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Energy loss of α-particles channeled in metal-hydrogen systems
N U C L E A R I N S T R U M E N T S AND METHODS
I32
(I976)
I25-I28;
©
NORTH-HOLLAND
PUBLISHING
CO.
ENERGY LOSS OF ~t-PARTICLES CHANNELED IN M E T A L - H Y D R O G E N SYSTEMS A N A N D P. P A T H A K
Section d'Etude des Solides Irradids, Centre d'Etudes Nucldaires, Fontenay-aux-Roses (92260), France A simple method is discussed to calculate the energy loss of MeV c~-particles channeled in a planar direction of a (transition) metal without and with a low concentration of hydrogen atoms. The contribution of additional electrons of hydrogen atoms to stopping power is obtained by assuming that in the general case these can go to d-band and to conduction band. In particular for ~-phase of P d - H system all the additional electrons tend to fill the holes in the d-band. The contribution due to additional protons is obtained by considering proton--~-particle scattering and using the experimental activation energy for proton diffusion (0.23 eV for Pd-H) as the m i n i m u m energy that the protons could take in this scattering process. This latter contribution is found to be extremely small compared to the electronic part.
1. Introduction
The problem of hydrogen diffusion in metals has been extensively studied, both experimentally as well as theoretically1). The study has been motivated from points of view of both fundamental as well as applied research, for various reasons such as extremely high diffusivity of hydrogen in metals, possibility to observe quantum effects at low temperatures, availability of three isotopes for hydrogen with largest possible relative mass difference and the technical interest of metal-hydrogen systems in various areas. Most of the studies have been made on the transition metal hydrides and out of these, the palladium-hydrogen system has been studied most extensively2), because even among transition metals, palladium is outstanding because of its unusual properties. Recently channeling studies have also been extended to such systems. The effects of a low concentration of hydrogen dissolved in palladium on the dechanneling of MeV c~-particles, initially channeled along (111) planes, have been clearly demonstrated3). The experimental results have been found to compare well with the calculations 4) based on the simple classical picture in which the scattering of the incoming or-particles from the protons (of dissolved hydrogen atoms) sitting halfway between consecutive (111) planes (i.e. octahedral sites), is described as of simple Rutherford type. In these calculations, the dechanneling is assumed to take place when the transverse energy of the incident particle during this scattering process becomes more than a critical value. Thus, as a first approximation it is reasonable to assume that for the ~t-phase of this system (i.e. for low concentration of H in Pd) the effect of the dissolved protons is only to obstruct the channel without producing any deformation or distortion and that the dechanneling is entirely "obstruc-
tion" type 5) as against distortion type encountered in other types of defects such as dislocations. Based on these simple ideas we give an estimate of the additional energy loss that the channeled particles will suffer due to the dissolved hydrogen in metals. A general procedure for such calculations has been discussed in section 2 and numerical values are given for the particular case of the P d - H system in section 3. 2. General formalism
It is well known that the electronic structure of metal hydrides (specially for transition metals) is very complicated. Depending upon the particular host metal and on the concentration of the dissolved hydrogen, the additional electrons of the H-atoms can go to different bands of the metal and can even induce some new bonding states as has been observed in photoemission studies on the Pd-H system 6, 7). The electronic part of the additional stopping power depends upon the band or shell of the host metal to which the added electrons go. With the point of view of stopping theories we will generally assume that the additional electrons first fill any holes present in for example the d-band of the transition host metal and the rest of the additional electrons tend to go to the conduction band. If there are no holes in the d-band we will assume that all the electrons go to the conduction band. It should be noted that this is only a simplified picture (and a crude one from the point of view of detailed electronic structure) and can be expected to be valid only for low H concentrations (a-phase). For example, in the Pd-H system additional low lying H-induced states have been shown to exist 6' 7) in the fl-phase and these are in fact energetically more favourable to be filled than the s-band. However, this simple picture can be expected to give sensible results because the moving energetic III. E L E C T R O N I C S T O P P I N G
126
ANAND
