is thought to arise from the single Mg2+ ion (a) furthest from the surface; the interaction with other Mg” ions (b) is small, probably about 0.9 gauss, and a nearby proton gives rise to a temperature independent splitting of 2 gauss. This surface centre can be compared with the F,+ centre
9 0
I 100
I 200 TEMPERATURE
I 300
I 400
I 500
( I(’
Fig. 2. The temperature dependence of the average hyperfine splitting for the FS+ centre. The solid line indicates the theoretical curve for L = 9.38 gauss and w = 1.7 x 101s Hz [see eq. (5)]. Ordinate: temperature (‘IQ. Abscissa: hfs (gauss).
in MgOs) which consists of an electron trapped in a bulk divacancy where axis of the vathe hyperfine interaction with Mg2+ along the tetragonal cancy pair is found to be 17.5 gauss, whereas the 4Mg2+ ions perpendicular to the axis of the divacancy have a hfs of -=z3 gauss. Et is interesting to note that the magnitude of the anisotropic splitting for the single Mg” ion is 0.7 gauss in good agreement with the values derived for the F,+ centre. With this model for the F,’ centre in mind, we now consider the behaviour of the ions at the surface. The increase in the hfs for the F,+ centre must arise from an increasing overlap of the unpaired electron wave function at the Mg nucleus as the temperature is raised. The exact geometric arrangement of the ions even in
TEMPERATURE
629
EFFECTS
defect free surface layers is not well known. Calculations on the alkali halideslr) indicate that cations in the first layer move inwards and those of the second layer outwards, whereas the anions all relax outwards. Low energy electron diffraction measurements on alkali halidesI 3l3) suggest that there is an overall expansion perpendicular to the plane of the surface of about 2%. There appears to be no evidence on the temperature dependence of these effects.
Mq
u ,,y y;.,,Mq u
Mq
(b)
‘\
0 Mq
Mq
0
0
Mq
Mb 7-L 0
M9
0
0
0
M9
Cal
M!3
Fig. 3. A model for the Fs+ centre where the open cicles represent 02- ions. The model is not to scale and for simplicity only an idealised {100} surface plane perpendicular to the paper is considered.
The isotropic hfs constant A is proportional to the square of the amplitude of the unpaired electron wave function at the nucleus of the ion. We can write the electronic wave function for an electron in a surface trap with the usual point-ion approximationl4) as the function $,,
(1) where !Ys is the envelope function of the surface centre and Yi refers to the ion-core orbitals of the Mg2+ ions. If we assume that the increase in hfs could be caused by the Mg2+ (a) moving into different regions of the wavefunction at higher temperatures then the isotropic hfs, which measures 14 (xMp)lz, must be averaged over the motion of the ion core. Assuming that the change in envelope wave function with the movement of the ions can be neglected, then
Since the ion-core terms in eq. (1) move with the Mg2 + ion we can use the envelope function (writing Y for YJ we have,
I y WI’ = I y (xo)12 + <(x - x0>‘>
2
aY
ax II Ii-l xo
’
A. J. TENCH
630
and (3) but ((x- x,,)~) is a measure of the change in the mean square displacement which is proportional to temperature in the high temperature approximationr5); this leads to the expression AA/A,
cc T,
(4)
which explains the linear region of the plot in fig. 2. A more complete description of the experimental results at low temperature can be obtained using the thermal average over the eigenfunctions v of the vibrationa); (vlA(Ri)lV)
=
LCOth2hkWT,
(5)
where o is the vibration frequency. The results in fig. 2 can be fitted by L=9.4+0.1 and cu = 1.7 x 1Ol3 Hz; this latter value is close to the frequency expected from the infra-red absorption of the lattice. The hfs constant at 4°K is found to be 9.35 gauss in good agreement with the value of L; the deviation of the data at higher temperatures from the line predicted using eq. (5) probably arises because the change in the envelope wave function with the movement of the ions cannot be neglected. This change would tend to reduce AA at higher temperatures. The mean square displacement of an ion is determined by the interionic forces acting on that ion. A surface ion is acted upon by fewer neighbours than an ion in the interior of a solid and this will generally cause the mean square