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Diffusion of thallous ions in single crystals of KBr
J. Phys.
Chem. Solidc
Pergamon
Press 1963. Vol. 24, pp. 129-133.
DIFFUSION
Printed in Great Britain.
OF THALLOUS IONS IN SINGLE CRYSTALS OF KBr* R. ILLINGWORTH
The Institute of Optics, University
of Rochester,
(Received
30 July
Rochester
20, New York
1962)
Abstract-DifIusion of thallous ions in single crystals of KBr has been investigated within the temperature range 350-7OO’C. After heating pure KBr crystals in TlBr vapour, the distribution of Tl+ in the crystals was found by measuring the absorption spectrum as successive layers were removed from the crystals by polishing. The activation energy for diffusion is 1.03 eV. The temperature dependence of the diffusion coefficient confirms the substitutional nature of the luminescence centre in KBr(TI).
INTRODUCTION
THE following pages will describe measurement of the rate of diffusion of thallium in single crystals of potassium bromide as a function of temperature. In work on the luminescence of the thalliumactivated alkali halide phosphors, it is usually assumed that the luminescence centre consists of a monovalent Tl+ ion occupying a substitutional cation site. This view has been confirmed theoretically(1) and experimentally(s) for KCl(T1). The present work was undertaken to give further confirmation of the substitutional nature of the luminescence centre in KBr(TI). In addition, from an experimental point of view, knowledge of the diffusion coefficient as a function of temperature would be useful in connection with experiments involving the annealing and cooling of the phosphor and, under certain circumstances, might enable crystals of the phosphor to be prepared from the pure alkali halide by controlled diffusion from the vapour phase. THEORY
AND
PREVIOUS
WORK
If the Tl+ ion in KBr(T1) occupies a substitutional K+ sitei simple theory(a) shows that (liffusion can only occur if the K+ site adjacent to a * Research supported by the National Aeronautics and Space Administration, Goddard Space Flight Center, under Contract NASW-107. I
129
particular Tl+ site is vacant. coefficient may be written
Then
the diffusion
D = D’n, exp( -qS/kT),
(1)
where D’ = constant, n, = fraction of cation sites which are unoccupied, 4 = activation energy required to move a Tlf ion into a neighbouring positive ion vacancy, k = Boltzmann’s constant, T = absolute temperature. In the pure crystal, n, is equal to the concentration of Schottky defects. If the crystal contains, in addition to the Schottky defects, a small amount of some divalent cation impurity, an equal number of positive ion vacancies are required to compensate for the additional charge on the impurity. Thus equation (1) may be written
D = D’(n++y
exp(-
W/2kT)) exp(+kT), (2)
where n, = mol. fraction of divalent impurities, W = energy required to create a Schottky defect at temperature T”K, y is the constant which enters into the well-known expression for the concentration of Schottky defects.@) Substituting D1 = D’y and Da = D’n,, equation (2) becomes D = & exp(-(+W++)/kT)+Dz
exp(-#kT). 13)
TAMAI@) has measured the diffusion coefficient
130
R.
