Calculation of heavy ion ranges in complex media

NUCLEAR INSTRUMENTS

AND METHODS

159 ( 1 9 7 9 )

421-431

; ©

NORTH-HOLLAND

P U B L I S H I N G CO.

CALCULATION OF HEAVY ION RANGES IN COMPLEX MEDIA SHANKAR MUKHERJI and A. K, NAYAK

Department o/ Chemistly, Indian Institute of Technology, Kanpur 208016, India

Received 10 April 1978 and in revised form 18 September 1978 Ranges of some heavy ions of initial energies of up to 10 MeV per nucleon in some complex media (Mylar, polyethylene, tissue - material, and G-5 nuclear emulsion) and in some elemental media (oxygen, nitrogen, argon, nickel and silver) have been computed on the basis of a set of stopping-power equations deduced from a basic stopping-power equation of Bohr. In the present work we have revised the earlier stopping-power equations of Mukherji and Srivastava and the formula of Mukherji for the calculation of the mean ionization potentials of the elements. The calculated ranges in Mylar, polyethylene, tissue-matter and in the elemental media are in excellent agreement with the available experimental data. In G-5 nuclear emulsion, the calculated ranges are higher than the experimental ones at the lower energies but agreement is good at the higher energies except in the case of 2°Ne ion.

I. Introduction The two basic stopping-power equations, one due to Bohr ~) and the other due to Bethe2), suffer from the drawback that each has its rather restricted region of validity3,4). Recently, Mikherji and Srivastava 5) deduced a set of stopping-power equations from a stopping-power equation put forward by Bohr 3) in 1948, which allows the calculation of the stopping powers of heavy ions at all energies at which electronic collision predominates. The good agreement between the experimental and calculated values of the, ranges of heavy ions in elemental media with the use of these equations 5) has logically led to the present investigation of their applicability in the case of complex media. However, before considering the complex media, we have first re-examined the earlier stopping power equations as well as the formula of Mukherji 6) for the calculation of the mean ionization potentials in an effort to discover the reasons for the disagreements between the experimental and calculated values noted earlier 5,6) in some cases. Since this has resulted in some changes in the equations, we have also re-calculated the ranges of some heavy ions in some elemental media before computing their ranges in the complex media.

2. Stopping power equations Bohr's 3) stopping-power equation, which gives the energy loss per unit path length d E I d X for a heavy ion of ionic charge ze and velocity V, may be written as (1)

dE 2nz2e4n ( j l + j ) ' dX mV ~

where e and m are the electronic charge and mass, n is the number of atoms of the medium per unit volume and 2 --2 A = Y~In (,7,[z]), (2) s

where ~/s = 2 V/Us, Z = 2 z V o / V , Us being the velocity of the sth orbital electron of the medium and V0 = e2/h. It is to be noted that the quantities within the square brackets, if less than unity, should be., replaced by unity. This condition allows J to be split into two terms J: and J3. Writing S for the ° sum of the two summation terms Us'

S=J,+J2+J3

=

X Us=O

2 VX - 1

ln(r/2[Z']-2) +

E Us=O

Us"

ln~/2+

E 2Vx -t

ln(~/~(-x)

(4)

422

S. MUKHERJI AND A. K. NAYAK

where U" and Us" are the upper cutoff values which render the logarithmic terms zero, indicating that the orbital electrons with velocities greater than these do not participate in the energy-loss process. Here, we would briefly go through the steps used earlier by Srivastavaand MukherjiS), simply setting down their derived results where no changes are required, and dwell on those points where modifications are needed. 2.1. X> 1 2.1.1. v > i ZVoz Under this condition S = 2Z ln(2rnV2/l),

(5)

where Z is the atomic number of the medium and i is its mean ionization potential. 2.1.2. ½ZV0z> V> ½( Z - 2) VoZ/f(Z) Srivastava and Mukherji 5) obtained the following expression for J3 in the velocity region 1Z J 2 V0Z> V> ~ZV0z'

