semi-empirical formula for photon energy absorption

16 Int. J. Nuclear Energy Science and Technology, Vol. 13, No. 1, 2019

Copyright © 2019 Inderscience Enterprises Ltd.

Semi-empirical formula for photon energy absorption buildup factors of elements and compounds

H.C. Manjunatha* Department of Physics, Government College for Women, Kolar-563101 Karnataka, India Email: [email protected] *Corresponding author

L. Seenappa Department of Physics, Government College for Women, Kolar-563101 Karnataka, India Email: [email protected]

K.N. Sridhar Department of Physics, Government First Grade College, Kolar-563101 Karnataka, India Email: [email protected]

Abstract: We have formulated a simple semi-empirical formulae for photon energy absorption buildup factors in the energy region 0.015–15 MeV, atomic number range 1 Z 92 and for mean free path up to 40 mfp. The results produced by the present formulae agree well with the data available in the literature. This semi-empirical formula may be extended to any compounds/mixtures/biological samples. This semi-empirical formula finds importance in the calculations of buildup factors of any materials which are required for radiation shielding, nuclear engineering, radiotherapy and nuclear medicine.

Keywords: fast reactors; sodium-cooled reactor; breed and burn; metallic fuel; ASTRID.

Reference to this paper should be made as follows: Manjunatha, H.C., Seenappa, L. and Sridhar, K.N. (2019) ‘Semi-empirical formula for photon energy absorption buildup factors of elements and compounds’, Int. J. Nuclear Energy Science and Technology, Vol. 13, No. 1, pp.16–26.

Biographical notes: H.C. Manjunatha is an Assistant Professor of Physics at Government College for Women, Kolar, Karnataka, India. His areas of research are radiation, nuclear and medical physics. He has published more than 100 research papers in reputable international journals. He has also served as reviewer for many international journals.

JID:NUPHA AID:21480 /FLA [m1+; v1.291; Prn:11/12/2018; 15:49] P.1 (1-15)

Available online at www.sciencedirect.com

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Nuclear Physics A ••• (••••) •••–•••

www.elsevier.com/locate/nuclphysa

Studies on the synthesis superheavy element Z = 120

K.N. Sridhar a,c, H.C. Manjunatha b,∗, H.B. Ramalingam d

a Department of Physics, Government First Grade College, Kolar-563101 Karnataka, Indiab Department of Physics, Government College for Women, Kolar-563101 Karnataka, India

c Research and Development Centre, Bharathiar University, Coimbatore-641046, Indiad Department of Physics, Government Arts College, Udumalpet-642126, Tamil Nadu, India

Received 23 October 2018; received in revised form 21 November 2018; accepted 26 November 2018

Abstract

We have identified the probable isotopes for superheavy element Z = 120 by comparing the alpha decay half-lives with that of spontaneous fission. The nuclei 290–304120 were found to have long half-lives and hence could be sufficient to detect them if synthesized in a laboratory. We have also studied the most possible projectile–target combination to synthesis those nuclei. The selected projectile–target system that can yield maximum production cross section is Ti+Cf.

2018 Elsevier B.V. All rights reserved.

Keywords: Superheavy element

1. Introduction

The synthesis of superheavy nuclei has received remarkable attention in recent years with

the advancement of modern accelerators and suitable detectors. Qiu et al. [1] determined macro-

scopic deformed potential energy for superheavy elements Z = 120 within a generalized liquid

drop model. Singh et al. [2] predicted the possible isotopes for superheavy element Z = 120

within RMF+BCS approach. Previous researchers [3] also calculated the energy levels of the

superheavy element Z = 120 using relativistic Hartree–Fock theory. Nasirov et al. [4] calculated

fusion and evaporation residue cross sections for the 50Ti + 249Cf and 54Cr + 248Cm reactions

* Corresponding author.

E-mail address: [email protected] (H.C. Manjunatha).

https://doi.org/10.1016/j.nuclphysa.2018.11.032

0375-9474/ 2018 Elsevier B.V. All rights reserved.


2 K.N. Sridhar et al. / Nuclear Physics A ••• (••••) •••–•••

using dinuclear model. Ghahramany et al. [5] studied different hot fusion reaction to synthe-

sis superheavy element Z = 119–122. The production cross sections decrease exponentially for

higher superheavy elements and now have reached the 1-pb limit [6].

Earlier researchers [7–10] studied the compound nucleus formation based on the dinuclear

system (DNS) concept. Previous researchers [11–13] also studied synthesis of superheavy nuclei

using the multi nucleon transfer reactions. The production cross section of superheavy nuclei

depends on the projectile–target combinations and the incident energy. The previous workers

studied the possible projectile target combinations to synthesis the superheavy nuclei and com-

petition between different decay modes of superheavy nuclei [14–27]. In this work, we have

studied the α-decay properties of superheavy nuclei Z = 120 in the range 265 ≤ A ≤ 316. By

studying the α-decay properties, we have identified the possible isotopes for superheavy element

Z = 120. After identifying the possible isotopes, we have selected the suitable projectile–target

combinations to synthesis super heavy element Z = 120.

2. Theoretical frame work

2.1. Competition between fission and alpha decay process

The interacting potential between two nuclei of fission fragments is taken as the sum of the

Coulomb potential and proximity potential. To study the alpha decay and fission, we have used

Denisov nuclear potential Vp(r) [28]. We have explained the detail procedure of calculation of

alpha decay half life and spontaneous fission half-life in the previous work [17].

2.2. Projectile–target combinations to synthesis SHN Z = 120 via fusion

The total interaction potential for fusion is considered as sum of coulomb and nuclear po-

tential. Coulomb potential is calculated using the expression given by previous workers [33].

The nuclear potential is calculated from the proximity potential. We have used the Myers and

Swiatecki [29] modified the proximity potential. The fusion barrier has two basic features: one

is the barrier position (RB ) and the other is barrier height (VB ). The knowledge of the analytical

form of the total interaction potential enables us to determine the exact values of these parame-

ters. Since fusion happens at a distance larger than the touching configuration of colliding pair,

the above form of the Coulomb potential is justified. One can extract the barrier height VB and

barrier position RB using the following conditions

dV(r)

dr

∣

∣

∣

∣

r=RB

= 0 andd2V (r)

dr2

∣

∣

∣

∣

r=RB

≤ 0 (1)

To study the fusion cross sections, we shall use the model given by Wong [30]. In this formalism,

the cross section for complete fusion is given by

σfus =πh2

2µ × Ecm

lmax∑

l=0

(2l + 1) × Tl(Ecm) · PCN(Ecm, l) (2)

where µ is the reduced mass. The centre of mass energy is denoted by Ecm. In the above formula,

lmax corresponds to the largest partial wave for which a pocket still exists in the interaction

potential and Tl(Ecm) is the energy-dependent barrier penetration factor. PCN is the probability

for the compound nucleus (CN) formation by two nuclei coming in contact. The probability of


K.N. Sridhar et al. / Nuclear Physics A ••• (••••) •••–••• 3

compound nucleus formation PCN suggested by previous workers [31–38] is used in the present

calculation. The calculation of PCN requires effective fissility which in turn depends on xthr and c.

xthr and c are adjustable parameters [14–16]. These parameters were suggested by Loveland

[39]. This form of energy dependence of fusion probability is similar to the one proposed by

Zargrebeav et al. [32].

After the fusion of two nuclei, the corresponding compound nuclei comes to ground state by

emitting neutrons. The evaporation residue cross section of SH element production in a heavy–

ion fusion reaction with subsequent emission of x neutrons is given by [32]

σ xnER =

π

k2

∞∑

l=0

(2l + 1)T (E, l)PCN(E, l)P xnsur

(

E∗, l)

(3)

Psur is the survival probability and it is the compound nucleus to decay to the ground state of the

final residual nucleus via evaporation of neutrons/light particles. The survival probability is the

probability that the fused system emits several neutrons followed by observing a sequence of α

decay from the residue. The survival probability under the evaporation of x neutrons is

Psur = Pxn

(

E∗CN

)

imax=x∏

i=1

(

Γn

Γn + Γf

)

i,E∗

(4)

where the index “i” is equal to the number of emitted neutrons. The calculation of Psur requires

the probability of evaporation of x neutrons from compound nucleus (Pxn). To calculate the Pxn,

we have adopted the procedure explained by the previous workers [38,40]. The term [Γn/(Γn +

Γf )] in equation (4) is calculated by the knowledge of the ratio of the emission width of a neutron

to the fission width (Γn/Γf ). In the present work, we have used the expression for Γn/Γf based

on the level densities of the Fermi-gas model [38].

3. Results and discussion

3.1. Decay properties of SHN Z = 120

We have identified the possible isotopes of superheavy element Z = 120 by comparing the

alpha decay half-lives with that of spontaneous half-lives. The energy released during the alpha

decay (Qα) is calculated using the procedure explained in previous work [18–20,41–43]. A com-

parison of alpha decay half-lives with the corresponding SF half-life makes it clear that the nuclei 265–277120 could not survive against fission. Even though the nuclei 278–289120 survive against

fission and shows alpha chains, but could not be detected due to shorter alpha decay half-lives

(<10−8 s). The nuclei 290–304120 will survive against fission. The nuclei 290–294120, 295–296120, 297–298120, 299–302120 and 303–304120 shows 6α, 5α, 4α, 3α and 2α chains, respectively. The

variation of log10(T1/2) against the mass number of the parent nuclei is shown in Figs. 1(a)–1(b).

Along with these, we have plotted the decay half-lives evaluated using the VSS formula [44]; the

UNIV [45] and the analytical formulas of Royer [46], Ni–Ren–Dong–Xu (NRDX) formula [47],

Denisov formula [48] and H.C. Manjunatha–K.N. Sridhar (HCM) semi-empirical formula [27].

Among the studied nuclei in the range 265 ≤ A ≤ 316, the nuclei 290–304120 were found to have

long half-lives and hence could be sufficient to detect them if synthesized in a laboratory. The

calculated alpha decay chains are also shown in Figs. 2(a), 2(b) and 2(c).

