Rodiat. Phys. Chem. Vol. 30, No. 4, pp. 279-284, 1987 lnt. 3. Radiat. AppL lnstrum. Part C Printed in Great Britain. All fights reserved

0146-5724/87 $3.00 + 0.00 Copyright © 1987 Pergamon Journals Lid

A PRESCRIBED DIFFUSION MODEL OF A MANY-RADICAL SYSTEM CONSIDERING ELECTRON TRACK

STRUCTURE IN WATER

Hmo~.n YAMAGUCHI Division of Physics, National Institute of Radiological Sciences, 9-1, Anasawa 4-chome, Chiba,

260, Japan

(Rece/ved 28 April 1987; /n revised form 1 June 1987)

Abstract--A detailed track structure of electrons in water is considered in the Schwarz modified prescribed diffu~on calculation for a meny-radical system. An interspur distance of an electron path in the primary electron as well as in subsequent 6 rays is introduced using the averase energy to produce a ~u r and a restricted energy loss by an electron along the path, which is characterized by a cutoff energy. The resultant G-values of free radicals for each electron path are summed up to give the integral G-values from the whole paths with the aid of an electron degradation spectrum. The concept of the interspur distance of an electron path segment and the use of the electron degradation spectrum improve the original Schwarz's calculation. The present method allows us to discuss related variables in the calculations such as the energy to produce spur, initial number of free radicals and their radii in the spur. The paper exemplifies this by comparison with experiments and results of other calculations.

INTRODUCTION

The Monte Carlo method has been used in a modeling of chemical reactions in pure water irradiated by electron/~> It simulates interactions between electrons and water molecules along electron paths and further diffusion controlled reaction kinetics. This microscopic, event-by-event approach visualizes the spatial pattern of reaction points as a function of time, and illuminates which elementary processes are important to interpret observable yields of free radicals.

On the other hand the traditional, macroscopic approach of diffusion controlled reaction kinetics is still useful to provide an overall view of the kinetics and estimate the yields of free radicals. ¢ ~ The basic idea is that ions and free radicals form a lump, often called a cluster or spur, and chemical reactions in the whole space are represented by those in an average lump: if some lumps overlap, the extent of overlap is effectively expressed by change in reaction rate con- stants. The track structure of electrons appears in the calculation in two ways. One is the size distribution of the lumps, ~21 the number of ions and free radicals in a lump. We call the ions and free radicals species. The other is the interspur distance, the average distance between two successive spurs over the whole electron path, if all lumps are regarded as spurs.

Mozumder and Magec ¢2) defined the lump, as spurs, blobs and short tracks and gave their fractions including the spur size distribution for the whole paths. Magee and C h a t t e r j ~ 31 defined the lump similarly and gave functions for yields of species, which include the approximate degradation spectrum of electrons. Their algorithms for a one-radical or a

279

two-radical system, however, are dimcult to expand to a many-radical system.

Kuppermann <41 and Schwarz ~ dealth with a many- radical system and solved 16 or 13 simultaneous partial differential equations for the realistic reaction mechanisms. The specific feature of their calculations with respect to the track structure is the average interspur distance over the whole electron paths, although Schwarz took the spur size distribution into account.

An electron track, if the energy of the electron is high enough, generally produces several generations of 6 rays during the degradation processes. Figure 1 shows schematically an electron track, down to the fourth generation of 6 rays. The average energy of 6 rays decreases as the generation of 6 rays becomes higher. Accordingly an average interspur distance Z~ on each 6 ray becomes shorter as the generation of 6 rays is higher, if we assume the size of the spherical spur is constant and they are in an equidistance being characterized by the energy of the 6 ray or the electron path. In this paper the term "path" is used to denote any electron trace in the primary electron as well as in 6 rays, and the term "track" denotes only that of the primary electron, since this distinction may avoid confusion which arises when an interspur distance for a path segment of an electron path is defined in the same way as that for the track segment of a beavy-particle track. The energy spectrum for these 6 rays is given as the electron degradation spectrum. The average interspur distance over the whole paths, which has been used in the calculations by Schwarz and others, means an average value of Zi over those of the primary electron and of all gener- ations of 6 rays.

280 Hmosm YAMAOU¢~II

o T,,:r ._,...

electron produces 6 rays down to the fourth generation. Spheres indicate spun which are constant size and in an equidistance Zm being characterized by the average emergy of

each electron path.

