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Proton diffusion in lutetium
Journal of the Less-Common Metals, @ Elsevier Sequoia S.A., Lausanne -
PROTON
DIFFUSION
68 (1979)
Printed
1
-5
in the Netherlands
IN LUTETIUM
S. PRAKASH* Physics
Department,
J. E, BONNET
Panjab
Chandigarh
(India)
and P. LUCASSON
k’quipe de Recherche 91405 Orsay (France)
(Received
University,
January
No.
720,
associke
au C.N.R.S.,
Bat.
350,
Universite’
Paris Sud,
8, 1978)
Summary The configurational energy of the proton is investigated in the lutetium matrix. The bare ion is represented by a model potential and the modified Hartree dielectric function is used for the screening due to conduction electrons. The octahedral (0) position is found to be the most probable position for the proton and the activation energy is found to be in good agreement with the experimental value for an effective charge on the proton of 0.23e. It is concluded that the proton diffuses by the hopping process and that the O-O, 0-T(tetrahedral)-0 and O-T-T-O paths are equally probable for the diffusion. The configurational energy obtained by varying the c/u ratio and the core radius, and that for the dilated lattice, are also calculated and discussed.
The thermal expansion [l] and the residual electrical resistivity [ 21 of lutetium-hydrogen solid solutions have been extensively studied in this laboratory. An understanding of these results demands the knowledge of the effective charge of the proton, its most probable position and its path of diffusion in the matrix. For this purpose, a simplified model has been adopted for the lutetium matrix and these properties have been studied. We represent the bare ions of the host lattice by the Ashcroft model potential [3] and adopt the modified Hartree dielectric function, which explicitly includes the exchange correlation corrections, for the screening due to the conduction electrons. The effective charge of the proton is taken as an adjustable parameter. Neglecting the lattice relaxation and the local field corrections, the configurational energy A E(z ) of the proton is written as follows [4] :
*Part of this work was performed
while this author
was at Orsay, France.
2
AE(ii)
= ‘$F
;,I!!$ I
-qj-dq
(1 +_)
‘OS “0s sin!/ 0
(1)
where R’ and z are the position coordinate and the effective charge of the proton, FI is the position vector of the Ith host atom with respect to proton position, zI the charge of the host ion and r. the potential parameter. co(q) is the modified Hartree dielectric function where exchange correlation after Vashishta and Singwi @] is used and q is the field wavevector. To calculate AE(R) in eqn. (l), we chose z = le, zr = -3e and r. = 1.6029 au which is the effective ionic radius [6]. The lattice parameters a and c are taken as 6.633 and 10.519 au, respectively, giving c/u = 1.586. The sum in eqn. (1) is carried out for up to twelve nearest neighbours; at this point the interaction energy becomes almost negligible. For comparative study the results are further normalized assuming that the interaction energy vanishes beyond the twelfth nearest neighbour. The integration over q is carried out numerically. The octahedral (0), tetrahedral (T) and host atom positions in the (1120) plane are shown in Fig. 1. The coordinates of the 0 and T positions are (l/3, l/3,2/3, l/4) and (2/3, 2/3, z/3, $ - +(a/~)‘), respectively. To decide the probable position and the path of the proton, AE(R) is calculated along the O-O, T-T and the O-T directions. These results are also shown in Fig. 1 (curve (a)). As t+heproton moves from one 0 position to another along the vertical axis, AE(R) increases and reaches a maximum halfway between the two 0 positions nearest to the host atom in the (1120) plane. However, as th_eproton moves from the 0 to the T position along an oblique axis, AE(R) shows a maximum at approximately 70% of the O-T distance from the 0 position and AE(R) decreases $.&her and shows a minimum at the T position. Another maximum of AE(R) is obtained between the two T positions. As the c/u ratio differs slightly from the ideal value, the heights of the potential barriers between the two T and the twz 0 positions are found to be only approximately the same. Evidently A&R) has a minimum at the 0 position where it is lower by 0.08 eV than its value at the T position; the 0 position is thus the most stable position for the proton in lutetium. The heights of the O-T and O-O barriers are 1.39 and 1.22 eV, respectively. Since the difference in the barrier heights is of the order of the zero point energy, the O-O and the O-T-O paths for the diffusion of the proton are equally probable. The height of the T-T barrier is only 10% of that of the O-O barrier; therefore the possibility of the O-T-T-O path cannot be ruled out. We varied the triangular configuration of the passage windows by changing the c/a ratio at constant atomic volume and recalculated AE(&) for the O-O, O-T and T-T paths. These results for the ideal value of c/u and for c/a = 1.68 are shown by curves (b) and (c) in Fig. 1. The configurational energy increases along the c axis except in a small region about the T and 0 positions. This is consistent with a geometrical deformation in which the crystal is extended along the c axis or contracted along the a axis. For the ideal c/u ratio, the O-O and O-T barrier heights are the same.