To check isotopes for the stability against the proton, neutron and beta emission, we have

calculated the corresponding separation energies. The calculated separation energies for different



(a)

(b)

Fig. 1. Comparison of the calculated alpha decay half-lives with the corresponding spontaneous fission half-lives. (a) Plot for the comparison of the calculated alpha decay half-lives [VSS, Royer, UNIV, Present AD, HCM, Denisov and NRDX] with the corresponding spontaneous fission half-lives [Present SF] of the isotopes 290–295120 and their decay products. (b) Plot for the comparison of the calculated alpha decay half-lives [VSS, Royer, UNIV, Present AD, HCM, Denisov and NRDX] with the corresponding spontaneous fission half-lives [Present SF] of the isotopes 296–304120 and their decay products.

isotopes of super heavy nuclei Z = 120 are shown in Fig. 3(a). From this calculation, it is found

that the one proton [S(1p)] and two-proton separation energy [S(2p)] are negative for isotopes

within the range 280 ≤ A ≤ 293. The nuclei 280–293120 comes outside the proton drip line and

thus may easily decay through proton emission. The nuclei 294–316120 were found to be stable

against neutron, proton and beta decay. The summary of the decay mode of isotopes of super

heavy elements SHN Z = 120 also shown in Fig. 3(b).



(a)

(b)

Fig. 2. Decay chain of the predicted probable isotopes for Z = 120. (a) Decay chain of the predicted probable isotopes for Z = 120 (A = 290–294). (b) Decay chain of the predicted probable isotopes for Z = 120 (A = 295–299). (c) Decay chain of the predicted probable isotopes for Z = 120 (A = 300–304).



(c)

Fig. 2. (continued)

(a)

(b)

Fig. 3. Stability of superheavy nuclei Z = 120. (a) Nucleon separation energies as a function of mass number for Z = 120. (b) Decay modes of super heavy nuclei for Z = 120.



Fig. 4. Reduced fusion barrier positions SB (fm) as a function ofZ1Z2

A1/31

+A1/32

.

Fig. 5. Fusion barrier heights VB (fm) as a function ofZ1Z2

RparB

(1 −1

RparB

).

3.2. Synthesis of predicted isotopes through fusion reaction

We have studied more than 2000 possible projectile target combinations to synthesis su-

perheavy nuclei 290–307120. For all projectile–target combinations, we have calculated the fu-

sion barrier heights (VB ) and positions (RB ). We have calculated the reduced fusion barrier

SB = RB − C1 − C2 and plotted reduced fusion barrier as a function of Z1Z2/(A1/31 + A

1/32 )

and it is shown in Fig. 4. We have parameterized reduced fusion barrier in terms of x =

Z1Z2/(A1/31 + A

1/32 ) as follows:

SParaB = −3.49 × 10−5

×[

Z1Z2/(

A1/31 + A

1/32

)]2+ 0.01 ×

[

Z1Z2/(

A1/31 + A

1/32

)]

+ 0.445 (5)

hence, fusion barrier position (RB ) becomes Rpara

B = Spara

B + C1 + C2.

The calculated fusion barrier height (VB ) is plotted as a function of (Z1Z2/Rpara

B )(1 −

1/Rpara

B ) and it is shown in Fig. 5. We have parameterized fusion barrier height as follows:



Fig. 6. Maximum evaporation residue cross section for different projectile–target combinations at maximum E* for different neutron evaporation channel (1 −46Ti+251Cf, 2 −46Ti+250Cf, 3 −46Ti+249Cf, 4 −47Ti+251Cf, 5 −48Ti+251Cf, 6 − 47Ti + 249Cf, 7 − 48Ti + 250Cf, 8 − 49Ti + 251Cf, 9 − 53Cr + 250Cm, 10 − 50Ti + 250Cf, 11 − 50V + 247Bk, 12 −51V+248Bk, 13 −50V+248Bk, 14 −50Cr+247Cm, 15 −50Cr+246Cm, 16 −50Cr+244Cm, 17 −50Cr+245Cm, 18 − 50Cr+ 243Cm, 19 − 52Cr+ 245Cm, 20 − 54Fe+ 242Pu, 21 − 54Fe+ 241Pu, 22 − 54Fe+ 240Pu, 23 − 55Fe+ 241Pu, 24 − 51V + 247Bk, 25 − 54Fe + 239Pu, 26 − 55Fe + 238Pu, 27 − 54Fe + 238Pu).

V ParaB = −1.34 × 10−3

×

[

Z1Z2

Rpara

B

(

1 −1

Rpara

B

)]2

+ 1.907 ×

[

Z1Z2

Rpara

B

(

1 −1

Rpara

B

)]

− 39.05 (6)

The constructed formula for the fusion barriers may be used to produce RB and VB of fusion

reactions to synthesis super heavy nuclei super heavy element Z = 120.

A comparison of maximum evaporation residue cross section among the studied the

projectile–target combinations are as shown in Fig. 6. Among the studied projectile–target com-

binations, the fusion reaction Ti+Cf is having maximum evaporation residue cross section at all

energies and at 1 to 6n evaporation channel. It is also observed that the projectile–target com-

bination Ti+Cf is having minimum driving potential, maximum fusion and evaporation residue

cross sections. Hence the selected most probable projectile–target combinations to synthesis su-

perheavy nuclei 290–304120 is Ti+Cf. The selected most probable projectile–target combinations

to synthesis the superheavy nuclei 290–304120 is shown in Fig. 7. The variation of evapora-

tion residue cross section as a function of E∗ for the possible reactions such as 46Ti + 251Cf, 54Cr + 250Cm, 54Fe + 238Pu and 58Ni + 232U are shown in Fig. 8. This figure enables us to

identify the maximum evaporation residue cross section for different channels (2n, 3n and 4n).

The pre-synthesis parameters such as compound nucleus fissility (χCN), charge product in the

entrance channel (ZpZt ), effective entrance channel fissility (χeff ), fusion barrier height (VB )

and fusion barrier width (RB ) for most probable fusion reactions are calculated using the pro-

cedure explained in our previous work [19] and it is shown in Table 1. To validate the present

work, we have compared the predicted halflives with that of experiments [49–56]. Fig. 9 shows

that nuclear chart indicating the comparison of predicted halflives with that of experiments for

superheavy nuclei in the atomic number range 110 < Z < 120.



Fig. 7. Evaporation residue cross section as a function of mass number of the compound nucleus.

Fig. 8. Variation of evaporation residue cross section as a function of energy.

Table 1

Pre-synthesis parameters of most probable fusion reactions to synthesis SHE Z = 120.

CN Most probable projectile–target combination

VB (MeV) RB (fm) ZpZt χCN Xeff × 10−3 N/A

290120

58Ni(S 68%) + 232U(68.9 y) 281.80 12.09 2576

1.030

1.48

0.58

81Kr(2.3e5 y) + 209Po(125.2 y) 327.52 12.11 3024 1.3882Kr(S 11.59%) + 208Po(2.89 y) 327.24 12.12 3024 1.3784Sr(S 0.56%) + 206Pb(S 24.1%) 337.66 12.06 3116 1.3986Sr(S 9.86%) + 204Pb(S 1.4%) 337.13 12.08 3116 1.37

291120

58Ni(S 68%) + 233U(1.59e5 y) 281.55 12.10 2576

1.028

1.46

0.58

59Ni(7.6e4 y) + 232U(68.9 y) 281.17 12.12 2576 1.4582Kr(S 11.59%) + 209Po(125.2 y) 326.93 12.14 3024 1.3683Kr(S 11.5%) + 208Po(2.89 y) 326.66 12.15 3024 1.3584Sr(S 0.56%) + 207Pb(S 22.1%) 337.33 12.07 3116 1.38

(continued on next page)



Table 1 (continued)



292120

54Fe(S 5.85%) + 238Pu(87.74 y) 267.17 12.13 2444

1.027

1.47

0.58

58Ni(S 68%) + 234U(2.45e5 y) 281.3 12.12 2576 1.4659Ni(7.6e4 y) + 233U(1.59e5 y) 280.92 12.14 2576 1.4460Ni(S 26.22%) + 232U(68.9 y) 280.55 12.16 2576 1.4264Zn(S 49.2%) + 228Th(S 1.91%) 293.82 12.14 2700 1.43

293120

44Ti(63 y) + 249Cf(351 y) 236.87 12.11 2156

1.026

1.51

0.59

50Cr(S 4.34%) + 243Cm(29.1 y) 251.93 12.15 2304 1.4654Fe(S 5.85%) + 239Pu(2.41e4 y) 266.94 12.14 2444 1.4655Fe(2.73 y) + 238Pu(87.74 y) 266.55 12.16 2444 1.4458Ni(S 68%) + 235U(7.04e8 y) 281.05 12.13 2576 1.45

294120

44Ti(63 y) + 250Cf(13.08 y) 236.67 12.12 2156

1.024

1.51

0.59

50Cr(S 4.34%) + 244Cm(18.1 y) 251.72 12.16 2304 1.4553Mn(3.74e6 y) + 241Am(432.2 y) 258.93 12.18 2375 1.4354Fe(S 5.85%) + 240Pu(6500 y) 266.7 12.15 2444 1.4555Fe(2.73 y) + 239Pu(2.41e4 y) 266.32 12.18 2444 1.43

295120

44Ti(63 y) + 251Cf(898 y) 236.48 12.13 2156

1.023

1.5

0.59

46Ti(S 8.25%) + 249Cf(351 y) 235.63 12.18 2156 1.4450Cr(S 4.34%) + 245Cm(8500 y) 251.5 12.18 2304 1.4452Cr(S 83.78%) + 243Cm(29.1 y) 250.45 12.24 2304 1.4053Mn(3.74e6 y) + 242Am(141 y) 258.71 12.19 2375 1.42

296120

44Ti(63 y) + 252Cf(2.64 y) 236.28 12.14 2156

1.021

1.49

0.59

46Ti(S 8.25%) + 250Cf(13.08 y) 235.43 12.19 2156 1.4447Ti(S 7.44%) + 249Cf(351 y) 235.03 12.22 2156 1.4150Cr(S 4.34%) + 246Cm(4730 y) 251.29 12.18 2304 1.4352Cr(S 83.78%) + 244Cm(18.1 y) 250.5 12.23 2304 1.39

297120

46Ti(S 8.25%) + 251Cf(898 y) 235.24 12.21 2156

1.019

1.43

0.60

48Ti(S 73.72%) + 249Cf(351 y) 234.44 12.25 2156 1.3847Ti(S 7.44%) + 250Cf(13.08 y) 234.83 12.23 2156 1.4050V(1.5e17 y) + 247Bk(1380 y) 242.48 12.24 2231 1.3850Cr(S 4.34%) + 247Cm(1.56e7 y) 251.08 12.19 2304 1.43

298120

46Ti(S 8.25%) + 252Cf(2.64 y) 235.04 12.22 2156

1.018

1.42

0.60

47Ti(S 7.44%) + 251Cf(898 y) 234.64 12.24 2156 1.4048Ti(S 73.72%) + 250Cf(13.08 y) 234.24 12.26 2156 1.3750V(1.5e17 y) + 248Bk(>300 y) 242.28 12.26 2231 1.3749Ti(S 5.41%) + 249Cf(351 y) 233.86 12.28 2156 1.35