I t is the motivation of the present work that if the interspur distance for each electron path segment is introduced as shown in Fig. 1 and the degradation spectrum of electrons is used more directly than in the calculation by Magce and Chatterjee, o) the Schwarz's calculation can be improved. We found this line of calculations more logical, and clarified the problems relevant to microscopic quantities, the energy re- quired to produce a spur, and the initial number and the initial radii of species in the spur.

M O D E L

(a) Mechanism

It is assumed that at a time about 1 0 - | 1 S after the pasmge of the electron the radiolysis of water is dem-ibed by,

H20 ,~,,-, e~, H, H30 +, OH, H2. (1)

The subsequent reactions of these species are listed in Table 1. o)

(b ) The prescribed diffusion model

An extension of the Schwarz calculation giving the differential yields, the yields of species for each electron path, is obvious following his derivation. The development with time of the ith species at any point within the spur can be described by,

dCJdt =ffi D, V2C, - - Zjk~/ClCj

+ Zj.k,JGCjC~-k,C,C,, (2)

where C~ and D~ are the concentration and the diffusion constant of species i, k u is the rate constant

Table I. Reactions and rote constants O~

Ructions k x lO-m(M-'s -.)

(I) e~ + e~ + 2H20 ~ H, + 2OH- 0.55 (2) ¢,~ + H + H00--*H2 + OH- 2.5 (3) e~ + H+--*H 1.7 (4) e,~ + OH -.-,OH- 2.5 (5) e~ + H202--,OH- + O H !.3 (6) H + H--*H 2 i .0 (7) H + O H --* H20 2 (8) H + H~O 2 - , H20 + O H 0.01 (9) H+ + O H - - ' H 2 0 10

(10) O H + O H --* H202 0.6

for the reaction between a species i and a species j, k,C~C, is the rate of the reaction of a species i with solute and C, is the concentration of the solute. Integration of equation (2) over all space and the assumption of a string of Gaussian spurs along a path lead to the equation of Nt, the number of species i in the spur,

dNj/dt ffi T.jkcNiN~fo+ T.j.t,,ik/tN/N~f~-k~C,N i, (3)

fit = [1 +{n(by+b2)yln)/ZJ/{xfb~ + bl)} °/2), (4)

where Zt is the interspur distance for a particular electron path: this concept was originally introduced for a track segment of heavy-particle tracks, (6) and b y is the width parameter given by twice the square of radius of the Gaussian: by = 2~ at t ffi 0, where r ° is the initial radius of species j. Schwarz °) added differential equations for by to improve accuracy.

db~/dt ffi 4Di - ~.jfli( i,j)

+~7.k,J$,(j,k)+ /J,(j,s), (5)

where the functions p,(i,j) #t(J, k) and Or(./, s) are derived (~ so that equation (5) avoids a cutoff con- dition in solving the original formula of Schwarz,

/~,(i,j) ffi - b 2 koNJ#b2,/(b 2 + by), (6)

ffi:: 2 2 2 fl~j, k) b, k~(NjNk/N,~/[b/bk • 2 2 2 =- {b,(bj + b~)} - q, (7)

fl~j, s) ffi - (by/N~)k,#Nj(l - b]/b~). (8)

There is a possibility of non-Ganssian distribution for the hydrated electrons. (8'9) This effect can be included in a slightly different formula from equation (4) (Appendix).

(c ) Interspur distance

Pagtamenta and Marshall (t°) proposed a method to calculate the electron degradation spectrum y(E, Q), the total path length per unit energy interval at Q, in water irradiated by electron of energy E. We calculated y(E, Q) following their algorithm down to the fourth generation of 6 rays (Fig. 2). The equation for the energy loss of the electron into primary 6 rays is useful to define a new concept of interspur distance for an electron path,

Lt ffi [dE/dx I ~

ffi --~:'~ NI(E, Q)(Q + B)dQ, (9)

where N~(E, Q) dQ, is the number of the first gener- ation electrons with energy between Q and Q + dQ generated per micrometer of electron track of energy E, B is the ionization potential of the outermost orbital (B ffi 12.6eV) and Qmu(E) is the maximum energy of 6 rays. By using equation (9) a ~ U ~ M energy loss Ll.¢ is defined by,

I; L~.o= N,(~,Q)(Q + B ) d Q +B,¢,(E), 00)