displacement to be larger for the surface ion. Such displacement may be anisotropic since the environment of a surface ion has a different symmetry than that of an interior ion. A number of calculations have shown that in the high temperature approximation the mean square displacement has a linear dependence on the absolute temperature and Wallis et a1.r5) have shown that (u~)~cT for surface ions and that the magnitude of (u2) is between 2 and 4 times higher for surface ions than in the bulk, but decreases within a few layers to that characteristic of the bulk. Low energy electron diffraction studiesls) confirm the large mean square displacements of the surface ions. No calculations appear to have been carried out for ionic solids but calculated valuess) for the temperature dependence of (u2) in nickel are available and these indicate an increase in (u2> by a factor of 4 as the temperature is raised from 100 to 500 “K. To see the magnitude of the effect, Y was taken to be a simple Gaussian and the maximum value of IY (O)j’ was put to 250 gauss (i.e., Mg+)17) and
TEMPERATE
EFFECTS
631
the vaIue of lulj(0)l’
was taken as 9 gauss at 2 A from the centre of the vacancy. With these assumptions the observed variation of 30% in the hfs between 100 and 500 “K could be obtained by a movement of the ion toward the vacancy by 4% of the bulk nearest-neighbour distance. For comparison, the calculated data of Wallis et a1.15) for Ni indicate that (~)* increases by 3.5% of the nearest neighbour distance over the same temperature range. The dipolar term B would also be expected to increase with temperature on this model and this is found experimentally. Similarly the Mg2+ ions (b) in fig.3 should also show considerable temperature effects but this could not be examined because the hfs was not clearly resolved. For comparison, the Mg 2+ ions of the bulk Ff centre should show a much smalier effect. A similar treatment to that used above and allowing for lattice expansion of 0.6% over 500°K indicates that the hfs should increase by about 2%; in fact a slight decrease of N 1% is observed and this agreement is regarded as reasonable in the light of this approximate treatment since the effects are small compared to the 30% increase observed with the F,+ centre. In conclusion, it appears that the abnormal vibrational amplitudes at surfaces can cause effects on the hyperfine constants of surface defects. A more quantitative theoretical treatment of surface defects is required to understand the behaviour of the system fully. Acknowledgements The author for comments computational
is indebted to Dr. M. Stoneham, Theoretical Physics Division, and suggestions on part of this paper and A. L. V. Tenth for help. References
1) 2) 3) 4) 5) 6) 7) 8)
W. M. Walsh, T. Jeener and N. Bloembergen, Phys. Rev. 139 (1965) A1338. E. Simanek and R. Orbach, Phys. Rev. 145 (1964) 191. W. Dreybolt, Phys. Status Solidi 21(1967) 99. W. Assmus and W. Dreybolt, Phys. Status Solidi 34 (1969) 183. R. L. Nelson, A. J. Tenth and B. J. Harmsworth, Trans. Faraday Sot. 63 (1967) 1427. A. J. Tenth and R. L. Nelson, J. Colloid Interface Sci. 26 (1968) 364. B. Henderson and J. E. Wertz, Advan. Phys. 17 (1968) 749. R. L. Nelson, J. W. Hale, B. J. Harmsworth and A. J. Tenth, Trans. Faraday Sot. 64 (1968) 2521. 9) A. J. Tenth and R. L. Nelson, Proc. Phys. Sot. (London) 92 (1967) 1055; S. M. Blinder, J. Chem. Phys. 33 (1960) 748. 10) K. C. To, A. M. Stoneham and B. Henderson, Phys. Rev. Sl(l969) 1237. 11) G. C. Benson, P. I. Freeman and E. Dempsey, Solid &r-faces and fhe Gas-Solid Interface, No. 33, Advances in Chemistry Series (1961) p. 26. 12) A. U. MacRae and C. W. Caldwell, Surface Sci. 2 (1964) 509. 13) I. Marklund and S. Anderson, Surface Sci. 5 (1966) 197. 14(a) B. S. Gourary and F. J. Adrian, Phys. Rev. 105(1957) 1180.
632
A. J.TENCH
(b) A. M. Stoneham, W. Hayes, P. H. S. Smith and J. P. Stott, Proc. Roy. Sot. (London) A 306 (1968) 369. 15) R. F. Wallis, B. C. Clark and R. Herman, in: The Structure and Chemistry of Solid Surfaces, Ed. G. Somorjai (Wiley, New York, 1969) p. 17-1. 16) A. U. MacRae, Surface Sci. 2 (1964) 522. 17) M. F. Crawford, F. M. Kelly, A. L. Shawlow and W. M. Grey, Phys. Rev. 76 (1949) 1527.
SCIENCE 25 (1971) 625-632 0 North-Holland
TEMPERATURE
EFFECTS
Publishing Co.
