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The temperature dependence of the trapping rate of positrons by vacancies in metals
Volume 41A, number 3
PHYSICS LETTERS
THE TEMPERATURE
DEPENDENCE
POSITRONS
25 September 1972
OF THE TRAPPING
BY VACANCIES
RATE
OF
IN METALS
A. SEEeER Max-Planck-Institut fir Metallforschung, Stuttgart, and Institut fiir theoretische und angewandte Physik der Universitiit Stuttgart, Stuttgart, Germany Received 14 August 1972 The predicted T-‘12 temperature dependence of the trapping rate of thermalized positrons by vacancies in metals agrees well with experiments of Connors, Crisp, and West on quenched cadmium. The theory is used to estimate the deformation potential for positrons in aluminium.
Positron annihilation has developed into a very promising tool for determinin the formation energy % [l-8] . A prerequisite of monovacancies in metals, E,, to the accurate measurement of EFv is the determination of the temperature dependence of the rate of trapping, u, of thermalized positrons by an individual vacancy. In an earlier note [9] it was shown that under fairly general conditions u should be proportional to the inverse square root of the absolute temperature T.The simplest way to test this theoretical prediction is to measure the trapping rate due to quenched-in vacancies as a function of Tattemperatures low enough for the vacancies to be immobile. Connors, Crisp, and West [lo] compared the angular correlation of the positron annihilation r-radiation of a quenched and an annealed cadmium sample. Fig. 1 gives the difference m(O) of the counting rates at zero angles as a function of temperature. LLlr(O) should be proportional to u and is seen to show indeed, within experimental error, the predicted T1/2-dependence. The deviation of the data point at 2 12 K is to be attributed to the annealing of vacancies, as may be seen as follows: The tracer self-diffusion coefficient of cadmium parallel to the hexagonal axis is given by [ 1 l]
DT = 0.12exp(-0.8 1 eV/kT) cm2 s-l , the equilibrium
vacancy concentration
(1) by [ 121
Cl? = 1.35 exp(-0.40 eV/kT) . The vacancy jump frequency
out of the basal plane
(2)
-
(T/K)
-“2
Fig. 1. Temperature dependence of the difference AN(O) of the counting rates at zero angle between a quenched and an annealed cadmium sample, according to ref. [lo] .
follows from (1) and (2) as (correlation f= 0.78, lattice constant c = 5.62 a)
factor (3)
vlV,B = 4
D;lc2fCrq = 5 X 1Ol3exp(-0.41 eV/kT)s- ‘.
From (3) we find viva = 5X lo3 se1 forT=212K andv 1V,B=4 X 10’ s-I for T= 171K. Since an angular correlation experiment lasts typically several hours, we have to conclude that the vacancies were strongly affected in the 212 K experiment but not in the 17 1 K experiment. The temperature dependence of u is particularly important when positron annihilation measurements at high and at low temperatures are compared with 267
Volume
4 1A,number
3
PHYSICS
(clR2/m+ed)2
(2n/kTm+)“’
,
(4)
(p = density, fl= atomic volume, r. = capture radius of a vacancy, R = Plan&s constant divided by 27~1, valid for an elastically isotropic cubic metal, permits us to estimate the parameter fd of the deformation potential for positrons. If we insert into (4) r0 = = Wigner-Seitz radius, m, = mo, we obtain from the life-time measurements of McKee et al. [S] on aluminium, with EFv = 0.66 eV and the monoan error in [ 81;’ m eq. (4) of be replaced by h .
* Eq. (4) corrects bol e should
268
1972
(3)
where m, is the positron mass and cI the longitudinal velocity of sound [ 131. For m, = free electron mass m. and cl = 5.1 X lo3 m/s (aluminium) we obtain To = 21 K. Measurements of the temperature variation of u in the neighbourhood of To, together with a theory of inelastic scattering of positrons by phonons, should permit to determine the effective positron mass m, experimentally. The high-temperature expression * u = (8npro/3Q)
25 September
vacancy formation entropy Syv = 0.8 kJ [ 141, for the positron deformation potential in aluminum fd = 50 eV. This is a rather large value, but is should be kept in mind that an effective positron mass of a few free electron masses would bring it. into the expected order of magnitude of the Fermi energy.