(Z-2)Vo J3 =

:2v,-i

In

tin(Us) + 21n

(6)

where (Z-2)Vo/f(Z) represents the velocity of the ( Z - 2 ) t h orbital electron of the medium as obtained from the relation n(U,) = f (Z) U,/Vo, (7) where n(U) is the number of electrons of the medium with velocity less than Us, f(Z) assumes the values 0.28 Zt and Z t, respectively, corresponding to Z~<45.5 and Z>~45.5. Eq. (7) has been considered to be valid for all the orbital electrons of the medium except for the two K-shell electrons for which the velocity has been taken as Z V o. In evaluating ,/3, as given by eq. (6), the possibility of the occurrence of the condition 2 VX-~ >~(Z-2)Vo/f(Z) was not taken into accountS). Since no term is permitted to assume a negative value, the integral in eq. (6) should be set equal to zero if 2 VZ- ~ >~( Z - 2) V0/f(Z), and further, if V<~½ZVoX~, then J3 itself should be replaced by zero. Thus, if ½ZVoZ> V>½ZVoZ ~ and ½ZV0z ~> k ( z - 2 ) Voz/f(Z), then J3 is given by J3 = 2In(2Zv--V-~-) 3.

(8)

Further, both in J~ and J2, the upper limit for U, is 2 VZ-~ , which, under the given conditions, is at least equal to the velocity of the ( Z - 2 ) t h orbital electron of the medium but less than that of the Kshell electrons. Thus 2vx-' s=z-2 {2mV2~ {2mV2~ Sl ~--- E lIl(qsX-2) = E In = ( Z - 2 ) ln Us=O s:l t/-~ --J Iti---x-~-J . (9) J2 =

2v., s=z_2 (2mv2 Z In,l~ = Y. In = ( Z - 2 ) ln . Us=0 s=l k Is ) t --'~1 )

(10)

The new quantity [I stands for the geometrical mean of the ionization potentials -of the outer ( Z - 2 ) orbital electrons of the medium

s=Z--2 ( Z - 2 ) lnil =

~ $=1

lnl,.

The value of [1 may be obtained, in units of eV, from Mukherji's formula 6)

(11)

HEAVY ION RANGES ["

Z--2

423

"]z

il =: 13.6 L2.717 f(Z) j •

(12)

Thus from eqs. (4) and (8)-(10), one obtains (2_mVq ( 2V ~3 S =:2(Z-2) l n \ i1Z /+21n\z----~oZ~] .

(13)

2.1.3. ZVoZ ~ > 2 V > ( Z - 2 ) VoZ/f(Z) In this region J3 = 0, while J~ and J2 are still given by eqs. (9) and (10). Therefore S =: 2(Z-2) In \ i1 z /].

(14)

2.1,,4. ( Z - 2 ) ~ X / . I ( Z ) > ZVoX ~ In this case, the following two regions have to be considered: a) (Z-2)VoZ/f(Z)>2 V > Z V o Z I. In this case J3 is given by eq. (6) and Ji requires an upper cutoff value Us' = 2 VZ -I . Thus s) 2Vx-t J, = ~ In ( 2 V z - ' / U O z = 4f(Z) Z-' V/Vo. (15) Us=O

2V;(- t J2 =

ln(2V/Us) 2 = 4 f ( Z ) z -~ (l+lnz) V/V o.

~

Us=O

S : = [ 3 ( Z - 2 ) + 3 ( Z - 2 ) I n ( 2 f ( Z ) V )-I ,2) , ( ZVo + 6 In (2~--~-o)+ 2f(Z) VVo~

(16) Z lnx].

19) Z V o z ~>2 V>(Z-2)Vog'l.l'(Z). This condition leads to (Z-2)Vo f JIZ, ( 2VIN~3 J3 = In ~ dn(U~), J2vT.-,

\U,~/

(17)

(18)

while Jl and J2 are given by eqs. (15) and (16). Thus

2I(z) v

S = 3 ( Z - 2) + Vo-----7 -

3(Z_2) ln{!Z-2) \

2 l/f (Z) ]"

(19)

2.1.5. 2 V < ( Z - 2 ) VoZ} In this case J~ and J2 would still be given by eqs. (15) and (16), while J3 would require an upper cutoff value U " = 2 VZ -~. Thus S = 2f(Z) (3Z_~+Z_I)" Vo

(20)

2.2. Z < I For Z< 1, Jl and J, given by eqs. (2) and (3), become identical s:, = s = Z In $

= Y. lr (2 v/ $

o 2,

(21)

and only three velocity regions need be considered. 2.2.1. 2 V> Z V o In this case s) S = 2 Z ln(2mV2/i).

(22)

424

s.