Prescribed diffusion model based on track structure 281

10s

10 4

103

f- >

~. I 0 a

E - 2 ._u E ,.2 t01

10 o

10- 10-z 10-¢ 10 ° 10¢ 10a 10~ 10 4

Deoroded energy, 0 (keV)

Fig. 2. The degradation spectrum defined as the total path length of all electrons per unit energy range. The curves are obtained by the algorithm of Pagoamenta and Marshall ~m) down to fourth generation 6 rays. Each curve correspomh to an electron of inithtl energy of El = 14.6 ~leV (i s 1,2 . . . . . 78), where

~o/4) = 1/2. The envelope of the spectra shows the inverse of the LET.

where c is the cutoff energy, B. is the mean excitation energy transfer, and ~c(E) is the cross section for excitation of primary electron energy E. (m We as- sume Bc = 13 eV, since the value of B. is about 13 eV in the energy range of 102-10SeV. m)

From this we can define an interspur distance Zt for an electron path of energy E,

Zi = Es/Li,c, (11)

where Es is the energy required to produce a spur, which is assumed to be constant throughout electron paths. We regard c and Es as variables when we compare our model with experiments, but the value of c would relate to the initial width of the spur and the value of Es to the initial number of species in the spur.

(d) G-values by electrons The differential G-value G'~(E) for species i, i th

radical and molecular products per 100 eV by the path segment of energy E, is calculated from the solutions Ni of equations (3) and (5) at a particular time,

G'~(E) = NJEs × 100. 02)

where Es is the energy to produce a spur.

The integral G-value GLT) for species i, an average yield of i th species per lOOeV over the whole paths from the electron of initial energy T is,

G,(T) = j/ G'~(E)y(T, E) d v

× L(E)dE/~;y(T.E)L(E)dE, (13)

where L(E) is the LET, the linear energy tran~er. For a radiation which has primary electron energy

spectrum P(T) the G-value Gi for species i is calcu- lated by,

G, = ~o P(T)G,(T) dT. (14)

Only these values can be compared with experimental values.

The G-value of the Fricke dosimeter under aerobic conditions is obtained by ~l)

G(Fe 3+) = 3G(e~q) + 3G(H)

+ G(OH) + 2G(H202). (15)

m U L T S

In equation (2) we assign s p e c ~ as e~q-1, H=2, H+=3, OH=4, OHm5, H20,m6 and

282 Hmosm Y~ht~oucm

Table 2. Diffusion constants, initial radii and numbers of species

Species Dxl0~(cm2s -I) r°(~) N°(50.SeV -I) oo(100eV -I)

i. e~ 4.5 24.58 2.54 5.03 2. H 7 ! 1.45 0.328 0.65 3. H + 9 11.45 2.54 5.03 4. OH 2.8 I !.45 3.03 6.0 5. OH- 5 I 1.45 0 0 6. H=O 2 2.2 I 1.45 0 0 7. H 2 8 11.45 0.080 0.158

H 2 = 7 (Table 1) and ignore the term koCoNi since we consider no solute in water. The diffusion constants for species are taken the same value as Schwarz's m (Table 2). Table 2 also gives the values of initial radii which result in the same values as Burns's ¢9) after several trials with different sets of ~ . The differential equations (3) and (5) are numerically solved f r o m t = 10 -12 to 3 x 10"Ts. T h e t ime o f

3 x 10 -~ s allows the results to be compared with the experimental G-value of Fricke dosimeter. (*) For comparison we used experimental G-values of Fricke dosimete~ m for four p rays and eeC, o 7 rays. The primary electron energy ~ c t r a of these p and 7 rays are tabulated 03-ts) as shown in Table 3.

The experimental base for the initial number of species in a spur is scarce. From the data observed in a cloud chamber by Wilson (1923) and Beckman (1949), an analytic expression for the distribution of species in a spur was found. °el This expression gives an average spur size for the number of radical pairs of 2.54. We assume this value for the number of species produced by ionization as N o e~ = N°s+ = 2.54. We set N O _ a= N~2o2 == 0

these species are absent at beginning. The values of A~s, Noos and N% are assumed to be those of Schwarz multiplied by a constant, which is a ratio of the present value 2.54 to Schwarz's value 4.78 for the hydrated electron. The values of G o in Table 2 mean N~JEs x 100.

