Oksana Katsuro-hopkins, Paper Journal Moscow Physics Society, Fission Degradation. 1997

J. Moscow Phys. Soc. 7 (1997) 153-162.

The space-time evolution of the track of an atomic nucleus fission fragment in a helium-metal vapour mixture Oks ana Katsuro-Hopkins, 2nd Autho r A P Budnik, I V Dobrovol'skaya, and 0 N Katsuro

State Scientific Center of the Russian Federation—Institute of Physics and Power Engineering, Bondarenko sq. 1, 249020 Obninsk, Kaluga Region, Russia

Received 25 September 1996

Abstract. Results of mathematical modelling of the space-time evolution of the track of a fission fragment are presented. The degradation of the energy of nuclear fission fragments in a helium-cadmium mixture are studied.

1. Introduction The degradation of the energy of fission fragments in inert gases is very important for the development of a device to generate nuclear energy at optical wavelengths (a "flash-lamp"— a nuclear-pumped laser). The theory of energy degradation for rapid electrons in gases is currently well developed. The state of the theory of energy degradation of multiply-charged ions, including fission fragments, is considerably worse. The interaction of fission fragments with matter is fundamentally different from the interactions of other charged particles with matter due to the strong influence of socalled track effects. Because of these effects, the kinetics of the energy degradation of fission fragments are, as a rule, significantly inhomogeneous. In recent years, a theory for the space–time evolution of the tracks of fission fragments in inert gases has been developed [1-3]. This theory is capable of rather correctly taking account of a whole series of processes: the diffusion of electrons in ions, the energy relaxation of electrons during collisions, electron–ion recombination, the creation of an electric field and the drift of charged particles under its influence. This paper concerns the theoretical study of the energy degradation of fission fragments in mixtures of helium and metal vapours. We present results of mathematical modelling of the space–time evolution of a fission fragment track in a helium–cadmium mixture. In contrast to previous studies [1-3], we take account of electron–electron

153



tho

The fission fragment track space-time evolution

and St ee (fo) is the electron-electron collision integral. In (5), the sum over a corresponds to various inelastic collisions between electrons and atoms, with U 2 l = U 2

+ 2ea rn

2ea — U 22 = U2 m

for elastic collisions

(5a)

for superelastic collisions

(5h)

where e a is the excitation energy for atoms in the a state. St ee (fo), representing the flux of electrons in velocity space due to electronelectron collisions, is determined by the formulas 4re 2 ) 2 1 a

rn St ee = (----

I n A

2u 2 eu

+ -i1u-

w 2 fo (r, w ,t) dw fo(r, u, t) 0

00

(u

w4/0(r,w,t)dw u 3 1 wfo (r, w, t).dw

0 fo (r u,t

Ou

(6)

where A is the ratio of the Debye radius ID to the Coulomb radius ro, defined as 1/2

kTe

(6a)

t)e 2

2e 2 3k7;

(6b)

where Te is the effective temperature. We write the boundary conditions for the system of equations (1)-(3) in the form 8 -07. fo(r,u,t)l„. 0 = 0 8

87 Nik(r,t)I r=0

(7)

= 0

(8)

E(r, t)ir=0 = 0 (9)

0

fo(r, u, t)l„. Nik (r,

(10)

—o 0

(11)

—0 4 .

(12'

In order not to excessively complicate the problem, we will describe the evolutio of a track at a moment in time when the electrons have experienced several collisiot after the flight of the fragment. In this time, the electrons diffuse over a distance the order of several mean free paths. Consequently, examining the evolution for mu longer times, we can with some small error specify the electron and ion concentratit to be Gaussian distributions. Then, the initial conditions can be written

Nii m der(e) fo(r, Olt=o = 4ru de

r2 /7 4%to

1

• k ir

2\\-1_,Ar

r 47e4""kUi

The fission fragment track space-time evolution Table 2 Plasma-chemical reactions in the He-Cd plasma model

No

Reaction

Reaction rate 0.3x 10 -31 cm 6 /s

1

He+ .+He+He=He: +He

2

He++He+He+He=He:+He+He 1.82x 10' 1 cm 9 /s

3

He: +CdsCd**+ +He+He

6.12x10' cm 3 /s

4

He: +Cd=Cds+ +Hei-He

7.92x 10 -1° cm 3 /s

5

He(20.6)+Cd=CdPs++Hq+e

0.35x 10' 3 cm3 /s

6

He(20.6)+Cd=CdP**++He+e

0.35x 10" I ° cm 3 /s

7

He(20.6)+Cd=CdS++He+e

2.55x 10' cm 3 /s

8

Cd**+=CdPs++hv i

2.9x 10+ 6 s"

9

Cds+ =Car"' +hv2

1.2x 10+ 6 s'