MUKHERJI

AND

A. K. N A Y A K

2.2.2. Zl/0>2 V > ( Z - 2 ) Vo/f(Z) Since (Z-2)Vo/l(z) is the velocity of the ( Z - 2 ) t h orbital electron of the medium, this condition implies that the heavy ion can ionize all the orbital electrons except the K-shell electrons. Hence s=Z-2

S = 2

~

ln(2V/Us) 2 = 2(Z-2) ln(2mVZ/il).

(23)

8=1

2.2.3. 2 V < ( Z - 2 ) Vo/f(Z) This condition necessitates an upper cutoff velocity 3,5) Us = 2 V. Hence 5) 2V

s = 2

~

ln(2ViU,) 2 = 8f(z)

V/Vo.

(24)

U~=O

2.3.

HYDROGEN AS THE MEDIUM

Since all the complex media we have considered contain hydrogen and as the general equations derived for multi-electron media would not hold for hydrogen which contains only one electron, this case requires a separate treatment. The summation terms have to be stripped of their summation signs and U, has to be replaced by I/0. Thus S = ln ls LZJ ) + i n = ln((2V~

\\v,/

2

r/2

[Z]-2) + In ((~oV)2 I ~ 1 - 1 ) .

(25)

Three different situations have to be considered at this point. 2.3.1. 2'<1 In this case, the logarithmic terms become identical 3) S = 2 In (2 m V 2/I),

(26)

where I is the ionization potential of the hydrogen atom. 2.3.2. Z > 1 The expression for S is given by S = In (2V'] 2

\Voz/ + l n

((~voV)2IV° Z] -' ) L2Vj

.

(27)

Since [½ VoZ/V]= 1 if 2 V~> V0Z, we have to consider two cases: a) 2 V> VoZ. This condition leads to S = ln\voZ) + l n \ V o J

= 2

\~].

(28)

b) VoZ>2 V> V0zi. Since ½Vox/V> 1, the square bracket may be removed from eq. (27) S = 2In (2V----~)+ 3 In ( 2V-~S) .

(29)

Further, since V< 1 VoZ, the first logarithmic term becomes zero and the second term remains positive as long as V>~½:VoZ ~. Thus S = 3In

(~) 2V

(2 m V2\ = ~ln/--r--r-].

\lZ,/

(30)

HEAVY ION RANGES

425

Since the maximum value of )C even for a very heavy ion like the 238U-ion is about 8, the lower velocity limit V= ½V0)r~ is very close to V0 at which eq. (30) remains valid, eqs. (26), (28), (29) and (30) would provide stopping-power equations which would be adequate for the calculation of the range of any heavy ion. Below V = I/0 one need not calculate the electronic stopping powers, since nuclear collisions become predominant in this velocity region. After substitution of the expressions for S from eqs. (5), (13), (14), (17), (19), (20), (22), (23), (24), (26), (28), (29), and (30) in eq. (1), conversion into proper units and simplifications, one obtains the following expressions for the stopping powers, in units of MeV.cm2/mg, valid under the stated conditions. 1) ~ > 1, V > ½ZVoZ: dE 63.65z2Z (11.39 V2"], d)f = AV 2 l ° g z o \ ~ ,/

(31)

where z is the ionic charge of the moving ion in units of the electronic charge and V is its velocity in units of 108 cm/s. l(in eV), A and Z are, respectively, the mean ionization potential, mass number and atomic number of the medium. These definitions are valid for the subsequent equations as well. The proper expressions for the calculation of z are given by Srivastava and Mukherji5). 2) Z > l, ½ZVo)~¢ > ½(Z-2) VoX/f(Z), ½ZVo)~ > V > ½ZVo)~¢: dX dE -

63.65 AI/2z2 I ( Z - 2 ) ,,Ogxo ~/'11.381/.2 -~IZ" )' + 3 loglo ( kZ-~oZ¢)l 2 ,

(32)

where [1(in eV) is the mean ionization potential of the medium excluding the K-shell electrons. 3) z > 1, ½ZVoz+ > ½ ( z - 2 )

de d'--X:=

Voz/f(z), ½ZVoz* > v > ½ ( z - 2 ) Voz/f(z):

63.65z 2 ( z - 2 ) tOg,o{~1._3932~ AV2

'

(33)

~, It Z //"

4) )~ > 1, ~ZVox + < ½(Z-2) Vo)Cff(Z), ½(Z-2) VoXff (Z ) < V < ½Zl%)~: eq. (33) would also apply in this region. :5) ~( > 1, ½Zl, oX-~ < ½(Z-2) Vo)~/f(Z), ½(Z-2) Voz/f(Z ) > V > iZl/oX+: d--~rdE= 13.79z 2AV 2

3 ( Z - 2 ) + 3 ( Z - 2 ) In !,,(Z-2) Vo,] + 6In

+ - - V oZ

Z lnz .