ion tends to overlook the detailed electronic structure and as we shall note, use of simple hydrogen like atomic orbitals in stopping power calculations gives fairly accurate results. The problem is thus reduced to calculate the stopping power due to the valence ( d ) b a n d with old and new electronic densities and to that due to the conduction band with old and new electronic densities. The conduction electron contribution is easily obtained using the Bethe-Bloch formula for an electron gas whose density is equal to that in the conduction band. For valence band and inner shells one needs to use specific electron wavefunctions. We will use here only the atomic wavefunctions and avoid the use of the complicated band structure. This simplification is implicit in most of the channel stopping power calculations and in any case it is not bad for filled or nearly filled d-band transition metals. In fact the use of hydrogen-like wavefunctions has given very good quantitative agreement between theory and experiments on channel stopping powers of semiconductors for energetic protonsS). The reason that such simplifications, too crude from the band theory point of view, do work, is that the fast incoming particle tends to overlook the detailed band structure. The electronic stopping power in the high velocity region may be written a s 9 l o ) : (1)
my 2
where
Lv
= Pval
(2r/Iv~/hcop),
in good agreement with the experiment1°). In fact one can easily see from eqs. (1) and ( 2 ) t h a t when P~oc becomes equal to Pva~, i.e. when the incoming particle encounters all of the valence electrons, one gets the formula of Dettmann and Robinson 8) for the valence shell. Ths last term L~ represents the contribution of inner shells, pj and Ij being corresponding densities and binding energies. First we consider the energy loss to those additional electrons which go to the valence d-shell and thus increasing the contribution Lv by a certain amount. If x is the atomic percentage of H atoms dissolved in the host metal, one has 0.01 x additional electrons per atom. Let us assume that out of these, 0.01 xx per atom go to the d-shell and rest, 0.01 x 2 ( = 0 . 0 1 x - 0 . 0 1 x l ) to the conduction band. If there are n h holes per atom in the d-band (for Pd, nh=0.36 ) then 0 . 0 1 x t < n h. The resulting change in Pva~ of eq. (2) is simply n~ = 0.01 xl N, where N is the atomic concentration in the host metal. The contribution to P~oc is obtained by using appropriate occupation numbers of the d-shell (with and without H atoms) in the planar averaged expression for change density due to t h e j t h shell 1° 11): ~(1)
f,j
-
N d p ~L:'~~,~ i)"2nJ - wjexp(-2~jy) 2 2nj
(2mv2/lj),
L I = ~ p j In
In (Vl/VF) + P,oc In (2my1VF/I )
,
(2)
where e and m are the electronic charge and mass, respectively and Z~ and v~, are charge and velocity of the incident particle. In the above equations L c represents the conduction electron contribution, Pc and cop being corresponding density and plasma frequency. Lv is the valence electron (d-shell in transition metals) contribution, having two terms: the first term corresponds to the contribution of collective excitations which uses the total density of valence electrons while the second term gives the contribution of local electron density in the channel, to single particle excitations, vF being the Fermi velocity and I, the binding energy of the d-shell. This separation of valence electron contribution into two parts 9) has been found to give the stopping power in the (l 11) planar channels of gold single crystals for energetic e-particles which is
2nj-t
~ × k=o
y2n~- 1 -k ×
4 nZ~ e 4 S~ -- - (Lc+Lv+LI),
L c = Pc In
P. PATHAK
,
(3)
(2 ~j)k (2 n i - 1 - k) ! where dp is the interplanar distance (so that Ndp is atomic density per unit area of the plane), n j, wj and ~j are the principal quantum number, the occupation number and the orbital exponent of the jth shell respectively, p}l) (y) represents the planar density due to one plane at a distance y from it, so that the total electron density due to both planes is: pj(y) = p~t)(y) + o(I) ,-J ( d p - y ) .
(4)
The change in Ploc due to diffused H atoms is obtained by using the change in occupation number wj, i.e. 0.01Xl in eqs. (3) and (4). Thus having found changes in Pva~ and Ploc due to the diffused H atoms, one can easily calculate the corresponding change in stopping power from eqs. (1) and (2). It may be added that eqs. (3) and (4) also give the contribution of other inner shells to the effective electron density responsible for stopping. Next, we consider the contribution to stopping power of any electrons going to the conduction band of the host metal. Here any change in the electron density also affects the plasma frequency cop appearing in the
ENERGY
logarithmic term of L c so that the change in the stoping power is obtained by calculating: dLc --- E c - L c = Pc' In -2mv~ - - p ¢ I n - 2my - , 2
(5)
where
and
,
dE = nm~m2v~ Qd. dx ( m 1 + 1Tl2) 2
(8)
where n is the density of protons ( = 0.01 xN), Qd is momentum transfer cross section for scattering of protons from the a-particle, given by:
Qa = 2n
P'c = Pc + / ' / 2 ~ Pc + 0 . 0 1 X2 N,
g% =
127
LOSS OF O ~ - P A R T I C L E S
f;
Io(0) (1 - cos0) sin0 dO.