The main task here is to determine Es, the energy to produce a spur, and the cutoff energy c, such that

Table 3. Primary electron energy spectra for the mdiatiom

3 H ~s S np JeCo TkeV P(T)t TkeV P(T) TkeV P(T) TMeV P(T)

1.0 0.909 4.92 0.130 2.0 0.986 14.92 0.122 3.0 1.0 29.87 0.221 4.0 0.973 65.58 0.430 5.0 0.927 117.53 0.097 6.0 0.855 " 7.0 0.768 ~Y 8.0 0.682 T keV P(T) 9.0 0.586 "

10.0 0.505 5.0 0.0034 I 1.0 0.409 15. I 0.0034 12.0 0.327 3 0 . 0 0.0071 ! 3.0 0.25 70.0 0.0229 14.0 0.172 205.3 0.094 15.0 0 .10 451.4 0.175 16.0 0.064 943.5 0.425 17.0 0,027 1606.0 0.269 18.0 0.009

5.1 0.0028 15.1 0.0030 30.2 0.0066 71.4 0.0245

207.3 0,124 456.3 0,263 903.8 0.52

I,~0.0 0.055

0.05 0.0975 0.15 0.0635 0.25 0.0g0$ 0.35 0.0785 0.45 0.0785 0.55 0.0795 0.65 0.0825 0.75 0.0915 0.65 0.1167 0.95 0.1197 i.05 0.0785 1.15 0.0231

t P(T) for ~H is the fraction to its maximum vahm, while the other P(T) is the fraction, the sum of which is normalized by one.

Table 4. Calculated G(F¢ J+) in comlmison with experiment o2)

Radiation Calculation Experiment

rays 1. 3H 12.91 12.9 + 0.2 2. 35S 14.80 14.3 + 0.4 3. ~ ' 15 .72 15.4+0.4 4. 32p 15.64 15.3 + 0.5

rays e°Co 15.55 15.5 + 0.3

the calculated values are consistent with the experi- mental values. First, we set values of Es and c, get the differential O-value of the ~ by solving equa- tions (3) and (5), and cak:ulate the integral G-value of the species ruing the degradation spectrum y ( T, El in equation (13). Then we calculate the G-value of the

species for each radiation truing its primary electron energy spectrum and compare with experimental values.

After having repeated this proc~s, we obtained the final result that Es . . 50.5 eV and c ffi 50 eV. Table 4 shows the calculated G-values o f Fricke dosimeters for the mdiatiom. Figure 3 shows the differential and integral G-valttm of Frieke dosimeter as a function of electron energy. Figure 4 shows the integral G-value of ~ of water radical= as a function of primary electron energy.

DISCUSSION

The calculated G-values for the Fricke dosimeter shown in Table 4 agree well with the experiments. This agreement is due to the proper choice of values of c and Es: the value of c sensitivity affects the energy dependence of the differential yields and Es affects the interspur distance Z~ and the absolute values of the yields.

The energy E, = 50.5 eV can be analyzed into some components. The calculated average energy needed to produce an ion pair at 10 - n S is 15.9 eV for a 5-keV electron and all of its secondaries. I" Thus the energy

/ . . . . . . . . . . ~-~i7--

i I I i i i l l / I i i i I l ~ i i i i 1 ~ i i i i 1 ~ i I I l ~

C 1 0 - 2 10-4 10 ° I0 ~ 402 405 104 Electron energy (mY)

Fig. 3. The dif~i~utial G-value G'(£) [dashed fine] and the integral O.value O(£) [mild fine] of the Fricke dmimetm

calculated by the present model.

Prescribed diffusion model based on track structure 283

3

O \ ~

..... ...............

i i i i i i I l l i i i i i i i l l i i i i i i I l l I I i n l l l l I i i i i i i i i I 1 I l U l ~

0-2 10-4 10 ° 10 4 10 2 10 s 10 4 Etectron enetqy (keV)

Fig. 4. The integral G-values o f var ious species in water i r radiated by electrons. Numbers indicate that 1--I e;q, 2=H, 3 =H +, 4=OH, 5 =OH-, 6=H202 and 7 = H 2.

