1. Introduction

WJNST

World Journal of Nuclear Science and Technology

2161-6795

Scientific Research Publishing

10.4236/wjnst.2022.124010

WJNST-121509

Articles

Engineering Physics&Mathematics

The Systematics Study of (n, p) Reaction Cross-Sections at 14.7 MeV Neutron Energy

Sadeem

Abdulrahman Alsuhaibani

¹Khalda

T. Osman

Department of Physics, College of Science, Qassim University, Buraydah, Saudi Arabia

28112022

120411313213, September 202228, October 2022 31, October 2022

2014

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Based on the statistical model and taking into account the Q-value dependence and odd-even effects, we proposed a new empirical formula to reproduce the cross sections of the (n, p) reactions at 14.7 MeV neutron energy and at the target mass number 14 ≤ A ≤ 198 for even A and 29 ≤ A ≤ 205 for odd A. All calculated results from the proposed empirical formula were compared to the experimental data as well as the available semi-empirical formula obtained by other authors. A high level of agreement has been found between the collected experimental data and the most of semiempirical formulae obtained by others.

(n p) Cross Section Neutron Energy Empirical Formula Statistical Model Odd Even Effect

1. Introduction

The data for gas formation via neutron induced reactions are critical in the field of fusion reactor technology, particularly in calculating nuclear transmutation rates, nuclear heating, and radiation damage due to gas formation. Unmeasured data can be estimated using model theory calculations and systematic predictions. However, because the 14.7 MeV neutron induced cross sections for different nuclei vary rather smoothly with their N and Z values, several semi-empirical relations to systematize the (n, p) reactions have been proposed [1]. In recent years, many experimental techniques for obtaining and detecting neutrons of various energies, as well as measuring the cross sections of various neutron-induced reactions, have been developed [2] [3]. In the present work a systematics is proposed to calculate the (n, p) reaction cross-sections based on the statistical model, with consideration of the Q-value dependence and odd-even effects. An empirical formula for odd-A and even-A nuclei is presented for the (n, p) reaction cross-sections at 14.7 MeV neutrons and the target mass range 14 ≤ A ≤ 198 for even A and 29 ≤ A ≤ 205 for odd A. The present formula is compared with recently proposed systematics based on the statistical model and the asymmetry parameter dependence.

2. Theoretical Framework

The aim of this work is to develop a semi empirical formula, which depends on the mass and charge numbers in order to calculate the (n, p) reaction for 14.7 MeV neutrons. The statistical evaporation model shows that the (n, p) cross section depends on the reaction energy Q, the nuclear temperature T and the Coulomb barrier V_p. The use of the effective reaction energy Q shows the important dependence of the (n, p) cross section on the (2Z−1)/A term describing the Coulomb barrier and also a dependence on an additional (N-Z+1)/A term that describes surface asymmetry effect. Thus, an analytical expression was derived and the parameters of the formula were determined with least-squared analysis of the existing cross section values for different nuclei. [4]

The evaluation of excitation functions with new theoretical calculations is at the forefront in reaction physics today. Thereby, semi-classical and quantum mechanics models, have been widely used for analysis of the new data of nuclear reactions [5] [6] [7].

In the present work the calculations of cross sections of (n, p) reactions at 14.7 MeV neutron energy based on the statistical model and taking into account the Q-value dependence and odd-even effects are discussed.

2.1. Empirical Formula

On the basis of statistical model, the (n, p) reaction cross-sections can be expressed as [8] [9] [10]:

σ n , p = σ R ( Γ p Γ n ) (1)

where σ R is the reaction or formation cross-section for 14 MeV neutrons.

Γ p is the decay width for a proton and Γ n is the decay width for a neutron.

The decay width for a proton can be written by means of principle of detailed balance as follows:

Γ p = ( 2 s p + 1 ) M p π 2 h 2 ρ a E a ∫ V p E a − B p − δ p ε p σ c ( ε p ) ρ ( E b ) d ε p (2)

where s p and M p are the spin statistical factor and mass of proton, respectively; B p and δ p are the separation energy of proton and odd even character of nucleus respectively; E a and E b are the excitation energies of compound and residual nuclei respectively; V p is the coulomb barrier of proton and ε p and σ c are the emitted proton energy and cross section of reverse process respectively.