299120

47Ti(S 7.44%) + 252Cf(2.64 y) 234.44 12.25 2156

1.017

1.39

0.60

48Ti(S 73.72%) + 251Cf(898 y) 234.05 12.27 2156 1.3749Ti(S 5.41%) + 250Cf(13.08 y) 233.67 12.29 2156 1.3450Ti(S 5.18%) + 249Cf(351 y) 233.29 12.32 2156 1.3251V(S 99.75%) + 248Bk(>300 y) 241.69 12.29 2231 1.35

300120

48Ti(S 73.72%) + 252Cf(2.64 y) 233.86 12.28 2156

1.016

1.36

0.60

49Ti(S 5.41%) + 251Cf(898 y) 233.47 12.31 2156 1.3450Ti(S 5.18%) + 250Cf(13.08 y) 233.1 12.33 2156 1.3150Cr(S 4.34%) + 250Cm(9000 y) 250.45 12.24 2304 1.4052Cr(S 83.78%) + 248Cm(3.4e5 y) 249.66 12.28 2304 1.36




Table 1 (continued)



301120

49Ti(S 5.41%) + 252Cf(2.64 y) 233.28 12.32 2156

1.014

1.33

0.60

50Ti(S 5.18%) + 251Cf(898 y) 232.91 12.34 2156 1.3153Cr(S 9.5%) + 248Cm(3.4e5 y) 249.08 12.31 2304 1.3354Cr(S 2.36%) + 247Cm(1.56e7 y) 248.71 12.33 2304 1.3157Fe(S 2.12%) + 244Pu(8.08e7 y) 263.99 12.3 2444 1.34

302120

50Ti(S 5.18%) + 252Cf(2.64 y) 232.72 12.36 2156

1.013

1.30

0.60

52Cr(S 83.78%) + 250Cm(9000 y) 249.25 12.30 2304 1.3554Cr(S 2.36%) + 248Cm(3.4e5 y) 248.51 12.34 2304 1.3158Fe(S 0.28%) + 244Pu(8.08e7 y) 263.41 12.33 2444 1.3160Fe(2.6e6 y) + 242Pu(3.73e5 y) 262.71 12.37 2444 1.28

304120

54Cr(S 2.36%) + 250Cm(9000 y) 248.1 12.36 2304

1.011

1.29

0.6060Fe(2.6e6 y) + 244Pu(8.08e7 y) 262.27 12.39 2444 1.2772Zn(S 46.5%) + 232Th(1.4e10 y) 288.17 12.42 2700 1.2376Ge(1.78e21 y) + 228Ra(5.75 y) 300.47 12.41 2816 1.23

(a)

Fig. 9. (a) Nuclear chart indicating the comparison of predicted halflives of superheavy nuclei (110 < Z < 114) with that of experiments. (b) Nuclear chart indicating the comparison of predicted halflives of superheavy nuclei (115 < Z < 120) with that of experiments.

4. Conclusion

We have identified the most possible isotopes for superheavy nuclei Z = 120 in the range

265 ≤ A ≤ 316. The nuclei 290–304120 were found to have long half-lives and hence could be

sufficient to detect them if synthesized in a laboratory. The selected most probable projectile–

target combinations to synthesis superheavy nuclei 290–304120 is Ti+Cf.



(b)

Fig. 9. (continued)

Appendix. Table A

Table A

Comparison of the alpha decay half-lives with SF half-lives of 290–304120 and their decay products. A prediction on the mode of decay is given by comparing the α decay half-lives with the SF half-lives.

Parent nuclei

Q

(MeV)

TSF (s)

T α1/2 (s) Mode of decayVSS Royer UNIV Present HCM Denisov NRDX

290120 14.03 1.2E+15 6.2E−08 2.4E−07 4.8E−08 1.7E−07 6.9E−04 5.4E−07 2.9E−07 α1286118 13.56 1.4E+09 1.5E−07 5.3E−07 1.1E−07 4.1E−07 1.0E−02 1.2E−06 5.8E−07 α2282116 13.13 3.9E+04 3.1E−07 1.0E−06 2.1E−07 8.5E−07 7.2E−02 2.1E−06 1.0E−06 α3278114 12.66 1.9E+01 7.5E−07 2.4E−06 5.0E−07 2.1E−06 2.7E−01 4.7E−06 2.1E−06 α4274112 11.80 1.3E−01 1.4E−05 4.1E−05 7.8E−06 5.2E−05 1.2E+00 9.3E−05 2.7E−05 α5270110 11.31 1.1E−02 4.6E−05 1.3E−04 2.6E−05 1.9E−04 2.0E+00 3.0E−04 8.0E−05 α6266108 10.55 8.9E−03 7.4E−04 2.1E−03 3.9E−04 3.7E−03 6.8E+00 5.4E−03 9.9E−04 SF262106 10.20 6.0E−02 1.4E−03 3.8E−03 7.4E−04 6.1E−03 5.6E+00 9.0E−03 1.8E−03 SF






(continued)

Parent nuclei

Q

(MeV)

TSF (s)


293120 13.64 1.5E+16 3.7E−06 1.1E−06 1.9E−07 9.4E−07 1.6E−03 1.1E−06 1.2E−06 α1289118 13.06 1.8E+10 1.6E−05 4.2E−06 7.5E−07 4.5E−06 2.7E−02 4.2E−06 4.0E−06 α2285116 12.51 4.3E+05 6.0E−05 1.5E−05 2.7E−06 1.7E−05 2.6E−01 1.4E−05 1.2E−05 α3281114 11.96 1.9E+02 2.5E−04 6.1E−05 1.1E−05 8.1E−05 1.5E+00 5.4E−05 4.2E−05 α4277112 11.49 1.2E+00 7.8E−04 1.8E−04 3.2E−05 2.6E−04 4.7E+00 1.5E−04 1.1E−04 α5273110 11.10 9.0E−02 1.7E−03 3.7E−04 6.9E−05 5.7E−04 8.0E+00 2.9E−04 2.3E−04 α6269108 10.13 6.7E−02 1.1E−01 2.3E−02 4.0E−03 3.0E−02 9.8E+01 1.7E−02 9.9E−03 SF265106 9.55 4.1E−01 9.1E−01 1.9E−01 3.3E−02 1.3E−01 4.5E+02 1.3E−01 7.2E−02 SF

294120 13.54 1.6E+16 4.9E−07 1.6E−06 2.8E−07 1.6E−06 2.0E−03 4.3E−06 1.8E−06 α1290118 12.93 1.8E+10 2.5E−06 7.4E−06 1.3E−06 7.9E−06 3.6E−02 2.1E−05 6.8E−06 α2286116 12.34 4.2E+05 1.2E−05 3.3E−05 5.6E−06 4.2E−05 3.8E−01 9.4E−05 2.5E−05 α3282114 11.76 1.8E+02 6.3E−05 1.7E−04 2.8E−05 2.4E−04 2.6E+00 5.0E−04 1.1E−04 α4278112 11.25 1.1E+00 2.4E−04 6.1E−04 1.0E−04 9.5E−04 1.0E+01 1.8E−03 3.5E−04 α5274110 11.07 8.0E−02 1.7E−04 4.2E−04 7.8E−05 6.5E−04 1.1E+01 1.1E−03 2.6E−04 α6270108 10.05 5.8E−02 1.5E−02 3.5E−02 5.9E−03 4.1E−02 1.8E+02 1.2E−01 1.5E−02 SF266106 9.40 3.4E−01 2.1E−01 4.9E−01 8.3E−02 2.3E−01 1.5E+03 1.8E+00 1.8E−01 SF

295120 13.45 1.1E+16 8.5E−06 3.5E−07 3.9E−07 2.3E−06 2.5E−03 2.4E−06 2.5E−06 α1291118 12.81 1.2E+10 4.9E−05 2.0E−06 2.0E−06 1.4E−05 4.7E−02 1.2E−05 1.1E−05 α2287116 12.19 2.7E+05 2.9E−04 1.1E−05 1.1E−05 8.9E−05 5.5E−01 6.3E−05 4.9E−05 α3283114 11.56 1.1E+02 2.0E−03 7.8E−05 7.0E−05 6.8E−04 4.4E+00 3.9E−04 2.7E−04 α4279112 11.03 6.7E−01 9.7E−03 3.7E−04 3.3E−04 3.4E−03 2.2E+01 1.7E−03 1.1E−03 α5275110 10.81 4.7E−02 8.0E−03 3.1E−04 2.8E−04 2.8E−03 3.2E+01 1.3E−03 9.2E−04 SF271108 10.00 3.3E−02 2.3E−01 9.1E−03 7.6E−03 5.0E−02 3.1E+02 3.4E−02 2.0E−02 SF

296120 13.37 5.1E+15 1.1E−06 3.2E−06 5.4E−07 3.2E−06 3.2E−03 9.3E−06 3.5E−06 α1292118 12.71 5.3E+09 6.9E−06 1.9E−05 3.1E−06 2.3E−05 6.2E−02 5.9E−05 1.7E−05 α2288116 12.06 1.2E+05 4.9E−05 1.3E−04 2.0E−05 1.7E−04 7.8E−01 4.1E−04 8.9E−05 α3284114 11.39 4.7E+01 4.4E−04 1.1E−03 1.6E−04 1.7E−03 7.4E+00 3.8E−03 6.1E−04 α4280112 10.81 2.7E−01 2.8E−03 6.6E−03 1.0E−03 1.1E−02 4.7E+01 2.4E−02 3.2E−03 α5276110 10.57 1.9E−02 2.8E−03 6.3E−03 1.0E−03 1.1E−02 9.0E+01 2.0E−02 3.2E−03 SF272108 9.97 1.3E−02 2.5E−02 5.5E−02 9.1E−03 5.7E−02 5.0E+02 1.9E−01 2.4E−02 SF

297120 13.29 1.5E+15 1.8E−05 4.4E−06 7.4E−07 4.8E−06 4.0E−03 4.7E−06 4.8E−06 α1293118 12.61 1.5E+09 1.3E−04 2.9E−05 4.6E−06 3.6E−05 7.9E−02 2.9E−05 2.5E−05 α2289116 11.94 3.4E+04 1.0E−03 2.2E−04 3.4E−05 3.3E−04 1.1E+00 2.1E−04 1.5E−04 α3285114 11.23 1.3E+01 1.2E−02 2.4E−03 3.6E−04 4.1E−03 1.2E+01 2.2E−03 1.3E−03 α4281112 10.62 7.2E−02 1.0E−01 2.0E−02 2.9E−03 3.0E−02 9.9E+01 1.7E−02 8.9E−03 SF277110 10.34 4.8E−03 1.3E−01 2.4E−02 3.8E−03 3.3E−02 2.6E+02 1.9E−02 1.1E−02 SF