lOSS by ionizations in the spur is 40.4eV (= 15.9 x 2.54). The average energy over the dis- sociative excitation, ~ l B i , ~ lA I and diffuse band weighted by their oscillator strengths is 13 eV. °~ The energy loss by dissociative excitations is 7.4eV (-- {No. of dissociation to (H + OH) + No. of dis- sociation to (H2+O) } x 13 = { ( N O a - N ? n ~ ) + AmH,} x 13={(3.03--2.54)+0.08} x 13--7.4). This means that the energy loss by excitations to other than dissociative bands is 2.7 eV (=Es - 4 0 . 4 - 7.4). This energy may correspond with those causing vibrational excitations. °s~ This analysis does not pro- vide any definite values of the spur yet, but it would exemplify that the values of Es must be well founded by the cross section data for elementary process. In this view the cross section for the dissociative pro- cesses should be given. °9~ These data must also be important in the Monte Carlo calculation for radical reactions, o)

Figure 3 shows difference between the differential and the integral G-values of Fricke dosimeter: the ratio G'(E)/G(E) are 1.19-1.15 from 100 to 1000keV. This relates to the value of Es. If we perform the calculation with the average interspur distance over the whole paths, like other calculations shown in Table 5, Es becomes larger than the present value of 50.5 eV. The above ratios may correspond to those of Es in Table 5 : 6 0 / 5 0 . 5 = 1 . 1 9 and 62.5/50.5 = 1.24. This implies that the analysis for Es from the result using the average interspur distance over the whole paths, is perhaps misleading.

The result c = 50 eV may relate to spread of the spur. The restricted energy loss L~,~ at the energy c -- 50eV is 9.1 keV//~m from equation (10) and the straight length of this path is 5.5 nm (=50/9.1). The length of this path roughly corresponds to the as- sumed diameter of the hydrated electron 4.91 nm (= 2.458 × 2). The Monte Carlo calculation in this range of electron energy may provide defmite data on the spread of the spur.

Figure 4 shows the results at a time of 3 x 10 - 7 S.

Table 5. The G-values of species in water irradiated by e°Co y rays in the ale-nee of scavenger

Calculations

Species Burns ~) Trumbore ca) Present work Experiment°9~ "

2.85 2.97 3.04 2.70 ~-~H 3.06 3.67 2.86 2.86 H 0.712 0.804 0.647 0.61 H 2 0.407 0.452 0.402 0.43 H202 0.659 0.503 0.82 0.61 G(Fe3+):I: 15.06 16.0 15.55 14.01

Es in eV 62.5 60 50.5

t Data in the presence of scavenger. $ Cak'ulated by equation (15).

The present method can estimate G-values of the species in water at any time after 10 -H s.

Figure 5 shows that the interspur distance Z~ varies with electron energy of the path segment from 2.7 run at about 100eV to about 1 ~m at energy above 1000 keV. This curve allows us to imagine that iso- lated spurs make a line through a high energy region, i.e. 1000 keV, and spurs overlap to form a cylindrical shape for electron energy 80-100 eV; spurs are again apart because of domination of elastic scattering below 80 eV: these spurs may form a larger aggre- gation like a blob32) The present method can provide a first approximation for these track structures. The more detailed treatment including the spur size distri- bution needs more rigorous definition of the spur: i.e. electron energy dependence of Es, number of species, and width of the spur.

Table 5 shows that there are differences among calculations of G-values of species in water irradiated by ~°Co 3' rays. With the Gaussian distribution for hydrated electrons we can fit the G-value of the Fricke dosimeter, but for the G-values of these components. The presence of scavangers in the experimental data 09) does not allow us rigorous com- parison with calculations, c9~ The realistic calculation of the diffusion process, which shows non-Gaussian distribution, through the diffusion, for the hydrated

10 4

10 s

10 4

i iiii rill i lllnlll i i lllllll i i iiiiiii i I lllllll i a iiiiiii

10 ~ ,I0_ 2 40-~ tO o 404 402 tO3

Eteclron e r~y IIwV)

Fig. 5. The interspur distance Z, in nanometers as a function of electron energy. The curve was calculated by equation

(11) with c --- 50 eV and Es = 50.5 eV.

284 Hmcem YA~L~OUCm

electron by Burns, ¢j) and the prescribed diffusion model with a non-Gaussian distribution for the hy- drated electron by Trumbore e ta / . (s~ are proposed. We studied the latter proposal. We used a non- Gaussian distribution in equation (A.2) which is slightly different from that of Trumbore et al/'~ and has zero concentration for the hydrated electron at the center of the spur. We also assured that some non-Gaussian distribution is necessary to obtain a better fit of the components with experiments. The present results in Table 5 were obtained by equation (A.2). If the non-Gausslan distribution for the hy- drated electron at the beginning is true, the question would arise what the underlying mechanism is. A study is in progress that the decrease in concentration of the hydrated electron at the center of the spur can be attributed partly to the initial ion recombination taking place before 10-" s, which has long been paid less attention in liquid water radiolysis.