(34)

6) Z > 1, ½Zl/oZ ~- < ½(Z-2) VoZ/f(Z), ½ZVoZ• > V > ½(Z-2) Voz+/f(Z): dE dX-

13"79z2f 3 ( Z - 2 ) + - AV2 L

Vo)¢

,,

3 ( Z - 2 ) in

ro ql ]j

\ 2f(z)v

(35)

7) x > 1, ½ZVo)~¢ > ½(z-2) Voz/f(z ), ½(z-2) Voz/f(z) > v > ½(z-2) Voz¢/f(z): eq. (35) would also apply in this region. 8) z > ~, v < ½( z - 2) Vo z*/l (z):

dE 12.68 ! (Z) z 2 1). d-~(- = AV (3Z-++Z-

(36)

9) Z < I , V~-{ZVo: d~ =

A V2

l°gl o

':

•

(37)

426

S.

MUKHERJI

AND

A.

K.

NAYAK

At relativistic energies of the ion 5) dE 63.65z 2 Z (11.39 V2 /?z ) d--X = A V 2 l°gl° ~,iO---fl-Yj 2.~3_'

(38)

where / / = V/c, c being the velocity of light. 10) Z < 1, ½ZVo > V > ½(Z-2) Vo/./(Z): dE 63.65z 2 ( Z - 2 ) (ll.39V2~. dX = V2 l°gl°, ~ ]

(39)

11) Z < 1, V < ½ ( Z - 2 ) Volf(Z): dE dX - -

=

50.6 I(Z) z 2 AV

(40)

The following three equations hold in the case of hydrogen as the medium. 12) z > 1, ½Vo)/ > V > ½Vo)(~: dE dX-

47.74z 2 (ll.39 V2~ 1/2 l ° g l ° . / 7 ]'

(4J)

where / ( = 13.6 eV) is the ionization potential of the hydrogen atoms. 13) Z > 1, V>½1%Z: dE dX

63.65z 2 =

V 2

(11.39 V2.) l°gl° \ ~ "

(42)

14) Z< 1, V > ½I%: dX =

V2

(43)

l°gl°

3. Range computation in complex media Assuming the validity of Bragg's additivity rule, one may express the stopping power of a complex medium [(dE/dX)c]E at the ion energy E in terms of the stopping powers of the constituent elements [(dE/dX)z]E at the same ion energy E dE

= 1

dE

,

where Ac (= ~ g, Ai) is the molecular mass number of the medium, Ai and Y~ are the mass number and the number bf atoms per molecule of the ith atomic species, respectively. Eq. (44) would directly give the stopping power of the complex medium in units of MeV.cm2/mg if the expression for (dE/dX)~ is taken from eqs. (31)-(43). To compute the range R for a heavy ion of initial energy E~ MeV, one has to calculate [(dE/dX)~.]e starting with E = E~, at each new ion energy decreasing by a small amount f E ( ~ O . O l MeV) over which the elemental stopping powers remain virtually constant, till the final ion energy Eo, corresponding to the velocity V0 7) is reached. The range is then given by Eo dE R (mg/cm 2) = ~ [(dE/dX)¢]e"

(45)

Ei

4. Mean ionization potentials Since many of the stopping-power equations presented here differ from those obtained earlierS), it was necessary to test their validity in the case of elemental media before applying them to complex media.

HEAVY ION RANGES

427

Accordingly, the range and stopping-power calculations reported earlier 5) were repeated by us using these revised equations. The new results, however, were found to differ negligibly from the earlier ones, leaving both the agreements and disagreements with experimental data unchanged. At this stage, using a computer (IBM 7044), ranges of some heavy ions (I°B, liB, t2C, 14N, 19F and 2°Ne) were calculated in G-5 nuclear emulsion with the help of eq. (45) using the procedure of Srivastava and MukherjiS). Since G-5 nuclear emulsion consists 8) of a suspension of fine grains of silver bromide of an average diameter of 0.275 # m in gelatin, it can not be regarded as a homogeneous medium at the molecular level. Two different computational procedures were, therefore, used in this case. In the first procedure, the average chemical composition has been taken as (cf. appendix).