298120 13.20 3.1E+14 2.3E−06 6.2E−06 1.0E−06 6.9E−06 5.1E−03 2.0E−05 6.8E−06 α1294118 12.53 3.0E+08 1.7E−05 4.3E−05 6.6E−06 5.5E−05 1.0E−01 1.4E−04 3.6E−05 α2290116 11.83 6.4E+03 1.6E−04 3.7E−04 5.6E−05 5.5E−04 1.5E+00 1.3E−03 2.5E−04 α3286114 11.09 2.4E+00 2.3E−03 5.1E−03 7.3E−04 9.1E−03 1.9E+01 2.1E−02 2.6E−03 α4282112 10.44 1.3E−02 2.6E−02 5.6E−02 7.9E−03 6.9E−02 2.0E+02 2.5E−01 2.3E−02 SF278110 10.12 8.2E−04 4.2E−02 8.8E−02 1.3E−02 8.9E−02 7.2E+02 3.6E−01 3.7E−02 SF

299120 13.11 4.1E+13 4.0E−05 8.9E−06 1.4E−06 9.9E−06 6.5E−03 9.5E−06 9.6E−06 α1295118 12.44 3.9E+07 3.0E−04 6.3E−05 9.5E−06 8.0E−05 1.3E−01 6.3E−05 5.3E−05 α2291116 11.73 8.0E+02 3.1E−03 6.0E−04 8.8E−05 9.4E−04 2.0E+00 5.7E−04 4.0E−04 α3287114 10.97 2.9E−01 5.3E−02 9.9E−03 1.4E−03 1.7E−02 2.9E+01 8.9E−03 4.9E−03 SF283112 10.28 1.5E−03 8.0E−01 1.4E−01 2.0E−02 1.4E−01 3.9E+02 1.2E−01 5.6E−02 SF




(continued)

Parent nuclei

Q

(MeV)

TSF (s)


300120 13.02 3.7E+12 5.2E−06 1.3E−05 2.1E−06 1.6E−05 8.3E−03 4.5E−05 1.4E−05 α1296118 12.35 3.3E+06 4.0E−05 9.3E−05 1.4E−05 1.3E−04 1.7E−01 3.3E−04 7.7E−05 α2292116 11.64 6.6E+01 4.3E−04 9.5E−04 1.4E−04 1.5E−03 2.7E+00 3.7E−03 6.1E−04 α3288114 10.86 2.3E−02 8.7E−03 1.8E−02 2.5E−03 2.9E−02 4.4E+01 8.3E−02 8.7E−03 SF284112 10.13 1.2E−04 1.7E−01 3.4E−01 4.5E−02 2.6E−01 7.3E+02 1.8E+00 1.2E−01 SF

301120 12.93 2.1E+11 9.6E−05 2.0E−05 3.0E−06 2.4E−05 1.1E−02 2.1E−05 2.1E−05 α1297118 12.26 1.9E+05 7.3E−04 1.4E−04 2.1E−05 2.0E−04 2.2E−01 1.4E−04 1.2E−04 α2293116 11.55 3.7E+00 8.2E−03 1.5E−03 2.1E−04 2.4E−03 3.6E+00 1.4E−03 9.4E−04 α3289114 10.75 1.2E−03 1.9E−01 3.2E−02 4.3E−03 4.7E−02 6.5E+01 2.9E−02 1.5E−02 SF285112 10.00 6.1E−06 4.5E+00 7.4E−01 9.7E−02 4.3E−01 1.3E+03 6.2E−01 2.6E−01 SF

302120 12.82 8.3E+09 1.4E−05 3.1E−05 4.7E−06 3.9E−05 1.4E−02 1.2E−04 3.2E−05 α1298118 12.16 7.2E+03 1.0E−04 2.2E−04 3.1E−05 3.1E−04 2.8E−01 8.6E−04 1.8E−04 α2294116 11.45 1.3E−01 1.2E−03 2.4E−03 3.3E−04 4.0E−03 4.9E+00 1.0E−02 1.5E−03 α3290114 10.66 4.4E−05 2.8E−02 5.5E−02 7.2E−03 7.2E−02 9.7E+01 2.8E−01 2.5E−02 SF286112 9.89 2.1E−07 8.2E−01 1.5E+00 2.0E−01 6.9E−01 2.3E+03 9.1E+00 5.1E−01 SF

303120 12.71 2.2E+08 2.7E−04 5.1E−05 7.4E−06 6.7E−05 1.9E−02 5.5E−05 5.1E−05 α1299118 12.06 1.8E+02 2.0E−03 3.5E−04 5.0E−05 5.3E−04 3.7E−01 3.6E−04 2.8E−04 α2295116 11.36 3.2E−03 2.3E−02 3.8E−03 5.2E−04 6.3E−03 6.6E+00 3.7E−03 2.3E−03 SF291114 10.56 1.0E−06 5.9E−01 9.3E−02 1.2E−02 1.1E−01 1.4E+02 8.4E−02 4.1E−02 SF

304120 12.59 3.7E+06 4.1E−05 8.7E−05 1.2E−05 1.1E−04 2.5E−02 3.5E−04 8.3E−05 α1300118 11.95 3.0E+00 3.0E−04 5.9E−04 8.1E−05 8.7E−04 5.0E−01 2.6E−03 4.6E−04 α2296116 11.26 5.2E−05 3.4E−03 6.4E−03 8.5E−04 1.1E−02 9.1E+00 3.0E−02 3.8E−03 SF292114 10.47 1.6E−08 9.1E−02 1.6E−01 2.0E−02 1.6E−01 2.1E+02 9.0E−01 7.0E−02 SF

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372 Int. J. Nuclear Energy Science and Technology, Vol. 13, No. 4, 2019


Empirical formula for bremsstrahlung cross-section in actinides

H.C. Manjunatha* and L. Seenappa

Department of Physics,

Government College for Women,

Kolar-563101 Karnataka, India

Email: [email protected]


*Corresponding author

K.N. Sridhar


Government First Grade College,

Kolar-563101 Karnataka, India


Abstract: We have formulated the empirical formula for bremsstrahlung cross section in actinides. This formula produces bremsstrahlung cross sections for electrons with energies from 1 keV to 10 GeV incident on atoms with atomic numbers Z = 89 to 103. This formula produces bremsstrahlung cross sections with simple inputs of energy of incident electron (T), energy of emitted bremsstrahlung photon and atomic number of the target.

Keywords: bremsstrahlung; electron.

Reference to this paper should be made as follows: Manjunatha, H.C., Seenappa, L. and Sridhar, K.N. (2019) ‘Empirical formula for bremsstrahlung cross-section in actinides’, Int. J. Nuclear Energy Science and Technology, Vol. 13, No. 4, pp.372–380.

Biographical notes: H.C. Manjunatha is an Assistant Professor of Physics at Government College for Women, Kolar, India. His areas of research are radiation, nuclear and medical physics. He has published more than 120 research papers in reputable international journals.

L. Seenappa is an Assistant Professor of Physics at Government College for Women, Kolar, India. His areas of research are radiation and medical physics. He has published many research papers in reputable international journals.

K.N. Sridhar is an Assistant Professor of Physics at Government First Grade College, Kolar, India. His areas of research are radiation and nuclear physics. He has published many research papers in reputable international journals.

Empirical formula for bremsstrahlung cross-section in actinides 373

1 Introduction

Literature survey shows that there were earlier works on the calculations of

bremsstrahlung cross sections such as Sommerfeld’s non-relativistic calculations, Sauter,

Bethe and Heitler and Racah’s relativistic calculations and Bethe–Heitler’s Born

approximations (Mangiarotti and Martins, 2017). Manjunatha and Rudraswamy (2017)

proposed a numerical method to evaluate bremsstrahlung cross-section in compounds

such as NaI, SiLi and GeLi using tabulated data given for elements. Haug (2008)

calculated bremsstrahlung cross-section with screening and Coulomb corrections at high

energies using cross-sections of Sommerfeld–Maue functions with additional higher-

order terms. Tessier and Kawrakow (2008) reported a numerical calculation of the

electron–electron bremsstrahlung cross-section in the field of atomic electrons. Omar

et al. (2018) evaluated bremsstrahlung cross sections using Monte Carlo method and

compared that with the different theories available in the literature. Manjunatha and

Rudraswamy (2007) estimated the theoretical data of bremsstrahlung radiation cross-

section of bone. Poškus (2018) proposed a program for calculating spectra and angular

distributions of bremsstrahlung at electron energies less than 3 MeV for exact screened

calculations of atomic-field bremsstrahlung.

Singh et al. (2017) measured the bremsstrahlung spectrum generated in thick targets

of oxides of lanthanides by 89Sr beta particles in the photon energy region 1–100 keV.

Sandrock et al. (2018) calculated radioactive corrections to the average bremsstrahlung

energy loss of high-energy muons using a modified Weizsäcker–Williams method. Singh

et al. (2018) measured angular distributions of bremsstrahlung photons produced by

10–25 keV electrons incident on thick Ti and Cu targets using a Si-PIN photodiode

detector. Jung (2014) studied the electron-exchange and quantum shielding effects on the

polarisation bremsstrahlung spectrum due to the electron-shielding sphere encounters in

quantum plasmas. Prajapati et al. (2018) studied the anisotropy of bremsstrahlung

photons emitted from 4.0 keV electrons in scattering by a free CH4 molecule by

performing the measurements of cross-sections of bremsstrahlung photons. Kunashenko

(2013) developed the theory of coherent bremsstrahlung from neutrons in crystals in the

framework of virtual photons.

Minter and Jenkins (1990) proposed a program which calculates the double differential

cross section for bremsstrahlung production by electrons interacting with a spinless point

target. Al-Beteri and Raeside (1989) calculated an electron bremsstrahlung cross-section

using Monte Carlo transport simulation. Manjunatha (2014) evaluated the beta-induced

bremsstrahlung spectra and other dosimetric parameters in bone. Kunashenko (2015)

studied the combined effect in coherent bremsstrahlung of channelled electrons. Mack

and Mitter (1973) calculated differential cross-section for bremsstrahlung emission in

electron-electron collisions using computer programs for formula manipulations.

