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Quantum diffusion model of light interstitial atoms in BCC metals
Volume 73A, number 5,6
PHYSICS LETTERS
15 October 1979
QUANTUM DIFFUSION MODEL OF LIGHT INTERSTITIAL ATOMS IN BCC METALS Joseph W. HAUS University of Essen GHS, 4300 Essen 1, West-Germany
and Kazuo KITAHARA Department of Liberal Arts, Shizuoka University, Shizuoka 422, Japan Received 11 July 1979
We present a model hamiltonian for quantum light-particle diffusion in a lattice which differs from that of previous investigations by the addition of a phonon coupling to the hopping term. In one dimension we derive a set of master equations which is valid in the high-temperature regime and calculate the diffusion coefficient.
In this letter we report the results of our recent investigations into the mechanism of hydrogen diffusion in the BCC metals (V, Nb, Ta). This work is the result of a succession of research efforts to understand and explain hydrogen diffusion. Hydrogen has a large mobility in these metals and it performs a hopping motion with residence times at interstitial sites. An important method used to study the properties of hydrogen in these systems has been neutral scatteringspectroscopy. The inelastic scattering has been examined and the structure factor has been correlated with phenomenological master equations [1] On the other hand, the quantum theories of hydrogen diffusion have only discussed the diffusion coefficient [2,31 and they have not stressed the usefulness of master equations in gaining insight into the diffusion mechanism. A master equation is derivable for the model hamiltonian given by Holstein [4] Holstein’s model describing a localized electron trapped by its deformation of the host lattice, provided a basis for the work of Flynn and Stoneham [4] (FS) on the diffusion of a hydrogen atom in a metal. In the “high temperature regime” (HTR), that is, the regime where temperature dependent hopping terms in the effective hamiltonian are negligible (typically T> 8D/ 10, where 8D is the Debye temperature) the density matrix in the master .
-
equation involves only diagonal components, p,~, where n is the site index of the interstitial lattice. The form of the master equation is that given by a classical random walk with nearest-neighbor hopping ~‘. Experimentally, this model is unsatisfactory at temperatures T> 0D [61.Lottner et al. found large deviations when this simple master equation was fit to the structure factor at these temperatures; whereas, at temperature T 8D the model provided a consistent description of the data. Kagan and Klinger (KK) have presented a set of master equations. In the HTR, the master equations of KK have the form of a set of classical master equations with nearest neighbor hopping between equivalent states. The master equations limited to two states are similar to one used in a recent neutron scattering investigation [71 (the difference lies only in the neglect of hopping via the ground state). In the evaluation of these experiments the temperature dependence of the coefficients of the master equations could not be interpreted in terms of an energy activated two-state model. The method used in this letter leads to a set of master equations for the density matrix with diagonal and off-diagonal components for the HTR in the site repre~ Such a model was considered for inelastic neutron scattering by Chudley and Elliott [5].
423
Volume 73A, number 5,6
PHYSICS LETTERS
15 October 1979 2kn,kI
sentation. The derivation and interpretation of these terms is novel. Although the work described here is not regarded as a final solution, the adequacy of this model is capable of being evaluated by neutron scattering data. Recently, research of Emin et a!. [8] has given a detailed description of the diffusion coefficient and pointed out further deficiencies of the FS model. Their work presents the diffusion coefficient for the isotopes of
acterization of the coupling is valid if 4jjJ
hydrogen which are very similar to the experimental
the host atoms straddling the jump path. For this case,
curves. Specifically, they were able to show that the Condon approximation, that is, the neglect of the phonon coupling to the particles bandwidth, is violated. On the other hand, the relation between the work of Emin eta!, and the calculations of FS and KK is not
we may introduce the following canonical transformation:
and present a set of master equations based on the analysis of Emin et a!. Our work starts with the hamiltonian
E
~
n, s +
\
~
The fifth term is a phonon coupling to the particle’s
excited state bandwidth. The amplitude ~ is assumed to be independent of the site for two reasons: the wavefunction of the excited state is more spread and does not experience the lattice position as sharply as the ground state; and the change in the overlap is assumed to be mainly affected by transverse motions of
r U 1
clear. Hence we shall now give a model hamiltonian
-~
Ji’ k
1
ij
exP[~
(b — b~k)Ea lam, (2) k n, m This shift of the phonon coordinates when the excited state is populated is a distortion due to the Idnetic energy of the particle. 6nk~ The is taken excited as a state small energy amcoupled to the phonons, plitude. The last term is the nonradiative coupling of the =
-~-~---
ground and excited state [91.~‘ stands for a sum over n,k,s,ands’withs~s’.