When the energy of the incident neutron is not too high, the inverse cross section remains approximately constant and can be taken as follows:

for neutrons:

σ c ( ε n ) = π R 2 (3)

for protons:

σ c ( ε p ) = { π R 2 ( 1 − V p ε p ) for ε p > V p 0 for ε p < V p (4)

where 1 − V p ε p is the probability for the barrier penetration for a proton in the classical limit, ε n is the emitted neutron energy and R is the nuclear radius.

The level density can be approximately expressed as the function of the entropy of the nuclear system [6]:

ρ b ( E b ) ρ a ( E a ) ≈ exp [ S b ( E b ) − S a ( E a ) ] (5)

With the entropy of the system given by

d s d E = 1 T (6)

Where is the nuclear temperature. Thus

S b ( E b ) − S a ( E a ) ≈ ( E b − E a ) / T = − ( ε p + B p + δ p ) / T (7)

By substituting the Relations (4)-(7) into Equation (2) the following expression can be obtained:

Γ p = ( 2 p + 1 ) M p R 2 π h 2 ∫ v p E a − B p − δ p ε p ( 1 − V p / ε p ) exp ( − ( ε p + B p + δ p ) / T ) d ε p (8)

Integration of Equation (8) gives for the decay width of a proton:

Γ p ≈ 2 S p + 1 π h 2 M n R 2 T 2 ( 1 − V p / ε p ) exp [ − ( B p + δ n + V p ) / T ] (9)

And similarly for the width of a neutron:

Γ n ≈ 2 S n + 1 π h 2 M n R 2 T 2 exp [ − ( δ n + B n ) / T ] (10)

where S_nand M_n are the spin statistical factor and mass of the neutron, respectively; B_n and δ_n are the separation energy of the neutron and the odd-even character of the nucleus, respectively.

Thus the (n, p) reaction cross-section will be:

σ n , p = σ R ( 2 S p + 1 2 S n + 1 ) M p M n ( 1 − V p / ε p ) exp ( Q n , p − V p T ) . (11)

σ n , p = σ R ( 2 S p + 1 2 S n + 1 ) M p M n ( 1 − V p / ε p ) exp ( a c ( 2 Z − 1 ) T A 1 / 3 − 4 a a ( A − 2 Z + 1 A T ) − V p T ) .

(12)

where T = ( E n a ) 1 / 2 , with a = A 15 MeV⁻¹ is the level density and E_n is the incident neutron energy; σ R = π r o 2 ( 1 + A 1 / 3 ) 2 mb is the reaction cross-section and r_o= 1.4 fm.

2.2. Fitting Procedure

For the fitting of Equation (11) it can be written as:

ln σ n , p M p M n σ R = a 0 + a 1 2 Z − 1 T A 1 / 3 − a 2 A − 2 Z + 1 T A − a 3 T (13)

where a 0 = ln c 3 ( 1 − V p ε p ) , a 1 = a c which is Coulomb constant, a 2 = 4 a a which is the symmetry, a 3 = V p which is Coulomb barrier and c 3 = 2 S p + 1 2 S n + 1 .

The Legendre method of least squares and Cramer’s rule can be applied to Equation (13) to obtain the values of a₀, a₁, a₂ and a₃.

The equation can be written in the following form:

X + a Y + b Z + c F − K = 0 (14)

where in case of Equation (12)

X = a 0 = ln c 3 ( 1 − V p / ε p ) , Y = a 1 = a c , Z = a 2 = 4 a a , F = a 3 = V p , K = ln ( σ n , p / ( σ R M p / M n ) ) , a = ( 2 Z − 1 ) / T A 1 / 3 , b = ( A − 2 Z + 1 ) / T A ; and c = 1 / T .

We choose X, Y, Z and F such that the sum of the squares of the error is least i.e. the quantity ∑ s = 1 n ( X + a Y + b Z + c F − K ) 2 is a minimum. The input data used in the present analysis of (n, p) reaction cross sections at 14.7 MeV are given for even and odd A nuclides respectively, All the data were taken from [3].

3. Results and DiscussionThe Best Fit Parameters

In this study, an even- and odd-target mass number semiempirical formula for computing the (n, p) reaction cross section at 14.7 MeV is established. The input parameters are the level density parameter, observed (n, p) cross sections at 14.7 neutron energy, and atomic number and mass number. For even and odd A nuclides, the values for the coefficients and their uncertainties derived from least square fit to the (n, p) reaction cross section with the obtained empirical formula are provided in Table 1 and Table 2, respectively.

