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The Landau-Zener model for electron capture by highly charged ions in collisions with atomic hydrogen
Nuclear Instruments and Methods in Physics Research B42 (1989) 475-478
475
North-Holland, Amsterdam
THE LANDAU-ZENER MODEL FOR ELECTRON IN COLLISIONS WITH ATOMIC HYDROGEN J.H. MACEK Department
CAPTURE
BY HIGHLY CHARGED
IONS
and X.Y. DONG
of Physics and Astronomy,
The University of Tennessee, Knoxville,
TN 37992-1200,
USA
The two-state Landau-Zener model with rotational and degenerate Stark-effect coupling has been formulated to yield cross sections Qn,m for populating Rydberg substates 1n/m) of the one electron projectile ion P (z-‘)+ in the transfer reaction PZ++H(ls)
+p(=-‘)+(nlm)+H+,
where Pz+ represents I-distributions-exhibit
a bare projectile ion. Total cross sections some discrepancies at high velocities.
1. Intruduction
Electron capture by highly charged projectiles at velocities below the mean electron velocity of the electron in the initial 1s state of the target has been described theoretically by Olson and Salop [l] in terms of a two-state curve crossing mechanism. The diabatic potential curve correlating to the initial state Pz+ + H(ls) crosses several diabatic potential curves correlating to the final state PCz-‘)+(nlm) + H+ formed by electron capture. For a given n-manifold of n(n + 1)/2 eigenstates only one particular substate interacts significantly with the initial state at the crossing. This particular insight has been further developed by Taulbjerg [2] and forms the basis for his successful reaction window analysis of low-energy ion-atom collisions. Despite this success, total cross sections in the original two-state theory are not in good accord with more elaborate calculations [3,4] and with experiment [5]. The multistate calculations of Briggs and Vaaben [3] and Green, Shipsey, and Browne [4] show that all symmetry-allowed nlm substates play a role in the capture process. A model of Abramov, Baryshnikov, and Lisitsa [6], which is similar to a model of Jaecks and co-workers [7], takes account of coupling between all of the degenerate substates approximately and obtains total cross sections in much better accord with more elaborate calculations than does the original two-state model [8]. The model of Abrabov et al. [6] was originally formulated only for total cross sections but was adapted to yield partial cross sections by Grozdanov and Belie [9]. This particular adaptation was unsuccessful, but it emphasized that partial cross sections, and in particular I-distributions, test the model more sensitively than do total cross sections. This model is also similar to that of ref. [7], which was used to compute orientation parame0168-583X/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
agree well with experiment
and more elaborate
calculations
but the
ters, but employs additional approximations. In the work reported here we employ the weaker approximations of ref. [7] to compute I-distributions and total cross sections. We find satisfactory agreement with the I-distributions computed by Green et al. [4] at low velocities, but find significant discrepancies as the relative velocity approaches the matching velocity.
2. Formulation We follow the general method of ref. [7] to write down the time evolution matrix U(t, 1’) for the collision system. The time domain is divided into three regions, labeled I, II, and III, as illustrated in fig. 1. The crossing between the initial state 10) with the single Stark state ] Q) coupling to 10) occurs on the incoming part of the trajectory at ti and on the outgoing part at t,. The times ti and t, separate the regions I, II, and III. The Stark eigenstates other than ] Q) are are labeled by P. This partition of the degenerate Rydberg states of the ion with principle quantum number n is such that the space spanned by P + Q includes all Irn substates of the level n. Alternatively, the eigenstates can be written
Fig. 1. Schematic adiabatic energy levels of the P+= + H(ls) system vs time.
476
J. H. Mocek, X. Y. Dong / Electron capture by
in the pseudo-spin
representation 1j,m, j,m,) with and m,+m,=m. Both sets are complete in the subspace of the n-level, but the latter is more convenient for computations since the long-range Stark coupling is diagonal in this representation. We write the U( - co, CX) matrix using the Landau-Zener model at curve crossings, and employ the model of ref. [6] to compute U(t, t’) in the regions I, II and III. We have
jl=j,=(n-1)/2
U(co,
-co)
= uI”(OO,tr)UU(lf)U’I(tf, xLT’(t,
-i (1)
a=b=
s*3
a = 0,
-s,
a=Q,
6 oh>
otherwise
l)[l-
(b,‘R)2]1’2
(3) ’
-nT/*+in/4+iy(lny-1)
where u represents the relative velocity. In addition the Landau-Zener probabi~ty p is given by e-2nY
(6)
and we have lsl2=1--p. The matrix given by
(7) U*(t,
t’),
where A = I, II, or III,
I uyt, l’)@= U”(t, UA(t,
t’)OO=exp
i
-i[‘E,(r”)
dt”
tyio=O,
is
(8)
1
i f 0,
(9)
and
U*(t,
t’)ij=u(t,
tr)ij
exp -ir’E,,(t”)
[ i+O,
j#O.