ILLINGWORTH
for Tl+ in single crystals of KBr in the tempera, ture region 520-700°C where the second term in equation (3) is assumed to be negligible. He finds that D is of the form D = 47 exp(- 1*98/U) cms/sec where Boltzmann’s constant is expressed in electron volts. GLASNER et d.(5) using a different technique, have investigated diffusion of Tl+ into powdered KBr, and find that, in the temperature range 140-205°C D = 0.23 exp( - 1~02/KT) cms/sec. In the present work, by means of an improved technique similar in principle to that used by TAMAI,(*) measurement of the diffusion coefficient of Tl+ in single crystals of KBr has been extended to lower temperatures where the second term on the right-hand side of equation (3) may be expected to predominate. EXPERIMENTAL
PROCEDURE
Crystals of KBr measuring about 10 x 5 x 3 mm were cut from a single ingot, grown by the writer, using the Kyropoulos technique. The diffusion treatment was carried out in a horizontal tube furnace with temperature control accurate to about _t 3°C. Each crystal was heated in a Pyrex tube to a diffusion temperature in the range 350-700°C. Thallous bromide was placed in another part of the tube and held at a lower temperature to provide a convenient vapour pressure of TlBr. To avoid reaction between the thallous bromide and oxygen, the Pyrex tube was flushed out with nitrogen, evacuated to forepump pressure, and sealed prior to the diffusion treatment. After heating for a time which varied from lhr to 2 days (depending on the rate of diffusion) and cooling quickly to room temperature, the crystals were cleaved in half and the absorption spectrum due to the Tl+ impurity measured with a Cary Recording Spectrophotometer. The distribution of Tlf in the crystal was then found by polishing layers of known thickness from the surface and measuring the optical absorption spectrum after each polish. Assuming that the maximum optical density in the A band in KBr(T1) is proportional to the number of Tlf ions in the crystal, the diffusion coefficient could be calculated from the dependence of the optical density on the thickness of the layer removed by polishing. Polishing was carried out with a silk cloth, using a paste of alumina powder and alcohol. Usually, each layer measured from 2 to 3 microns in thickness. The
thickness of the removed layers was measured with a simple optical comparator consisting of a fixed glass plate A with the upper surface polished to an optical flat, on one edge of which was pivoted another optical flat B. The crystal was cemented into a holder which was placed on a third optical flat C fixed in position with respect to A. One end of B rested on a steel ball, which in turn’ rested on the surface of the crystal. Thus there was a wedge-shaped air film between A and B, the wedge angle being determined by the height of the crystal surface above C. The initial wedge angle before polishing vas measured by recording the separation of interference fringes set up between A and B with parallel monochromatic light. The crystal (cemented in its holder was removed, the absorption spectrum measured, a, layer polished from the surface, the crystal replaced in the comparator, and the new fringe separation measured. From the change in fringe separation, the change in total thickness of the crystal was calculated. This simple device gave the thickness of the layer removed by polishing accurate to better than f O*8 microns in the range 5-50 microns.
If the dimensions of the KBr crystal are large compared with the depth of penetration of the Tl+ ions, the impurity concentration C(x, t) after heating the crystal for time t set at some steady temperature is given by the diffusion equation
where it is assumed that the diffusion coefficient D is independent of the impurity concentration, and x is the distance into the crystal measured from the surface. With the boundary conditions C(0, t) = CO = constant, and C(co, t) = 0,
(5)
(6)
the solution to (4) is C(x, t) =Cs(l-erfX),
(7)
where X = x . (4Dt)-l/2 and erf signifies Gauss’s error function. The magnitudes of CO and D were found from the experimental data by the following procedure. The maximum optical density in the A band due to absorption by Tl+ ions for which x > 6x is given by m
d&z = K
J C(x, t) ax,
(8)
DIFFUSION
OF THALLOUS
IONS
where K is a constant, the value of which will be calculated below, and da, is the optical density of a crystal when a layer of thickness Sx has been removed from the surface by polishing. Substituting (7) into (8), equation (8) becomes *
IN
SINGLE
CRYSTALS
SMAKULA’Sformula(‘)
OF KBr
131
it was found that
K = 1.50 x 10-a (ems)-1 (mol. per cent)-1.