In Figure 1 and Figure 2 the effect of (2Z− 1)/TA¹^/³ on the cross-section ln ( σ n , p ( M p / M n ) σ R ) for even and odd-A nuclides are shown and we noticed that the cross-section decreases with this parameter for even and odd A nuclides.

We also note that the decreasing of cross-sections with the parameter

Table 1 Present equation for even-A nuclides

Parameters	a 0 = ln c 3 ( 1 − V p ε p )	a₁ = a_c (MeV)	a₂ = 4a_a (MeV)	a₃ = V_p (MeV)	No. of data points
Parameters	0.697 ± 0.442	0.302 ± 0.036	65.544 ± 2.358	0.494 ± 1.342	88
Equation	σ n , p = exp ( 0.697 + 0.302 2 Z − 1 T A 1 3 − 65.544 A − 2 Z + 1 T A − 0.494 1 T ) M p M n σ R

Table 2 Present equation for odd-A nuclides

Parameters	a 0 = ln c 3 ( 1 − V p ε p )	a₁ = a_c (MeV)	a₂ = 4a_a (MeV)	a₃ = V_p (MeV)	No. of data points
Parameters	2.153 ± 0.968	0.001 ± 0.064	11.316 ± 5.320	5.079 ± 2.898	53
Equation	σ n , p = exp ( 2.153 + 0.001 2 Z − 1 T A 1 3 − 11.316 A − 2 Z + 1 T A − 5.079 1 T ) M p M n σ R

(A− 2Z+ 1)/TA for even and odd A nuclides as shown in Figure 3 and Figure 4. As illustrated in Figure 5 and Figure 6, the cross-section also decreases with temperature 1/T. An illustration of the relationship between (n, p) cross sections for different elemental isotopes and mass number A for both even and odd A is shown in Figure 7 and Figure 8, the plot shows a decrease in (n, p) cross sections as isotope mass number increases.

The odd-even effect correction as indicated in Figure 9 and Figure 10 and given by the following formula provides a satisfactory fit for the cross-section values obtained using the current formula.

σ n , p = ( 1 + A 1 3 ) 2 α exp [ β N − Z + δ A ] (14)

where α and β are fitting parameters and δ is odd-even character. They have the following values:

For even-A nuclides: α = 4.54 ± 0.1169, β = −35.95 ± 0.784, δ = 1.

and for odd-A nuclides: α = 3.06 ± 0.1492, β = −28.029 ± 0.784, δ = 0.

4. Comparison with Others Systematics

As listed in Table 3, we have compared the present equation with the equations

Table 3 Comparison of (n, p) systematics at 14.7 MeV [<xref ref-type="bibr" rid="scirp.121509-ref3">3</xref>]

Author	Formula, σ (mb)		Mass region
Levkovskii	σ = { 55.3 ( A 1 3 + 1 ) 2 exp ( − 38.4 N − z + 1 A + 1 ) 49.4 ( A 1 3 + 1 ) 2 exp ( − 35.1 N − Z A )		19 ≤ A ≤ 40 40 ≤ A ≤ 188
Eder et al.	σ = exp ( 1.31 + 0.806 A 1 2 − 10.3 N − Z A 2 3 )		19 ≤ A ≤ 188
Bychkov et al.	σ = exp ( A 1 3 + 1 ) 2 exp { A 140 ( − 53.3 N − Z + 1 A + 0.622 Z − 1 A 1 3 − 3.20 ) }		40 ≤ A ≤ 188
Forrest	σ = 11.23 exp ( A 1 3 + 1 ) 2 exp ( − 32.73 N − Z A − 46.57 ( N − Z A ) 2 + 0.218 A 1 / 2 )		40 ≤ A ≤ 188
Kumabe and Fukuda	σ = { 27.9 A exp ( − 39.1 N − z A ) 0.58 A 2 exp ( − 42.3 N − Z A ) 0.94 A 2 exp ( − 47.8 N − Z A )		40 ≤ A ≤ 62 63 ≤ A ≤ 89 90 ≤ A ≤ 188
Ait-Tahar	σ = 140.2 ( A 1 / 3 + 1 ) 2 exp ( − 39.1 N − Z + 1 A )		40 ≤ A ≤ 188
Kasugai et al.	σ = 1830 ( N − Z + 1 ) exp ( − 50.7 N − Z + 1 A )	19 ≤ A ≤ 188
Korovin et al.	σ = π r o 2 ( A 1 / 3 + 1 ) 2 × { A 1.1128 ( ( N − Z + 1 A ) 2 − 0.73212 ( N − Z + 1 A ) + 0.11707 ) 3 + 0.4936 exp ( − 194.69 ( N − Z + 1 A ) 2 − 5.3778 ( N − Z + 1 A ) ) }	11 ≤ A ≤ 209
Dόczi et al.	σ = 23.659 ( A 1 / 3 + 1 ) 2 exp ( − 23.041 ( N − Z A ) + ( N − Z A ) 2 )	11 ≤ A ≤ 209
Present (Even)	σ = exp ( 0.697 + 0.302 2 Z − 1 T A 1 3 − 65.544 A − 2 Z + 1 T A − 0.494 1 T ) M p M n σ R	14 ≤ A ≤ 198
Present (Odd)	σ n , p = exp ( 2.153 + 0.001 2 Z − 1 T A 1 3 − 11.316 A − 2 Z + 1 T A − 5.079 1 T ) M p M n σ R	29 ≤ A ≤ 205