f
dt”
I
dt - iJmE,(t)
dt
I*
= C1 -P){[
P + (l-
fi)’
I UQ,Q(tf>
ti> I’]
h> I *
ti)
+
(174
c*c.]),
where C.C. denotes complex conjugate. It remains to determine the matrix u(t, t’). In the work of Jaecks et al. 171 this matrix was just a rotation matrix mixing degenerate eigenstates at the united atom limit. In the present model it is also necessary to take into account the electrostatic interaction of the H+ ion with the electron in a Rydberg level of the P(‘-‘)+ ion. This is done approximately in the model of Abramaov et al. [6]. We simply summarize their results. Let O( t, t’) represent the angle through which the internuclear axis turns in the time interval (t, t’), and let b represent the impact parameter in the straight line approximation for the relative motion of the nuclei. Sets of Euler angles a, /3 and a complex angle 6 are defined according to t’) = [(3n,‘2buZ)2+
tan[ a( t, t’)] (10)
Here Ea( t) and E,,(t) denote the energies of the diabatic states correlating to the 1s state of hydrogen and the Rydberg level of the one-electron ion, respectively. The submatrix u( t, t’) which couples the Rydberg levels is determined by the dynamics of the collision in an approximation which represents the wave function of the system as a superposition of the degen-
” E,(t)
= I&,(b)
XU*nlm,Q (m>
w(t,
,
-02 fi 1. (11)
X I Gn,Q(~*
(5)
3
1
Upon squaring and dropping terms with the rapidly varying factor cos{ j:r[E,(t) - E,(t)] dt} we have P,r,(b)
e-2.648R/fi
r(l+i~)~
p = g2 =
-i I
(4)
(2ay)“2
I
Xexp
g = e--ny,
‘=
dt
(2)
b=O,
(9.13R)’ oZ(Z-
mE,(t)
/‘1
Q!
b=Q,
and iJ”( r,) = ULz( ti)‘. In the reaction window model of Taulbjerg [2] the Landau-Zener parameters g and s are given by
‘=
erate substates of the Rydberg level n. A specific model for u(t, t’) will be given later. We are interested in the element Un,m,O(~, - 00) equal to the amplitude A,,,(b) for the populating the state nfm by capture at an impact parameter b. It is found by carrying out the matrix multiplication in eq. (1). The result is
t;)uk(r;)
-co),
a=b=O,
ULZ(ti)&=g,
highfy churged ions
l]%(t,
tan[ w( t, t/)/2]
= -
[(3n,‘2buZ)‘+
sin[p(r,
(13)
t’)
(14)
111’*
sin[w(t, t’)/2] t’)] = [1 + (2buZ,3n)2]1/2
cos[ 6( t, t’)] = 2/ 1 + ei2a(‘,r’) cot
i
05) W(t, f’) 2
1_ (16) * 1
J.H. Macek, X. Y. Dong / Electron capture by highly charged ions
The evolution operator u(t, t’) is then given in the pseudo-spin representation by
477
B
where Jr and J, are pseudo-spin operators. Because the matrix u( t, t’) is given in terms of angular momentum operators, the manipulations necessary to obtain the l-distributions can be carried out analytically. After a considerable amount of angular momentum algebra we obtain a relatively simple expression for P”,(b) = To express our result we introduce angles Q and B with subscripts i, f and fi corresponding to time intervals (co, t,), (co, tr) and (tr, ti) respectively. In addition we introduce the complex angles tii, trr and t,r given by
Then our expression
Fig. 2. Total cross section Q, vs velocity for electron capture into the n = 4 level by fully stripped carbon ions. The solid curve is the present calculation, the dotted curve is the 33-state calculation of ref. [4], and the points with error bars are experimental data of ref. [5].