AgBro.994 I0.006 C1.371 N0.314 So.ol 3 H2.54.O0.62, and the ranges were computed assuming the medium to be homogeneous. In the second procedure, the heavy ion was considered as passing alternately through segments of 0.1833 u m of silver bromide and 0.1888/~m of gelatin, each retaining its own bulk density (cf. appendix). For computational purposes, each such segment was theoretically subdivided into several layers so that the energy lost by the ion in passing through a single layer does not exceed 0.01 MeV. The range was obtained from the number of such segments through which the ion had to pass till the final velocity Vo was reached. Both procedures yielded practically identical ranges which were significantly shorter, to the extent of 10-15 ~m, from the experimental ranges9), the deviation increasing with increasing mass number of the ion. Since the present stopping-power equations predict correct energy losses in the case of light media and deviations from the experimental data appear with increasing mass number of the medium, as in the case of nickelS), we assumed that a modification was needed in the prescription for obtaining the mean ionization potentials of the elemental media for the following reasons. Firstly, there is a large difference between the experimental values of the mean ionization potentials and those calculated by Mukherji's 6) prescription in the case of elements of atomic number Z in the region 7 4 > Z > 2 6 . Secondly, from the theoretical point of view, as has been stressed earlier6), the expressions f ( Z ) = 0 . 2 8 Z] and f ( Z ) = Z ~ corresponding to Z<~45.5 and Z>~45.5 were obtained from a consideration of the interaction between a moving ion and only a few of the outer orbital electrons of the medium, except in the case of very light media, and, hence, should be valid only for the outermost electrons, whereas their use in Mukherji's 6) formula

F. Z--2

-]2

Z In[ -- ( Z - 2 ) In 13.6 L2.717f(Z)_] + 21n(13.6Z 2)

(46)

implies their validity for all the orbital electrons except the two K-shell electrons. Lastly, since f(Z) largely represents the "effective quantum number" of the electrons3,5), which, as Bohr argued3), would be unity for the innermost and the outermost orbital electrons, the values given by the earlier expressions for f(Z) are possibly overestimations for the medium-weight and heavy-weight elements, particularly at high ion energies when a majority of the orbital electrons is involved in the interaction. Thus, one should take values o f f ( Z ) averaged over all the orbital electrons; these would be smaller than the earlier ones, would lead to larger values of ], and would, therefore, be consistent with the experimental values6). To obtain a new expression for f(Z), we assumed that eq. (46) is basically valid and that by substituting the experimental values of the mean excitation potentials in place of I in that equation one would obtain the averaged or new values off(Z). The experimental values were taken from Bakker and Segr61°), as calculated by Bethe and Ashkin 4) and from the compilation of Barkas and Berger ~1) for this purpose. Fig. 1 shows a logarithmic plot of the new values o f f ( Z ) against Z. A least-squares fit yields the following relation

f(Z) = 0.3634Z °'5s5,

(47)

which, on substitution in eq. (46), gives , //13.95 (Z--2) 2) Z ]tni = ( Z - 2 ) m ~ Z1.1o + 21n(13.6Z2),

(48)

428

S. M U K H E R J I

AND

A. K. N A Y A K

9

0.60.4

0.2

"~" 0.0

o~

-0.2

-0.4

I

-0.6 0

I

0.4

i

I

0.8

i

tog

I

1.2 Z

i

[

1.6

,

I

2.0

Fig. 1. Plot of Iogto f(Z) versus log10 Z. Points shown by plusses are based on the data from Bakker and Segr6 (ref. 10) and those shown by dots are based on the data from Barkas and Berger (ref. 11).

and il =

13.95 ( Z - 2 ) 2 Z1.1o

(49)

For Z < 13 and Z > 7 4 , the values o f f ( Z ) from eq. (47) differ very little from the ones obtained from the earlier expressions, but for 1 3 < Z < 7 4 , there is 3-17% lowering in the values off(Z). Table 1 lists the newly calculated values of I for some elements along with the corresponding accurate experimental values from Andersen et al. 12) and Sorensen and Andersen ~3) which have not been used as inputs in obtaining the relation given by eq. (47). Since the earlier expressions for f(Z) were deduced from the ranges of fission products, the new expression forf(Z) from eq. (47) would yield considerably larger values for calculated fission product ranges. As fission products are essentially heavy ions with initial kinetic energies in the range of 0.5-1.0 MeV/nucleon, in all our computations, we have used eq. (47) at ion energies greater than 1.0 MeV/nucleon while below that energy we have used the older expressions 5) for f(Z). Similarly, in the expressions 5) for the effective charge z for the ion, we have used for ./(Z~) eq. (47) with Z replaced by Zj where Z~ is the atomic number of the ion, at ion energies above TABLE 1