+
The special form of our hamiltonian (1), allows a +
E
m I k (b~+ b k)afl 1am 1
~‘,i —
‘
‘
simple ex~ressionfor the canonically transformed ham-
iltonian,~.The site energies are shifted due to the deformation of the metal atoms. The ground state shift
‘
÷E ~ k(’~’k+ b_k)a~ 1a~,1 n,k
+E
is due to the usual elastic dipole term of the polaron problem
‘
iW~k(b~—b_k)a,san,s~.
(1)
~~o= e0_~Ienkl2!~)k,
(3)
k The first term is the lattice hamiltonian, denoted by HL, it has the usual harmonic oscillator form, wk bkbk, where b~and bk denote the phonon creation and annihilation operators, respectively. a~,5and are similarly defmed operators for the particle on an,s the nth
the excited state energy is shifted by:
lattice site in a state s. Two states are considered and denoted by s = 0, 1. The second and third term contam the particle’s energies and bandwidths in the absence of lattice coupling. The fourth term is a phonon coupling to the ground state energy. This coupling is considered to be strong. We will therefore treat it nonperturbatively by shifting the local phonon coordinates. The canonical
where Z is the number of nearest neighbors, and a new overlap term is introduced into ~(which has a topological second-neighbor extension of the wavefunction, In m I = 2. By topological we mean that these overlap terms are extended to all nearest neighbors of the neighboring overlap site (fig. 1) *2 The transfer between second-neighbor sites is given by
transformation is well known from the small polaron theory [41 and it was used by FS and KK. This char-
*2
424
=
—
Z ~k
~
210)k
‘
(4)
I
—
That this leads to other than second neighbor terms is first realized in two-dimensions.
Volume 73A, number 5,6
PHYSICS LETTERS
/ ,,
ap~jat=—2l°pn°n
_11
2
/
15 October 1979
~
0 010 101 ln—1P117 Pnn +7 Pnn
for the ground state, where 71 Fig. 1. Schematic of a two dimensional lattice with first neighbor overlap contributions, J’, and topological second neigh1. Since J~is an overlap induced from nearbor contributions, J est-neighbor bandwidth 2 fluctuations, it is shown branching from the first neighbor site given by open circles. In this figure the true second neighbor overlap, given by a dashed line, has a strength, 2,4, due to the two paths leading via first neighbors to this site.
=
(5)
—
k
=
Re f6e~(t)6e~dt, 0
0
=
1” (6J0
Re
~J
—m
~
~t~6J0
n—mi’’
in —m I
>dt = 0 7
0
J (6W01(t)6W10)dt 00
701
=
Re
701
=
0 e~~y10 ,
(9)
and
~e=&1_&0.
The strength of this overlap term is related to the shift of the excited state energy. The particle can find itself
moving coherently over a larger distance in the effective hamiltonian 9C. Three terms are treated as small perturbations in
Note that detailed balance is satisfied between 701 and 710 The time dependence of the coefficients in eq. ~) is given by the unperturbed Hamilton operator in ~.
~,
The square brackets in eq. (7) denote the commuta-
tors of the matrices appearing in the arguments.
‘6e,~which represents the energy fluctuations of the excited state; &J~ m I which contains the overlap fluctuations of the ground state and 6 W~’which denotes the phonon coupling to the nonradiative contribution. These terms contain phonon operators under the operation of the canonical transformations discussed above, We may now write the Liouville equation for this hamiltonian
The master equations of the form in eqs. (7) and (8) are the starting point for further investigations of hydrogen diffusion. A similar model has also been developed by Efrima and Metiu to describe surface diffusion of atoms [12]. First let us discuss several aspects which make it a plausible model for hydrogen diffusion. It has a coherent term, i [J1, p1] + i [~J1, p1] ,in the excited state which tends to spread the particle over
aP /at r&~
nearest and topological second nearest A phenomenological hopping model withneighbors. such a jump
—
=
‘6’
L
‘~.‘
Since we will only be concerned with the diffusing particle coordinates, the phonon states are summed over using the canonical distribution [10] The three terms, 6p~, ~ m I and 6 W~’are handled in secondorder perturbation theory. The model hamiltonian is similar to that used in phenomenological models of exciton transport [11]. The HTR is assumed, and we have master equations for one dimension given by -
1
,
apnm,at ~
1
—.
1
-
1
1
1 1
~ ‘p ]nm +i[L~J ‘P ]nm 2y Pnm 6 7 Pnm nm ~o1oz 7 Pnn mn 7 Ppm mn 1
for the excited state and
—
‘
process has been introduced by Gissler and Rother [13] and extended for hydrogen in metals by Lottner and coworkers #3 [6] Although their model is classical, the effect of our coherent term is similar, that is, the partide jumps further. A recent neutron scattering study of the topological jump model and the two-state model was inconclusive as to the correctness of one picture or the other [7]. On the other hand, the rates determined from this study had features which could not be clearly reconciled with -
3
The model of ref. [12] was shown to be equivalent to a master equation in ref. [14]. 425
Volume 73A, number 5,6
PHYSICS LETTERS
an underlying microscopic view. We must await a more detailed experimental analysis of this mode! before a conclusion can be made. One drawback of the proposed master equation is that it has five parameters. However, two parameters are 1 and ~J1, and the latter temperature independent, J one is correlated with the phonon frequencies. 701 and 710 also satisfy detailed balance,
The diffusion coeffIcient is given by: D =P~[2(J’)2 + 8(~J1)2]/ 1+P~q270 7 where
P~q= 710/(710 + 701),
,
P~q= 701/(710 + 701),(lo)
15 October 1979
factor of our model has a more complicated frequency and wave number dependence which is not lorentzian. We would like to thank Dr. K.W. Kehr, for support and criticism of this work and Professor K. Binder for his hospitality in the Kernforschungsanlage JUlich, where this work was begun. One of us, (JWH), also thanks Professors F. Haake, H. Haken, and M. Wagner
and Dr. D. Emin for interesting discussions concerning the work. We are also grateful to Dr. D. Emin and coworkers andY. Lottner and coworkers for sending us preprints of their work prior to publication. One of us (KK) was partially supported by the Sakkokai Foundation and by a grant from the Ministry of Education of the Japanese Government. References
Concerning the isotope dependence of the diffusion
coefficient, we can only say that the isotope dependence of the FS diffusion coefficient without coherent terms is consistent with data in Ta and Nb for T< 8D [15]. In this regime, the prefactor can be expected to have an isotope dependence,whil~the polaron binding energy has only a weak isotope dependence. This last statement is inferred from the weak isotope dependence of the force dipole tensor [161. To qualitatively explain the isotope dependence of the regime T> 8D’ we can infer that the main isotope dependence of the activation energy arises from the change in the zero point energy of the ground state. The overlap integral and the site energy fluctuations 6e~ should be less isotope dependent~ In our model we have neglected the effect of the electron. We could handle these couplings just as we did the phonon coupling and in the end we could project such operators out of the master equation. Recent phenomenological calculations by Lottner et a!. [17] have suggested that this is an important mechanism. However, we find their suggestion of classical diffusion for all temperatures to be inconclusive. For instance, the high temperature diffusion data is not explained
by their calculations. In a future publication, we shall present more explicit calculations for the above model and compare it with the models proposed by other authors [2,3]. We only mention here that even the dynamic structure
426
[1]For example, J.M. Rowe, J.J. Ruth and H.E. Flotow,
Phys. Rev. B9 (1974) 5039. [2]C.P. Flynn and A.M, Stoneham, Phys, Rev, Bi (1970)
3966. [3] Yu, Kagan and M,l. Klinger, J. Phys, C7 (1974) 2791. [4] T. Holstein, Ann, Phys. (NY) 8 (1959) 325. [51C.T. Chudley and R.J. Elliott, Proc. R, Soc. 77 (1961) 353
[6] V. Lottner, A. Heim, K.W. Kehr and T, Springer, IAEA Symp, on Neutron inelastic scattering, Vienna 2 (1978)
J,W. Haus, A. Heim and K.W, Chem. Solids, to be published.
[7]V. Lottner,
Kehr, J. Phys.
[8] D. Emin, M,I, Baskes and W.D, Wilson, Phys, Rev. Lett, 42 (1979) 791. [9] See, Y. Weissmann and J. Jortner, Phil. Mag. B37 (1978) 21, and references therein. [101 This method is of comthon use: R. Silbey, Ann. Rev. Phys. Chem, 27 (1976) 203 and references therein. [11] H. Haken and G. Strobl, Z. Phys. 262 (1972) 135; K. Kitahara and J.W. Haus, Z. Phys. B32 (1979) 419, and
references therein. [12]S. Efrima and H. Metiu, J. Chem. Phys. 69 (1978) 2286. [13] W. Gissler and H. Rother, Physics 50 (1970) 380.
[14] J.W. Haus and K.W. Kehr, Solid State Commun. 26 (1978) 753. [15] See J. Völkl and G. Alefeld, in: Hydrogen in metals I, eds. G Alefeld and J. Völkl (Springer, Berlin, 1978). [16] Further comment is given by: K.W, Kehr and H. Peisi, in: Hydrogen metals I, eds. G. Alefeld and J. VOlkl (Springer, in Berlin, 1978), [17] V. Lottner, H,R. Schober and W.J. Fitzgerald, Phys. Rev. Lett. 42 (1979) 1162.
PHYSICS LETTERS
15 October 1979
QUANTUM DIFFUSION MODEL OF LIGHT INTERSTITIAL ATOMS IN BCC METALS Joseph W. HAUS University of Essen GHS, 4300 Essen 1, West-Germany
and Kazuo KITAHARA Department of Liberal Arts, Shizuoka University, Shizuoka 422, Japan Received 11 July 1979
We present a model hamiltonian for quantum light-particle diffusion in a lattice which differs from that of previous investigations by the addition of a phonon coupling to the hopping term. In one dimension we derive a set of master equations which is valid in the high-temperature regime and calculate the diffusion coefficient.
In this letter we report the results of our recent investigations into the mechanism of hydrogen diffusion in the BCC metals (V, Nb, Ta). This work is the result of a succession of research efforts to understand and explain hydrogen diffusion. Hydrogen has a large mobility in these metals and it performs a hopping motion with residence times at interstitial sites. An important method used to study the properties of hydrogen in these systems has been neutral scatteringspectroscopy. The inelastic scattering has been examined and the structure factor has been correlated with phenomenological master equations [1] On the other hand, the quantum theories of hydrogen diffusion have only discussed the diffusion coefficient [2,31 and they have not stressed the usefulness of master equations in gaining insight into the diffusion mechanism. A master equation is derivable for the model hamiltonian given by Holstein [4] Holstein’s model describing a localized electron trapped by its deformation of the host lattice, provided a basis for the work of Flynn and Stoneham [4] (FS) on the diffusion of a hydrogen atom in a metal. In the “high temperature regime” (HTR), that is, the regime where temperature dependent hopping terms in the effective hamiltonian are negligible (typically T> 8D/ 10, where 8D is the Debye temperature) the density matrix in the master .