Acknowledgements--The author wishes to thank Dr M. Imamura, Institute of Physical and Chemical Research, for a number of useful comments on this paper. This work was supported by a special grant for Tritium Research from the Science and Technolosy Allency, Japan.

I C E S

1. J. E. Turner, J. L. Magee, H. A. Wright, A. Chatterjee and R. N. Ritchie, Rad/at. ~es. 1983, 96, 437.

2. (a) A. Mozumder and J. L. Magee, ~_ad~. Res. 1966, 28, 203 and 215; (b) A. Mozumder and J. L. Matp~, J. Chem. Phys. 1966, 45, 3332.

3. J. L. Magee and A. Chatterj~, J. Phys. Chem. 1976, 82, 2219.

4. (a)A. Kuppermann, In Radiation Research 1966, p. 212. North Holland Amsterdam, 1967; (b)A. Kupparmaon, In Physical MechanLnn in Radiation Biology, p. 155. CONF-721001, 1974.

5. H. A. Schwarz, J. Phys. Chem. 1969, 73, 1928. 6. (a) J. L. Magee and A. Chatterjee, J. Phys. Chem. 1980,

3529; (b) A. Chntterjee and J. L. Magee, J. Phys. Chem. 1980, 84, 3537.

7. W. G. Burns and A. R. Curtis, J. Phys. Chem. 1972, 76, 3008.

8. C. N. Trumhnre, D. R. Short, J. E. Fanning and J. H. Olsen, J. Phys. Chem. 1978, 82, 2762.

9. W. G. Burns, H. E. Sims and J. A. B. Goodall, Rad/at. Phys. Chem. 1984, 23, 143.

10. A. Pemnamenta and J. H. Marshall, Radiat. Res. 1986, 106, 1.

1 I. H. G. Paretzke and M. J. Berger, Proc. Vlth Syrup. Microdos. p. 749. Harwood, London, 1978.

12. A. O. Fregene, Radiat. Res. 1967, 31, 256. 13. T. Salgnsa, Ryukokukiyo, 1981, 2, 123. 14. Table of Rad~tion Isotopes (Edited by E. Browne, R. B.

Firestone and V. S. Shirley). John Wiley, New York, 1986.

15. D.V. Cormack and H. E. Johns, Br. J. Radial. 1952, 25, 369.

16. A. H. Samuel and J. L. Magce, J. Chem. Phys. 1953, 21, 1080.

17. A. E. S. Green and J. H. Miller, In Physical Mechanism in P~di.a~ion B/ology, p. 68, CONF-721001, 1974.

18. J. J. Olivero, R. W. Stqat and A. E. S. Green, J. Geophys. Res. 1972, 77, 4797.

19. A. Appleby and H. A. Schwarz, J. Phys. Chem. 1969, 73, 1937.

APPENDIX

Trumbore et al.(8) assumed the following non-Ganssian distribution for the hydrated electron,

C.(r) = n,(r/b,)3exp{ -(rib,)2}, (A.I)

where b, is the radial width of the concentration and n e is the normalization constant. We assume the following distri- bution in cylindrical coordinate system,

C,(r, z) = ne(r/be)exp[- {r 2 + (z - z~)2}/b2,], (A.2)

where z, is the position of a spur along the path. Following the similar derivation of equation (3) with

equation (A.2), we get different expressions for f¢ of equa- tion (4).

fu =" gJx, for i ffi e~q, (A.3)

f¢ " gc{b]/(b2~ + b~}°/2)/4,

for i = e~q, j ffi the other, (A.4)

fu ffi equation (4), for i,j ~t e~q, (A.5)

where

gu ffi [1 + {2 ~(b 2 + b~)}~t/2)/Zt]/{n(b ~ + by)} °/2~, (A.6)

where symbols have the same meaning as equation (4). The function g¢ is very similar to equation (4). We found that the higher the order of factorial to produce a non-Gansslan distribution is applied, the larger the constant reduction of f..~i, equation (A.3) comes out. For the form of the

bution, which we assumed, the order of factorial 1 for radial direction was the best.