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Information-theoretical description of charge-state distributions in heavy ion-atom collisions
Volume 90A, number 9
PHYSICS LETTERS
9 August 1982
INFORMATION-THEORETICAL DESCRIPTION OF CHARGE-STATE DISTRIBUTIONS IN HEAVY ION-ATOM COLLISIONS 0. GOSCINSKI Department of Quantum Chemistry, Uppsala University, P.O. Box 518, 75120 Uppsala, Sweden
and T. ABERG and M. PELTONEN Laboratory of Physics, Helsinki University of Technology, 02150 Espoo 15, Finland Received 27 April 1982
Universal equations for asymmetric equilibrium charge-state distributions are derived using the maximum-entropy principle. They are tested on chlorine, bromine, and iodine projectiles in several gases at variousincident energies. A single energyindependent parameter relates the rate of charge formation to the observed asymmetries.
M often made statement is that the explanation of charge distributions requires detailed knowledge of individual cross sections of the numerous single and multiple collision and decay processes involved [1—5]. In this letter, the opposite view is taken by using the maximum entropy principle [6] * 1 (MEP) which from the very beginning takes advantage of the prevailing chaotic conditions. From this will follow a systematization of the asymmetry of observed equilibriurn charge distributions [4,5] which are not gaussian as predicted by the Bohr—Undhard theory [1]. Recent studies by Baudinet-Robinet et al. indicate that a modified chi-squared (x2) distribution can sometimes be used in connection with the variable x = q + q 0, where q is the net charge in units of e and where q0 is either two [8] or an adjustable parameter [9]. However, no motivation was given, except that the first choice of x was dictated by the fact that the distribution must vanish at q = —2 [8]. The general equation for the variation of the charge q = x 2 as a function of the distance z travelled by the ions is —
dx/dz ~vG(x)
464
—
which vanishes when x is approaching a saturation constant k ( and a = —1 in eqs. (1) and (2). The resulting distribution was considered to be essentially gaussian. However, as shown below, a will determine the asymmetry of the distribution. The interpretation of x as a stochastic variable and the requirement that the average of the saturation function with respect to the final distribution P0(x) should vanish lead to the moment condition:
(1)
* 1 A related information-theoretical description of beam—
foil experiments has recently been given in ref. [7].
where r (>0) is the net electron loss or capture per unit of distance and charge. The fact that the charge of a colliding ion is limited to the range (—1,Z) and that it becomes independent ofz and of the initial charge prompts the introduction of a saturation-inducing function [10,11] (2) G(x) = a4 [1 (x/k)a]
G(x)
=
f P (x) [1
-~—
—
(x/k)°~] dx
=
0
.
(3)
a 0 03l-9163/82/0000—0000/$02.75 © 1982 North-Holland
Volume 90A,
number 9
PHYSICS LETTERS
According to eq. (1) this requirement is equivalent to the condition that (d(logx)/dz) = 0 with respect to the final distribution which assures that x has on the average proper asymptotic behaviour. Here, invoking MEP [6] we use condition (3) as a constraint for obtaming the most probable charge distribution, cornpatible with the assumption of random excitation. In order to fulfil the proper boundary condition at x = 0 we introduce in addition to eq. (3) a family of density functions, proportional tox~1 (p> 1). This assures that the distribution vanishes at x = 0 and would in absence of capture describe the charge distribution due to losses. Using the MEP [6] the resulting chargestate fraction is
9 August 1982
whereas the left-hand side can be evaluated from experimentally available quantities. Hence eq. (5) provides the testing ground of our theory. We have compared our universal ‘I~(x)curves with experimental equilibrium charge-state distributions for I to 14 MeV Cl, Br and I projectiles in various gases [4,5] and assigned by inspection an a value to each projectile-target combination. The points in figs. 1—3 correspond to the experimental P(q) values in N
-
/
o
C1—He
/ /
o=2
/
P~(x)=a(x/k)~exp[—X(x/k)°] X [kX~/°FO.t/a)]1
(4)
where A is a Lagrange parameter and I’ the gamma function. According to eq. (3) A must satisfy the relation p Xa. This general derivation should be contrasted to the one which would follow from treating eq. (1) as a stochastic dynamical equation. The latter involvestheintroductionineq.(1)ofaz-dependent gaussian random function. Consequently it is based on unwarranted assumptions about details of the quantum mechanical processes leading to charge formation, which are not available. Using the maximum value of Pa(x) corresponding tOXmax qmax + 2, eq. (4) takes form
><
E (MeV)
/
(0
/
7 J
0 N
i .o 2.0
‘
4.0 6 .0 8.0
‘
-
10.0
0 ,
0
000
1.0
.5
2.0
~
2.5
.0
3.0
X/XmOX
Fig. 1. The 4~function, defined by eq. (5), as a function of x/xmax (x = q ÷ 2) for Cl—He collisions (ref. [5]) corresponding to a = 2.