Manjunatha (2013) computed the beta-induced bremsstrahlung spectra produced by high-

energy beta particles (>1 MeV) in bone. Manjunatha and Rudraswamy (2013) measured

bremsstrahlung spectrum produced by the beta particles from 90Sr to 90Y, 147Pm and 204Tl

in nuclear radiation detection compounds such as caesium iodide and sodium iodide.

From the literature survey, it is found that there is no simple relation to calculate the

bremsstrahlung cross section. Also, bremsstrahlung production is important in the

actinide region of elements. Hence in the present work, we have constructed the new

empirical formula for bremsstrahlung cross-section in actinides.

374 H.C. Manjunatha, L. Seenappa and K.N. Sridhar

2 Theory

We have studied the variation of bremsstrahlung cross section with incident electron

energy (T) and emitted photon energy (k). It has been observed from the tabulations of

the previous workers (Seltzer and Berger, 1986) that the mass of bremsstrahlung cross

section does not vary linearly with incident electron energy (T) and emitted photon

energy (k). Because of nonlinear variations of bremsstrahlung cross-sections, we have

performed the nonlinear regressions/nonlinear least squares fittings. We have tried more

than 100 functions such as gaussians, sigmoidals, rationals, sinusoidals, etc. that fit a

given set of data points. We have tried suitable functions such as ,42αα

1 3α E α E

32 41 2 3

,E E E

2

1 2

1,

3E E

2

4

1 3

5

,E

E

2

1 3,E

3

1 2 4exp ,E

2

1 2 3 4α exp α lnE α α ,

3

14

2

2

,

1E

21 3exp ln ,E

E

1 2exp E

3 4exp ,E 1 2 3 4expE E , 31 2 4

ββ E β β lnEE

, 1 2 3 4δ E δ exp δ E δ

and polynomial function 4 3 2

1 2 3 4 5E E E E . Using the procedure of linear

and non-linear regressions, we have constructed the formula for bremsstrahlung cross

sections of actinides.

2.1 Empirical formula for bremsstrahlung cross-section of actinides in the low energy region

The constructed formula for bremsstrahlung cross-section of actinides in the incident

electron energy region from 1 keV to 20 keV is as follows:

2

2

22

2

2

1 1 10.511

n n ni i i

i i i

i o i o i o

n n ni i i

i i i

i o i o i o

n n ni i i

i i i

i o i o i o

kZ T Z T Z

T

d Z kZ T Z T Z

dk TTk

Z T Z T Z

(1)

Here /d dk total bremsstrahlung cross-section in mb/MeV. Z is the atomic number of

the target. T and k are energies of the incident electron and emitted photon. α, β, Φ, δ, η, θ, Δ, χ and Ω are fitting parameters and these are given in Table 1.


Table 1 Fitting parameters for bremsstrahlung cross-section formula in the low-energy region (1–10 keV)

i=3 i=2 i=1 i=0

α –38.27707225 1.073892000E+04 –1.004941338E+06 3.136872184E+07

β –8.000219055E-02 19.82470608 –1.602500610E+03 4.184574414E+04

φ –6.704485149E-04 0.1886013616 –17.68659538 5.530536079E+02

δ 48.27882266 –1.346737183E+04 1.252906297E+06 –3.889753100E+07

η 0.1203986146 –31.42383766 2.713063232E+03 –7.701312109E+04

θ 5.151270616E-04 –0.1450826274 13.61417139 –4.254458790E+02

Δ –15.45719290 4.282464905E+03 –3.953166172E+05 1.214534738E+07

χ 4.180468619E-03 –1.353872299 1.396878662E+02 –4.202003906E+03

Ω –1.236739301E-04 3.448326769E-02 –3.206330664 99.43986988

2.2 Empirical formula for bremsstrahlung cross-section of actinides in the medium and high-energy region

The constructed formula for bremsstrahlung cross section of actinides in the incident

electron energy region 20 keV–10 GeV is as follows:

3

2

2

2

2

2

21 1 10.511

n n ni i i

i i i

i o i o i o

n n ni i i

i i i

i o i o i o

n n ni i i

i i i

i o i o i o

kZ T Z T Z

T

kZ T Z T Z

Td Z

dk Tk Z T Z T Z

2n n n

i i i

i i i

i o i o i o

k

T

Z T Z T Z

(2)

α, β, Φ, δ, η, θ, Δ, χ, Ω, Ψ, ρ and τ are fitting parameters. The fitting parameters for

energy region 20 keV–2 MeV are given in the Table 2. The fitting parameters for energy

region 3 MeV–40 MeV are given in the Table 3. The fitting parameters for energy region

50 MeV–10 GeV are given in the Table 4.

Table 2 Fitting parameters for bremsstrahlung cross-section formula in the energy region (20 keV–2 MeV)

i=2 i=1 i=0

α –8.389672615×10–3 1.577550071 –67.4416445

β 2.687623004×10–2 –5.042842638 212.4298057

φ –2.022796124×10–2 3.774238761 –151.1318791

δ 1.315138999×10–3 –0.2547085708 11.71463826

η 8.192031457×10–3 –1.487102785 52.78763464

θ –2.793864593×10–2 5.070271558 –178.0250456


Table 2 Fitting parameters for bremsstrahlung cross-section formula in the energy region (20 keV–2 MeV) (continued)

i=2 i=1 i=0

Δ 2.112064696×10–2 –3.794624337 117.0865392

χ –1.643137995×10–3 0.3310715245 –15.90256524

Ω –8.466408974×10–5 0.04695247729 4.749503303

Ψ 9.20783717×10–4 –0.2801477572 –9.865315599

ρ –1.216059224×10–3 0.3242415865 5.652432231

τ 1.295587723×10–4 –0.05318550403 9.26255849

Table 3 Fitting parameters for bremsstrahlung cross-section formula in the low-energy region (3 MeV–40 MeV)

i=2 i=1 i=0

α –8.421094829×10–8 1.646466647×10–5 –1.095703381×10–3

β 2.548829005×10–5 –4.834815635×10–3 0.2550092021

φ –9.166403809×10–4 0.175198528 –9.152784317

δ 1.515040153×10–3 –0.3120318698 23.52034341

η –2.175318584×10–7 4.023367755×10–5 –1.542215354×10–3

θ –1.866361082×10–5 3.522501189×10–3 –0.1944975754

Δ 1.068648754×10–3 –0.2029888979 10.53530948

χ –3.256341962×10–3 0.6804117758 –51.95781901

Ω –5.687060291×10–9 –1.386293227×10–6 1.16074304×10–4

Ѱ 3.600464826×10–6 –4.903407257×10–4 1.962273791×10–2

ρ –1.703440475×10–4 2.81452991×10–2 –1.286400949

τ 1.096265591×10–3 –0.2253302141 22.84002807

Table 4 Fitting parameters for bremsstrahlung cross-section formula in the low-energy region (50 MeV–10 GeV)

i=2 i=1 i=0

α –5.994727129×10–13 1.116198646×10–10 –5.132114343×10–9

β 1.005263986×10–8 –1.870639732×10–6 8.595539594×10–5

φ –4.539871835×10–5 8.442188101×10–3 –0.3875010602

δ –4.292568451×10–2 8.54144394 –422.5360436

η 4.696577657×10–13 –8.767725362×10–11 4.066237614×10–9

θ –7.873950963×10–9 1.468813528×10–6 –6.805321392×10–5

Δ 3.559461072×10–5 –6.63384251×10–3 0.3068728318

χ 3.356801335×10–2 –6.65256432 321.7391008

Ω –4.802100951×10–14 8.990756482×10–12 –4.188722009×10–10

Ѱ 8.075762118×10–10 –1.51047811×10–7 7.028782832×10–6

ρ –3.674076445×10–6 6.862915901×10–4 –3.18752827×10–2

τ –3.413039019×10–3 0.6447866737 –20.35799717


Figure 1 Variation of total scaled bremsstrahlung energy weighted cross-section (TSBEW) with energy (MeV) for k/T=0

10-2

100

102

104

0

4

8

12

10-2

100

102

104

0

4

8

12

10-2

100

102

104

0

4

8

12

10-2

100

102

104

0

4

8

12

10-2

100

102

104

0

4

8

12

10-2

100

102

104

0

4

8

12

10-2

100

102

104

0

4

8

1295

10-2

100

102

104

0

4

8

1296

10-2

100

102

104

0

4

8

1297

10-2

100

102

104

0

4

8

1298

10-2

100

102

104

0

4

8

12 99

10-2

100

102

104

0

4

8

12

Data[21]

Formula

FermiumEinsteinium

Berkelium

Californium

Amerecium Curium

PlutoniumNeptuniumUranium

ProtactiniumThoriumActinium

100

TS

BE

W (

mb

pe

r a

tom

)

T (MeV)

Z=89

k/T=0

90 91

92 93 94

Figure 2 Variation of total scaled bremsstrahlung energy weighted cross-section (TSBEW) with energy (MeV) for k/T=0.3

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

Data[21]

Formula

95

10-2

100

102

104

0

4

896

10-2

100

102

104

0

4

8 97

10-2

100

102

104

0

4

898

10-2

100

102

104

0

4

8 99

10-2

100

102

104

0

4

8

TS

BE

W (

mb

pe

r a

tom

)

FermiumEinsteinium

Berkelium

Californium

Amerecium Curium



100

T (MeV)

Z=89

k/T=0.3

90 91

92 93 94


Figure 3 Variation of total scaled bremsstrahlung energy weighted cross-section (TSBEW) with energy (MeV) for k/T=0.6

10-2

100

102

104

0

4

10-2

100

102

104

0

4

10-2

100

102

104

0

4

10-2

100

102

104

0

4

10-2

100

102

104

0

4

10-2

100

102

104

0

4

10-2

100

102

104

0

4

Data[21]

Formula

95

10-2

100

102

104

0

496

10-2

100

102

104

0

4

97

10-2

100

102

104

0

498

10-2

100

102

104

0

4

99

10-2

100

102

104

0

4

TS

BE

W (

mb

pe

r a

tom

)

FermiumEinsteinium

Berkelium

Californium

Amerecium Curium



100

T (MeV)

Z=89

k/T=0.6

90 91

92 93 94

Figure 4 Variation of total scaled bremsstrahlung energy weighted cross-section (TSBEW) with energy (MeV) for k/T=1

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

10-2

100

102

104

0

4

8

Data[21]

Formula

95

10-2

100

102

104

0

4

896

10-2

100

102

104

0

4

897

10-2

100

102

104

0

4

898

10-2

100

102

104

0

4

899

10-2

100

102

104

0

4

8

FermiumEinsteinium

Berkelium

Californium

Amerecium Curium



100

TS

BE

W (

mb

pe

r a

tom

)

T (MeV)

Z=89k/T=1

90 91

92 93 94


3 Results and discussion

The present formula produces bremsstrahlung cross section in mb/MeV. This formula

produces bremsstrahlung cross sections for electrons with energies from 1 keV to 10

GeV incident on atoms with atomic numbers Z = 89 to 103. In addition to the

bremsstrahlung in the Coulomb field of the atomic nucleus, bremsstrahlungs in the field

of the atomic electrons are also included in the present formula. To validate the present

formula, we have calculated the Total Scaled Bremsstrahlung Energy Weighted

(TSBEW) cross-section using present formula and then it is compared with the data

available in the literature (Seltzer and Berger, 1986).