(18)
a, b = i, f.
5&=%-q,
VELOCITY (lo7 cm/set)
for P,,(b)
is
P,,(b)=(l-p)(M-j]10)2 x ([ P + (l - fi)’ X (1 -
cos”“-1’(PS/2)]
sin’& sin’s f ) “-‘~,(cos
+(l
- sin2& sin2cri)n-1p,(cos
-(l
- fi)[ei2(n-1)nfi
S,,) .$,) L-A _.a
cos”“-‘)(p,/2)
X [ ei2ar cos2( &/2)
+ sin2 (/I/2)]
X [eei2011 cos’( /3,/2) + sin’( p/2)] XP,(COS &) + C.C.
1)
& n-1 ‘-l
,
where P,(x) denotes Legendre’s (j,m, j2mz 1Im) is a Clebsch-Gordan
-
0.5
1
(19) polynomial coefficient.
and
IO
VELOCITY (lo7 cm/set) Fig. 3. Partial cross sections Q4, for capture into the n = 4 I-sublevels by fully sthpped carbon ions.
3. Numerical results With our analytical results for P,,(b) and the corresponding cross sections Q,,, the l-distributions are readily computed. As a check on our computations we compute Q, by summing over 1 and compare this with the cross section obtained by Abramov et al. [6]. The results are identical. Fig. 2 shows the total cross section as a function of velocity for C6+ projectiles. The solid curve is our result, the dotted curve the 33-state calculations of Green et al. [4], and the points are the experimental data of ref. [5]. The agreement is satisfactory, but the comparison does not provide a stringent test of the model. Fig. 3 shows the I-distributions which are the main result of this work. The most pronounced feature is the strong peaking of the I = 3 partial cross section near
WY P0 c
a
a
6
11”’ 0.5
1
IO
VELOCITY (lo7 cm/see) Fig. 4. Partial cross section Q.,s for capture into the n = 4, I = 3 sublevels by fully stripped carbon ions in collisions with H(k). The solid curve is the present calculation and the dotted curve is the 33-state calculation of ref. [4].
478
J. H. Macek, X. Y. Dong / Electron capture by highly charged ions
u = 5 x 10’ cm/s. Fig. 4 compares this I = 3 partial cross section with the multi state computations of ref. [4]. Note that the rise towards a large I = 3 cross section at 5 x 10’ cm/s is in agreement with the 33-state calculations but the rapid decrease with increasing velocity on the high energy side of the peak is not. This indicates that the model of Abramov et al. [6] is not reliable for i-distributions above a velocity of OS au. even though total cross sections are in good agreement with the multistate calculations. The model is readily extended to treat crossings of many different n-levels. It is only necessary to include more factors in the product eq. (1) representing the evolution matrix U( t, t’). Total cross sections in such a model have been reviewed elsewhere [8]. Here we emphasize that I and m distributions are also easily computed.
4. Summary In summary, the model of Abramov et al. [6] represents a very good first approximation for total electron capture cross sections within the Landau-Zener framework. Partial cross sections can also be computed using the method of ref. [7] and provide a more stringent test of the model. Our results show that I-distributions
depart from those obtained by more exact calculations, but that the agreement is still good at velocities near to and below the cross section maximum. At higher velocities discrepancies appear, as would be expected in any Landau-Zener treatment. Support for this research by the National Science Foundation under grant number PHY-8602988 is gratefully acknowledged.
References PI BE. Olsen and A. Salop, Phys. Rev. Al4 (1976) 579. PI K. Taulbjerg, J. Phys. B19 (1986) L367. [31 J.S. Briggs and J. Vaaben, J. Phys. B15 (1982) L520. [41 T.A. Green, E.J. Shipsey and J.C. Browne, Phys. Rev. A25 (1982) 1364. [A R.A. Phanueuf, I. Alvarez, F.W. Meyer and D.H. Crandall, Phys. Rev. A26 (1982) 1892. [61 V.A. Abramov, F.F. Baryshnikov and V.S. Lisitsa, Zh. Exp. Teor. Fiz. 74 (1978) 897. 171 D.H. Jaecks, F.W. Erikson, W. de Rijk and J.H. Macek, Phys. Rev. Lett. 35 (1975) 723. 181 R.K. Janev and H. Winter, Phys. Rep. 117 (1985) 265. [91 T.P. Grozdanov and D.S. Belie, Phys. Scripta 30 (1984)
194.