N
~a(x)
=
—N 0(~)ln[Pa(X)IPa(Xmax)1 2 [ya a In y 1], —
—
(5)
= 2a wherey=(x/xmax)andx ax[2+(a1)2] X [v + I a + (a l)2]~ to second order in 1T’. The parameter v is related to the mean value (1 A 21n(1 calculation shows that this is the+2/v) most asymmetric representation. Fora>Othe family ~a(x)becomes more gaussian-like as a increases from zero to two which is the highest a value we shall consider in our analysis of the asymmetry. Note that the right-hand side of eq. (5) describes a family of universal functions —
—
—
-
\
Br-He
~ \
o
//
=1
/
a~__ 0
—
x
~
-
0
/
\ f/‘
-
E (MeV) 2 .0 ‘
0
:
0 —
000
0.5
1.0
1.5
I 2.0
4.0 6 .0 8 .0
10.0 12.0 I
2.5
3.0
X/Xmox .
Fig. 2. The ‘a function, defmed by eq. (5), as a function of = q + 2) for Br—He collisions (ref. [SD corresponding to a = 1. X/Xm~,(x
465
Volume 90A, number 9
PHYSICS LETTERS
-
/
—
~=o
I ~02
/
-
>~0
‘~
0
• -
2
.95
~ .s
/
o
/
(MeV) 1 05
o7
-
/
6.0 8 .0
0
o00.0
0
0.5
1 .0
1
2
10 .0 12 0
3~
2
X/Xçnax
Fig.
3. The ~
X/Xmaj~(x =
ing to a
=
q
function, defined by eq. (5) as a function of + 2) for 1—02 collisions (ref. [41)correspond-
o.
Na(V) ln[P(q)/P(q~~~)] plotted as a function of (q + 2)/Xmax for all measured energies. It is seen that
the Cl—He, Br—He and 1—02 data are well reproduced by a 2, 1 and 0, showing the increasing asymmetry. Similar trends are also found in all the other cases with the optimum a values listed in table 1. It is remarkable that a is independent of the projectile yelocity and that it has not been necessary to use noninteger a values. Note also that the charge distributions predicted by the 40(x) curves extend over four orders of magnitude in P(q). It can be seen from figs.! —3 that 40(x) is, in accordance with its functional form a sensitive function ofa for both small and large y. Hence our analysis of the asymmetry is superior to P(q)2-plots versus q[8] fittingsfollow [3,5,8,9], theq0 2 relates x which fromincluding eq. (4) for a = 1.= It the Table 1 The universal asymmetry parameter a for various targets as a function of projectile atomic number Z. The experimental charge distributions are from refs. [4,51. Z
9 August 1982
charge formation to the asymmetry of the final charge-state distribution since according to eqs. (1) and (2) the increasing asymmetry is associated via a with a faster rate of electron loss. This conclusion is in agreement with experiments since according to table 1 heavy targets have systematically smaller a values than light targets. The information-theoretical analysis could also be extended to charge distributions obtained under single-collision conditions [12]. This requires a consideration of the appropriate boundary conditions. In conclusion, we have obtained a universal representation of equilibrium gas-target charge distributions. Their asymmetry can be described by a single energy-independent parameter a which according to eq. (2) characterizes the rate of saturation. It has been shown that the corresponding distributions follow from MEP without any specific assumptions regarding the collision mechanisms. The connection between the present approach and other stochastic methods as well as further applications will be discussed elsewhere. Stimulating discussions with J. Lindhard, D. Maor, E. Merzbacher and K~Taulbjerg are gratefully acknowledged. References [1] N. Bohr and J. Lindhard, K. Dansk. Vidensk. Seisk. Mat. Fys. Medd. 28 (1954) No. 7. [2] I.S. Dmitriev and V.S. Nikolaev, Zh. Eksp. Teor. Fiz. 47 (1964) 615 [Sov.Phys.JETP2O(1965)409I. [3] H.-D. Betz, Rev. Mod. Phys. 44 (1972) 465. [41 G. 185Ryding, (1969) A.B. 129. Wittkower and P.H. Rose, Phys. Rev.