ON THE HYPERFINE
SURFACE
COUPLING
OF A
CENTRE
A. J. TENCH Chemistry
Received 2 November
Division,
A.E.R.E.,
Harwell,
Berks.,
England
1970; revised manuscript received 20 November 1970
The ESR spectra of an electron trapped at a surface anion vacancy (F+ centre) show temperature dependent hypertine splitting constants. This temperature dependence is proportional to the absolute temperature and is much larger for surface centres than the equivalent bulk centres. The effect of lattice vibrations at the surface is discussed and the change in the hfs-constants is explained in terms of the variation of the mean square displacements of surface ions with temperature.
1. Introduction The temperature dependence of hyperfine splitting (hfs) constants can give information on the coupling of lattice vibrations to point defects in crystals. Two kinds of defects have been reported, the first where lattice vibrations couple to excited states of an impurity ion1v2) and the second where both lattice and to a lesser extent molecular vibrations are important3,4). A number of measurements of the temperature dependence for Mn2+ in alkaline earth oxides have been reported and the temperature effect has been ascribed to the mixing of higher configurational states into the 3d5 ground state of the Mn2+ ion by non-cubic lattice vibrations. Another type of temperature dependence has been measured for the hfs constants of the diatomic halogen centres where it has been shown that both lattice and molecular vibrations contribute to the temperature dependence. In such cases, the hfs is found to decrease with increasing temperature. The temperature dependence of the hfs constant for a point defect on the surface would be expected to give information about the lattice dynamics at the surface of the solid. No previous work of this kind on surfaces has been found in the literature. In this paper, the temperature dependence of the hfs for the F: centre in the surface of MgO is investigated and found to increase with increasing temperature. 2. Experimental High surface
area samples
of MgO were prepared 625
by a thermal
decom-
626
A. J. TENCH
position of the hydroxide or carbonate in vacuo5) and the samples sealed off in silica ampoules under 10 Torr of hydrogen and y-irradiated to form surface centres6). The centres will be referred to as Fz following the nomenclature of Henderson and Wertz’), the subscript referring to the surface. Similar samples were n-irradiated in oxygen to form F+ centres in the bulk, in this case the oxygen reacted with the F,f centres leaving only the bulk centres *). The samples were measured on a Varian 4502 spectrometer, modulated at 100 kHz, at a frequency of 9.3 GHz. The magnetic field was controlled to 1 part in lo6 by a Mark II Field-dial. Low microwave powers were used to prevent saturation; the sample temperature was varied using a system of flowing nitrogen gas and the actual temperature measured using a copperconstantan thermocouple. The temperature gradient across the sample was found not to exceed 3 “K and the maximum errors in the measured temperatures are given as f 1.5”K. Additional measurements were carried out at 77 “K and 4°K using a dewar with a cold finger inserted into the microwave cavity. 3. Results 3.1. ANALYSIS The investigation of the properties of defects at the surface of a solid is more difficult than in the bulk since in general single crystals cannot be used because the number of defects formed is too small. When details of hyperfine interactions are studied it is necessary to use high surface area materials; these powders give an ESR spectrum which is an envelope of all the single crystal orientations. The hyperfine energy for a trapped electron interacting with one nucleus in the nearest neighbour position is given by,
for IAl $ ]B], where A and B are the isotropic and anisotropic components. Turning points should be observed in the powder spectrums) corresponding (A- B)m, and (A +2B)m, which allow the magnitudes of A and B to be determined. 3.2. F,+ CENTRE The ESR spectrum of the F,+ centre [described previously as the S, centrev)] shows 6 hyperfine lines arising from the interaction of a single 25M g nucleus (I=+). The Mg hyperfine splitting constant is found to increase with increasing temperature, fig. 1, but the small splitting of 2 gauss which has been attributed to a nearly proton remained constant. The average “Mg splitting constant measured from peak to peak for the +$ to -3 and + 5
TEMPERATURE
EFFECTS
627
to -_5 lines was found to increase almost linearly with temperature (fig. 2) from 100 to 450°K. The spectra were analysed using the treatment outlined above and values of A = 8.9 and B=0.4gauss were obtained at 104”K, while at 470 “K analysis gave A = 11.0 and B=0.5 gauss. These are maximum values for B since no clear structure was visible and the line extremities were taken as the turning points.
100 GAUSS , 1
Fig, 1. The ESR spectrum of the Fs+ centre at (a) 100°K and (b) 42O’K. Only the four outer hyperfine lines are shown because of overlap with 24Mg line in the centre of the spectrum.
A similar series of measurements were carried out on the bulk in a similar sample. In this sample the hype&e splitting constant to decrease by - 1% as the temperature was raised from < 100°K This variation was too small to measure accurately on the powder
F” centre was found to 300°K. spectrum.
