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Positron annihilation in liquid mercury
Volume 49A, number 4
PHYSICS LEITERS
23 September 1974
POSITRON ANNIHILATION IN LIQUID MERCURY L.D. BURTON and W.F. HUANG Department of Physics, University of Louisville, Louisville, Kentucky 40208, USA Received 31 May 1974 A positron wave function has been calculated for a vacancy and applied to the angular correlation data of liquid mercury. Our results indicate that 19% of all annihilations occur at vacancy sites.
Two photon angular correlation measurements from positrons annihilating in metals are in general bell-shaped, with an inverted parabola superimposed on a broader Gaussian distribution. The parabolic part of the distribution is attributed to annihilation of positrons with free electrons, and the Gaussian part to annihilation with core electrons. The intercept of the parabola gives the corresponding Fermi ener~’of the metal. For some metals the parabolic part of the angular correlation distribution becomes blurred on melting. Gustafson et al. [1] showed that there is a perfect parabola for the solid mercury, but the liquid mercury exhibits a blurring of as much as 20% at the Fermi limit. It has been considered unlikely that this is due to scattering of the electrons from the disordered lattice and the consequent blurring of the Fermi surface [2]. It was therefore suggested that the blurring is caused by positrons annihilating in vacancies of the liquid mercury where the positron wave function has been distorted. Temperature dependence of positron lifetime data for mercury and other metals lends further support to this interpretation [3—6]. As an attempt to account for the angular correlation distribution for liquid mercury, we used a simple model for the calculation of positron-vacancy interaction. The total angular correlation data is considered for our calculation to consist of three components, i.e., annihilation with conduction, vacancy and core electrons. The vacancy was taken to be spherical and have a volume of the average volume of a liquid mercury atom which has a radius of 3.41 Bohr units. Inside the spherical vacancy the potential energy for the p0sitron is issumed to be from that of a uniformly charged sphere with a charge density corresponding
I0 9.8
96
~9.2 ~9.O
I °
I
I
DISTANCE IN BOHR RADIUS
I
30
Fig. 1. Calculated radial positron wave function.
to Z = 1. The positron wave function was evaluated by solving numerically the Schrodinger equation with a boundary condition that the slope of the positron wave function vanishes at the vacancy boundary. Fig. 1 shows the calculated radial positron wave function. Our experimental angular correlation data for solid and liquid mercury confirms that reported in ref. [1]. A Gaussian distribution curve was first subtracted from the experimental data. Contribution from annihilation in the vacancies was then computed by evaluating the usual Fourier transform of the positron and electron wave function product and fitted to the tail part of the data (fig. 2). The remaining part of the data was fitted to a parabola. The fractionsof areas of the three components thus determined are % —
Parabola Vacancy Core
23 19 58 323
Volume 49A, number 4
PHYSICS LETTERS
1.0
0.8 ‘I)
z 0 U
W0.6 U z w 0
z “0.4 ‘Ii
> 4 -J U
02
_______________________________
Fig. 2. Open circles are from experimental data, — . — for parabolic fit, — — — — for annihilation in vacancies, and for the sum of the two curves.
We further checked our results with positron lifetime for liquid mercury. The total annihilation rate corresponds to the total area of the angular correlation curve, with each component corresponding to each respective fraction of area. The annihilation rate in conduction electrons can be given by [7] Xcond
=
where c~is the fine structure constant, n = 2 for the number of free electrons per atom and ~2is the average atomic volume for liquid mercury. The total annihilation rate of positrons in liquid mercury is then
324
2.63 X i09 sec~ which gives a mean positron lifetime of about 380 psec as compared to about 270 psec as reported by direct measurement [3]. Closer agreement could be obtained provided that more information about the charge density distribution at vacancy sites becomes available. The model seems to be able to account for the liquid mercury angular correlation data satisfactorily. The total area of all three components was determined to be within 1% of that of the experimental data. And the intercept of the parabolic fit is found to be 5.31 mrad. Within experimental uncertainty this value is in good agreement with the theoretical Fermi momentum of 5.17 mrad with two conduction electrons per mercury atom. =
0
0
23 September 1974
We are grateful to Dr. S.Y.Wu for helpful discussions. References [1] D.R.
Gustafson, A.R. Mackintosh and D.J. Zaffarano, Phys. Rev. 130 (1963) 1455. [2] N.H. March, Liquid metals (Pergamon Press, 1968). [3] M.H. Chu, G.J. Jan, P.K. Tseng and W.F. Huang, Phys. Lett. 43A (1973) 423. [4] I.K. MacKenzie, T.L. Khov, A.B. McDonald and B.T.A. McKee, Phys. Rev. Lett. 19 (1967) 946. [5] B.T.A. McKee, A.G.D. Jost and 1.K. MacKenzie, Can. J. P113’S. 50 (1972) 415.Stott, Solid State Comm. 7 (1969) [6] B. Bergersen and M.J. 1203. [7] S. Berko and J.S. Plaskett, Phys. Rev. 112 (1958) 1877.