151
A.B. Wittkower and G. Ryding, Phys. Rev. A4 (1971) 226. [61 See, e.g. R.D. Levine and M. Tribus, eds., The maximum entropy formalism (MIT Press, Cambridge, 1979) for a current review. [7] T. Aberg and 0. Goscinski, Phys. Rev. A24 (1981) 801. [81 Y. Baudinet-Robinet, P.D. Dumont and H.P. Garnir, I. Phys.B1l (1978) 1291. [9] Y. Baudinet-Robinet and D. Lamotte, J. Phys. B12 (1979) 3329.
Target H 2
17 35 53
2 1 1
He
N2
02
Ar
Kr
2 1 1
1 0 a)
1 0 0
0 0 0
0 0 0
a) No experimental data available.
466
[101 N.S. Goel and N. Richter-Dyn, Stochastic models in biology (Academic Press, New York, 1974). [11] E.W. Montroll and B.J. West, Fluctuation phenomena (North-Holland, Amsterdam, 1979) Ch. 2. [121 D. Maor, B. Rosner, M. Meson, H. Schmidt-Bocking and R. Schuch, J. Phys. B14 (1981) 693.
PHYSICS LETTERS
9 August 1982
INFORMATION-THEORETICAL DESCRIPTION OF CHARGE-STATE DISTRIBUTIONS IN HEAVY ION-ATOM COLLISIONS 0. GOSCINSKI Department of Quantum Chemistry, Uppsala University, P.O. Box 518, 75120 Uppsala, Sweden
and T. ABERG and M. PELTONEN Laboratory of Physics, Helsinki University of Technology, 02150 Espoo 15, Finland Received 27 April 1982
Universal equations for asymmetric equilibrium charge-state distributions are derived using the maximum-entropy principle. They are tested on chlorine, bromine, and iodine projectiles in several gases at variousincident energies. A single energyindependent parameter relates the rate of charge formation to the observed asymmetries.
M often made statement is that the explanation of charge distributions requires detailed knowledge of individual cross sections of the numerous single and multiple collision and decay processes involved [1—5]. In this letter, the opposite view is taken by using the maximum entropy principle [6] * 1 (MEP) which from the very beginning takes advantage of the prevailing chaotic conditions. From this will follow a systematization of the asymmetry of observed equilibriurn charge distributions [4,5] which are not gaussian as predicted by the Bohr—Undhard theory [1]. Recent studies by Baudinet-Robinet et al. indicate that a modified chi-squared (x2) distribution can sometimes be used in connection with the variable x = q + q 0, where q is the net charge in units of e and where q0 is either two [8] or an adjustable parameter [9]. However, no motivation was given, except that the first choice of x was dictated by the fact that the distribution must vanish at q = —2 [8]. The general equation for the variation of the charge q = x 2 as a function of the distance z travelled by the ions is —
dx/dz ~vG(x)
464
—
which vanishes when x is approaching a saturation constant k (
(1)
* 1 A related information-theoretical description of beam—
foil experiments has recently been given in ref. [7].