-
equation involves only diagonal components, p,~, where n is the site index of the interstitial lattice. The form of the master equation is that given by a classical random walk with nearest-neighbor hopping ~‘. Experimentally, this model is unsatisfactory at temperatures T> 0D [61.Lottner et al. found large deviations when this simple master equation was fit to the structure factor at these temperatures; whereas, at temperature T 8D the model provided a consistent description of the data. Kagan and Klinger (KK) have presented a set of master equations. In the HTR, the master equations of KK have the form of a set of classical master equations with nearest neighbor hopping between equivalent states. The master equations limited to two states are similar to one used in a recent neutron scattering investigation [71 (the difference lies only in the neglect of hopping via the ground state). In the evaluation of these experiments the temperature dependence of the coefficients of the master equations could not be interpreted in terms of an energy activated two-state model. The method used in this letter leads to a set of master equations for the density matrix with diagonal and off-diagonal components for the HTR in the site repre~ Such a model was considered for inelastic neutron scattering by Chudley and Elliott [5].
423
Volume 73A, number 5,6
PHYSICS LETTERS
15 October 1979 2kn,kI
sentation. The derivation and interpretation of these terms is novel. Although the work described here is not regarded as a final solution, the adequacy of this model is capable of being evaluated by neutron scattering data. Recently, research of Emin et a!. [8] has given a detailed description of the diffusion coefficient and pointed out further deficiencies of the FS model. Their work presents the diffusion coefficient for the isotopes of
acterization of the coupling is valid if 4jjJ
hydrogen which are very similar to the experimental
the host atoms straddling the jump path. For this case,
curves. Specifically, they were able to show that the Condon approximation, that is, the neglect of the phonon coupling to the particles bandwidth, is violated. On the other hand, the relation between the work of Emin eta!, and the calculations of FS and KK is not
we may introduce the following canonical transformation:
and present a set of master equations based on the analysis of Emin et a!. Our work starts with the hamiltonian
E
~
n, s +
\
~
The fifth term is a phonon coupling to the particle’s
excited state bandwidth. The amplitude ~ is assumed to be independent of the site for two reasons: the wavefunction of the excited state is more spread and does not experience the lattice position as sharply as the ground state; and the change in the overlap is assumed to be mainly affected by transverse motions of
r U 1
clear. Hence we shall now give a model hamiltonian
-~
Ji’ k
1
ij
exP[~
(b — b~k)Ea lam, (2) k n, m This shift of the phonon coordinates when the excited state is populated is a distortion due to the Idnetic energy of the particle. 6nk~ The is taken excited as a state small energy amcoupled to the phonons, plitude. The last term is the nonradiative coupling of the =
-~-~---
ground and excited state [91.~‘ stands for a sum over n,k,s,ands’withs~s’.
+
The special form of our hamiltonian (1), allows a +
E
m I k (b~+ b k)afl 1am 1
~‘,i —
‘
‘
simple ex~ressionfor the canonically transformed ham-
iltonian,~.The site energies are shifted due to the deformation of the metal atoms. The ground state shift
‘
÷E ~ k(’~’k+ b_k)a~ 1a~,1 n,k
+E
is due to the usual elastic dipole term of the polaron problem
‘
iW~k(b~—b_k)a,san,s~.
(1)
~~o= e0_~Ienkl2!~)k,
(3)
k The first term is the lattice hamiltonian, denoted by HL, it has the usual harmonic oscillator form, wk bkbk, where b~and bk denote the phonon creation and annihilation operators, respectively. a~,5and are similarly defmed operators for the particle on an,s the nth
the excited state energy is shifted by:
lattice site in a state s. Two states are considered and denoted by s = 0, 1. The second and third term contam the particle’s energies and bandwidths in the absence of lattice coupling. The fourth term is a phonon coupling to the ground state energy. This coupling is considered to be strong. We will therefore treat it nonperturbatively by shifting the local phonon coordinates. The canonical
where Z is the number of nearest neighbors, and a new overlap term is introduced into ~(which has a topological second-neighbor extension of the wavefunction, In m I = 2. By topological we mean that these overlap terms are extended to all nearest neighbors of the neighboring overlap site (fig. 1) *2 The transfer between second-neighbor sites is given by
transformation is well known from the small polaron theory [41 and it was used by FS and KK. This char-
*2
424
=
—
Z ~k
~
210)k
‘
(4)
I
—
That this leads to other than second neighbor terms is first realized in two-dimensions.