each other, e.g., for establishing a quantitative relationship with the quenched in resistivity due to vacancies. According to the T-‘j2 - law u decreases by a factor of 7 between 20 K and 1000 K. The T’j2-law ceases to be valid when the positronphonon collisions are no longer elastic. This is below a temperature of about & = 12m,cflk,
LETTERS
[ 81 the sym-
References 111 I.K. MacKenzie,
T.L. Khoo, A.B. McDonald and B.T.A. McKee, Phys.Rev.Lett. 19 (1967) 946. 121 B. Bergersen and M.J. Stott, Solid State Commun. 7 (1969) 1203. 30A (1969) 24. [31 D.C. Connors and R.N. West, Phys.Lett. 141 D.C. Connors, V.H.C. Crisp and R.N. West, J.Physics F (Metal Phys.) 1 (1971). 355. on Positron [51 B.T.A. MacKee et al., Second Intern.Conf. Annihilation, Kingston, Ontario (1971). A.T. Stewart, Phys. [61 B.T.A. McKee, W.Triftshiuser, Rev.Lett. 28 (1972) 358. 171 B.T.A. McKee, A.G.D. Jost and 1.K. MacKenzie, Can.J.Phys.50 (1972) 417. iSI A. Seeger, Comments on Solid State Physics IV (1972), in press. 40A (1972) 135. 191 A. Seeger, Phys.Letters IlO1 D.C. Connors, V.H.C. Crisp and R.N. West, Phys.Letters 33A (1970) 180. ill1 Chin-wen Mao, Phys.Rev. BS (1972) 4693. 1121 R. Feder and A.S. Nowick, Phys.Rev.BS (1972) 1244. FestkGrperphysik, Band II [I31 A. Haug, Theoretische (Franz Deuticke, Wien 1970) 5 21b. 1141 A. Seeger and H. Mehrer, Vacancies and Interstitials in Metals, A. Seeger, D. Schumacher, W. Schilling and J. Diehl, eds, (North-Holland Publ. Comp., Amsterdam. 1970) p.
1.
PHYSICS LETTERS
THE TEMPERATURE
DEPENDENCE
POSITRONS
25 September 1972
OF THE TRAPPING
BY VACANCIES
RATE
OF
IN METALS
A. SEEeER Max-Planck-Institut fir Metallforschung, Stuttgart, and Institut fiir theoretische und angewandte Physik der Universitiit Stuttgart, Stuttgart, Germany Received 14 August 1972 The predicted T-‘12 temperature dependence of the trapping rate of thermalized positrons by vacancies in metals agrees well with experiments of Connors, Crisp, and West on quenched cadmium. The theory is used to estimate the deformation potential for positrons in aluminium.
Positron annihilation has developed into a very promising tool for determinin the formation energy % [l-8] . A prerequisite of monovacancies in metals, E,, to the accurate measurement of EFv is the determination of the temperature dependence of the rate of trapping, u, of thermalized positrons by an individual vacancy. In an earlier note [9] it was shown that under fairly general conditions u should be proportional to the inverse square root of the absolute temperature T.The simplest way to test this theoretical prediction is to measure the trapping rate due to quenched-in vacancies as a function of Tattemperatures low enough for the vacancies to be immobile. Connors, Crisp, and West [lo] compared the angular correlation of the positron annihilation r-radiation of a quenched and an annealed cadmium sample. Fig. 1 gives the difference m(O) of the counting rates at zero angles as a function of temperature. LLlr(O) should be proportional to u and is seen to show indeed, within experimental error, the predicted T1/2-dependence. The deviation of the data point at 2 12 K is to be attributed to the annealing of vacancies, as may be seen as follows: The tracer self-diffusion coefficient of cadmium parallel to the hexagonal axis is given by [ 1 l]
DT = 0.12exp(-0.8 1 eV/kT) cm2 s-l , the equilibrium
vacancy concentration
(1) by [ 121
Cl? = 1.35 exp(-0.40 eV/kT) . The vacancy jump frequency
out of the basal plane
(2)
-
(T/K)
-“2
Fig. 1. Temperature dependence of the difference AN(O) of the counting rates at zero angle between a quenched and an annealed cadmium sample, according to ref. [lo] .
follows from (1) and (2) as (correlation f= 0.78, lattice constant c = 5.62 a)
factor (3)
vlV,B = 4
D;lc2fCrq = 5 X 1Ol3exp(-0.41 eV/kT)s- ‘.