475
North-Holland, Amsterdam
THE LANDAU-ZENER MODEL FOR ELECTRON IN COLLISIONS WITH ATOMIC HYDROGEN J.H. MACEK Department
CAPTURE
BY HIGHLY CHARGED
IONS
and X.Y. DONG
of Physics and Astronomy,
The University of Tennessee, Knoxville,
TN 37992-1200,
USA
The two-state Landau-Zener model with rotational and degenerate Stark-effect coupling has been formulated to yield cross sections Qn,m for populating Rydberg substates 1n/m) of the one electron projectile ion P (z-‘)+ in the transfer reaction PZ++H(ls)
+p(=-‘)+(nlm)+H+,
where Pz+ represents I-distributions-exhibit
a bare projectile ion. Total cross sections some discrepancies at high velocities.
1. Intruduction
Electron capture by highly charged projectiles at velocities below the mean electron velocity of the electron in the initial 1s state of the target has been described theoretically by Olson and Salop [l] in terms of a two-state curve crossing mechanism. The diabatic potential curve correlating to the initial state Pz+ + H(ls) crosses several diabatic potential curves correlating to the final state PCz-‘)+(nlm) + H+ formed by electron capture. For a given n-manifold of n(n + 1)/2 eigenstates only one particular substate interacts significantly with the initial state at the crossing. This particular insight has been further developed by Taulbjerg [2] and forms the basis for his successful reaction window analysis of low-energy ion-atom collisions. Despite this success, total cross sections in the original two-state theory are not in good accord with more elaborate calculations [3,4] and with experiment [5]. The multistate calculations of Briggs and Vaaben [3] and Green, Shipsey, and Browne [4] show that all symmetry-allowed nlm substates play a role in the capture process. A model of Abramov, Baryshnikov, and Lisitsa [6], which is similar to a model of Jaecks and co-workers [7], takes account of coupling between all of the degenerate substates approximately and obtains total cross sections in much better accord with more elaborate calculations than does the original two-state model [8]. The model of Abrabov et al. [6] was originally formulated only for total cross sections but was adapted to yield partial cross sections by Grozdanov and Belie [9]. This particular adaptation was unsuccessful, but it emphasized that partial cross sections, and in particular I-distributions, test the model more sensitively than do total cross sections. This model is also similar to that of ref. [7], which was used to compute orientation parame0168-583X/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
agree well with experiment
and more elaborate
calculations
but the
ters, but employs additional approximations. In the work reported here we employ the weaker approximations of ref. [7] to compute I-distributions and total cross sections. We find satisfactory agreement with the I-distributions computed by Green et al. [4] at low velocities, but find significant discrepancies as the relative velocity approaches the matching velocity.
2. Formulation We follow the general method of ref. [7] to write down the time evolution matrix U(t, 1’) for the collision system. The time domain is divided into three regions, labeled I, II, and III, as illustrated in fig. 1. The crossing between the initial state 10) with the single Stark state ] Q) coupling to 10) occurs on the incoming part of the trajectory at ti and on the outgoing part at t,. The times ti and t, separate the regions I, II, and III. The Stark eigenstates other than ] Q) are are labeled by P. This partition of the degenerate Rydberg states of the ion with principle quantum number n is such that the space spanned by P + Q includes all Irn substates of the level n. Alternatively, the eigenstates can be written
Fig. 1. Schematic adiabatic energy levels of the P+= + H(ls) system vs time.
476
J. H. Mocek, X. Y. Dong / Electron capture by
in the pseudo-spin
representation 1j,m, j,m,) with and m,+m,=m. Both sets are complete in the subspace of the n-level, but the latter is more convenient for computations since the long-range Stark coupling is diagonal in this representation. We write the U( - co, CX) matrix using the Landau-Zener model at curve crossings, and employ the model of ref. [6] to compute U(t, t’) in the regions I, II and III. We have
jl=j,=(n-1)/2
U(co,
-co)
= uI”(OO,tr)UU(lf)U’I(tf, xLT’(t,
-i (1)
a=b=
s*3
a = 0,
-s,
a=Q,
6 oh>
otherwise
l)[l-
(b,‘R)2]1’2
(3) ’
-nT/*+in/4+iy(lny-1)
where u represents the relative velocity. In addition the Landau-Zener probabi~ty p is given by e-2nY
(6)
and we have lsl2=1--p. The matrix given by
(7) U*(t,
t’),
where A = I, II, or III,
I uyt, l’)@= U”(t, UA(t,
t’)OO=exp
i
-i[‘E,(r”)
dt”
tyio=O,
is
(8)
1
i f 0,
(9)
and
U*(t,
t’)ij=u(t,
tr)ij
exp -ir’E,,(t”)
[ i+O,
j#O.
f
dt”
I
dt - iJmE,(t)
dt
I*
= C1 -P){[
P + (l-
fi)’
I UQ,Q(tf>
ti> I’]
h> I *
ti)
+
(174
c*c.]),
where C.C. denotes complex conjugate. It remains to determine the matrix u(t, t’). In the work of Jaecks et al. 171 this matrix was just a rotation matrix mixing degenerate eigenstates at the united atom limit. In the present model it is also necessary to take into account the electrostatic interaction of the H+ ion with the electron in a Rydberg level of the P(‘-‘)+ ion. This is done approximately in the model of Abramaov et al. [6]. We simply summarize their results. Let O( t, t’) represent the angle through which the internuclear axis turns in the time interval (t, t’), and let b represent the impact parameter in the straight line approximation for the relative motion of the nuclei. Sets of Euler angles a, /3 and a complex angle 6 are defined according to t’) = [(3n,‘2buZ)2+
tan[ a( t, t’)] (10)
Here Ea( t) and E,,(t) denote the energies of the diabatic states correlating to the 1s state of hydrogen and the Rydberg level of the one-electron ion, respectively. The submatrix u( t, t’) which couples the Rydberg levels is determined by the dynamics of the collision in an approximation which represents the wave function of the system as a superposition of the degen-
” E,(t)
= I&,(b)
XU*nlm,Q (m>
w(t,
,
-02 fi 1. (11)
X I Gn,Q(~*
(5)
3
1
Upon squaring and dropping terms with the rapidly varying factor cos{ j:r[E,(t) - E,(t)] dt} we have P,r,(b)
e-2.648R/fi
r(l+i~)~
p = g2 =
-i I
(4)
(2ay)“2
I
Xexp
g = e--ny,
‘=
dt
(2)
b=O,
(9.13R)’ oZ(Z-
mE,(t)
/‘1
Q!
b=Q,
and iJ”( r,) = ULz( ti)‘. In the reaction window model of Taulbjerg [2] the Landau-Zener parameters g and s are given by
‘=
erate substates of the Rydberg level n. A specific model for u(t, t’) will be given later. We are interested in the element Un,m,O(~, - 00) equal to the amplitude A,,,(b) for the populating the state nfm by capture at an impact parameter b. It is found by carrying out the matrix multiplication in eq. (1). The result is
t;)uk(r;)
-co),
a=b=O,
ULZ(ti)&=g,
highfy churged ions
l]%(t,
tan[ w( t, t/)/2]
= -
[(3n,‘2buZ)‘+
sin[p(r,
(13)
t’)
(14)
111’*
sin[w(t, t’)/2] t’)] = [1 + (2buZ,3n)2]1/2
cos[ 6( t, t’)] = 2/ 1 + ei2a(‘,r’) cot
i
05) W(t, f’) 2
1_ (16) * 1
J.H. Macek, X. Y. Dong / Electron capture by highly charged ions
The evolution operator u(t, t’) is then given in the pseudo-spin representation by
477
B
where Jr and J, are pseudo-spin operators. Because the matrix u( t, t’) is given in terms of angular momentum operators, the manipulations necessary to obtain the l-distributions can be carried out analytically. After a considerable amount of angular momentum algebra we obtain a relatively simple expression for P”,(b) = To express our result we introduce angles Q and B with subscripts i, f and fi corresponding to time intervals (co, t,), (co, tr) and (tr, ti) respectively. In addition we introduce the complex angles tii, trr and t,r given by
Then our expression
Fig. 2. Total cross section Q, vs velocity for electron capture into the n = 4 level by fully stripped carbon ions. The solid curve is the present calculation, the dotted curve is the 33-state calculation of ref. [4], and the points with error bars are experimental data of ref. [5].
(18)
a, b = i, f.