Comparison between the calculated values of f with the experimental values of the mean excitation potential /. Element

Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ag Au Pb U

Atomic no. Z

20 21 22 23 24 25 26 27 28 29 30 47 79 82 92

I(eV)

/(eV) experimental

calculated

195.0 ~ 217.0 a 229.0 a 238.7 a 259.1 a 273.8 a 280.8 a 299.3 a 303.4 a 321.7 a 323,4 b 466.8 b 750.0 b 766.0 b 831.0 b

231.2 241.0 250.7 260.2 269.8 279.2 288.6 297.9 307.1 316.3 325.4 474.0 733.8 757.4 835.0

a H. H. A n d e r s e n , H. Sorensen and P. Vajda (ref. 12). b Sorensen and H. H. Andersen (ref. 13).

HEVAY I O N

429

RANGES 2

1.0MeV/nucleon; below that energy we have used ./'(Z0= 0.28 Z? for Z1~<45.5 and Z 1>/45.5.

.f(Z1)--Zi

for

5. Results and discussion Using the present stopping-power equations along with eqs. (47)-(49) and the general procedure outlined earlierS), we calculated the ranges of different heavy ions in elemental oxygen, nitrogen, argon and nickel. Since G-5 nuclear emulsion contains silver as one of the main constituent elements, we have also calculated the ranges of t2C ions in silver. Fig. 2 shows the calculated energy-loss curves for I°B, lIB, 12C, IaN, 160 and 19F ions in oxygen gas and the corresponding experimental values from Roll and Steigert14). Fig. 3 shows the calculated energy-loss curves for 12C and 4°Ar ions in nitrogen and argon as the media and the corresponding experimental data from Martin and Northcliffe~S). Fig. 4 shows the calculated energy-loss curves for l°B, 11B, 12C, 14N, 160, 19F and 2°Ne ions in nickel along with the experimental data from Roll and Steigert~4). Fig. 5 shows the calculated range--energy curve for ~C ions in silver and the corresponding experimental data from Walton and Hubbardt6). In the cases of both silver and nickel we also calculated the ranges using the earlier values o f f ( Z ) at all ion energies. Fig. 5 includes these for silver, while for nickel these values differ from the experimental values by more than 10% but could not be shown in fig. 4 due to lack of space. After obtaining satisfactory agreement between the calculated and experimental values as shown by figs. 2-5, we calculated the ranges and stopping powers of several heavy ions in polyethylene, Mylar and simulated tissue materials having, respectively, the chemical compositions 17) (CH2),, CmHsO 4 and C 7 H 7 0 N 2 0 3 2 . In the case of G-5 nuclear --r

200 180 160

~ 140

I , I, ~k

\

, I f I , I , I , I , I , 1,1,11 [, 1607 19F7 o 12CI E.xpt. o 14NI Expt. \ .

%

~i . ~ " e

• c11B] oBc],

>- 120 L.b nuJ 100

I

I

I

T

I

'--

/*OAr in N2~ o 4OAr in ArJ a 12 C in Ngl Expt /

9 E 8 o

I ~

'

•

'~ 12C in ArJ

7

w 4

Lu 3

52

40 20 I i I , I ,~I\I ,-'~, 0 0 10 20 30 L,O 50 60

I

, I, I,,, ~ iI,, Ix, I i 10 20 30 40 50 60 70

, t , l I0

o

20

THICKNESS(rag/era 2)

Fig. 2. Energy-loss curves for I°B, 11B, 12C, 14N, 160 and 19F ions in oxygen gas. The experimental data are from Roll and Steigert 14).

,i

,i

r I'l

'1~1

• 20Ne7 o

'1'

16o ~ x p t .

• 11B

, I , ] , I , i , i-, i , i , i T I ' \ ° mF7 .~ • 14NI

30 40 50 60 GAS THICKNESS (mg/cm 2)

70

80

Fig. 3. Energy-loss curves for 12C and 4°Ar ions in gaseous nitrogen and argon. The e×perimental data are from Martin and Northcliffe]5). 10

i

i

,

i

1

i

9 =~

,

/r

i

i

//

8

J

~w 8o oz

I

z

6o

20O , i , i 180 ~. ",, 160 X'k~ 140 ~(_~12o lOO

'

6

uJ 80

z

'

~5

z

o

I

IO

- - - Cole.