10

CdP**+=CdS++hv3

5.15x10+" s"

II

CdPs+ sCdS++hv4

4.314x 10+" s-1

cross-sections for collisions between electrons and helium atoms from and crosssections for the excitation of helium atoms by electrons from [81. Data on associative ionization, the conversion of atomic to molecular helium ions, and dissociative recombination were adopted from [9-111. The probabilities for transitions between excited states of helium were calculated using the oscillator strengths presented in [121. In the model, we took account of such plasma-chemical reactions determining the character of processes in the helium-cadmium plasma as Penning ionization of the cadmium atoms, the conversion of atomic helium and cadmium ions to molecular ions, the recharging of molecular helium ions on cadmium atoms, and processes involving the plasma electrons—ionization and recombination processes, and excitation and deexcitation of the helium atoms by electrons ; etc. In addition, we allowed for radiative transitions between excited states of the helium atoms and cadmium ions. Table 1 presents all components of the plasma considered with their formation energies, and Table 2 presents the plasma-chemical reactions in which they take part, together with the corresponding reaction rates and their dependence on the temperature of thf medium. 3. Results

We carried out mathematical modelling of the evolution of a track of a fission fragr in a neutral gas—a mixture of helium (NH. = 2.68 x 10 19 cm -3) and satui cadmium vapours at a temperature of 650 K. We take the parameter (r 2 ) 0 equal to ( r2 ) = 4 x 10 -8 cm 2 . Figures 1-9 show the results of our calculations radial dependence of the concentrations of electrons, atomic and molecular 7 and cadmium ions in the ground (CdS) and excited 4d 9 5s 2 atoms (He+, state (Cd*+) at various times after the passage of the fission fragment are sl

The fission fragment track space-time evolution N

Cd .*

159

, CITC 3

10 9

t= 0.13 ns t= 1.2 ns

10' 10 5



1111 t=30 t = 45 ns

10 3 10

'

0

200

400

600

800

1000 r, 10 5 cm -

Figure 4. Radial dependence of the concentration of cadmium ions Cd in the exited state 4d 9 58 2 2 D s/3 (Ce+ ) at various times.

N CdS cm -3 9

10 6

10 4

10 2 10° 0

200

400

600

800

1000 r, 10 5 CM -

Figure 5. Radial dependence of the concentration of cadmium ions in the ground state (CdS) at various times.

Figures 1-5. It is clear (Figure 1) that the electrons rapidly leave the region near the track axis due to diffusion, so that a rather strong electric field 10 2 V/cm) arises there. The electrons create a significant number of He+ ions via inelastic collisions; the concentration of these ions rapidly grows, especially far from the track axis (Figure 2). The concentration of atomic helium ions He+ then decreases due to diffusion and drift, and, especially, due to conversion into molecular helium ions Het (Figure 2). The concentration of molecular helium ions thus grows due to the cerversion process (Figure 3); due to recharging, the concentration of atomic cadmium io ns also grows (Figures 4-5). Figure 6 shows the concentrations of various plasma components at time 45 ns. We can see that near the track, the component with the largest concentration is molecular helium ions Het ; the concentration of positive ions significantly exceeds the concentration of electrons near the track axis. Superelastic collisions significantly affect the evolution of the track. In particular, even at large times, these collisions determine the electron energy distribution near

161

The fission fragment track space-time evolution f(s) , eV -312

10-1 4 10 - ' 7

10 -20 10 -2 3 10 -26 10 -29 10 42 10 -35 ....■■■•■■1

0



5



10

15

Figure 8. Electron energy distribution function

20 f(E)

25 e, eV

a distance r from the track axis

at time t.45 ns.

E, V / cm 200 150 100 50 0 0

100

200

300

400

500 r, 10 - scm

Figure 9. Radial dependence of the electric field strength at the track at various times.

complex composition (helium with cadmium vapours). Our results may be useful for the development of the theory of microscopic kinetic processes in nuclear-pumped lasers based on metal vapours, taking account of the track structure of the plasma.

References [1] Budnik A P, Sokolov Yu V, and Vakulovskiy A S 1994 Hyperfine Interactions 88 185 [2] Budnik A P, Vakulovskiy A S, and Sokolov Yu V 1992 Proc. Conf. "LYaN-92" vol 1 (Obninsk: Physical-Energetic Institute) p 178 [3] Budnik A P, Vakulovskiy A S, Dobrovol'skaya I V, and Sokolov Yu V 1993 Physical-Energetic Institute Preprint-2315 [4] Georjuoy E 1966 Phys. Rev. 148 54 [5] Reymann K, Schartner K-H, Sommer B, and Trabert E 1988 Phys. Rev. A 38 2290 [6] Nikerov V A and Sholin G V 1985 Kinetika Degradatsionnykh Protsessov (Moscow: