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The variational upper and lower bounds for stopping powers of atoms and molecules
%olume 6; ntimber 3 _.
1 August 1970
CHEMICALPHYSICSLXTTERS
THE FOR
VARIATIONAL
STOPPING
UPPER‘AND
-POWERS
OF
BOUNDS
AND
MOLECULES
..
M. S. YURIFV
--
LOWER
ATOMS
Department of Theoretical Physics, Physical Institute, University of Leningrad, Leningrad, USSR
Received
19
May 1970
A variational procedure for calculating upper and lower bounds for the logarithm of the average energy cf excitation InZ in the theory of the stopping power of :Itoms and molecules is described. The upper and lower bounds for lnlof the hydrogen atom are calculated.
The knowledge of the average energy loss by the fast charged particles during inelastic collisions with atoms (or molecules) is important in such fields as plasma physics, nuclear physics and astrophysics [l]. In the Born-B&he approximation, the average energy loss K is given by [2] K=?@-ln
4
fiqrnax I
7X4
where
v is the velocity
pulse,
transferred
(1)
’
of the incident
in t&e collision,
particle, 2 is the nuclear change, Mrnax is the maximum imand I is an average excitation energy, defined by
(2) in a conventional notation. Eq. ( 1) contains only one constant, InI; wh ch characterizes the stopping matter. Dalgarno and collaborators (see for example ref. [ ;3] ) ca.lcula:sd lni for many.atoms, using the sum rules for the oscillator strengths. Besides, Chan and Dalgarno suggested a variational procedure for the evaluation of lnd[4]. ‘In this paper a variational metnod for calculating upper and lower bounds for 1nI is suggested. Our method is based on ,an integral representation for 1nI analogous to the Dmitriev-Labsovski represen.tation for 1nKo (the so-called “Bethe logarithm” in the theory of the Lamb shift [6]). We may write . _
ld=i_6’
@b(H-EO)&~jO)drt
The integrand in (3) can be performed
(3)
+ In!-q). in the following form
(4) -All terms’in the right-hand side of (4), except the last, may be calculated exactly. This term may be prritten as a stationary value pf a variational expression that permits the calculation of upper and lower bounds. This expression is a sIight modification of Rebane’s stationary expressjon [?I. .L(u;A,.X)
As ‘&a~ demonstrated it.iS given by
in ief. (71 the stationary value L&;A;
: 1’2
1
-.
-(5)
=$(Sv(~-Eo-A)(~-ao)ud7+2Su(~-Eo-A)x‘Yod~rl~gx2~oO.d7).
_,I’
-. ._
:
._--_:-
; .
‘,I..,_ _. ,_.._
A) does not depend on the parameter I ::.,
,_
.I_ ~._,A~__~~-,~;-~. _,.. -.
A and .. _
‘___ .-
CHEMICAL PHYSICS LETTERS
‘Volume 6, number 3
&
= (0 1%
L&;A,A)
x/O>.
(6)
In other words, the stationary value of L(u;A,X) equals the last term in the right-hand side of (3). HOWever, we.cannot substitute the right-hand side of (4) into the integral (3) directly because of the divergences. It is possible to remove the divergences by the choice of trial function in (5). If we choose u =Ixg
‘XV,
EO where cpis a new trial function,
(7)
we come to the following result = e;(l -A) L;(;p;A,X)
,
(8)
where 1;; is the stationary value of functional L1 i = -&q(XH-eo-
&J;A,X) Finally,
A)(XH-~9;
pdr+$
~-&H-E~-
A)Hx’?+)dT+ 2
j.(H*912
ck).
(9)
we have 2
lnZ=>a
(1-A)LA(q;A,X)dA
+ ln(-q).