60

L.0 20 0

, i, I ~ L=I, ~,I~,i'~,i , 10 20 30 t,O 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 THICKNESS1 mg/cm 2 )

Fig. 4. Energy-loss curves for l°B, lIB, 12C, 14N, i 6 0 , 19F and 2°Ne ions in nickel. The experimental data are from Roll and, Steigert 1 4 ) .

2

xpt.

1

O~ 0

t

i

2LO : ; 0

t

6'0

i

8tO

i

L

100

A

120

RANGE (mg/cm 2 )

Fig. 5. Range-energy curve for 12C ions in silver. The experimental data are from ref. 16. Dashed line indicates results with old values o f f ( Z ) .

430

S.

MUKHERJI

AND

A.

K.

NAYAK

MYLARTHICKNESS(mgcrff 2) 10

10

20

0

10

20

,

30

40

50

a~uEE~1

'Ne 160 0

10 20 30 40 POLYETHYLENE THICKNESS(mgcrff 2)

30

12C + 40

I 50

O

Fig. 6. Energy-loss curves for 12C and 160 ions in polyethylene and for 12C, 160 and 2°Ne ions in Mylar. The experimen-

tal data are from Schambra et al.18). 2607, I ~ I ' r ' I ~ I ' I ' p ' I ' 'r ' i ' 240~ o10B7Expt 220~~ 160] /o 200~- - C a l c ' / 'a0 r

/

;2 ,:oF

// / ,

'l'l'['l'l']'r'r'r'p a 11B ° 12~7Expt

6

0 0

__i" '~J

/

'

10

'

]

'

I

'

I

0:~o~

20 30 40 50 60 PENETRATED THICKNESS X(mg/crr~)

'

70

Fig. 7. Plots of the stopping powers of lIB, uC, 160 and 2°Ne ions against their penetration depths in tissue material. The experimental data are from Brustadl9). ~'

/

~

I ' I ' J ' I , I , I ' I r I , I

200L-

o 14 N 7

la0~-

~, 20mejExpt

160I

Catc.

/

140

'2°t ,oo~

//

Zy

6o~

x~

/ / ,, / x o

~

-

60 40

4O

~

20 200

0 1 2

I J J , J , I

3 4 5 6 7 8 9 10 1 2 3 4 5 6 1ON ENERGY(MeV/omu)

7

8 9 10

Fig. 8. Range-energy curves for l°B, liB, 12C, 160 and 19F ions in G-5 nuclear emulsion. The experimental data are from Roll and Steigert9).

0

,1~

1

I , I , I ~

2 3 4 5 6 7 8 ION ENERGY (MeV/amu)

9 10

Fig. 9. Range-energy curves for 14N and 2°Ne ions in G-5 nuclear emulsion. The experimental data are from Roll and Steigert9).

emulsion, we used the second procedure mentioned earlier to calculate the ranges. Figs. 6, 7, 8 and 9 show our calculated results and the experimental data from Schambra et al.18) for polyethylene and Mylar, from Brustad m) for tissue material and from Roll and Steigert 9) for G-5 nuclear emulsion. It may be seen that although there is good agreement between the calculated and experimental values in the cases of polyethylene, Mylar and tissue material, there is some disagreement in the case of G-5 nuclear emulsion. Figs. 8 and 9 show that the calculated ranges are systematically larger than the experimental ones by - 3 # m at ion energies below - 4 . 5 M e V per nucleon for all the ions except 2°Ne for which there is almost a constant difference of ~ 5 # m at all energies. There are some plausible reasons for this discrepancy. For the heavier ions, the uncertainty in their charge state, and, hence, in their measured kinetic energy, increases as the ion energy decreases below 9) ~ 4 MeV/nucleon. Further, the density of the emulsion, while it is under vacuum during bombardment with the ions, is not known precisely and the various stages of development of the emulsion track, involving heating and humidification, may cause sufficient surface distortion to account for small discrepancies of the order of 3-5 #m. We are thankful to Dr. B. K. Srivastava, Radiochemistry Division, Bhabha Atomic Research Centre, Trombay, Bombay, for his help during the initial stages of this work.