3
Fig. 1. Configurational energy of a proton in lutetium. Curves (a), (b) and (c) represent the configurational energies for the c/a ratios 1.586, 1.633 and 1.68, respectively. The broken and broken-dotted lines represent the configurational energy for 2.5% and 5% expansions of the lattice. In the inset, open circles represent the octahedral position, X represents the tetrahedral position and solid circles represent the host atoms. Note that for clarity the path lengths T-T, O-O, T-O are normalized; their absolute values are different in each case since the calculations have been performed with different lattice parameters (see text).
The values of AE(g) at the 0 and T positions remain the same. The width of the potential well also remains almost the same. The heights of the O-O and T-T barriers increase with increasing c/u ratio while the height of the O-T barrier decreases. Therefore the O-T-O path of the proton becomes most probable and faster diffusion is expected. We also investigated the configurational energy of the proton in the expanded lattice along the O-O, O-T and T-T paths. These results are shown in Fig. 1 by the broken and broken-dotted lines for the 2.5% and 5% expansions, respectively. The configurational energy decreases by approximately 5 and lo%, respectively; the widths of the O-O, T-T and O-T potential wells also decrease by approximately 2 - 4%, but the relative heights of the barriers remain the same. This predicts faster diffusion of hydrogen in the expanded lattice. The lutetium ion has a partially filled d shell, so that the single parameter Ashcroft model potential is a rather oversimplified picture of the lutetium ion. However, the partial localization of d and f electrons around
4
the rigid core may effectively increase the core radius. We therefore increased r. from 1.603 to 1.65 A and recalculated the configurational energy along various possible paths in the (1120) plane, The configurational energy increased by approximately 8% but no substantial change was found in the relative heights of the potential barriers. As the calculated curve is very close to curve (a) it is not shown in the figure. If we define the activation energy as the minimum height of the potential barrier to be crossed by the proton from one 0 position to another, it is calculated to be 1.22 eV. The indirectly measured [ 71 activation energy is 0.28 * 0.02 eV. The experimental value involves many uncertainties, but the calculated value is almost four times as large as the experimental value. The calculated and the experimental values agree closely for an effective charge of 0.23e on the proton. This conclusion is similar to that found for noble metals [4] ; it suggests that the proton is near the atomic state in the lutetium matrix. The potential wells around the 0 and T positions are neither symmetric nor harmonic. However, if we fit the harmonic potential well in the vicinity of the 0 position along the c axis, the vibrational frequency and the vibrational energy are found to be 3.29 X 1013 Hz and 0.136 eV, respectively. The vibrational energy is smaller than the activation energy by approximately an order of magnitude, so that the tunnelling process is least probable and the hopping process is most probable for proton diffusion in the metal. The analysis of the neutron scattering data for LuH,~~ and ZrH, in the CYphase [l, 81 shows that hydrogen is located at the T position while our calculations predict the 0 position, which is in the minimum ionic density region, as the most probable position. The hydrogen concentrations in these compounds, however, were rather high. One should keep in mind that our calculations are for a single proton in a perfect matrix. If the crystal is not perfect and contains point or extended defects or impurities, the conclusions will change altogether. For example, the proton will always be trapped in a vacancy if one is available. If the crystal contains a higher concentration of hydrogen, the results may be altered further. The present calculations may be improved if one takes a more realistic description of the lutetium matrix where partially filled d bands are considered explicitly. However, this would involve a prohibitively difficult computation.
Acknowledgments The authors benefited from informative discussions with Dr. J. N. Daou. One of us (S. P.) is grateful to Professor A. Lucasson for providing laboratory facilities and to the C.N.R.S. (France) for financial assistance.
References 1 2 3 4 5 6
J. E. Bonnet, J. Less-Common Met., 49 (1976) 123. J. N. Daou, A. Lucasson and P. Lucasson, Solid State Commun., 19 (1976) 895. M. L. Cohen and V. Heine, Solid State Phys., 24 (1970) 38. S. Prakash, Phys. Rev., Sect. B, 18 (8) (1978) 3980. P. Vashishta and K. S. Singwi, Phys. Rev., Sect. B, 6 (1972) 875. K. N. R. Taylor and M. I. Darby, Physics of Rare Earth Solids, Chapman and Hall, 1972, p. 8. 7 J. N. Daou and R. Viallard, Congre‘s International L’hydroge‘ne duns les Mktaux, T. 1, 123, Edition Sciences et Industrie, Paris, 1972. 8 P. P. Narang, G. L. Paul and K. N. R. Taylor, J. Less-Common Met., 56 (1977) 125.