(9)
For fast a-particles, the scattering is essentially from the bare charge Z1 ( = 2) so that Io (0) is given by:
(4n_~e2) ~ ~
(
= COp 1 + -
Z~ e 4 1 I 0(0) = m 2/)14 ( l - c o s 0 f "
nzf,
(10)
Using eq. (10) in eq. (9) and performing the indicated integration we get
Pc
so that
dE
2 n n m 2l Z- 2l e 4-
2m2m2vl2
-
dx
h~op For n 2 ~ P ¢ , this can be put in a closed form as (7)
h~p Thus in this limit, the conduction electron contribution to the change in stopping power is also directly proportional to the change in the electron density like in the previous case of change in the valence electron contribution, although an additional factor of one half appears in eq. (7). Finally, we consider the contribution of protons themselves to the stopping of incoming particles. Since the proton mass is about three orders of magnitude larger than the electron mass, it is to be expected that this contribution will be smaller by about the same order than the electronic contribution. However, for completeness, we give the detailed formula and show that this is in fact quite a small contributon. Let us consider the scattering of a proton from the incoming a-particle. It is reasonable to assume that any proton velocity due to its jumping around in the host metal is negligible small compared to the velocity of the incoming a-particle. Also for low H concentrations in the a-phase of the system, the interaction between the protons themselves can be neglected. The energy taken away by the proton (mass m~) from the a-particle (mass m2) in such a scattering process is given by 13):
.
Of)
( m l + m 2 ) 2 m 2 v2 I n ( m l + m 2 ) 2 I s
The divergences in the integration have been taken care of by recalling 13) that 0 = n in eq. (9) corresponds to maximum energy transfer from a-particle to proton, T~nax = 2m~m 2 v2/(ml + m 2 ) 2 and minimum 0 is decided by the minimum energy that the proton could take 0.e. its diffusion energy I2 = 0.23 eV for the P d - H system) T mi, 2 -----I 2 . 3. Results and discussion
We now use the above formulae to calculate the stopping power for 4 MeV a-particles channeled along (111) planes of a palladium crystal without and with a small concentration of H atoms (in a-phase) introduced in it. It is well established, both experimentally 14) and theoretically 15) that there are 0.36 holes per atom in the 4d-band of palladium and the same number of electrons in the conduction band. Thus Pc = 0 . 3 6 N and the occupation number w4a = 9.64 so that Pval = 9.64 N -- 0.097 (in a.u.). As long as the atomic percentage (x) of H atoms is less than 36, we can safely assume that all the additional electrons go to the d-band and none to the conduction band 022 ~ 0 ) . Thus only the valence electron contribution to the stopping power will change and nl=n=O.OlxN=O.OOOlx (a.u.) because for Pd, N = 0.01007 a.u. Using eqs. (3) and (4) to calculate Ploc at a point half-way between two consecutive (111) planes we get P~oc = 0.00857 in pure palladium, i.e. with w4a = 9.64, and P~oc= 0.00857+9 x 1 0 - 6 x for the P d - H system, i.e. with u,~d = 9 . 6 4 + 0 . 0 1 x . Similarly calculating the planar electron density due to other inner shells, and III. E L E C T R O N I C
STOPPING
128
A N A N D P. P A T H A K
then using it in the stopping power expressions (l) and (2), we get for 4 MeV ~-particles16): dE -
1.331 eV/A
(conduction electrons)
+ 14.59 eV/A
(valence electrons)
+ 0.015x eV/A
(due to x ~o H atoms)
+ 0.112 eV/A
(4p electrons)
+ 0.015 eV/A
(4s electrons)
dx
= (16.048+0.015x) eV/A.