RESULTS AND DISCUSSION Figure 1 shows the ultraviolet absorption spectrum of a typical crystal of KBr which had been da3 = KC0 (1 -erfX)dx. (9) heated for 1.5 x 105 set at 389°C in thallous bromI ide vapour and then polished. Figure 2 shows the 6x maximum optical density of the A band, plotted against 6x. The spectrum in Fig. 1 is identical to Then, making the substitutions that obtained from crystals of the phosphor preF(p) = (KCa(4Dt)rls)-1 dsz (10) pared by adding the impurity during growth from the melt,@) except that, in addition to the A, B and and C thallium bands, a very small unidentified band ,8 = 6x . (4Dt)-112, (11) is present at 3000 A. Absorption in this band, which did not occur in the undoped crystals, was (9) may be written in the form uniform throughout the crystals, whereas absorption in the thallium bands was confined to regions f(p) = T(l-erfX) dX. (12) of the crystals within 50 microns of the surface. B Since the magnitude of the 3000 A band bore no apparent relation to the quantity of thallium in In practice, the function f(p) was obtained from the crystal, and since the absorption coefficient at tables(s) and plotted against p. Then the experi3000 A was never greater than O*Ol ems-1, its premental values for d,, were plotted against 6x on a sence has been ignored. different sheet of paper. For convenience, logIn Fig. 2, the theoretical curve fits the experiarithmic graph paper was used. Then, as shown mental points for (4Dt)l/a = 5.75 x 10-4 ems and in Fig. 2, the theoretical curve (plotted with respect KCo(4Dt)lI2 = 76 Thus, for this particular crysto thef(lB> and/J axes) was fitted to the experimental tal, D = 5.5 x lo-13 cma/sec and Co = 0.88 mol. points (plotted with respect to the dgGand 6x axes) per cent. Figure 3 shows the concentration profile by adjusting the relative positions of the two sets for this crystal, calculated using these values for of axes. Since the experimental points in Fig. 2 D and Co. now fitted a curve of the form of equation (12), The magnitude of CO in the various crystals equations (10) and (11) were valid. Then, as normally lay within the limits of O-4 mol. per cent indicated in Fig. 2, when fl andf(@ were equal to to 1 mol. per cent. The theoretical curve f(p) unity, the corresponding values of Sx and dds could always be fitted accurately to the expericould be read directly from the superimposed mental points, indicating that D was independent graphs and substituted into equations (10) and of the impurity concentration, and that boundary (11) to give the magnitudes of KCa(4Dt)l/a and conditions (5) and (6) were obeyed. The accuracy (4Dt)1/2. of the individual measurements of D was estimated The constant K was evaluated in the following in several crystals by measuring the diffusion coway. From (8) the optical density d’ of a crystal efficient at the opposite faces of each crystal. The of KBr(T1) of thickness h ems in which the Tl+ two values for D obtained in this way for each impurity concentration C’ is uniform throughout crystal, agreed to within 5 per cent. the crystal is In Fig. 4, log D is plotted against the reciprocal a of the absolute temperature. The line drawn d’ = K C’ dx = KC’h. (13) through the experimental points satisfies a relation s of the form of equation (3), 0 Using TAMAI’S value@) of 0.073 for the oscillator strength of the A band in KBr(T1) and applying
D = 50 exp(-2*01/KT) +4-l x 10-s exp( - 1*03/M)
(14)
132
R.
ILLINGWORTH
WAVELENGTH
(1)
FIG. 1. Absorption spectrum of a crystal of KBr previously heated for 1.5 x 105 set at 389°C in TlBr vapour. A layer of thickness 6.3 x 1O-4 cm has been removed from the surface of the crystal.
where D is expressed in cmz/sec, and k is Boltzmann’s constant expressed in eV. In fitting a curve of the form of equation (3) to the experimental points, there is an uncertainty of about 5 3 per cent in the gradients of the high and low temperature regions (proportional to (QW+ #) and I$ respectively). This relatively small uncertainty in the gradients leads to large errors of about + 80 per cent in the values for D1 and Dz which are obtained by extrapolation to l/T = 0. It is apparent that the experimental results confirm the simple model for the diffusion of a substitutional Tl+ impurity. Comparison of (3) with (14) gives a value of 1.96 eV for Win the temperature range of the experiment. This value for W may be compared with the theoretical value of 1.92 eV calculated by MOTT and LITTLETON@)for zero degrees Kelvin. In addition, from equations (2) (3), and (14) assuming@) that y is of the order of 50, the concentration of divalent impurities is of the order of 5 x 10-5 mol. fraction, a reasonable
concentration for crystals grown from the melt without previous recrystallization. Examination of Fig. 4 shows that the scatter of the experimental points about the solid line is considerably greater in the low-temperature region than in the high-temperature region, and, at low temperatures, is somewhat larger than expected from random errors in the measurement of the temperature and diffusion coefficient. It is thought that these deviations of the diffusion coefficient from the mean value are caused by variations in the concentration of positive ion vacancies throughout the ingot of KBr, presumably arising during growth from the melt. The results given above are in agreement with TAMAI’S data@) in the temperature range 520700%, and the magnitude of the activation energy + agrees well with the value obtained in the temperature range 140°C to 30°C by GLASNERet u&(s) using powdered KBr instead of single crystals. Thus, for the single crystals used in the present
DIFFUSION
OF
THALLOUS
IONS
IN
SINGLE
CRYSTALS
OF
KBr
133
1
experiment, it is probable that the diffusion coefficient may be represented by equation (14) for all temperatures at which significant diffusion occurs.