2

2

1 1 10.511

Tk

dTSBEW

Z dk

(3)

The comparison of the values produced by the present formula with that of the data

available in the literature (Seltzer and Berger, 1986) is as shown in Figures 1–4. From

these figures, it is found that presented formula agrees well with the data available in the

literature. To judge the present formula we have also calculated the percentage of

difference of bremsstrahlung cross sections using the following relation:

var

% 100

formula exact

exact

d d

d dk dk

dk d

dk

(4)

Here formulad dk is bremsstrahlung cross-section values produced by the present

formula and exactd dk is standard data available in the literature. The calculated

percentage of difference of bremsstrahlung cross-sections is less than 2.5%.

4 Conclusion

We have formulated the empirical formula for bremsstrahlung cross section in actinides.

This formula produces bremsstrahlung cross-sections for electrons with energies from

1 keV to 10 GeV incident on atoms with atomic numbers Z = 89 to 103 with simple

inputs of atomic number of the target (Z) and energies of the incident electron (T) and

emitted photon (k).


References

Al-Beteri, A.A. and Raeside, D.E. (1989) ‘An improved electron bremsstrahlung cross-section formula for Monte Carlo transport simulation’, Nuclear Instruments and Methods in Physics Research Section B, Vol. 44, No. 2, pp.149–157.

Haug, E. (2008) ‘Bremsstrahlung cross-section with screening and coulomb corrections at high energies’, Radiation Physics and Chemistry, Vol. 77, No. 3, pp.207–214.

Jung, Y. (2014) ‘Polarization bremsstrahlung process in quantum plasmas including electron-exchange and shielding effects’, Physics Letters A, Vol. 378, Nos. 30/31, pp.2176–2180.

Kunashenko, Y.P. (2013) ‘Coherent bremsstrahlung from neutrons’, Nuclear Instruments and Methods in Physics Research Section B, Vol. 309, pp.88–91.

Kunashenko, Y.P. (2015) ‘Combined effect in coherent bremsstrahlung of channeled electrons’, Nuclear Instruments and Methods in Physics Research Section B, Vol. 355, pp.110–113.

Mack, D. and Mitter, H. (1973) ‘Calculation of electron-electron-bremsstrahlung cross-sections’, Physics Letters A, Vol. 44, No. 1, pp.71–72.

Mangiarotti, A. and Martins, M.N. (2017) ‘A review of electron-nucleus bremsstrahlung cross sections between 1 and 10 MeV’, Radiation Physics and Chemistry, Vol. 141, pp.312–338.

Manjunatha, H.C. (2013) ‘Bremsstrahlung dose induced by high energy beta particles (>1 MeV) in bone’, Annals of Nuclear Energy, Vol. 59, pp.53–62.

Manjunatha, H.C. (2014) ‘A dosimetric study of Beta induced bremsstrahlung in bone’, Applied Radiation and Isotopes, Vol. 94, pp.282–293.

Manjunatha, H.C. and Rudraswamy, B. (2007) ‘Theoretical data of external bremsstrahlung radiation cross-section of bone’, Applied Radiation and Isotopes, Vol. 65, No. 4, pp.397–400.

Manjunatha, H.C. and Rudraswamy, B. (2013) ‘External bremsstrahlung of 90Sr–90Y, 147Pm and 204Tl in detector compounds’, Radiation Physics and Chemistry, Vol. 85, pp.95–101.

Manjunatha, H.C. and Rudraswamy, B. (2017) ‘Numerical method for external Bremsstrahlung cross sections in compounds’, Radiation Measurements, Vol. 42, No. 2, pp.251–255.

Minter, A. and Jenkins, D. (1990) ‘Bremsstrahlung cross section for a point, spin less target’, Computer Physics Communications, Vol. 59, No. 3, pp.499–505.

Omar, A., Andreo, P. and Poludniowski, G. (2018) ‘Performance of different theories for the angular distribution of bremsstrahlung produced by keV electrons incident upon a target’, Radiation Physics and Chemistry, Vol. 148, pp.73–85.

Poškus, A. (2018) ‘BREMS: a program for calculating spectra and angular distributions of bremsstrahlung at electron energies less than 3 MeV’, Computer Physics Communications, Vol. 232, pp.237–255.

Prajapati, S., Singh, B., Singh, B.K. and Shanker, R. (2018) ‘Study of anisotropy of bremsstrahlung radiation emitted from 4.0 keV electrons in scattering by CH4 molecule’, Radiation Physics and Chemistry, Vol. 153, pp.92–97.

Sandrock, A., Kelner, S.R. and Rhode, W. (2018) ‘Radiative corrections to the average bremsstrahlung energy loss of high-energy muons’, Physics Letters B, Vol. 776, pp.350–354.

Seltzer, S.M. and Berger, M.J. (1986) ‘Bremsstrahlung energy spectra from electrons with kinetic energy 1 keV–10 GeV incident on screened nuclei and orbital electrons of neutral atoms with Z = 1–100’, Atomic Data and Nuclear Data Tables, Vol. 35, 3, pp.345–418.

Singh, A., Singh, T. and Dhaliwal, A.S. (2017) ‘Studies of total bremsstrahlung spectra in the oxides of lanthanides’, Radiation Physics and Chemistry, Vol. 141, pp.207–212.

Singh, B. et al. (2018) ‘Measurement of the angular distribution of thick target bremsstrahlung produced by 10–25 keV electrons incident on Ti and Cu targets’, Radiation Physics and Chemistry, Vol. 150, pp.82–89.

Tessier, F. and Kawrakow, I. (2008) ‘Calculation of the electron-electron bremsstrahlung cross-section in the field of atomic electrons’, Nuclear Instruments and Methods in Physics Research Section B, Vol. 266, No. 4, pp.625–634.

Int. J. Nuclear Energy Science and Technology, Vol. 13, No. 4, 2019 325


Selection of shielding materials for gamma/X-ray and neutron radiations among the commonly used polymers

N. Nagaraja



Kolar-563101, Karnataka, India

and

Research and Development Centre,

Bharathiar University,

Coimbatore-641046, Tamil Nadu, India


H.C. Manjunatha* and L. Seenappa


Government College for Women,




*Corresponding author

K.N. Sridhar





H.B. Ramalingam


Government Arts College,

Udumalpet-642126, Tamil Nadu, India


Abstract: We have studied the X-ray and gamma shielding parameters like mass attenuation coefficient, mean free path, half value layer, tenth value layer, effective atomic numbers, electron density, exposure buildup factors, and specific gamma ray constant in commonly used polymers such as polysterene, polypropylene, polytetrafluroethylene (PTFE), polyvinylchloride (PVC),

326 N. Nagaraja et al.

polychlorotetrafluroethylene (PCTFE). We have also measured X-ray and gamma shielding parameters at different energies such as 170Tm (0.084 MeV), 137Cs (0.662 MeV) and 60Co (1.170, 1.330 MeV) in the same polymers. The measured values agree well with the theoretical values. Neutron shielding parameters such as coherent neutron scattering length, incoherent neutron scattering length, coherent neutron scattering cross section, incoherent neutron scattering cross section, total neutron scattering cross section and neutron absorption cross section are in the same polymers. PCTFE is found to be good shielding material for the gamma/X-ray and neutron radiation.

Keywords: attenuation; shielding; gamma.

Reference to this paper should be made as follows: Nagaraja, N., Manjunatha, H.C., Seenappa, L., Sridhar, K.N. and Ramalingam, H.B. (2019) ‘Selection of shielding materials for gamma/X-ray and neutron radiations among the commonly used polymers’, Int. J. Nuclear Energy Science and Technology, Vol. 13, No. 4, pp.325–339.

Biographical notes: N. Nagaraja is an Assistant Professor of Physics at Government First Grade College, Kolar, India. His areas of research are radiation and nuclear physics. He has published many research papers in reputable international journals.

H.C. Manjunatha is an Assistant Professor of Physics at Government College for Women, Kolar, India. His areas of research are radiation, nuclear and medical physics. He is a recipient of “Dr. Tarun Datta Memorial Award-2018” for outstanding contributions in the field of nuclear and radiochemistry by Indian Association of Nuclear Chemists and Allied Scientists (IANCAS) and also recipient of “Nucleonix Best Research Paper Award” by Indian Society for Radiation Physics. He has published more than 110 research papers in the reputed international journals.

L. Seenappa is an Assistant Professor of Physics at Government College for Women, Kolar, India. His areas of research are radiation and medical physics. He has published many research papers in reputable international journals.

K.N. Sridhar is an Assistant Professor of Physics at Government First Grade College, Kolar, India. His areas of research are radiation and nuclear physics. He has published many research papers in reputable international journals.

H.B. Ramalingam is an Associate Professor of Physics at Government Arts College, Udumalpet, India. His areas of research are radiation and nuclear physics. He has published many research papers in reputable international journals.

I Introduction

Study of attenuation parameters of X-rays, gamma rays and neutrons in composite

materials such as polymers, alloys and different compounds are important in selecting the

new shielding materials. There is a need to develop new shielding materials which are

Selection of shielding materials for gamma/X-ray and neutron radiations 327

low cost and non-toxic. Earlier studies (Vahabi et al., 2017; Mirji and Lobo, 2017; Singh

et al., 2015) measured the mass attenuation coefficient of some polymers and compared

with the theoretical values generated by MCNP4C model. Kaur et al. (2017) suggested

the maximum thickness required to measure the mass attenuation coefficients of Sn-Pb

alloy. Chaiphaksa et al. (2016) measured the mass attenuation coefficients, effective

atomic number, and effective electron density of scintillator materials at different

energies. Limkitijaroenporn et al. (2013) measured the attenuation properties in some

super alloys. Un and Caner (2014) developed Zeff software to calculate effective atomic

number. Tekin et al. (2017) studied mass attenuation coefficients of concrete

using MCNPX simulation. Sharanabasappa et al. (2010) proposed a new method to

measure the X-ray mass attenuation using HPGe detector. İçelli et al. (2011) studied the

mass attenuation coefficients and the effective atomic numbers in colemanite and

Emetcolemanite clay. Sayyed et al. (2018) studied the exposure buildup factors of

different types of polymers using GP fitting method. Sathyaraj et al. (2017) calculated

effective atomic number and buildup factor for nanoparticles doped in polymer gel.