(12)
The contribution due to protons, which are known to take octahedral positions in Pd and are hence situated half-way between consecutive (11 I) planes is obtained by using eq. (11) with I 2 = 0.23 eV and one gets for 4 MeV a-particles: d E / d x = 1.84 × 10- 5 x eV//~ which is about three orders of magnitude smaller than the corresponding electronic contribution shown in eq. (12), and hence can always be neglected. Therefore the net energy loss contribution from the additional H atoms is the electronic part given by the second term of eq. (12). This itself is too small to be detected at low concentrations (i.e. small x). For example for 10 % H atoms in Pd, the additional energy loss is only about 1% of the total electronic energy loss. This also shows that it is quite safe to neglect this effect in the dechanneling calculations for low H concentration, in this system. However, if by suitably adjusting temperature and pressure, one introduces a high concentration of H atoms within the a-phase, the additional energy loss of well channeled particles could be measured. Such an experiment would consist in measuring the minimum energy loss of well channeled particles ]Pal (111) planar channel in the present case] without and with H atoms. The measurements will have to be made at low temperature (say 77 K) because otherwise the dissolved H atoms will anneal out. The energy loss due to dissolved H atoms will be larger in a system where the additional electrons go to the conduction band. For example, if even in the P d - H system the additional electrons were to go to the conduction band, then from eqs. (1) and (7), the additional energy loss would have been 0.0355xeV/,~, which is more than double what is obtained in eq. (12).
It would be equally interesting to study the more complicated fl-phase with higher H concentration. In this case the theory will also have to be extended, because the electrons of H atoms will start filling the new H induced bonding states 7), the number depending upon the H/Pd ratio and approaching 0.5 electron state per P d f o r H/Pd = 1. In addition the interaction between protons themselves will also play an important role, and the whole problem will have to be treated from the point of view of a binary alloy. Another case of theoretical as well as experimental interest is that of heavy atoms being dissolved in metals. In this case many more additional electrons will be added to various bands of the host metal and therefore the additional energy loss will be easily measurable. Dechanneling experiments on one such system (Pd-C) are planned in this laboratory and we hope to be able to study this system theoretically, as regards the channel stopping power in the near future. References
1) For a review and detailed references, see for example Diffusion in solids: recent developments (eds. A. S. Mowick and J. J. Burton; Academic Press, New York, 1974). 2) F . A . Lewis, The palladium/hydrogen system. (Academic Press, New York, 1967). ~) J.J. Quillico and J.C. Jousset, Phys. Rev. BI1 (1975) 1791. 4) j. C. Jousset, J. Mory and J. J. Quillico, J. Physique (Letters) 35 (1974) L 229. 5) y . Qu6r6, Ann. Phys. (Paris) 5 (1970) 105; J. Nucl. Mat. 53 (1974) 262. 6) D. E. Eastman, J. K. Cashion and A. C. Switendick, Phys. Rev. Lett. 27 (1971) 35. 7) E.J. Miller and C.B. Satterthwaite, Phys. Rev. Lett. 34 (1975) 144. s) K. Dettmann and M. T. Robinson, Phys. Rev. B10 (1974) 1. 9) B. R. Appleton, C. Erginsoy and W. M. Gibson, Phys. Rev. 161 (1967) 330. 10) A. P. Pathak, Phys. Stat. Sol. (b) 71 (1975) K 35. ix) A. P. Pathak, J. Phys. C (Sol. Stat. Phys.) 8 (1975) L 341. 12) E. Clementi, D. L. Raimondi and W. P. Reinhardt, J. Chem. Phys. 47 (1967) 1300. 13) j. S. Briggs and A. P. Pathak, J. Phys. C (Sol. State Phys.) 6 (1973) L 153; and in Atomic collisions in solids (eds. S. Datz, B, R. Appleton and C. D. Moak; Plenum Press, New York, 1975) vol 1, p. 15. 14) j . j . Vuillemin and M . G . Priestley, Phys. Rev. Lett. 14 (1965) 307; J. J. Vuillemin, Phys. Rev. 144 (1966) 396. 15) F. M. Mueller, A . J . Freeman, J.O. Dimmock and A. M. Furdyna, Phys. Rev. B1 (1970) 4617. 16) A. P. Pathak, Phys. Rev. BI2 (1975).