PROTON
DIFFUSION
68 (1979)
Printed
1
-5
in the Netherlands
IN LUTETIUM
S. PRAKASH* Physics
Department,
J. E, BONNET
Panjab
Chandigarh
(India)
and P. LUCASSON
k’quipe de Recherche 91405 Orsay (France)
(Received
University,
January
No.
720,
associke
au C.N.R.S.,
Bat.
350,
Universite’
Paris Sud,
8, 1978)
Summary The configurational energy of the proton is investigated in the lutetium matrix. The bare ion is represented by a model potential and the modified Hartree dielectric function is used for the screening due to conduction electrons. The octahedral (0) position is found to be the most probable position for the proton and the activation energy is found to be in good agreement with the experimental value for an effective charge on the proton of 0.23e. It is concluded that the proton diffuses by the hopping process and that the O-O, 0-T(tetrahedral)-0 and O-T-T-O paths are equally probable for the diffusion. The configurational energy obtained by varying the c/u ratio and the core radius, and that for the dilated lattice, are also calculated and discussed.
The thermal expansion [l] and the residual electrical resistivity [ 21 of lutetium-hydrogen solid solutions have been extensively studied in this laboratory. An understanding of these results demands the knowledge of the effective charge of the proton, its most probable position and its path of diffusion in the matrix. For this purpose, a simplified model has been adopted for the lutetium matrix and these properties have been studied. We represent the bare ions of the host lattice by the Ashcroft model potential [3] and adopt the modified Hartree dielectric function, which explicitly includes the exchange correlation corrections, for the screening due to the conduction electrons. The effective charge of the proton is taken as an adjustable parameter. Neglecting the lattice relaxation and the local field corrections, the configurational energy A E(z ) of the proton is written as follows [4] :
*Part of this work was performed
while this author
was at Orsay, France.
2
AE(ii)
= ‘$F
;,I!!$ I
-qj-dq
(1 +_)
‘OS “0s sin!/ 0
(1)
where R’ and z are the position coordinate and the effective charge of the proton, FI is the position vector of the Ith host atom with respect to proton position, zI the charge of the host ion and r. the potential parameter. co(q) is the modified Hartree dielectric function where exchange correlation after Vashishta and Singwi @] is used and q is the field wavevector. To calculate AE(R) in eqn. (l), we chose z = le, zr = -3e and r. = 1.6029 au which is the effective ionic radius [6]. The lattice parameters a and c are taken as 6.633 and 10.519 au, respectively, giving c/u = 1.586. The sum in eqn. (1) is carried out for up to twelve nearest neighbours; at this point the interaction energy becomes almost negligible. For comparative study the results are further normalized assuming that the interaction energy vanishes beyond the twelfth nearest neighbour. The integration over q is carried out numerically. The octahedral (0), tetrahedral (T) and host atom positions in the (1120) plane are shown in Fig. 1. The coordinates of the 0 and T positions are (l/3, l/3,2/3, l/4) and (2/3, 2/3, z/3, $ - +(a/~)‘), respectively. To decide the probable position and the path of the proton, AE(R) is calculated along the O-O, T-T and the O-T directions. These results are also shown in Fig. 1 (curve (a)). As t+heproton moves from one 0 position to another along the vertical axis, AE(R) increases and reaches a maximum halfway between the two 0 positions nearest to the host atom in the (1120) plane. However, as th_eproton moves from the 0 to the T position along an oblique axis, AE(R) shows a maximum at approximately 70% of the O-T distance from the 0 position and AE(R) decreases $.&her and shows a minimum at the T position. Another maximum of AE(R) is obtained between the two T positions. As the c/u ratio differs slightly from the ideal value, the heights of the potential barriers between the two T and the twz 0 positions are found to be only approximately the same. Evidently A&R) has a minimum at the 0 position where it is lower by 0.08 eV than its value at the T position; the 0 position is thus the most stable position for the proton in lutetium. The heights of the O-T and O-O barriers are 1.39 and 1.22 eV, respectively. Since the difference in the barrier heights is of the order of the zero point energy, the O-O and the O-T-O paths for the diffusion of the proton are equally probable. The height of the T-T barrier is only 10% of that of the O-O barrier; therefore the possibility of the O-T-T-O path cannot be ruled out. We varied the triangular configuration of the passage windows by changing the c/a ratio at constant atomic volume and recalculated AE(&) for the O-O, O-T and T-T paths. These results for the ideal value of c/u and for c/a = 1.68 are shown by curves (b) and (c) in Fig. 1. The configurational energy increases along the c axis except in a small region about the T and 0 positions. This is consistent with a geometrical deformation in which the crystal is extended along the c axis or contracted along the a axis. For the ideal c/u ratio, the O-O and O-T barrier heights are the same.