4. Discussion The origin of this large increase in hfs with increasing temperature is not obvious and it is clear that none of the mechanisms discussed previouslyr-4) are able to account for this effect. Comparison with the results obtained for the bulk F+ centre suggests that in order to explain the behaviour of the FS+ centre we must look in detail at the surface properties of the solid. Any theoretical explanation must account for both the magnitude of the increase in hfs and its dependence on temperature.
628
A. 1. TENCH
The F,+ centre is an electron trapped at a surface anion vacancy (fig. 3) and the observed hfs is thought to arise from the single Mg2+ ion (a) furthest from the surface; the interaction with other Mg” ions (b) is small, probably about 0.9 gauss, and a nearby proton gives rise to a temperature independent splitting of 2 gauss. This surface centre can be compared with the F,+ centre
9 0
I 100
I 200 TEMPERATURE
I 300
I 400
I 500
( I(’
Fig. 2. The temperature dependence of the average hyperfine splitting for the FS+ centre. The solid line indicates the theoretical curve for L = 9.38 gauss and w = 1.7 x 101s Hz [see eq. (5)]. Ordinate: temperature (‘IQ. Abscissa: hfs (gauss).
in MgOs) which consists of an electron trapped in a bulk divacancy where axis of the vathe hyperfine interaction with Mg2+ along the tetragonal cancy pair is found to be 17.5 gauss, whereas the 4Mg2+ ions perpendicular to the axis of the divacancy have a hfs of -=z3 gauss. Et is interesting to note that the magnitude of the anisotropic splitting for the single Mg” ion is 0.7 gauss in good agreement with the values derived for the F,+ centre. With this model for the F,’ centre in mind, we now consider the behaviour of the ions at the surface. The increase in the hfs for the F,+ centre must arise from an increasing overlap of the unpaired electron wave function at the Mg nucleus as the temperature is raised. The exact geometric arrangement of the ions even in
TEMPERATURE
629
EFFECTS
defect free surface layers is not well known. Calculations on the alkali halideslr) indicate that cations in the first layer move inwards and those of the second layer outwards, whereas the anions all relax outwards. Low energy electron diffraction measurements on alkali halidesI 3l3) suggest that there is an overall expansion perpendicular to the plane of the surface of about 2%. There appears to be no evidence on the temperature dependence of these effects.
Mq
u ,,y y;.,,Mq u
Mq
(b)
‘\
0 Mq
Mq
0
0
Mq
Mb 7-L 0
M9
0
0
0
M9
Cal
M!3
Fig. 3. A model for the Fs+ centre where the open cicles represent 02- ions. The model is not to scale and for simplicity only an idealised {100} surface plane perpendicular to the paper is considered.
The isotropic hfs constant A is proportional to the square of the amplitude of the unpaired electron wave function at the nucleus of the ion. We can write the electronic wave function for an electron in a surface trap with the usual point-ion approximationl4) as the function $,,
(1) where !Ys is the envelope function of the surface centre and Yi refers to the ion-core orbitals of the Mg2+ ions. If we assume that the increase in hfs could be caused by the Mg2+ (a) moving into different regions of the wavefunction at higher temperatures then the isotropic hfs, which measures 14 (xMp)lz, must be averaged over the motion of the ion core. Assuming that the change in envelope wave function with the movement of the ions can be neglected, then
Since the ion-core terms in eq. (1) move with the Mg2 + ion we can use the envelope function (writing Y for YJ we have,
I y WI’ = I y (xo)12 + <(x - x0>‘>
2
aY
ax II Ii-l xo
’
A. J. TENCH
630
and (3) but ((x- x,,)~) is a measure of the change in the mean square displacement which is proportional to temperature in the high temperature approximationr5); this leads to the expression AA/A,
cc T,
(4)
which explains the linear region of the plot in fig. 2. A more complete description of the experimental results at low temperature can be obtained using the thermal average over the eigenfunctions v of the vibrationa); (vlA(Ri)lV)
=
LCOth2hkWT,
(5)
