- Email: [email protected]
Activation energy of a proton in α-Zr
Journal of the Less-Common
ACTIVATION
Metals 77 (1981)
ENERGY OF A PROTON IN c&r
NATTHI SINGH, SUDHIR MAHAJAN* Department
265
265 - 267
and SATYA PRAKASH
of Physics, Panjab University, Chandigarh-160014
(India)
(Received June 10,198O)
Summary The configurational and activation energies of a proton in cr-Zr are investigated. The Ashcroft model potential is used for the bare ions and a modified Hartree dielectric function is used for screening due to the conduction electrons. The octahedral (0) position is found to be the most probable position for the proton. The cakulated activation energy is found to correspond to the experimental value if the effective charge of the proton is assumed to be O.le. The O-O, O-T-O and O-T-T-O paths (T, tetrahedral) are found to be almost equally probable for proton movement through hopping processes.
In an earlier paper we reported the activation energy and the path of proton movement in lutetium [l f . Since much experimental [ 2J information exists for (w-Y& and a simplified model potential is also available [3], we present in this communication the results of a calculation of the configurational and activation energies for cu-Zr.The method and the formula adopted are the same as those discussed in ref. 1. For ready reference the configurational energy AE(s) is given as AE($
= !!??. 71
f:
.$1:
-
,,&+-
_?.&cos*r;sinqr,j
(1)
1
where 2 is the position coordinate of the proton,T1 is the position vector of the Ith host atom with respect to the proton, r. is the potential parameter, Ze is the effective charge of the proton, 2, is the ionicity of the Ith ion at & EoGi) is th e modified Hartree dielectric function in which the exchange and correlation corrections of Vashishta and Singwi [ 41 are used and 3 is the field wavevector. In this derivation lattice relaxation and local field corrections are neglected. *Permanent address: Department of Physics, Himachal Pradesh Agricultural Univer sity, Solan-173223, India. 0022-508S/81/0000-0000/.$02.50
@ Elsevier Sequoia/Printed
in The Netherlands
266
Fig. 1. The configurational energy of a proton in a-Zr. Inset: 0, 0 positions; tions; 0, host atoms.
x , T posi-
The lattice parameters Q and c are taken as 6.1038 au and 9.7226 au respectively; the protonic and the bare ionic charges are taken as le and -4e respectively. The sum in eqn. (I) is carried out for up to 12 nearest neighbours and the results are normalized assuming that the interaction energy vanishes beyond the twelfth nearest neighbour. The calculated configurational energy AE($) along O-O, T-T and O-T directions (0, octahedral; T, tetrahedral) in the (1120) plane is shown in Fig. 1, The general ~havio~ is the same as that found previously for lutetium. As the proton moves from an 0 position to another 0 position along the c axis, AE$) increases and reaches a maximum exactly half-way between the O-O positions nearest to the host atom. Along an O-T path, A&s) is a maximum at about 70% of the distance from the 0 site and then decreases to the value appropriate to the T site. Evidently the configurational energy at the 0 site is lower by 2.706 eV than the energy of its adjacent T site, so that the 0 position is the most probable position of the proton in at-Zr. Since the difference between the O-O and O-T barrier heights is of the order of the zero point energy (0.16 eV), the O-O and O-T-O paths are almost equally probable for diffusion of the proton. Because the T-T barrier height is only of the order of 10% of the O-O barrier height, the possibility of an O-T-T-0 path for diffusion cannot be ruled out. The activation energy, defined as the minimum height of the potential barrier to be crossed by the proton from one 0 position to another 0 position, is 3,060 eV in the present calculations. The experimental value [ 21 of the activation energy is 0.3 eV in the temperature range 305 - 610 K, and the calculated activation energy will therefore agree with the experimental value only if the effective charge of the proton is approximately O.le. This suggests
267
that there is a large build-up of charge around the proton in the e-Zr matrix and that the proton is nearly in its atomic state. Assuming that there is a harmonic potential well in the vicinity of the 0 position, the calculated frequency is found to be 3.96 X 1013 Hz which gives a vibrational energy of 0.016 eV for an effective charge of O.le for the ,proton. The vibrational energy is smaller by an order of magnitude than the activation energy, so that hopping diffusion is most probable for a proton in cr-Zr. From the neutron diffraction measurements of Narang et al. [ 21 it is predicted that hydrogen randomly occupies the tetrahedral positions in e-Zr. The experiment does not give any evidence for preferential occupation of 0 or T positions by the proton. However, from our calculations we predict that the preferred sites are 0 positions which are the regions of minimum ionic density in the crystal. The experiments were carried out on polycrystalline a-Zr with 3.2 at.% H, whereas our calculations are for a single proton in a crystal without internal stresses. The presence of point defects, extended defects, impurities or higher concentrations of hydrogen may altogether change the conclusions. Experiments on single crystals of cr-Zr with a few parts per million of hydrogen are still required to verify the theoretical predictions.
Acknowledgments The authors are grateful to Professor H. S. Hans and Professor V. B. Bhanot for encouragement throughout this work. One of us (N. S.) acknowledges financial support from the Council of Scientific and Industrial Research, New Delhi, India.
References 1 S. Prakash, J. E. Bonnet and P. Lucasson, J. Less-Common Met., 68 (1979) 1. 2 P. P. Narang, G. L. Paul and K. N. R. Taylor, J. Less-Common Met., 56 (1977) 3 M. L. Cohen and V. Heine, Solid State Phys., 24 (1970) 54. A. 0. E. Animalu, Phys. Rev. B, 8 (1973) 3542. 4 P. Vashishta and K. S. Singwi, Phys. Rev. B, 6 (1972) 875.