I32
(I976)
I25-I28;
©
NORTH-HOLLAND
PUBLISHING
CO.
ENERGY LOSS OF ~t-PARTICLES CHANNELED IN M E T A L - H Y D R O G E N SYSTEMS A N A N D P. P A T H A K
Section d'Etude des Solides Irradids, Centre d'Etudes Nucldaires, Fontenay-aux-Roses (92260), France A simple method is discussed to calculate the energy loss of MeV c~-particles channeled in a planar direction of a (transition) metal without and with a low concentration of hydrogen atoms. The contribution of additional electrons of hydrogen atoms to stopping power is obtained by assuming that in the general case these can go to d-band and to conduction band. In particular for ~-phase of P d - H system all the additional electrons tend to fill the holes in the d-band. The contribution due to additional protons is obtained by considering proton--~-particle scattering and using the experimental activation energy for proton diffusion (0.23 eV for Pd-H) as the m i n i m u m energy that the protons could take in this scattering process. This latter contribution is found to be extremely small compared to the electronic part.
1. Introduction
The problem of hydrogen diffusion in metals has been extensively studied, both experimentally as well as theoretically1). The study has been motivated from points of view of both fundamental as well as applied research, for various reasons such as extremely high diffusivity of hydrogen in metals, possibility to observe quantum effects at low temperatures, availability of three isotopes for hydrogen with largest possible relative mass difference and the technical interest of metal-hydrogen systems in various areas. Most of the studies have been made on the transition metal hydrides and out of these, the palladium-hydrogen system has been studied most extensively2), because even among transition metals, palladium is outstanding because of its unusual properties. Recently channeling studies have also been extended to such systems. The effects of a low concentration of hydrogen dissolved in palladium on the dechanneling of MeV c~-particles, initially channeled along (111) planes, have been clearly demonstrated3). The experimental results have been found to compare well with the calculations 4) based on the simple classical picture in which the scattering of the incoming or-particles from the protons (of dissolved hydrogen atoms) sitting halfway between consecutive (111) planes (i.e. octahedral sites), is described as of simple Rutherford type. In these calculations, the dechanneling is assumed to take place when the transverse energy of the incident particle during this scattering process becomes more than a critical value. Thus, as a first approximation it is reasonable to assume that for the ~t-phase of this system (i.e. for low concentration of H in Pd) the effect of the dissolved protons is only to obstruct the channel without producing any deformation or distortion and that the dechanneling is entirely "obstruc-
tion" type 5) as against distortion type encountered in other types of defects such as dislocations. Based on these simple ideas we give an estimate of the additional energy loss that the channeled particles will suffer due to the dissolved hydrogen in metals. A general procedure for such calculations has been discussed in section 2 and numerical values are given for the particular case of the P d - H system in section 3. 2. General formalism
It is well known that the electronic structure of metal hydrides (specially for transition metals) is very complicated. Depending upon the particular host metal and on the concentration of the dissolved hydrogen, the additional electrons of the H-atoms can go to different bands of the metal and can even induce some new bonding states as has been observed in photoemission studies on the Pd-H system 6, 7). The electronic part of the additional stopping power depends upon the band or shell of the host metal to which the added electrons go. With the point of view of stopping theories we will generally assume that the additional electrons first fill any holes present in for example the d-band of the transition host metal and the rest of the additional electrons tend to go to the conduction band. If there are no holes in the d-band we will assume that all the electrons go to the conduction band. It should be noted that this is only a simplified picture (and a crude one from the point of view of detailed electronic structure) and can be expected to be valid only for low H concentrations (a-phase). For example, in the Pd-H system additional low lying H-induced states have been shown to exist 6' 7) in the fl-phase and these are in fact energetically more favourable to be filled than the s-band. However, this simple picture can be expected to give sensible results because the moving energetic III. E L E C T R O N I C S T O P P I N G
126
ANAND