Singh and Badiger (2014) studied the exposure buildup factor as a function of penetration

depth and incident photon energy. Oto et al. (2015) studied the gamma ray buildup

factors of concrete mixture containing limonite ore. Kumar et al. (2017) studied energy

absorption buildup factors of materials containing amines. Levet and Ozdemir (2017)

measured the mass attenuation coefficient using energy dispersive X-ray fluorescence

spectroscopy in some compounds. Other publications give further specific information on

X-ray and gamma interaction parameters in some alloys (Seenappa et al., 2018;

Manjunatha et al., 2018). Interaction of the gamma rays also influences the conductivity

property of the material (Manjunatha, 2015). Previous researchers also measured the

attenuation properties in the polymethacrylate and kapton (Manjunatha, 2017). Earlier

investigators also measured attenuation properties in lead and barium compounds

(Rudraswamy et al., 2010; Manjunatha et al., 2017a). Previous studies formulated new

empirical formula for mass attenuation coefficient and energy absorption coefficient

(Manjunatha et al., 2017b).

In the present paper, we have studied the X-ray and gamma shielding parameters like

mass attenuation coefficient, mean free path, half value layer (HVL), tenth value layer

(TVL), effective atomic numbers, electron density, exposure buildup factors, and specific

gamma ray constant in commonly used polymers such as polysterene, polypropylene,

PTFE, PVC and PCTFE. We also measured X-ray and gamma shielding parameters

at different energies such as 170Tm (0.084 MeV), 137Cs (0.662 MeV) and 60Co

(1.170, 1.330 MeV) of the same polymers, and neutron shielding parameters such as

coherent neutron scattering length, incoherent neutron scattering length, coherent neutron

scattering cross section, incoherent neutron scattering cross section, total neutron

scattering cross section and neutron absorption cross section of the same polymers

2 Theory

2.1 Gamma/X-ray interaction parameters

In this work, the Mass Attenuation Coefficients (MACs) are generated using WinXCom

(Gerward et al., 2004). The derivatives of mass attenuation coefficients such as linear

attenuation coefficient (μ), Half Value Layer (HVL), Tenth Value Layer (TVL), mean


free path (λ) and specific gamma ray constants (Γ) are evaluated using the procedure

explained in our previous studies (Seenappa et al., 2018; Manjunatha et al., 2018).

Effective atomic number is computed by using the following equation:

aeff

e

Z

, here

1i i

im ca

i i

i i

E n AN

n n

,

1 i ie

i i i

f A

N Z

where a and e are atomic and electronic cross sections. Here, ni is the number of

atoms of the i-th element in a given molecule, c

is the mass attenuation coefficient

for the compound, N is the Avogadro’s number and Ai is the atomic weight of the

element i. fi is the fractional abundance (a mass fraction of the i-th element in the

molecule) and Zi is the atomic number of the i-th element in the molecule. The effective

electron density (Ne) is computed using the following equation:

1

e eff i

ii i

i

NN g Z n

n A

(1)

2.2 Secondary radiation from interactions of gamma/X-rays with a medium

When gamma/X-rays interact with the medium, it degrades their energy and produces

secondary radiations by scattering and absorption. During this process energy is

deposited in the medium. This can be studied by calculating the buildup factor. In the

present work, we have calculated energy absorption buildup factors enB using the

Geometric Progression (GP) fitting method (Seenappa et al., 2018; Manjunatha et al.,

2018; Manjunatha, 2015).

2.3 Neutron shielding parameters

Neutrons undergo scattering and absorption while penetrating the medium. It undergoes

coherent, incoherent or scattering and absorption process. The study of scattering and

absorption of neutrons in the medium is required for the selection of good shielding

material. We have calculated neutron shielding parameters (NSP)polymers, such as the

coherent neutron scattering length (bcoh), incoherent neutron scattering length (binc),

coherent neutron scattering cross-section (σcoh), incoherent neutron scattering cross-

section (σinc), total neutron scattering cross-section (σtot) and neutron absorption cross-

section (σabs) in the studied polymers using the mixture rule:

NSP NSPicompound if (2)

Here (NSP)i is neutron shielding parameter of the i-th element (Sears, 1992) in the

compound and fi is the fractional abundance (a mass fraction of the i-th element in the

molecule).


2.4 Experiment

We have conducted transmission experiments with the narrow beam geometry to

measure the incident and transmitted intensities. The details of experimental arrangement

are explained in the previous report (Manjunatha et al., 2017a). We have used NaI(Tl)

crystal detector mounted on a photomultiplier tube housed in a lead chamber and a

sophisticated PC based MCA for the detection purpose. The gamma sources such as 170Tm (0.084 MeV), 137Cs (0.662 MeV) and 60Co (1.170, 1.330 MeV) are used. Polymers

such as polysterene, polypropylene, PTFE, PVC and PCTFE are used as target samples.

The sample was directly attached to the opening of the lead shield where source is

placed. The integral intensities, I0 and I of the beam before and after passing through the

sample are measured for sufficient time. c

of the sample is then estimated using

the relation:

01ln

c

I

t I

(3)

Experimental values of elN and effZ of polymers were obtained from c

using the

procedure explained in our previous work (Seenappa et al., 2018; Manjunatha et al.,

2018; Manjunatha, 2015).

3 Results and discussions

Computed mass attenuation coefficient is plotted as a function of energy and shown in

Figure 1. Comparison of half value layer (HVL), tenth value layer (TVL) and mean free

path (λ) among the studied polymers are as shown in Figures 2–4. From these figures it is

found that half value layer, tenth value layer and mean free path are small for

polychlorotetrafluroethylene (PCTFE) compared to the other polymers. Comparisons of

specific gamma ray constant (Γ) among studied polymers are shown in Figure 5. From

the figure it is found that specific gamma ray constant is large for PCTFE compared to

the other polymers. Variation of effective atomic number (Zeff) and effective electron

density with the energy for studied polymers is shown in Figure 6. First and second

layers of Figure 6 show the variation of effective atomic number (Zeff) and effective

electron density with the energy. Third and fourth layers of the Figure 6 show

comparison of effective atomic number (Zeff) among studied polymers at 1 MeV and

1000 MeV. From this figure, it is found that the value of effective atomic number is large

for PCTFE compared to the other polymers. Fifth and sixth layers of the Figure 6 show

the comparison of effective electron density among studied polymers at 1 MeV and

1000 MeV. From this figure, it is found that the value of effective electron density is

small for PCTFE compared to the other polymers. Figure 7 shows the variation of

buildup factor with energy (MeV) for different mean free paths. Energy absorption

buildup factor increases with increasing energy up to Epe and then decreases. Epe is the

energy at which the photoelectric interaction coefficients get equal to the Compton

interaction coefficients for a given value of the effective atomic number (Zeff).


Figure 1 Variation of mass attenuation coefficient with energy (MeV)

10-1

102

105

10-1

101

103

10-1

102

105

10-1

101

103

Polypropylene

10-1

102

105

10-1

101

103

PTFE

10-1

102

105

10-2

100

102

Cl X-ray

PVC

cm

2/g

Polysterene

E (MeV)

10-1

102

105

10-1

101

103

Cl X-ray

PCTFE

Figure 2 Comparison of half value layer (HVL) with polymers (P1-Polysterene, P2-Polypropylene, P3-PTFE, P4-PVC and P5-PCTFE) @ different energies (MeV)

P1 P2 P3 P4 P5

1x10-4

2x10-4

3x10-4

4x10-4

P1 P2 P3 P4 P5

0.0

0.1

0.2

0.3

0.4

P1 P2 P3 P4 P5

2

3

4

P1 P2 P3 P4 P54

6

8

10

P1 P2 P3 P4 P5

18

24

30

36

P1 P2 P3 P4 P5

20

30

40

50

60

P1 P2 P3 P4 P5

20

30

40

50

P1 P2 P3 P4 P510

20

30

40

P1 P2 P3 P4 P510

20

30

40

E=10-3 MeV 10

-2MeV 10

-1MeV

102MeV10 MeV1MeV

103 MeV 10

4 MeV 10

5 MeV

HV

L


Figure 3 Comparison of tenth value layer (TVL) with polymers (P1-Polysterene, P2-Polypropylene, P3-PTFE, P4-PVC and P5-PCTFE) @ different energies (MeV)

P1 P2 P3 P4 P5

5x10-4

1x10-3

2x10-3

P1 P2 P3 P4 P50.0

0.5

1.0

P1 P2 P3 P4 P56

9

12

15

P1 P2 P3 P4 P5

21

28

35

P1 P2 P3 P4 P5

60

90

120

P1 P2 P3 P4 P5

50

100

150

200

P1 P2 P3 P4 P5

50

100

150

P1 P2 P3 P4 P5

50

100

150

P1 P2 P3 P4 P5

50

100

150

E=10-3 MeV

10-2MeV 10

-1MeV

102MeV10 MeV1MeV

103 MeV 10

4 MeV 10

5 MeV

TV

L

Figure 4 Comparison of mean free path (λ) with polymers (P1-Polysterene, P2-Polypropylene, P3-PTFE, P4-PVC and P5-PCTFE) @ different energies (MeV)

P1 P2 P3 P4 P5

2x10-4

4x10-4

6x10-4

P1 P2 P3 P4 P5

0.0

0.3

0.6

P1 P2 P3 P4 P52

4

6

P1 P2 P3 P4 P5

8

12

16

P1 P2 P3 P4 P5

20

40

60

P1 P2 P3 P4 P5

30

60

90

P1 P2 P3 P4 P5

25

50

75

P1 P2 P3 P4 P5

25

50

75

P1 P2 P3 P4 P5

20

40

60

E=10-3 MeV

10-2MeV 10

-1MeV

102MeV10 MeV1MeV

103 MeV 10

4 MeV 10

5 MeV


Figure 5 Comparison of specific gamma ray constant (Γ) with polymers (P1-Polysterene, P2-Polypropylene, P3-PTFE, P4-PVC and P5-PCTFE) @ different energies (MeV)

P1 P2 P3 P4 P51x10

4

2x104

3x104

P1 P2 P3 P4 P5

0

4x103

8x103

P1 P2 P3 P4 P5

0

2x102

4x102

P1 P2 P3 P4 P5

2x102

4x102

6x102

P1 P2 P3 P4 P5

8x102

2x103

2x103

P1 P2 P3 P4 P5

1x104

2x104

3x104

(

Rm

2/C

i.h)

E=0.01 MeV 0.1 MeV

1 MeV 10 MeV

100 MeV 1000 MeV

Figure 6 Variation of effective atomic number (Zeff) with energy (MeV) and comparison effective atomic number (Zeff) and NEL with polymers (P1-Polysterene, P2-Polypropylene, P3-PTFE, P4-PVC, P5-PCTFE and P6-LDPE)


Figure 7 Variation of buildup factor with energy (MeV) for different mean free paths

0.01 0.1 1 10

102

105

PTFE

PCTFEPVC

PolypropylenePolysterene

0.01 0.1 1 10

101

103

105

0.01 0.1 1 10

101

103

105

0.01 0.1 1 10

101

103

105

0.01 0.1 1 10

101

103

105

E (MeV)

Ben

Figure 8 shows the variation of buildup factor with mean free path for different energies.