From (3) we find viva = 5X lo3 se1 forT=212K andv 1V,B=4 X 10’ s-I for T= 171K. Since an angular correlation experiment lasts typically several hours, we have to conclude that the vacancies were strongly affected in the 212 K experiment but not in the 17 1 K experiment. The temperature dependence of u is particularly important when positron annihilation measurements at high and at low temperatures are compared with 267
Volume
4 1A,number
3
PHYSICS
(clR2/m+ed)2
(2n/kTm+)“’
,
(4)
(p = density, fl= atomic volume, r. = capture radius of a vacancy, R = Plan&s constant divided by 27~1, valid for an elastically isotropic cubic metal, permits us to estimate the parameter fd of the deformation potential for positrons. If we insert into (4) r0 = = Wigner-Seitz radius, m, = mo, we obtain from the life-time measurements of McKee et al. [S] on aluminium, with EFv = 0.66 eV and the monoan error in [ 81;’ m eq. (4) of be replaced by h .
* Eq. (4) corrects bol e should
268
1972
(3)
where m, is the positron mass and cI the longitudinal velocity of sound [ 131. For m, = free electron mass m. and cl = 5.1 X lo3 m/s (aluminium) we obtain To = 21 K. Measurements of the temperature variation of u in the neighbourhood of To, together with a theory of inelastic scattering of positrons by phonons, should permit to determine the effective positron mass m, experimentally. The high-temperature expression * u = (8npro/3Q)
25 September
vacancy formation entropy Syv = 0.8 kJ [ 141, for the positron deformation potential in aluminum fd = 50 eV. This is a rather large value, but is should be kept in mind that an effective positron mass of a few free electron masses would bring it. into the expected order of magnitude of the Fermi energy.
each other, e.g., for establishing a quantitative relationship with the quenched in resistivity due to vacancies. According to the T-‘j2 - law u decreases by a factor of 7 between 20 K and 1000 K. The T’j2-law ceases to be valid when the positronphonon collisions are no longer elastic. This is below a temperature of about & = 12m,cflk,
LETTERS
[ 81 the sym-
References 111 I.K. MacKenzie,
T.L. Khoo, A.B. McDonald and B.T.A. McKee, Phys.Rev.Lett. 19 (1967) 946. 121 B. Bergersen and M.J. Stott, Solid State Commun. 7 (1969) 1203. 30A (1969) 24. [31 D.C. Connors and R.N. West, Phys.Lett. 141 D.C. Connors, V.H.C. Crisp and R.N. West, J.Physics F (Metal Phys.) 1 (1971). 355. on Positron [51 B.T.A. MacKee et al., Second Intern.Conf. Annihilation, Kingston, Ontario (1971). A.T. Stewart, Phys. [61 B.T.A. McKee, W.Triftshiuser, Rev.Lett. 28 (1972) 358. 171 B.T.A. McKee, A.G.D. Jost and 1.K. MacKenzie, Can.J.Phys.50 (1972) 417. iSI A. Seeger, Comments on Solid State Physics IV (1972), in press. 40A (1972) 135. 191 A. Seeger, Phys.Letters IlO1 D.C. Connors, V.H.C. Crisp and R.N. West, Phys.Letters 33A (1970) 180. ill1 Chin-wen Mao, Phys.Rev. BS (1972) 4693. 1121 R. Feder and A.S. Nowick, Phys.Rev.BS (1972) 1244. FestkGrperphysik, Band II [I31 A. Haug, Theoretische (Franz Deuticke, Wien 1970) 5 21b. 1141 A. Seeger and H. Mehrer, Vacancies and Interstitials in Metals, A. Seeger, D. Schumacher, W. Schilling and J. Diehl, eds, (North-Holland Publ. Comp., Amsterdam. 1970) p.
1.