ion tends to overlook the detailed electronic structure and as we shall note, use of simple hydrogen like atomic orbitals in stopping power calculations gives fairly accurate results. The problem is thus reduced to calculate the stopping power due to the valence ( d ) b a n d with old and new electronic densities and to that due to the conduction band with old and new electronic densities. The conduction electron contribution is easily obtained using the Bethe-Bloch formula for an electron gas whose density is equal to that in the conduction band. For valence band and inner shells one needs to use specific electron wavefunctions. We will use here only the atomic wavefunctions and avoid the use of the complicated band structure. This simplification is implicit in most of the channel stopping power calculations and in any case it is not bad for filled or nearly filled d-band transition metals. In fact the use of hydrogen-like wavefunctions has given very good quantitative agreement between theory and experiments on channel stopping powers of semiconductors for energetic protonsS). The reason that such simplifications, too crude from the band theory point of view, do work, is that the fast incoming particle tends to overlook the detailed band structure. The electronic stopping power in the high velocity region may be written a s 9 l o ) : (1)
my 2
where
Lv
= Pval
(2r/Iv~/hcop),
in good agreement with the experiment1°). In fact one can easily see from eqs. (1) and ( 2 ) t h a t when P~oc becomes equal to Pva~, i.e. when the incoming particle encounters all of the valence electrons, one gets the formula of Dettmann and Robinson 8) for the valence shell. Ths last term L~ represents the contribution of inner shells, pj and Ij being corresponding densities and binding energies. First we consider the energy loss to those additional electrons which go to the valence d-shell and thus increasing the contribution Lv by a certain amount. If x is the atomic percentage of H atoms dissolved in the host metal, one has 0.01 x additional electrons per atom. Let us assume that out of these, 0.01 xx per atom go to the d-shell and rest, 0.01 x 2 ( = 0 . 0 1 x - 0 . 0 1 x l ) to the conduction band. If there are n h holes per atom in the d-band (for Pd, nh=0.36 ) then 0 . 0 1 x t < n h. The resulting change in Pva~ of eq. (2) is simply n~ = 0.01 xl N, where N is the atomic concentration in the host metal. The contribution to P~oc is obtained by using appropriate occupation numbers of the d-shell (with and without H atoms) in the planar averaged expression for change density due to t h e j t h shell 1° 11): ~(1)
f,j
-
N d p ~L:'~~,~ i)"2nJ - wjexp(-2~jy) 2 2nj
(2mv2/lj),
L I = ~ p j In
In (Vl/VF) + P,oc In (2my1VF/I )
,
(2)
where e and m are the electronic charge and mass, respectively and Z~ and v~, are charge and velocity of the incident particle. In the above equations L c represents the conduction electron contribution, Pc and cop being corresponding density and plasma frequency. Lv is the valence electron (d-shell in transition metals) contribution, having two terms: the first term corresponds to the contribution of collective excitations which uses the total density of valence electrons while the second term gives the contribution of local electron density in the channel, to single particle excitations, vF being the Fermi velocity and I, the binding energy of the d-shell. This separation of valence electron contribution into two parts 9) has been found to give the stopping power in the (l 11) planar channels of gold single crystals for energetic e-particles which is
2nj-t
~ × k=o
y2n~- 1 -k ×
4 nZ~ e 4 S~ -- - (Lc+Lv+LI),
L c = Pc In
P. PATHAK
,
(3)
(2 ~j)k (2 n i - 1 - k) ! where dp is the interplanar distance (so that Ndp is atomic density per unit area of the plane), n j, wj and ~j are the principal quantum number, the occupation number and the orbital exponent of the jth shell respectively, p}l) (y) represents the planar density due to one plane at a distance y from it, so that the total electron density due to both planes is: pj(y) = p~t)(y) + o(I) ,-J ( d p - y ) .
(4)
The change in Ploc due to diffused H atoms is obtained by using the change in occupation number wj, i.e. 0.01Xl in eqs. (3) and (4). Thus having found changes in Pva~ and Ploc due to the diffused H atoms, one can easily calculate the corresponding change in stopping power from eqs. (1) and (2). It may be added that eqs. (3) and (4) also give the contribution of other inner shells to the effective electron density responsible for stopping. Next, we consider the contribution to stopping power of any electrons going to the conduction band of the host metal. Here any change in the electron density also affects the plasma frequency cop appearing in the
ENERGY
logarithmic term of L c so that the change in the stoping power is obtained by calculating: dLc --- E c - L c = Pc' In -2mv~ - - p ¢ I n - 2my - , 2
(5)
where
and
,
dE = nm~m2v~ Qd. dx ( m 1 + 1Tl2) 2
(8)
where n is the density of protons ( = 0.01 xN), Qd is momentum transfer cross section for scattering of protons from the a-particle, given by:
Qa = 2n
P'c = Pc + / ' / 2 ~ Pc + 0 . 0 1 X2 N,
g% =
127
LOSS OF O ~ - P A R T I C L E S
f;
Io(0) (1 - cos0) sin0 dO.