--I
d-.
0
2’
4
6 X
6
IO
I
12
i
-I
(MICRONS)
FIG. 3. Concentration profile calculated from the data given by Fig. 2. C is the Tl+ concentration in mol. per cent and x is the distance into the crystal measured from the surface.
-*
I
II
I
VFIG. 2. The experimental values for de% are plotted against 6x. The solid line is the theoretical function j(g) plotted against b and is fitted to the experimental points by adjusting the relative positions of the two sets of axes. Since j(B) = 1 when de% = 7.6, and fi = 1 when 6x = 5.75 x 10-4 cm, the values of (4Dt)l12 and (KCs(4Dt)r/s can be found immediately from equations (10) and (11).
i6-
REFERENCES
Ti-
1. BRAUERP. 2. Nuturj. 6a, 560 (1951); ibid 7a, 372 (1952). 2. RUNCIEMAN W. A. and STEWARDE. G., Proc. phys. Sot. Lond. A66,484 (1953). 3. MOTT N. F. and GURNEY R. W., Electronic Processes in Ionic Crystals, p. 30. University Press, Oxford (1957). 4. TAMAI T., J. phys. Sot. Japan 16, 2459 (1961). 5. GLA~NER A., REJOANA. and REISFELDR., J. Phys. Chews. Solids 19, 332 (1961). 6. cf. JOST W., D@uion in solids, liquids, and gases. Academic Press, New York (1960). 7. SMAKULA A., 2. Phys. 59, 603 (1930). 8. TAMAI T., J. phys. Sot. Japan 15, 2116 (1960). 9. MOTT N. F. and LITTLETON M. J., Trans. Faraday Sot. 34, 685 (1938).
1
H-
E19
FIG. 4. Log&
plotted against the reciprocal absolute temperature.
of the
Chem. Solidc
Pergamon
Press 1963. Vol. 24, pp. 129-133.
DIFFUSION
Printed in Great Britain.
OF THALLOUS IONS IN SINGLE CRYSTALS OF KBr* R. ILLINGWORTH
The Institute of Optics, University
of Rochester,
(Received
30 July
Rochester
20, New York
1962)
Abstract-DifIusion of thallous ions in single crystals of KBr has been investigated within the temperature range 350-7OO’C. After heating pure KBr crystals in TlBr vapour, the distribution of Tl+ in the crystals was found by measuring the absorption spectrum as successive layers were removed from the crystals by polishing. The activation energy for diffusion is 1.03 eV. The temperature dependence of the diffusion coefficient confirms the substitutional nature of the luminescence centre in KBr(TI).
INTRODUCTION
THE following pages will describe measurement of the rate of diffusion of thallium in single crystals of potassium bromide as a function of temperature. In work on the luminescence of the thalliumactivated alkali halide phosphors, it is usually assumed that the luminescence centre consists of a monovalent Tl+ ion occupying a substitutional cation site. This view has been confirmed theoretically(1) and experimentally(s) for KCl(T1). The present work was undertaken to give further confirmation of the substitutional nature of the luminescence centre in KBr(TI). In addition, from an experimental point of view, knowledge of the diffusion coefficient as a function of temperature would be useful in connection with experiments involving the annealing and cooling of the phosphor and, under certain circumstances, might enable crystals of the phosphor to be prepared from the pure alkali halide by controlled diffusion from the vapour phase. THEORY
AND
PREVIOUS
WORK
If the Tl+ ion in KBr(T1) occupies a substitutional K+ sitei simple theory(a) shows that (liffusion can only occur if the K+ site adjacent to a * Research supported by the National Aeronautics and Space Administration, Goddard Space Flight Center, under Contract NASW-107. I
129
particular Tl+ site is vacant. coefficient may be written
Then
the diffusion
D = D’n, exp( -qS/kT),
(1)
where D’ = constant, n, = fraction of cation sites which are unoccupied, 4 = activation energy required to move a Tlf ion into a neighbouring positive ion vacancy, k = Boltzmann’s constant, T = absolute temperature. In the pure crystal, n, is equal to the concentration of Schottky defects. If the crystal contains, in addition to the Schottky defects, a small amount of some divalent cation impurity, an equal number of positive ion vacancies are required to compensate for the additional charge on the impurity. Thus equation (1) may be written
D = D’(n++y
exp(-
W/2kT)) exp(+kT), (2)
where n, = mol. fraction of divalent impurities, W = energy required to create a Schottky defect at temperature T”K, y is the constant which enters into the well-known expression for the concentration of Schottky defects.@) Substituting D1 = D’y and Da = D’n,, equation (2) becomes D = & exp(-(+W++)/kT)+Dz
exp(-#kT). 13)
TAMAI@) has measured the diffusion coefficient
130
R.