3
Fig. 1. Configurational energy of a proton in lutetium. Curves (a), (b) and (c) represent the configurational energies for the c/a ratios 1.586, 1.633 and 1.68, respectively. The broken and broken-dotted lines represent the configurational energy for 2.5% and 5% expansions of the lattice. In the inset, open circles represent the octahedral position, X represents the tetrahedral position and solid circles represent the host atoms. Note that for clarity the path lengths T-T, O-O, T-O are normalized; their absolute values are different in each case since the calculations have been performed with different lattice parameters (see text).
The values of AE(g) at the 0 and T positions remain the same. The width of the potential well also remains almost the same. The heights of the O-O and T-T barriers increase with increasing c/u ratio while the height of the O-T barrier decreases. Therefore the O-T-O path of the proton becomes most probable and faster diffusion is expected. We also investigated the configurational energy of the proton in the expanded lattice along the O-O, O-T and T-T paths. These results are shown in Fig. 1 by the broken and broken-dotted lines for the 2.5% and 5% expansions, respectively. The configurational energy decreases by approximately 5 and lo%, respectively; the widths of the O-O, T-T and O-T potential wells also decrease by approximately 2 - 4%, but the relative heights of the barriers remain the same. This predicts faster diffusion of hydrogen in the expanded lattice. The lutetium ion has a partially filled d shell, so that the single parameter Ashcroft model potential is a rather oversimplified picture of the lutetium ion. However, the partial localization of d and f electrons around
4
the rigid core may effectively increase the core radius. We therefore increased r. from 1.603 to 1.65 A and recalculated the configurational energy along various possible paths in the (1120) plane, The configurational energy increased by approximately 8% but no substantial change was found in the relative heights of the potential barriers. As the calculated curve is very close to curve (a) it is not shown in the figure. If we define the activation energy as the minimum height of the potential barrier to be crossed by the proton from one 0 position to another, it is calculated to be 1.22 eV. The indirectly measured [ 71 activation energy is 0.28 * 0.02 eV. The experimental value involves many uncertainties, but the calculated value is almost four times as large as the experimental value. The calculated and the experimental values agree closely for an effective charge of 0.23e on the proton. This conclusion is similar to that found for noble metals [4] ; it suggests that the proton is near the atomic state in the lutetium matrix. The potential wells around the 0 and T positions are neither symmetric nor harmonic. However, if we fit the harmonic potential well in the vicinity of the 0 position along the c axis, the vibrational frequency and the vibrational energy are found to be 3.29 X 1013 Hz and 0.136 eV, respectively. The vibrational energy is smaller than the activation energy by approximately an order of magnitude, so that the tunnelling process is least probable and the hopping process is most probable for proton diffusion in the metal. The analysis of the neutron scattering data for LuH,~~ and ZrH, in the CYphase [l, 81 shows that hydrogen is located at the T position while our calculations predict the 0 position, which is in the minimum ionic density region, as the most probable position. The hydrogen concentrations in these compounds, however, were rather high. One should keep in mind that our calculations are for a single proton in a perfect matrix. If the crystal is not perfect and contains point or extended defects or impurities, the conclusions will change altogether. For example, the proton will always be trapped in a vacancy if one is available. If the crystal contains a higher concentration of hydrogen, the results may be altered further. The present calculations may be improved if one takes a more realistic description of the lutetium matrix where partially filled d bands are considered explicitly. However, this would involve a prohibitively difficult computation.
Acknowledgments The authors benefited from informative discussions with Dr. J. N. Daou. One of us (S. P.) is grateful to Professor A. Lucasson for providing laboratory facilities and to the C.N.R.S. (France) for financial assistance.
References 1 2 3 4 5 6
J. E. Bonnet, J. Less-Common Met., 49 (1976) 123. J. N. Daou, A. Lucasson and P. Lucasson, Solid State Commun., 19 (1976) 895. M. L. Cohen and V. Heine, Solid State Phys., 24 (1970) 38. S. Prakash, Phys. Rev., Sect. B, 18 (8) (1978) 3980. P. Vashishta and K. S. Singwi, Phys. Rev., Sect. B, 6 (1972) 875. K. N. R. Taylor and M. I. Darby, Physics of Rare Earth Solids, Chapman and Hall, 1972, p. 8. 7 J. N. Daou and R. Viallard, Congre‘s International L’hydroge‘ne duns les Mktaux, T. 1, 123, Edition Sciences et Industrie, Paris, 1972. 8 P. P. Narang, G. L. Paul and K. N. R. Taylor, J. Less-Common Met., 56 (1977) 125.