5&=%-q,
VELOCITY (lo7 cm/set)
for P,,(b)
is
P,,(b)=(l-p)(M-j]10)2 x ([ P + (l - fi)’ X (1 -
cos”“-1’(PS/2)]
sin’& sin’s f ) “-‘~,(cos
+(l
- sin2& sin2cri)n-1p,(cos
-(l
- fi)[ei2(n-1)nfi
S,,) .$,) L-A _.a
cos”“-‘)(p,/2)
X [ ei2ar cos2( &/2)
+ sin2 (/I/2)]
X [eei2011 cos’( /3,/2) + sin’( p/2)] XP,(COS &) + C.C.
1)
& n-1 ‘-l
,
where P,(x) denotes Legendre’s (j,m, j2mz 1Im) is a Clebsch-Gordan
-
0.5
1
(19) polynomial coefficient.
and
IO
VELOCITY (lo7 cm/set) Fig. 3. Partial cross sections Q4, for capture into the n = 4 I-sublevels by fully sthpped carbon ions.
3. Numerical results With our analytical results for P,,(b) and the corresponding cross sections Q,,, the l-distributions are readily computed. As a check on our computations we compute Q, by summing over 1 and compare this with the cross section obtained by Abramov et al. [6]. The results are identical. Fig. 2 shows the total cross section as a function of velocity for C6+ projectiles. The solid curve is our result, the dotted curve the 33-state calculations of Green et al. [4], and the points are the experimental data of ref. [5]. The agreement is satisfactory, but the comparison does not provide a stringent test of the model. Fig. 3 shows the I-distributions which are the main result of this work. The most pronounced feature is the strong peaking of the I = 3 partial cross section near
WY P0 c
a
a
6
11”’ 0.5
1
IO
VELOCITY (lo7 cm/see) Fig. 4. Partial cross section Q.,s for capture into the n = 4, I = 3 sublevels by fully stripped carbon ions in collisions with H(k). The solid curve is the present calculation and the dotted curve is the 33-state calculation of ref. [4].
478
J. H. Macek, X. Y. Dong / Electron capture by highly charged ions
u = 5 x 10’ cm/s. Fig. 4 compares this I = 3 partial cross section with the multi state computations of ref. [4]. Note that the rise towards a large I = 3 cross section at 5 x 10’ cm/s is in agreement with the 33-state calculations but the rapid decrease with increasing velocity on the high energy side of the peak is not. This indicates that the model of Abramov et al. [6] is not reliable for i-distributions above a velocity of OS au. even though total cross sections are in good agreement with the multistate calculations. The model is readily extended to treat crossings of many different n-levels. It is only necessary to include more factors in the product eq. (1) representing the evolution matrix U( t, t’). Total cross sections in such a model have been reviewed elsewhere [8]. Here we emphasize that I and m distributions are also easily computed.
4. Summary In summary, the model of Abramov et al. [6] represents a very good first approximation for total electron capture cross sections within the Landau-Zener framework. Partial cross sections can also be computed using the method of ref. [7] and provide a more stringent test of the model. Our results show that I-distributions
depart from those obtained by more exact calculations, but that the agreement is still good at velocities near to and below the cross section maximum. At higher velocities discrepancies appear, as would be expected in any Landau-Zener treatment. Support for this research by the National Science Foundation under grant number PHY-8602988 is gratefully acknowledged.
References PI BE. Olsen and A. Salop, Phys. Rev. Al4 (1976) 579. PI K. Taulbjerg, J. Phys. B19 (1986) L367. [31 J.S. Briggs and J. Vaaben, J. Phys. B15 (1982) L520. [41 T.A. Green, E.J. Shipsey and J.C. Browne, Phys. Rev. A25 (1982) 1364. [A R.A. Phanueuf, I. Alvarez, F.W. Meyer and D.H. Crandall, Phys. Rev. A26 (1982) 1892. [61 V.A. Abramov, F.F. Baryshnikov and V.S. Lisitsa, Zh. Exp. Teor. Fiz. 74 (1978) 897. 171 D.H. Jaecks, F.W. Erikson, W. de Rijk and J.H. Macek, Phys. Rev. Lett. 35 (1975) 723. 181 R.K. Janev and H. Winter, Phys. Rep. 117 (1985) 265. [91 T.P. Grozdanov and D.S. Belie, Phys. Scripta 30 (1984)
194.