ILLINGWORTH
for Tl+ in single crystals of KBr in the tempera, ture region 520-700°C where the second term in equation (3) is assumed to be negligible. He finds that D is of the form D = 47 exp(- 1*98/U) cms/sec where Boltzmann’s constant is expressed in electron volts. GLASNER et d.(5) using a different technique, have investigated diffusion of Tl+ into powdered KBr, and find that, in the temperature range 140-205°C D = 0.23 exp( - 1~02/KT) cms/sec. In the present work, by means of an improved technique similar in principle to that used by TAMAI,(*) measurement of the diffusion coefficient of Tl+ in single crystals of KBr has been extended to lower temperatures where the second term on the right-hand side of equation (3) may be expected to predominate. EXPERIMENTAL
PROCEDURE
Crystals of KBr measuring about 10 x 5 x 3 mm were cut from a single ingot, grown by the writer, using the Kyropoulos technique. The diffusion treatment was carried out in a horizontal tube furnace with temperature control accurate to about _t 3°C. Each crystal was heated in a Pyrex tube to a diffusion temperature in the range 350-700°C. Thallous bromide was placed in another part of the tube and held at a lower temperature to provide a convenient vapour pressure of TlBr. To avoid reaction between the thallous bromide and oxygen, the Pyrex tube was flushed out with nitrogen, evacuated to forepump pressure, and sealed prior to the diffusion treatment. After heating for a time which varied from lhr to 2 days (depending on the rate of diffusion) and cooling quickly to room temperature, the crystals were cleaved in half and the absorption spectrum due to the Tl+ impurity measured with a Cary Recording Spectrophotometer. The distribution of Tlf in the crystal was then found by polishing layers of known thickness from the surface and measuring the optical absorption spectrum after each polish. Assuming that the maximum optical density in the A band in KBr(T1) is proportional to the number of Tlf ions in the crystal, the diffusion coefficient could be calculated from the dependence of the optical density on the thickness of the layer removed by polishing. Polishing was carried out with a silk cloth, using a paste of alumina powder and alcohol. Usually, each layer measured from 2 to 3 microns in thickness. The
thickness of the removed layers was measured with a simple optical comparator consisting of a fixed glass plate A with the upper surface polished to an optical flat, on one edge of which was pivoted another optical flat B. The crystal was cemented into a holder which was placed on a third optical flat C fixed in position with respect to A. One end of B rested on a steel ball, which in turn’ rested on the surface of the crystal. Thus there was a wedge-shaped air film between A and B, the wedge angle being determined by the height of the crystal surface above C. The initial wedge angle before polishing vas measured by recording the separation of interference fringes set up between A and B with parallel monochromatic light. The crystal (cemented in its holder was removed, the absorption spectrum measured, a, layer polished from the surface, the crystal replaced in the comparator, and the new fringe separation measured. From the change in fringe separation, the change in total thickness of the crystal was calculated. This simple device gave the thickness of the layer removed by polishing accurate to better than f O*8 microns in the range 5-50 microns.
If the dimensions of the KBr crystal are large compared with the depth of penetration of the Tl+ ions, the impurity concentration C(x, t) after heating the crystal for time t set at some steady temperature is given by the diffusion equation
where it is assumed that the diffusion coefficient D is independent of the impurity concentration, and x is the distance into the crystal measured from the surface. With the boundary conditions C(0, t) = CO = constant, and C(co, t) = 0,
(5)
(6)
the solution to (4) is C(x, t) =Cs(l-erfX),
(7)
where X = x . (4Dt)-l/2 and erf signifies Gauss’s error function. The magnitudes of CO and D were found from the experimental data by the following procedure. The maximum optical density in the A band due to absorption by Tl+ ions for which x > 6x is given by m
d&z = K
J C(x, t) ax,
(8)
DIFFUSION
OF THALLOUS
IONS
where K is a constant, the value of which will be calculated below, and da, is the optical density of a crystal when a layer of thickness Sx has been removed from the surface by polishing. Substituting (7) into (8), equation (8) becomes *
IN
SINGLE
CRYSTALS
SMAKULA’Sformula(‘)
OF KBr
131
it was found that
K = 1.50 x 10-a (ems)-1 (mol. per cent)-1.
RESULTS AND DISCUSSION Figure 1 shows the ultraviolet absorption spectrum of a typical crystal of KBr which had been da3 = KC0 (1 -erfX)dx. (9) heated for 1.5 x 105 set at 389°C in thallous bromI ide vapour and then polished. Figure 2 shows the 6x maximum optical density of the A band, plotted against 6x. The spectrum in Fig. 1 is identical to Then, making the substitutions that obtained from crystals of the phosphor preF(p) = (KCa(4Dt)rls)-1 dsz (10) pared by adding the impurity during growth from the melt,@) except that, in addition to the A, B and and C thallium bands, a very small unidentified band ,8 = 6x . (4Dt)-112, (11) is present at 3000 A. Absorption in this band, which did not occur in the undoped crystals, was (9) may be written in the form uniform throughout the crystals, whereas absorption in the thallium bands was confined to regions f(p) = T(l-erfX) dX. (12) of the crystals within 50 microns of the surface. B Since the magnitude of the 3000 A band bore no apparent relation to the quantity of thallium in In practice, the function f(p) was obtained from the crystal, and since the absorption coefficient at tables(s) and plotted against p. Then the experi3000 A was never greater than O*Ol ems-1, its premental values for d,, were plotted against 6x on a sence has been ignored. different sheet of paper. For convenience, logIn Fig. 2, the theoretical curve fits the experiarithmic graph paper was used. Then, as shown mental points for (4Dt)l/a = 5.75 x 10-4 ems and in Fig. 2, the theoretical curve (plotted with respect KCo(4Dt)lI2 = 76 Thus, for this particular crysto thef(lB> and/J axes) was fitted to the experimental tal, D = 5.5 x lo-13 cma/sec and Co = 0.88 mol. points (plotted with respect to the dgGand 6x axes) per cent. Figure 3 shows the concentration profile by adjusting the relative positions of the two sets for this crystal, calculated using these values for of axes. Since the experimental points in Fig. 2 D and Co. now fitted a curve of the form of equation (12), The magnitude of CO in the various crystals equations (10) and (11) were valid. Then, as normally lay within the limits of O-4 mol. per cent indicated in Fig. 2, when fl andf(@ were equal to to 1 mol. per cent. The theoretical curve f(p) unity, the corresponding values of Sx and dds could always be fitted accurately to the expericould be read directly from the superimposed mental points, indicating that D was independent graphs and substituted into equations (10) and of the impurity concentration, and that boundary (11) to give the magnitudes of KCa(4Dt)l/a and conditions (5) and (6) were obeyed. The accuracy (4Dt)1/2. of the individual measurements of D was estimated The constant K was evaluated in the following in several crystals by measuring the diffusion coway. From (8) the optical density d’ of a crystal efficient at the opposite faces of each crystal. The of KBr(T1) of thickness h ems in which the Tl+ two values for D obtained in this way for each impurity concentration C’ is uniform throughout crystal, agreed to within 5 per cent. the crystal is In Fig. 4, log D is plotted against the reciprocal a of the absolute temperature. The line drawn d’ = K C’ dx = KC’h. (13) through the experimental points satisfies a relation s of the form of equation (3), 0 Using TAMAI’S value@) of 0.073 for the oscillator strength of the A band in KBr(T1) and applying
D = 50 exp(-2*01/KT) +4-l x 10-s exp( - 1*03/M)
(14)
132
R.