(13)
It is easy to see from [7], that if we choose A in the interval 0 < A < ]XE~- ~9 1, EI being the first excited state of hamiltonian H, then the stationary value of the expression (10) is a minimum and we get an upper bound for Inl. If A < 0, then (11)
= maxL’(cp;A,X)
,?$p:h,X)
and we get a lower bound. The best value of A in this case is A = - ~0and Ll( cp;A, X) transforms following expression, analogous to the well-known Hylleraas functional &;-co
,~)=-~I’P(XH-EO)~d~+$I~~~Od~).
to the 02)
With the help of this method we have calculated the logarithm of the average excitation energy for the hydrogen atom. We choose the following trial function cp = e+ 5 a,Fcosi3 With the conclusion
(13)
.
of 10 terms in (13), we have for lnZ(in atomic units) (14)
-0.596 165 < 1nZ < -0.596 148 and for the average energy Z 0.550920
05)
< I < 0.55093p.
This calculation shows the rapid convergence of the suggested method bath for the upper and lower bounds. Upper and lower bounds may be calculated also for the total cross section crof inelastic scattering. In the Born-tiethe approximation (Tis proportional to lnZ_1, where In&l =.${O)xln]H-eO]x]O).
(16)
Upper and lower bounds for lnZ_l are given by the formula 260 lnZ_1 = yJo where &q;A,X)
1
h Li (cpiA,h)&
(17)
is defined by (9). _>.
_._
+ ln(-et)),
173
-VoItie _I
6, raqber
3
CHEMICAL
PRYSXCS LETTERi
l.‘August .
.._
‘?he author is’greatly indebted to T. K. Rebane and Yu. 131.Dmitrisv for &eful’ .. discussions. .._ 1 -.
,.
REFERENkES ed., in: Atomic and molecutar processes (Academic Press, New York, Ann. Phys. 5 (1930) 325. [3] A. Dalgarno, Proc. Phys. Sot. (London) 76 (1960) 422; A. Dalgarno and R. ;1. Bell. Proc. Phys.Soc. (London) 86 (lq65) 375; 89 (1966) 55. [+I .y. M. Ghan and A. Dalgarno, -Proc. Roy. Sot. A285 11967) 457. [5] C. Schwartz, P$x. Rev. 123 0961) 1700. 16) Yu. Yu. Dmitriev and L. N. Labsovski, Phys. Letters 23A (l96S) 153.
(11 D. R. Bates.
.p] Ii. A.EeW.
[?],T.K.Rebane.
Opt.Spectry.
21 (1966) 118.
1962).
1970
1 August 1970
CHEMICALPHYSICSLXTTERS
THE FOR
VARIATIONAL
STOPPING
UPPER‘AND
-POWERS
OF
BOUNDS
AND
MOLECULES
..
M. S. YURIFV
--
LOWER
ATOMS
Department of Theoretical Physics, Physical Institute, University of Leningrad, Leningrad, USSR
Received
19
May 1970
A variational procedure for calculating upper and lower bounds for the logarithm of the average energy cf excitation InZ in the theory of the stopping power of :Itoms and molecules is described. The upper and lower bounds for lnlof the hydrogen atom are calculated.
The knowledge of the average energy loss by the fast charged particles during inelastic collisions with atoms (or molecules) is important in such fields as plasma physics, nuclear physics and astrophysics [l]. In the Born-B&he approximation, the average energy loss K is given by [2] K=?@-ln
4
fiqrnax I
7X4
where
v is the velocity
pulse,
transferred
(1)
’
of the incident
in t&e collision,
particle, 2 is the nuclear change, Mrnax is the maximum imand I is an average excitation energy, defined by
(2) in a conventional notation. Eq. ( 1) contains only one constant, InI; wh ch characterizes the stopping matter. Dalgarno and collaborators (see for example ref. [ ;3] ) ca.lcula:sd lni for many.atoms, using the sum rules for the oscillator strengths. Besides, Chan and Dalgarno suggested a variational procedure for the evaluation of lnd[4]. ‘In this paper a variational metnod for calculating upper and lower bounds for 1nI is suggested. Our method is based on ,an integral representation for 1nI analogous to the Dmitriev-Labsovski represen.tation for 1nKo (the so-called “Bethe logarithm” in the theory of the Lamb shift [6]). We may write . _
ld=i_6’
@b(H-EO)&~jO)drt
The integrand in (3) can be performed
(3)
+ In!-q). in the following form
(4) -All terms’in the right-hand side of (4), except the last, may be calculated exactly. This term may be prritten as a stationary value pf a variational expression that permits the calculation of upper and lower bounds. This expression is a sIight modification of Rebane’s stationary expressjon [?I. .L(u;A,.X)
As ‘&a~ demonstrated it.iS given by
in ief. (71 the stationary value L&;A;
: 1’2
1
-.