(9)
For fast a-particles, the scattering is essentially from the bare charge Z1 ( = 2) so that Io (0) is given by:
(4n_~e2) ~ ~
(
= COp 1 + -
Z~ e 4 1 I 0(0) = m 2/)14 ( l - c o s 0 f "
nzf,
(10)
Using eq. (10) in eq. (9) and performing the indicated integration we get
Pc
so that
dE
2 n n m 2l Z- 2l e 4-
2m2m2vl2
-
dx
h~op For n 2 ~ P ¢ , this can be put in a closed form as (7)
h~p Thus in this limit, the conduction electron contribution to the change in stopping power is also directly proportional to the change in the electron density like in the previous case of change in the valence electron contribution, although an additional factor of one half appears in eq. (7). Finally, we consider the contribution of protons themselves to the stopping of incoming particles. Since the proton mass is about three orders of magnitude larger than the electron mass, it is to be expected that this contribution will be smaller by about the same order than the electronic contribution. However, for completeness, we give the detailed formula and show that this is in fact quite a small contributon. Let us consider the scattering of a proton from the incoming a-particle. It is reasonable to assume that any proton velocity due to its jumping around in the host metal is negligible small compared to the velocity of the incoming a-particle. Also for low H concentrations in the a-phase of the system, the interaction between the protons themselves can be neglected. The energy taken away by the proton (mass m~) from the a-particle (mass m2) in such a scattering process is given by 13):
.
Of)
( m l + m 2 ) 2 m 2 v2 I n ( m l + m 2 ) 2 I s
The divergences in the integration have been taken care of by recalling 13) that 0 = n in eq. (9) corresponds to maximum energy transfer from a-particle to proton, T~nax = 2m~m 2 v2/(ml + m 2 ) 2 and minimum 0 is decided by the minimum energy that the proton could take 0.e. its diffusion energy I2 = 0.23 eV for the P d - H system) T mi, 2 -----I 2 . 3. Results and discussion
We now use the above formulae to calculate the stopping power for 4 MeV a-particles channeled along (111) planes of a palladium crystal without and with a small concentration of H atoms (in a-phase) introduced in it. It is well established, both experimentally 14) and theoretically 15) that there are 0.36 holes per atom in the 4d-band of palladium and the same number of electrons in the conduction band. Thus Pc = 0 . 3 6 N and the occupation number w4a = 9.64 so that Pval = 9.64 N -- 0.097 (in a.u.). As long as the atomic percentage (x) of H atoms is less than 36, we can safely assume that all the additional electrons go to the d-band and none to the conduction band 022 ~ 0 ) . Thus only the valence electron contribution to the stopping power will change and nl=n=O.OlxN=O.OOOlx (a.u.) because for Pd, N = 0.01007 a.u. Using eqs. (3) and (4) to calculate Ploc at a point half-way between two consecutive (111) planes we get P~oc = 0.00857 in pure palladium, i.e. with w4a = 9.64, and P~oc= 0.00857+9 x 1 0 - 6 x for the P d - H system, i.e. with u,~d = 9 . 6 4 + 0 . 0 1 x . Similarly calculating the planar electron density due to other inner shells, and III. E L E C T R O N I C
STOPPING
128
A N A N D P. P A T H A K
then using it in the stopping power expressions (l) and (2), we get for 4 MeV ~-particles16): dE -
1.331 eV/A
(conduction electrons)
+ 14.59 eV/A
(valence electrons)
+ 0.015x eV/A
(due to x ~o H atoms)
+ 0.112 eV/A
(4p electrons)
+ 0.015 eV/A
(4s electrons)
dx
= (16.048+0.015x) eV/A.