ILLINGWORTH
WAVELENGTH
(1)
FIG. 1. Absorption spectrum of a crystal of KBr previously heated for 1.5 x 105 set at 389°C in TlBr vapour. A layer of thickness 6.3 x 1O-4 cm has been removed from the surface of the crystal.
where D is expressed in cmz/sec, and k is Boltzmann’s constant expressed in eV. In fitting a curve of the form of equation (3) to the experimental points, there is an uncertainty of about 5 3 per cent in the gradients of the high and low temperature regions (proportional to (QW+ #) and I$ respectively). This relatively small uncertainty in the gradients leads to large errors of about + 80 per cent in the values for D1 and Dz which are obtained by extrapolation to l/T = 0. It is apparent that the experimental results confirm the simple model for the diffusion of a substitutional Tl+ impurity. Comparison of (3) with (14) gives a value of 1.96 eV for Win the temperature range of the experiment. This value for W may be compared with the theoretical value of 1.92 eV calculated by MOTT and LITTLETON@)for zero degrees Kelvin. In addition, from equations (2) (3), and (14) assuming@) that y is of the order of 50, the concentration of divalent impurities is of the order of 5 x 10-5 mol. fraction, a reasonable
concentration for crystals grown from the melt without previous recrystallization. Examination of Fig. 4 shows that the scatter of the experimental points about the solid line is considerably greater in the low-temperature region than in the high-temperature region, and, at low temperatures, is somewhat larger than expected from random errors in the measurement of the temperature and diffusion coefficient. It is thought that these deviations of the diffusion coefficient from the mean value are caused by variations in the concentration of positive ion vacancies throughout the ingot of KBr, presumably arising during growth from the melt. The results given above are in agreement with TAMAI’S data@) in the temperature range 520700%, and the magnitude of the activation energy + agrees well with the value obtained in the temperature range 140°C to 30°C by GLASNERet u&(s) using powdered KBr instead of single crystals. Thus, for the single crystals used in the present
DIFFUSION
OF
THALLOUS
IONS
IN
SINGLE
CRYSTALS
OF
KBr
133
1
experiment, it is probable that the diffusion coefficient may be represented by equation (14) for all temperatures at which significant diffusion occurs.
--I
d-.
0
2’
4
6 X
6
IO
I
12
i
-I
(MICRONS)
FIG. 3. Concentration profile calculated from the data given by Fig. 2. C is the Tl+ concentration in mol. per cent and x is the distance into the crystal measured from the surface.
-*
I
II
I
VFIG. 2. The experimental values for de% are plotted against 6x. The solid line is the theoretical function j(g) plotted against b and is fitted to the experimental points by adjusting the relative positions of the two sets of axes. Since j(B) = 1 when de% = 7.6, and fi = 1 when 6x = 5.75 x 10-4 cm, the values of (4Dt)l12 and (KCs(4Dt)r/s can be found immediately from equations (10) and (11).
i6-
REFERENCES
Ti-
1. BRAUERP. 2. Nuturj. 6a, 560 (1951); ibid 7a, 372 (1952). 2. RUNCIEMAN W. A. and STEWARDE. G., Proc. phys. Sot. Lond. A66,484 (1953). 3. MOTT N. F. and GURNEY R. W., Electronic Processes in Ionic Crystals, p. 30. University Press, Oxford (1957). 4. TAMAI T., J. phys. Sot. Japan 16, 2459 (1961). 5. GLA~NER A., REJOANA. and REISFELDR., J. Phys. Chews. Solids 19, 332 (1961). 6. cf. JOST W., D@uion in solids, liquids, and gases. Academic Press, New York (1960). 7. SMAKULA A., 2. Phys. 59, 603 (1930). 8. TAMAI T., J. phys. Sot. Japan 15, 2116 (1960). 9. MOTT N. F. and LITTLETON M. J., Trans. Faraday Sot. 34, 685 (1938).
1
H-
E19
FIG. 4. Log&
plotted against the reciprocal absolute temperature.
of the