Volume 73A, number 5,6
PHYSICS LETTERS
/ ,,
ap~jat=—2l°pn°n
_11
2
/
15 October 1979
~
0 010 101 ln—1P117 Pnn +7 Pnn
for the ground state, where 71 Fig. 1. Schematic of a two dimensional lattice with first neighbor overlap contributions, J’, and topological second neigh1. Since J~is an overlap induced from nearbor contributions, J est-neighbor bandwidth 2 fluctuations, it is shown branching from the first neighbor site given by open circles. In this figure the true second neighbor overlap, given by a dashed line, has a strength, 2,4, due to the two paths leading via first neighbors to this site.
=
(5)
—
k
=
Re f6e~(t)6e~dt, 0
0
=
1” (6J0
Re
~J
—m
~
~t~6J0
n—mi’’
in —m I
>dt = 0 7
0
J (6W01(t)6W10)dt 00
701
=
Re
701
=
0 e~~y10 ,
(9)
and
~e=&1_&0.
The strength of this overlap term is related to the shift of the excited state energy. The particle can find itself
moving coherently over a larger distance in the effective hamiltonian 9C. Three terms are treated as small perturbations in
Note that detailed balance is satisfied between 701 and 710 The time dependence of the coefficients in eq. ~) is given by the unperturbed Hamilton operator in ~.
~,
The square brackets in eq. (7) denote the commuta-
tors of the matrices appearing in the arguments.
‘6e,~which represents the energy fluctuations of the excited state; &J~ m I which contains the overlap fluctuations of the ground state and 6 W~’which denotes the phonon coupling to the nonradiative contribution. These terms contain phonon operators under the operation of the canonical transformations discussed above, We may now write the Liouville equation for this hamiltonian
The master equations of the form in eqs. (7) and (8) are the starting point for further investigations of hydrogen diffusion. A similar model has also been developed by Efrima and Metiu to describe surface diffusion of atoms [12]. First let us discuss several aspects which make it a plausible model for hydrogen diffusion. It has a coherent term, i [J1, p1] + i [~J1, p1] ,in the excited state which tends to spread the particle over
aP /at r&~
nearest and topological second nearest A phenomenological hopping model withneighbors. such a jump
—
=
‘6’
L
‘~.‘
Since we will only be concerned with the diffusing particle coordinates, the phonon states are summed over using the canonical distribution [10] The three terms, 6p~, ~ m I and 6 W~’are handled in secondorder perturbation theory. The model hamiltonian is similar to that used in phenomenological models of exciton transport [11]. The HTR is assumed, and we have master equations for one dimension given by -
1
,
apnm,at ~
1
—.