From this variation it is found that buildup factors increases with increase in the mean

free path. This is due to the reason that, as thickness of the target medium increases,

gamma/X-ray photons collide more frequently with the atoms in the target medium, as a

result of this, energy deposited in the medium increases and buildup factors increase with

increase in the mean free path. Variation of buildup factors with mass attenuation

coefficient at different energies for different studied polymers are shown in Figure 9.

From this variation, it is observed that there is no systematic variation of buildup factors

with the mass attenuation coefficient. Comparison of neutron shielding parameters

among the studied polymers at different energies is shown in Figure 10. First and second

layers of the Figure 10 show the comparison of coherent neutron scattering length (bcoh),

incoherent neutron scattering length (binc) and among the studied polymers. From this

comparison, it is found that the value of scattering lengths is small for PCTFE compared

to the other polymers. Third, fourth, fifth and sixth layers of Figure 10 show the

comparison of coherent neutron scattering cross section (σcoh), incoherent neutron

scattering cross section (σinc), total neutron scattering cross section (σtot) and neutron

absorption cross section (σabs) among the studied polymers. From this comparison, it is

found that the values of these cross sections are large for PCTFE compared to the other

polymers. The measured mass attenuation coefficients and their derivatives derivable

such as linear attenuation coefficient (μ), half value layer (HVL), tenth value layer (TVL)

and mean free path (λ), effective atomic number (Zeff), effective electronic density (Ne)

and specific gamma ray constant (Γ) are compared with the theoretical values as shown

in Table 1. Measured values agrees well with the theoretical values.


Figure 8 Variation of buildup factor with mean free path for different energies

1 10

102

105

PTFE

PCTFEPVC

PolypropylenePolysterene

1 10

101

103

105

1 10

101

103

105

1 10

101

103

105

1 10

101

103

105

E=0.1 MeV E=1 MeV

E=5 MeV E=10 MeV

E=15 MeV

Ben

Figure 9 Comparison of buildup factor with mass attenuation coefficient at different energies

0 2 4 6 8 10 120

2

4

0.2 0.3 0.4 0.5

150

300

450

0.16 0.18200

400

600

0.13 0.14 0.15

200

300

400

0.09 0.1055

60

65

70

0.06 0.07

25

26

27

0.050 0.055 0.060

16.4

16.6

16.8

0.028 0.0306.05

6.10

6.15

6.20

0.018 0.020 0.022

3.05

3.10

3.15 15 MeV1.5 MeV 5 MeV

1MeV0.5MeV0.15MeV

0.05MeVE=0.015 MeV

Ben

cm2/g)

0.1MeV


Figure 10 Comparison of neutron shielding parameters with polymers (P1-Polysterene, P2-Polypropylene, P3-PTFE, P4-PVC and P5-PCTFE) at different energies (MeV)

P1 P2 P3 P4 P53

4

5

6

P1 P2 P3 P4 P5

0

2

4

P1 P2 P3 P4 P5

4

6

P1 P2 P3 P4 P5

0

7

14

P1 P2 P3 P4 P5

8

16

24

P1 P2 P3 P4 P5

0.1

0.2

bco

h

coh

to

t

abs

in

c b

inc

Table 1 Comparison of the mass attenuation coefficient (μ/ρ), linear attenuation coefficient (μ), half value layer (HVL), tenth value layer (TVL) and mean free path (λ), effective atomic number (Zeff), effective electronic density (Ne) and specific gamma ray constant (Γ) with the experiment values

Polymer E(keV) μ/ρ μ HVL TVL λ Zeff Ne Γ

Polysterene

511 Theory 0.0931 0.096 7.11 23.74 10.37

3.507(T)

3.338(E)

3.2E23(T)

3.0E23(E)

4.57E6

Expt. 0.0932 0.094 7.08 23.65 10.22 4.23E6

661.6 Theory 0.0836 0.086 8.05 26.65 11.55 5.45E6

Expt. 0.0825 0.084 8.10 26.23 11.72 5.24E6

835 Theory 0.0747 0.078 8.86 29.65 12.86 6.43E6

Expt. 0.0724 0.071 8.39 29.28 12.48 6.14E6

1173 Theory 0.0635 0.066 10.54 35.02 15.22 7.22E6

Expt. 0.0611 0.061 10.24 34.77 14.92 6.93E6

1274 Theory 0.0608 0.063 10.97 36.53 15.86 7.51E6

Expt. 0.0592 0.058 10.71 36.32 15.68 7.26E6

1332 Theory 0.0595 0.062 11.22 37.33 16.23 7.59E6

Expt. 0.0569 0.057 11.05 37.12 16.04 7.38E6


Table 1 Comparison of the mass attenuation coefficient (μ/ρ), linear attenuation coefficient (μ), half value layer (HVL), tenth value layer (TVL) and mean free path (λ), effective atomic number (Zeff), effective electronic density (Ne) and specific gamma ray constant (Γ) with the experiment values (continued)


Polypropylene

511 Theory 0.0989 0.093 7.38 24.55 10.69

2.677(T)

2.434(E)

3.5E23(T)

3.3E23(E)

68.82

Expt. 0.0966 0.088 7.16 24.28 10.66 67.84

661.6 Theory 0.0892 0.083 8.32 27.51 11.96 81.87

Expt. 0.0877 0.077 8.11 27.44 11.87 80.05

835 Theory 0.0795 0.075 9.20 30.62 13.30 97.58

Expt. 0.0775 0.070 9.02 30.33 13.12 95.38

1173 Theory 0.0676 0.063 10.88 36.23 15.73 109.84

Expt. 0.0665 0.058 10.67 36.11 15.58 91.02

1274 Theory 0.0648 0.061 11.33 37.78 16.37 114.36

Expt. 0.0617 0.057 11.13 37.48 16.08 99.95

1332 Theory 0.0633 0.059 11.64 38.58 16.74 115.16

Expt. 0.0599 0.055 11.43 38.29 16.53 94.84

PTFE

511 Theory 0.0830 0.182 3.73 12.67 5.48

8.003(T)

7.835(E)

2.9E23(T)

3.0E23(E)

70.38

Expt. 0.0801 0.177 3.24 12.34 5.24 66.33

661.6 Theory 0.0748 0.164 4.21 14.11 6.12 77.48

Expt. 0.0702 0.134 4.01 14.02 5.92 67.88

835 Theory 0.0668 0.147 4.68 15.68 6.80 87.76

Expt. 0.0647 0.138 4.48 15.48 6.58 74.39

1173 Theory 0.0566 0.124 5.55 18.61 8.06 94.15

Expt. 0.0538 0.114 5.32 18.48 7.98 88.49

1274 Theory 0.0541 0.119 5.82 19.33 8.39 97.79

Expt. 0.0521 0.101 5.53 19.03 8.32 89.49

1332 Theory 0.0531 0.116 5.92 19.71 8.57 98.51

Expt. 0.0521 0.105 5.59 19.45 8.45 79.40

PVC

511 Theory 0.0891 0.122 5.62 18.75 8.16

5.37(T)

4.948(E)

3.1E23(T)

2.8E23(E)

670.52

Expt. 0.0874 0.104 5.34 18.56 8.02 620.40

661.6 Theory 0.0796 0.109 6.36 20.92 9.10 393.9

Expt. 0.0768 0.102 6.05 20.78 8.98 369.40

835 Theory 0.0714 0.098 7.04 23.41 10.14

5.362(T)

5.131(E)

339.93

Expt. 0.0704 0.089 6.89 23.06 10.04 322.33

1173 Theory 0.0605 0.084 8.32 27.67 12.08 218.49

Expt. 0.0598 0.079 8.06 27.38 11.78 202.44

1274 Theory 0.0578 0.079 8.71 28.85 12.54 218.49

Expt. 0.0559 0.069 8.57 28.78 12.43 204.38

1332 Theory 0.0565 0.078 8.89 29.49 12.82 211.74

Expt. 0.534 0.067 8.48 29.44 12.65 199.49


Table 1 Comparison of the mass attenuation coefficient (μ/ρ), linear attenuation coefficient (μ), half value layer (HVL), tenth value layer (TVL) and mean free path (λ), effective atomic number (Zeff), effective electronic density (Ne) and specific gamma ray constant (Γ) with the experiment values (continued)


PCTFE

511 Theory 0.0834 0.175 3.94 13.21 5.75 9.361(T)

9.088(E)

2.9E23(T)

3.2E23(E)

388.5

Expt. 0.0823 0.154 3.65 12.89 5.45 364.48

661.6 Theory 0.0749 0.157 4.48 14.74 6.40

9.348(T)

9.301(E)

243.97

Expt. 0.0717 0.135 4.29 14.58 6.05 202.44

835 Theory 0.0670 0.140 4.94 16.46 7.14 224.44

Expt. 0.059 0.09 4.78 16.24 6.97 213.243

1173 Theory 0.0570 0.119 5.82 19.39 8.43 165.85

Expt. 0.048 0.102 5.76 19.08 8.13 143.45

1274 Theory 0.0545 0.114 6.09 20.24 8.76 169.76

Expt. 0.0512 0.989 5.95 20.02 8.58 144.40

1332 Theory 0.0530 0.112 6.20 20.62 8.99 169.76

Expt. 0.0502 0.098 5.97 20.23 8.78 138.89

3 Conclusion

Among the studied polymers, PCTFE has large values of μ/ρ, Zeff and Γ. PCTFE also has

smaller values of HVL, TVL and λ. Hence, PCTFE is considered as good shielding

material for gamma/X-ray radiation. It is also observed that neutron scattering and

absorption cross sections are large for PCTFE. PCTFE is also found to be good shielding

material for neutrons.

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semi-empirical formula for photon energy absorption

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