PHYSICS LEITERS
23 September 1974
POSITRON ANNIHILATION IN LIQUID MERCURY L.D. BURTON and W.F. HUANG Department of Physics, University of Louisville, Louisville, Kentucky 40208, USA Received 31 May 1974 A positron wave function has been calculated for a vacancy and applied to the angular correlation data of liquid mercury. Our results indicate that 19% of all annihilations occur at vacancy sites.
Two photon angular correlation measurements from positrons annihilating in metals are in general bell-shaped, with an inverted parabola superimposed on a broader Gaussian distribution. The parabolic part of the distribution is attributed to annihilation of positrons with free electrons, and the Gaussian part to annihilation with core electrons. The intercept of the parabola gives the corresponding Fermi ener~’of the metal. For some metals the parabolic part of the angular correlation distribution becomes blurred on melting. Gustafson et al. [1] showed that there is a perfect parabola for the solid mercury, but the liquid mercury exhibits a blurring of as much as 20% at the Fermi limit. It has been considered unlikely that this is due to scattering of the electrons from the disordered lattice and the consequent blurring of the Fermi surface [2]. It was therefore suggested that the blurring is caused by positrons annihilating in vacancies of the liquid mercury where the positron wave function has been distorted. Temperature dependence of positron lifetime data for mercury and other metals lends further support to this interpretation [3—6]. As an attempt to account for the angular correlation distribution for liquid mercury, we used a simple model for the calculation of positron-vacancy interaction. The total angular correlation data is considered for our calculation to consist of three components, i.e., annihilation with conduction, vacancy and core electrons. The vacancy was taken to be spherical and have a volume of the average volume of a liquid mercury atom which has a radius of 3.41 Bohr units. Inside the spherical vacancy the potential energy for the p0sitron is issumed to be from that of a uniformly charged sphere with a charge density corresponding
I0 9.8
96
~9.2 ~9.O
I °
I
I
DISTANCE IN BOHR RADIUS
I
30
Fig. 1. Calculated radial positron wave function.
to Z = 1. The positron wave function was evaluated by solving numerically the Schrodinger equation with a boundary condition that the slope of the positron wave function vanishes at the vacancy boundary. Fig. 1 shows the calculated radial positron wave function. Our experimental angular correlation data for solid and liquid mercury confirms that reported in ref. [1]. A Gaussian distribution curve was first subtracted from the experimental data. Contribution from annihilation in the vacancies was then computed by evaluating the usual Fourier transform of the positron and electron wave function product and fitted to the tail part of the data (fig. 2). The remaining part of the data was fitted to a parabola. The fractionsof areas of the three components thus determined are % —
Parabola Vacancy Core
23 19 58 323
Volume 49A, number 4
PHYSICS LETTERS
1.0
0.8 ‘I)
z 0 U
W0.6 U z w 0
z “0.4 ‘Ii
> 4 -J U
02
_______________________________
Fig. 2. Open circles are from experimental data, — . — for parabolic fit, — — — — for annihilation in vacancies, and for the sum of the two curves.
We further checked our results with positron lifetime for liquid mercury. The total annihilation rate corresponds to the total area of the angular correlation curve, with each component corresponding to each respective fraction of area. The annihilation rate in conduction electrons can be given by [7] Xcond
=
where c~is the fine structure constant, n = 2 for the number of free electrons per atom and ~2is the average atomic volume for liquid mercury. The total annihilation rate of positrons in liquid mercury is then
324
2.63 X i09 sec~ which gives a mean positron lifetime of about 380 psec as compared to about 270 psec as reported by direct measurement [3]. Closer agreement could be obtained provided that more information about the charge density distribution at vacancy sites becomes available. The model seems to be able to account for the liquid mercury angular correlation data satisfactorily. The total area of all three components was determined to be within 1% of that of the experimental data. And the intercept of the parabolic fit is found to be 5.31 mrad. Within experimental uncertainty this value is in good agreement with the theoretical Fermi momentum of 5.17 mrad with two conduction electrons per mercury atom. =
0
0
23 September 1974
We are grateful to Dr. S.Y.Wu for helpful discussions. References [1] D.R.
Gustafson, A.R. Mackintosh and D.J. Zaffarano, Phys. Rev. 130 (1963) 1455. [2] N.H. March, Liquid metals (Pergamon Press, 1968). [3] M.H. Chu, G.J. Jan, P.K. Tseng and W.F. Huang, Phys. Lett. 43A (1973) 423. [4] I.K. MacKenzie, T.L. Khov, A.B. McDonald and B.T.A. McKee, Phys. Rev. Lett. 19 (1967) 946. [5] B.T.A. McKee, A.G.D. Jost and 1.K. MacKenzie, Can. J. P113’S. 50 (1972) 415.Stott, Solid State Comm. 7 (1969) [6] B. Bergersen and M.J. 1203. [7] S. Berko and J.S. Plaskett, Phys. Rev. 112 (1958) 1877.