1
-
1
1
1 1
~ ‘p ]nm +i[L~J ‘P ]nm 2y Pnm 6 7 Pnm nm ~o1oz 7 Pnn mn 7 Ppm mn 1
for the excited state and
—
‘
process has been introduced by Gissler and Rother [13] and extended for hydrogen in metals by Lottner and coworkers #3 [6] Although their model is classical, the effect of our coherent term is similar, that is, the partide jumps further. A recent neutron scattering study of the topological jump model and the two-state model was inconclusive as to the correctness of one picture or the other [7]. On the other hand, the rates determined from this study had features which could not be clearly reconciled with -
3
The model of ref. [12] was shown to be equivalent to a master equation in ref. [14]. 425
Volume 73A, number 5,6
PHYSICS LETTERS
an underlying microscopic view. We must await a more detailed experimental analysis of this mode! before a conclusion can be made. One drawback of the proposed master equation is that it has five parameters. However, two parameters are 1 and ~J1, and the latter temperature independent, J one is correlated with the phonon frequencies. 701 and 710 also satisfy detailed balance,
The diffusion coeffIcient is given by: D =P~[2(J’)2 + 8(~J1)2]/ 1+P~q270 7 where
P~q= 710/(710 + 701),
,
P~q= 701/(710 + 701),(lo)
15 October 1979
factor of our model has a more complicated frequency and wave number dependence which is not lorentzian. We would like to thank Dr. K.W. Kehr, for support and criticism of this work and Professor K. Binder for his hospitality in the Kernforschungsanlage JUlich, where this work was begun. One of us, (JWH), also thanks Professors F. Haake, H. Haken, and M. Wagner
and Dr. D. Emin for interesting discussions concerning the work. We are also grateful to Dr. D. Emin and coworkers andY. Lottner and coworkers for sending us preprints of their work prior to publication. One of us (KK) was partially supported by the Sakkokai Foundation and by a grant from the Ministry of Education of the Japanese Government. References
Concerning the isotope dependence of the diffusion
coefficient, we can only say that the isotope dependence of the FS diffusion coefficient without coherent terms is consistent with data in Ta and Nb for T< 8D [15]. In this regime, the prefactor can be expected to have an isotope dependence,whil~the polaron binding energy has only a weak isotope dependence. This last statement is inferred from the weak isotope dependence of the force dipole tensor [161. To qualitatively explain the isotope dependence of the regime T> 8D’ we can infer that the main isotope dependence of the activation energy arises from the change in the zero point energy of the ground state. The overlap integral and the site energy fluctuations 6e~ should be less isotope dependent~ In our model we have neglected the effect of the electron. We could handle these couplings just as we did the phonon coupling and in the end we could project such operators out of the master equation. Recent phenomenological calculations by Lottner et a!. [17] have suggested that this is an important mechanism. However, we find their suggestion of classical diffusion for all temperatures to be inconclusive. For instance, the high temperature diffusion data is not explained
by their calculations. In a future publication, we shall present more explicit calculations for the above model and compare it with the models proposed by other authors [2,3]. We only mention here that even the dynamic structure
426
[1]For example, J.M. Rowe, J.J. Ruth and H.E. Flotow,
Phys. Rev. B9 (1974) 5039. [2]C.P. Flynn and A.M, Stoneham, Phys, Rev, Bi (1970)
3966. [3] Yu, Kagan and M,l. Klinger, J. Phys, C7 (1974) 2791. [4] T. Holstein, Ann, Phys. (NY) 8 (1959) 325. [51C.T. Chudley and R.J. Elliott, Proc. R, Soc. 77 (1961) 353
[6] V. Lottner, A. Heim, K.W. Kehr and T, Springer, IAEA Symp, on Neutron inelastic scattering, Vienna 2 (1978)
J,W. Haus, A. Heim and K.W, Chem. Solids, to be published.
[7]V. Lottner,
Kehr, J. Phys.
[8] D. Emin, M,I, Baskes and W.D, Wilson, Phys, Rev. Lett, 42 (1979) 791. [9] See, Y. Weissmann and J. Jortner, Phil. Mag. B37 (1978) 21, and references therein. [101 This method is of comthon use: R. Silbey, Ann. Rev. Phys. Chem, 27 (1976) 203 and references therein. [11] H. Haken and G. Strobl, Z. Phys. 262 (1972) 135; K. Kitahara and J.W. Haus, Z. Phys. B32 (1979) 419, and
references therein. [12]S. Efrima and H. Metiu, J. Chem. Phys. 69 (1978) 2286. [13] W. Gissler and H. Rother, Physics 50 (1970) 380.
[14] J.W. Haus and K.W. Kehr, Solid State Commun. 26 (1978) 753. [15] See J. Völkl and G. Alefeld, in: Hydrogen in metals I, eds. G Alefeld and J. Völkl (Springer, Berlin, 1978). [16] Further comment is given by: K.W, Kehr and H. Peisi, in: Hydrogen metals I, eds. G. Alefeld and J. VOlkl (Springer, in Berlin, 1978), [17] V. Lottner, H,R. Schober and W.J. Fitzgerald, Phys. Rev. Lett. 42 (1979) 1162.