Appendix From the composition of the " s t a n d a r d " G-5 emulsion, given 8) in percentage by weight, one obtains the empirical formula AgBr0 994I0.006Ci.371 N0.314S 0 , 0 1 3 H3.180 0 . 9 4 for G-5 nuclear emulsion of density 3.815.

HEAVY ION RANGES

431

If the density is taken as 4.0 g/ml when the emulsion is under vacuum during bombardment, then one has to calculate the changed empirical formula due to the evaporation of some water. Assuming that x g of water has evaporated per ml of the emulsion under vacuum and each gram of water removed causes a volume reduction of 0.73 ml as obtained from the de-humidification curvesS), we have 3.815--x

1.0-0.73x

= 4.0,

(50)

from which we get x---0.0962 g. This leads to the new empirical formula given in the text. In the second computational procedure, the empirical formula of gelatin is taken as C1.371N0,31450.013H2.5400.62 and neglecting the small atomic proportion of iodine in the silver bromide phase, the path lengths in gelatin and silver bromide have been obtained by closely following the treatment of BarkasS). The average grain diameter (D) of silver bromide in G-5 nuclear emulsion is 8) 0.275 ~tm with a = 0.0022. Assuming random traversal of the silver bromide grains by the heavy ion, the average path-segment 6 in a silver bromide grain is given by 6 = ~(D)=0.1833/~m. The average distance / b e t w e e n the centres of two successive grains of silver bromide is given by l = l/n,

(51)

where n is the number of silver bromide grains the ion will pass through, on the average, when it traverses a unit distance in the emulsion. For n one has 8) n =-- 2(D--~

I +

I -(--D-~]

J'

(52)

where c is the volume concentration of silver bromide in the emulsion. On the basis of 6.473 g/ml as the', bulk density of silver bromide and 3.1407 g/ml as the weight per unit volume of silver bromide in the original emulsion s) of density 3.815 g/ml, the value of c, after taking into account the reduction in the volume of the emulsion by a factor of (1-0.73×0.0962), is 0.5239. Thus, from eqs. (51) and (52), one obtains /=0.3721 pm. The average path length through gelatin is ( 0 . 3 7 2 1 - 6 ) = 0 . 1 8 8 8 p m . Thus the', heavy ion would alternately pass through an average distance of 0.1888pm of gelatin and 0.1833 p m of silver bromide. References I) N. Bohr, Phil. Mag. 25 (1913) I0. 2) H. A. Bethe, Ann Phys. (Leipzig) 5 (1930) 325. 3) N. Bohr, Kgl. Dansk. Vidensk. Selksk. Mat:-Fys. Medd. 18, no. 8 (1948). 4) H. A. Bethe and J. Ashkin, in Experimental nuclear physics (ed. E. Segr6; Wiley, New York, 1953) vol. 1, ch. 2. 5) B. K. Srivastava and S. Mukherji, Phys. Rev. AI4 (1976) 718. 6) S. Mukherji, Phys. Rev. B12 (1975) 3530. 7) S. Mukherji and B. K. Srivastava, Phys. Rev. B9 (1974) 3708. 8) W. H. Barkas, in Nuclear research emulsions, vol. 1 (Academic Press, New York and London, 1963). 9) p. G. Roll and F. E. Steigert, Nucl. Phys. 16 (1960) 534. 10) C. J. Bakker and E. Segr6, Phys. Rev. 81 (1951) 489. 11) W. H. Barkas and M. J. Berger, in Studies in penetration o f charged particles in matter (National Academy of Sciences-National Research Council, Washington, D.C., 1964) Publ. 1133, p. 103. 12) H. H. Andersen, H. Sorensen and P. Vajda, Phys. Rev. 180 (1969) 373. 13) H. Sorensen and H. H. Andersen, Phys. Rev. B8 (1973) 1854. 14) p. G. Roll and F. E. Steigert, Nucl. Phys. 17 (1960) 54. 15) F. W. Martin and L. C. Northcliffe, Phys. Rev. 128 (1962) 1166. 16) j. R. Walton and E. L. Hubbard, quoted in L. C. Northcliffe, Ann. Rev. Nucl. Sci. 13 (1963) 69. 17) D. E. Lea, in Action of radiations on living cells (Cambridge University Press, London, New York, 1946). 18) p. E. Schambra, A. M. Rauth and L. C. Northcliffe, Phys. Rev. 126 (1962) 61. 19) T. Brustad, Adv. Biol, Med. Phys. 8 (1962) 161.