-(5)
=$(Sv(~-Eo-A)(~-ao)ud7+2Su(~-Eo-A)x‘Yod~rl~gx2~oO.d7).
_,I’
-. ._
:
._--_:-
; .
‘,I..,_ _. ,_.._
A) does not depend on the parameter I ::.,
,_
.I_ ~._,A~__~~-,~;-~. _,.. -.
A and .. _
‘___ .-
CHEMICAL PHYSICS LETTERS
‘Volume 6, number 3
&
= (0 1%
L&;A,A)
x/O>.
(6)
In other words, the stationary value of L(u;A,X) equals the last term in the right-hand side of (3). HOWever, we.cannot substitute the right-hand side of (4) into the integral (3) directly because of the divergences. It is possible to remove the divergences by the choice of trial function in (5). If we choose u =Ixg
‘XV,
EO where cpis a new trial function,
(7)
we come to the following result = e;(l -A) L;(;p;A,X)
,
(8)
where 1;; is the stationary value of functional L1 i = -&q(XH-eo-
&J;A,X) Finally,
A)(XH-~9;
pdr+$
~-&H-E~-
A)Hx’?+)dT+ 2
j.(H*912
ck).
(9)
we have 2
lnZ=>a
(1-A)LA(q;A,X)dA
+ ln(-q).
(13)
It is easy to see from [7], that if we choose A in the interval 0 < A < ]XE~- ~9 1, EI being the first excited state of hamiltonian H, then the stationary value of the expression (10) is a minimum and we get an upper bound for Inl. If A < 0, then (11)
= maxL’(cp;A,X)
,?$p:h,X)
and we get a lower bound. The best value of A in this case is A = - ~0and Ll( cp;A, X) transforms following expression, analogous to the well-known Hylleraas functional &;-co
,~)=-~I’P(XH-EO)~d~+$I~~~Od~).
to the 02)
With the help of this method we have calculated the logarithm of the average excitation energy for the hydrogen atom. We choose the following trial function cp = e+ 5 a,Fcosi3 With the conclusion
(13)
.
of 10 terms in (13), we have for lnZ(in atomic units) (14)
-0.596 165 < 1nZ < -0.596 148 and for the average energy Z 0.550920
05)
< I < 0.55093p.
This calculation shows the rapid convergence of the suggested method bath for the upper and lower bounds. Upper and lower bounds may be calculated also for the total cross section crof inelastic scattering. In the Born-tiethe approximation (Tis proportional to lnZ_1, where In&l =.${O)xln]H-eO]x]O).
(16)
Upper and lower bounds for lnZ_l are given by the formula 260 lnZ_1 = yJo where &q;A,X)
1
h Li (cpiA,h)&
(17)
is defined by (9). _>.
_._
+ ln(-et)),
173
-VoItie _I
6, raqber
3
CHEMICAL
PRYSXCS LETTERi
l.‘August .
.._
‘?he author is’greatly indebted to T. K. Rebane and Yu. 131.Dmitrisv for &eful’ .. discussions. .._ 1 -.
,.
REFERENkES ed., in: Atomic and molecutar processes (Academic Press, New York, Ann. Phys. 5 (1930) 325. [3] A. Dalgarno, Proc. Phys. Sot. (London) 76 (1960) 422; A. Dalgarno and R. ;1. Bell. Proc. Phys.Soc. (London) 86 (lq65) 375; 89 (1966) 55. [+I .y. M. Ghan and A. Dalgarno, -Proc. Roy. Sot. A285 11967) 457. [5] C. Schwartz, P$x. Rev. 123 0961) 1700. 16) Yu. Yu. Dmitriev and L. N. Labsovski, Phys. Letters 23A (l96S) 153.
(11 D. R. Bates.
.p] Ii. A.EeW.
[?],T.K.Rebane.
Opt.Spectry.
21 (1966) 118.
1962).
1970