where r (>0) is the net electron loss or capture per unit of distance and charge. The fact that the charge of a colliding ion is limited to the range (—1,Z) and that it becomes independent ofz and of the initial charge prompts the introduction of a saturation-inducing function [10,11] (2) G(x) = a4 [1 (x/k)a]
G(x)
=
f P (x) [1
-~—
—
(x/k)°~] dx
=
0
.
(3)
a 0 03l-9163/82/0000—0000/$02.75 © 1982 North-Holland
Volume 90A,
number 9
PHYSICS LETTERS
According to eq. (1) this requirement is equivalent to the condition that (d(logx)/dz) = 0 with respect to the final distribution which assures that x has on the average proper asymptotic behaviour. Here, invoking MEP [6] we use condition (3) as a constraint for obtaming the most probable charge distribution, cornpatible with the assumption of random excitation. In order to fulfil the proper boundary condition at x = 0 we introduce in addition to eq. (3) a family of density functions, proportional tox~1 (p> 1). This assures that the distribution vanishes at x = 0 and would in absence of capture describe the charge distribution due to losses. Using the MEP [6] the resulting chargestate fraction is
9 August 1982
whereas the left-hand side can be evaluated from experimentally available quantities. Hence eq. (5) provides the testing ground of our theory. We have compared our universal ‘I~(x)curves with experimental equilibrium charge-state distributions for I to 14 MeV Cl, Br and I projectiles in various gases [4,5] and assigned by inspection an a value to each projectile-target combination. The points in figs. 1—3 correspond to the experimental P(q) values in N
-
/
o
C1—He
/ /
o=2
/
P~(x)=a(x/k)~exp[—X(x/k)°] X [kX~/°FO.t/a)]1
(4)
where A is a Lagrange parameter and I’ the gamma function. According to eq. (3) A must satisfy the relation p Xa. This general derivation should be contrasted to the one which would follow from treating eq. (1) as a stochastic dynamical equation. The latter involvestheintroductionineq.(1)ofaz-dependent gaussian random function. Consequently it is based on unwarranted assumptions about details of the quantum mechanical processes leading to charge formation, which are not available. Using the maximum value of Pa(x) corresponding tOXmax qmax + 2, eq. (4) takes form
><
E (MeV)
/
(0
/
7 J
0 N
i .o 2.0
‘
4.0 6 .0 8.0
‘
-
10.0
0 ,
0
000
1.0
.5
2.0
~
2.5
.0
3.0
X/XmOX
Fig. 1. The 4~function, defined by eq. (5), as a function of x/xmax (x = q ÷ 2) for Cl—He collisions (ref. [5]) corresponding to a = 2.
N
~a(x)
=
—N 0(~)ln[Pa(X)IPa(Xmax)1 2 [ya a In y 1], —
—
(5)
= 2a wherey=(x/xmax)andx ax
—
—
-
\
Br-He
~ \
o
//
=1
/
a~__ 0
—
x
~
-
0
/
\ f/‘
-
E (MeV) 2 .0 ‘
0
:
0 —
000
0.5
1.0
1.5
I 2.0
4.0 6 .0 8 .0
10.0 12.0 I
2.5
3.0
X/Xmox .
Fig. 2. The ‘a function, defmed by eq. (5), as a function of = q + 2) for Br—He collisions (ref. [SD corresponding to a = 1. X/Xm~,(x
465
Volume 90A, number 9
PHYSICS LETTERS
-
/
—
~=o
I ~02
/
-
>~0
‘~
0
• -
2
.95
~ .s
/
o
/
(MeV) 1 05
o7
-
/
6.0 8 .0
0
o00.0
0
0.5
1 .0
1
2
10 .0 12 0
3~
2
X/Xçnax
Fig.