125.
ACTIVATION
Metals 77 (1981)
ENERGY OF A PROTON IN c&r
NATTHI SINGH, SUDHIR MAHAJAN* Department
265
265 - 267
and SATYA PRAKASH
of Physics, Panjab University, Chandigarh-160014
(India)
(Received June 10,198O)
Summary The configurational and activation energies of a proton in cr-Zr are investigated. The Ashcroft model potential is used for the bare ions and a modified Hartree dielectric function is used for screening due to the conduction electrons. The octahedral (0) position is found to be the most probable position for the proton. The cakulated activation energy is found to correspond to the experimental value if the effective charge of the proton is assumed to be O.le. The O-O, O-T-O and O-T-T-O paths (T, tetrahedral) are found to be almost equally probable for proton movement through hopping processes.
In an earlier paper we reported the activation energy and the path of proton movement in lutetium [l f . Since much experimental [ 2J information exists for (w-Y& and a simplified model potential is also available [3], we present in this communication the results of a calculation of the configurational and activation energies for cu-Zr.The method and the formula adopted are the same as those discussed in ref. 1. For ready reference the configurational energy AE(s) is given as AE($
= !!??. 71
f:
.$1:
-
,,&+-
_?.&cos*r;sinqr,j
(1)
1
where 2 is the position coordinate of the proton,T1 is the position vector of the Ith host atom with respect to the proton, r. is the potential parameter, Ze is the effective charge of the proton, 2, is the ionicity of the Ith ion at & EoGi) is th e modified Hartree dielectric function in which the exchange and correlation corrections of Vashishta and Singwi [ 41 are used and 3 is the field wavevector. In this derivation lattice relaxation and local field corrections are neglected. *Permanent address: Department of Physics, Himachal Pradesh Agricultural Univer sity, Solan-173223, India. 0022-508S/81/0000-0000/.$02.50
@ Elsevier Sequoia/Printed
in The Netherlands
266
Fig. 1. The configurational energy of a proton in a-Zr. Inset: 0, 0 positions; tions; 0, host atoms.
x , T posi-
The lattice parameters Q and c are taken as 6.1038 au and 9.7226 au respectively; the protonic and the bare ionic charges are taken as le and -4e respectively. The sum in eqn. (I) is carried out for up to 12 nearest neighbours and the results are normalized assuming that the interaction energy vanishes beyond the twelfth nearest neighbour. The calculated configurational energy AE($) along O-O, T-T and O-T directions (0, octahedral; T, tetrahedral) in the (1120) plane is shown in Fig. 1, The general ~havio~ is the same as that found previously for lutetium. As the proton moves from an 0 position to another 0 position along the c axis, AE$) increases and reaches a maximum exactly half-way between the O-O positions nearest to the host atom. Along an O-T path, A&s) is a maximum at about 70% of the distance from the 0 site and then decreases to the value appropriate to the T site. Evidently the configurational energy at the 0 site is lower by 2.706 eV than the energy of its adjacent T site, so that the 0 position is the most probable position of the proton in at-Zr. Since the difference between the O-O and O-T barrier heights is of the order of the zero point energy (0.16 eV), the O-O and O-T-O paths are almost equally probable for diffusion of the proton. Because the T-T barrier height is only of the order of 10% of the O-O barrier height, the possibility of an O-T-T-0 path for diffusion cannot be ruled out. The activation energy, defined as the minimum height of the potential barrier to be crossed by the proton from one 0 position to another 0 position, is 3,060 eV in the present calculations. The experimental value [ 21 of the activation energy is 0.3 eV in the temperature range 305 - 610 K, and the calculated activation energy will therefore agree with the experimental value only if the effective charge of the proton is approximately O.le. This suggests
267
that there is a large build-up of charge around the proton in the e-Zr matrix and that the proton is nearly in its atomic state. Assuming that there is a harmonic potential well in the vicinity of the 0 position, the calculated frequency is found to be 3.96 X 1013 Hz which gives a vibrational energy of 0.016 eV for an effective charge of O.le for the ,proton. The vibrational energy is smaller by an order of magnitude than the activation energy, so that hopping diffusion is most probable for a proton in cr-Zr. From the neutron diffraction measurements of Narang et al. [ 21 it is predicted that hydrogen randomly occupies the tetrahedral positions in e-Zr. The experiment does not give any evidence for preferential occupation of 0 or T positions by the proton. However, from our calculations we predict that the preferred sites are 0 positions which are the regions of minimum ionic density in the crystal. The experiments were carried out on polycrystalline a-Zr with 3.2 at.% H, whereas our calculations are for a single proton in a crystal without internal stresses. The presence of point defects, extended defects, impurities or higher concentrations of hydrogen may altogether change the conclusions. Experiments on single crystals of cr-Zr with a few parts per million of hydrogen are still required to verify the theoretical predictions.
Acknowledgments The authors are grateful to Professor H. S. Hans and Professor V. B. Bhanot for encouragement throughout this work. One of us (N. S.) acknowledges financial support from the Council of Scientific and Industrial Research, New Delhi, India.
References 1 S. Prakash, J. E. Bonnet and P. Lucasson, J. Less-Common Met., 68 (1979) 1. 2 P. P. Narang, G. L. Paul and K. N. R. Taylor, J. Less-Common Met., 56 (1977) 3 M. L. Cohen and V. Heine, Solid State Phys., 24 (1970) 54. A. 0. E. Animalu, Phys. Rev. B, 8 (1973) 3542. 4 P. Vashishta and K. S. Singwi, Phys. Rev. B, 6 (1972) 875.
125.