(12)
The contribution due to protons, which are known to take octahedral positions in Pd and are hence situated half-way between consecutive (11 I) planes is obtained by using eq. (11) with I 2 = 0.23 eV and one gets for 4 MeV a-particles: d E / d x = 1.84 × 10- 5 x eV//~ which is about three orders of magnitude smaller than the corresponding electronic contribution shown in eq. (12), and hence can always be neglected. Therefore the net energy loss contribution from the additional H atoms is the electronic part given by the second term of eq. (12). This itself is too small to be detected at low concentrations (i.e. small x). For example for 10 % H atoms in Pd, the additional energy loss is only about 1% of the total electronic energy loss. This also shows that it is quite safe to neglect this effect in the dechanneling calculations for low H concentration, in this system. However, if by suitably adjusting temperature and pressure, one introduces a high concentration of H atoms within the a-phase, the additional energy loss of well channeled particles could be measured. Such an experiment would consist in measuring the minimum energy loss of well channeled particles ]Pal (111) planar channel in the present case] without and with H atoms. The measurements will have to be made at low temperature (say 77 K) because otherwise the dissolved H atoms will anneal out. The energy loss due to dissolved H atoms will be larger in a system where the additional electrons go to the conduction band. For example, if even in the P d - H system the additional electrons were to go to the conduction band, then from eqs. (1) and (7), the additional energy loss would have been 0.0355xeV/,~, which is more than double what is obtained in eq. (12).
It would be equally interesting to study the more complicated fl-phase with higher H concentration. In this case the theory will also have to be extended, because the electrons of H atoms will start filling the new H induced bonding states 7), the number depending upon the H/Pd ratio and approaching 0.5 electron state per P d f o r H/Pd = 1. In addition the interaction between protons themselves will also play an important role, and the whole problem will have to be treated from the point of view of a binary alloy. Another case of theoretical as well as experimental interest is that of heavy atoms being dissolved in metals. In this case many more additional electrons will be added to various bands of the host metal and therefore the additional energy loss will be easily measurable. Dechanneling experiments on one such system (Pd-C) are planned in this laboratory and we hope to be able to study this system theoretically, as regards the channel stopping power in the near future. References
1) For a review and detailed references, see for example Diffusion in solids: recent developments (eds. A. S. Mowick and J. J. Burton; Academic Press, New York, 1974). 2) F . A . Lewis, The palladium/hydrogen system. (Academic Press, New York, 1967). ~) J.J. Quillico and J.C. Jousset, Phys. Rev. BI1 (1975) 1791. 4) j. C. Jousset, J. Mory and J. J. Quillico, J. Physique (Letters) 35 (1974) L 229. 5) y . Qu6r6, Ann. Phys. (Paris) 5 (1970) 105; J. Nucl. Mat. 53 (1974) 262. 6) D. E. Eastman, J. K. Cashion and A. C. Switendick, Phys. Rev. Lett. 27 (1971) 35. 7) E.J. Miller and C.B. Satterthwaite, Phys. Rev. Lett. 34 (1975) 144. s) K. Dettmann and M. T. Robinson, Phys. Rev. B10 (1974) 1. 9) B. R. Appleton, C. Erginsoy and W. M. Gibson, Phys. Rev. 161 (1967) 330. 10) A. P. Pathak, Phys. Stat. Sol. (b) 71 (1975) K 35. ix) A. P. Pathak, J. Phys. C (Sol. Stat. Phys.) 8 (1975) L 341. 12) E. Clementi, D. L. Raimondi and W. P. Reinhardt, J. Chem. Phys. 47 (1967) 1300. 13) j. S. Briggs and A. P. Pathak, J. Phys. C (Sol. State Phys.) 6 (1973) L 153; and in Atomic collisions in solids (eds. S. Datz, B, R. Appleton and C. D. Moak; Plenum Press, New York, 1975) vol 1, p. 15. 14) j . j . Vuillemin and M . G . Priestley, Phys. Rev. Lett. 14 (1965) 307; J. J. Vuillemin, Phys. Rev. 144 (1966) 396. 15) F. M. Mueller, A . J . Freeman, J.O. Dimmock and A. M. Furdyna, Phys. Rev. B1 (1970) 4617. 16) A. P. Pathak, Phys. Rev. BI2 (1975).