where o is the vibration frequency. The results in fig. 2 can be fitted by L=9.4+0.1 and cu = 1.7 x 1Ol3 Hz; this latter value is close to the frequency expected from the infra-red absorption of the lattice. The hfs constant at 4°K is found to be 9.35 gauss in good agreement with the value of L; the deviation of the data at higher temperatures from the line predicted using eq. (5) probably arises because the change in the envelope wave function with the movement of the ions cannot be neglected. This change would tend to reduce AA at higher temperatures. The mean square displacement of an ion is determined by the interionic forces acting on that ion. A surface ion is acted upon by fewer neighbours than an ion in the interior of a solid and this will generally cause the mean square displacement to be larger for the surface ion. Such displacement may be anisotropic since the environment of a surface ion has a different symmetry than that of an interior ion. A number of calculations have shown that in the high temperature approximation the mean square displacement has a linear dependence on the absolute temperature and Wallis et a1.r5) have shown that (u~)~cT for surface ions and that the magnitude of (u2) is between 2 and 4 times higher for surface ions than in the bulk, but decreases within a few layers to that characteristic of the bulk. Low energy electron diffraction studiesls) confirm the large mean square displacements of the surface ions. No calculations appear to have been carried out for ionic solids but calculated valuess) for the temperature dependence of (u2) in nickel are available and these indicate an increase in (u2> by a factor of 4 as the temperature is raised from 100 to 500 “K. To see the magnitude of the effect, Y was taken to be a simple Gaussian and the maximum value of IY (O)j’ was put to 250 gauss (i.e., Mg+)17) and
TEMPERATE
EFFECTS
631
the vaIue of lulj(0)l’
was taken as 9 gauss at 2 A from the centre of the vacancy. With these assumptions the observed variation of 30% in the hfs between 100 and 500 “K could be obtained by a movement of the ion toward the vacancy by 4% of the bulk nearest-neighbour distance. For comparison, the calculated data of Wallis et a1.15) for Ni indicate that (~)* increases by 3.5% of the nearest neighbour distance over the same temperature range. The dipolar term B would also be expected to increase with temperature on this model and this is found experimentally. Similarly the Mg2+ ions (b) in fig.3 should also show considerable temperature effects but this could not be examined because the hfs was not clearly resolved. For comparison, the Mg 2+ ions of the bulk Ff centre should show a much smalier effect. A similar treatment to that used above and allowing for lattice expansion of 0.6% over 500°K indicates that the hfs should increase by about 2%; in fact a slight decrease of N 1% is observed and this agreement is regarded as reasonable in the light of this approximate treatment since the effects are small compared to the 30% increase observed with the F,+ centre. In conclusion, it appears that the abnormal vibrational amplitudes at surfaces can cause effects on the hyperfine constants of surface defects. A more quantitative theoretical treatment of surface defects is required to understand the behaviour of the system fully. Acknowledgements The author for comments computational
is indebted to Dr. M. Stoneham, Theoretical Physics Division, and suggestions on part of this paper and A. L. V. Tenth for help. References
1) 2) 3) 4) 5) 6) 7) 8)
W. M. Walsh, T. Jeener and N. Bloembergen, Phys. Rev. 139 (1965) A1338. E. Simanek and R. Orbach, Phys. Rev. 145 (1964) 191. W. Dreybolt, Phys. Status Solidi 21(1967) 99. W. Assmus and W. Dreybolt, Phys. Status Solidi 34 (1969) 183. R. L. Nelson, A. J. Tenth and B. J. Harmsworth, Trans. Faraday Sot. 63 (1967) 1427. A. J. Tenth and R. L. Nelson, J. Colloid Interface Sci. 26 (1968) 364. B. Henderson and J. E. Wertz, Advan. Phys. 17 (1968) 749. R. L. Nelson, J. W. Hale, B. J. Harmsworth and A. J. Tenth, Trans. Faraday Sot. 64 (1968) 2521. 9) A. J. Tenth and R. L. Nelson, Proc. Phys. Sot. (London) 92 (1967) 1055; S. M. Blinder, J. Chem. Phys. 33 (1960) 748. 10) K. C. To, A. M. Stoneham and B. Henderson, Phys. Rev. Sl(l969) 1237. 11) G. C. Benson, P. I. Freeman and E. Dempsey, Solid &r-faces and fhe Gas-Solid Interface, No. 33, Advances in Chemistry Series (1961) p. 26. 12) A. U. MacRae and C. W. Caldwell, Surface Sci. 2 (1964) 509. 13) I. Marklund and S. Anderson, Surface Sci. 5 (1966) 197. 14(a) B. S. Gourary and F. J. Adrian, Phys. Rev. 105(1957) 1180.
632
A. J.TENCH
(b) A. M. Stoneham, W. Hayes, P. H. S. Smith and J. P. Stott, Proc. Roy. Sot. (London) A 306 (1968) 369. 15) R. F. Wallis, B. C. Clark and R. Herman, in: The Structure and Chemistry of Solid Surfaces, Ed. G. Somorjai (Wiley, New York, 1969) p. 17-1. 16) A. U. MacRae, Surface Sci. 2 (1964) 522. 17) M. F. Crawford, F. M. Kelly, A. L. Shawlow and W. M. Grey, Phys. Rev. 76 (1949) 1527.