3. The ~
X/Xmaj~(x =
ing to a
=
q
function, defined by eq. (5) as a function of + 2) for 1—02 collisions (ref. [41)correspond-
o.
Na(V) ln[P(q)/P(q~~~)] plotted as a function of (q + 2)/Xmax for all measured energies. It is seen that
the Cl—He, Br—He and 1—02 data are well reproduced by a 2, 1 and 0, showing the increasing asymmetry. Similar trends are also found in all the other cases with the optimum a values listed in table 1. It is remarkable that a is independent of the projectile yelocity and that it has not been necessary to use noninteger a values. Note also that the charge distributions predicted by the 40(x) curves extend over four orders of magnitude in P(q). It can be seen from figs.! —3 that 40(x) is, in accordance with its functional form a sensitive function ofa for both small and large y. Hence our analysis of the asymmetry is superior to P(q)2-plots versus q[8] fittingsfollow [3,5,8,9], theq0 2 relates x which fromincluding eq. (4) for a = 1.= It the Table 1 The universal asymmetry parameter a for various targets as a function of projectile atomic number Z. The experimental charge distributions are from refs. [4,51. Z
9 August 1982
charge formation to the asymmetry of the final charge-state distribution since according to eqs. (1) and (2) the increasing asymmetry is associated via a with a faster rate of electron loss. This conclusion is in agreement with experiments since according to table 1 heavy targets have systematically smaller a values than light targets. The information-theoretical analysis could also be extended to charge distributions obtained under single-collision conditions [12]. This requires a consideration of the appropriate boundary conditions. In conclusion, we have obtained a universal representation of equilibrium gas-target charge distributions. Their asymmetry can be described by a single energy-independent parameter a which according to eq. (2) characterizes the rate of saturation. It has been shown that the corresponding distributions follow from MEP without any specific assumptions regarding the collision mechanisms. The connection between the present approach and other stochastic methods as well as further applications will be discussed elsewhere. Stimulating discussions with J. Lindhard, D. Maor, E. Merzbacher and K~Taulbjerg are gratefully acknowledged. References [1] N. Bohr and J. Lindhard, K. Dansk. Vidensk. Seisk. Mat. Fys. Medd. 28 (1954) No. 7. [2] I.S. Dmitriev and V.S. Nikolaev, Zh. Eksp. Teor. Fiz. 47 (1964) 615 [Sov.Phys.JETP2O(1965)409I. [3] H.-D. Betz, Rev. Mod. Phys. 44 (1972) 465. [41 G. 185Ryding, (1969) A.B. 129. Wittkower and P.H. Rose, Phys. Rev.
151
A.B. Wittkower and G. Ryding, Phys. Rev. A4 (1971) 226. [61 See, e.g. R.D. Levine and M. Tribus, eds., The maximum entropy formalism (MIT Press, Cambridge, 1979) for a current review. [7] T. Aberg and 0. Goscinski, Phys. Rev. A24 (1981) 801. [81 Y. Baudinet-Robinet, P.D. Dumont and H.P. Garnir, I. Phys.B1l (1978) 1291. [9] Y. Baudinet-Robinet and D. Lamotte, J. Phys. B12 (1979) 3329.
Target H 2
17 35 53
2 1 1
He
N2
02
Ar
Kr
2 1 1
1 0 a)
1 0 0
0 0 0
0 0 0
a) No experimental data available.
466
[101 N.S. Goel and N. Richter-Dyn, Stochastic models in biology (Academic Press, New York, 1974). [11] E.W. Montroll and B.J. West, Fluctuation phenomena (North-Holland, Amsterdam, 1979) Ch. 2. [121 D. Maor, B. Rosner, M. Meson, H. Schmidt-Bocking and R. Schuch, J. Phys. B14 (1981) 693.