of other authors. To compare their results with the measured and computed data for (n, p) reaction cross-sections at 14.7 MeV neutron energy for both even-A and odd-A nuclides, reported in Table 4 and Table 5 respectively, we chose the authors Eder et al., Kasugai et al., Korovin et al., and Dόczi et al. We observed that although the four systematics had a reasonable level of agreement, the present finding was the closest to the experimental value.

As can be seen from Figure 11 for the even-A nuclides, we have plotted the effect of mass number A on the ratio ( σ e x p σ c a l c ) . We note that both authors Eder et al. and Kasukai et al. obtained results that are closer to the experimental results at high A, and in the present result, we obtained results that are closer to the experimental results at low A. On the contrary, we note that the results of the author Dόczi et al. as being closer to the experimental results in a low A, and the present results are closer to the experimental results at a high Abut in general, we see a good agreement between the present results with the results obtained by others. For the even-A nuclides, the discrepancy between the author’s results (Korovin et al.) and the present results (Figure 12) can be clearly seen and the latter results are much closer to the experimental findings.

In Table 4, the comparison between the authors’ results and the present result for even-A nuclides are given and shown in Figure 11 for the even-A nuclides. In Figure 11 the plot of mass number A versus the ratio ( σ e x p σ c a l c ) are shown and we observed that both authors Eder et al. and Kasukai et al. obtained results closer to the experimental results in region of the middle A. And also, we can see that the results of the author Dόczi et al. are closer to the experimental results in

Table 4 Measured and calculated data for reaction cross-sections at 14.7 MeV (for even-A nuclides)

Reaction		(n, p) Cross-Section (mb) at 14.7 MeV
Reaction		Experimental	Eder et al.	Ratio	Kasugai et al.	Ratio	Korovin et al.	Ratio	Dόczi et al.	Ratio	Present	Ratio
1	¹⁴N(n,p)¹⁴C	45.2 ± 6	-	-	-	-	9.39	4.81	275.13	0.16	58.74	0.77
2	¹⁸O(n,p)¹⁸N	2.3 ± 0.5	-	-	-	-	0.11	21.03	18.04	0.13	10.16	0.23
3	²⁴Mg(n,p)²⁴Na	183.6 ± 9	192.21	0.96	221.31	0.83	28.36	6.47	357.00	0.51	142.27	1.29
4	²⁶Mg(n,p)²⁶Na	39 ± 11	21.60	1.81	15.81	2.47	2.31	16.87	55.08	0.71	27.26	1.43
5	²⁸Si(n,p)²⁸Al	238.7 ± 30	263.74	0.91	299.27	0.80	35.11	6.80	385.50	0.62	190.72	1.25
6	³⁰Si(n,p)³⁰Al	70.04 ± 20	36.27	1.93	34.49	2.03	5.04	13.90	77.54	0.90	40.73	1.72
7	³²S(n,p)³²P	237 ± 25	354.05	0.67	375.29	0.63	41.42	5.72	412.35	0.57	251.22	0.94
8	³⁴S(n,p)³⁴P	78 ± 8	57.22	1.36	62.62	1.25	8.75	8.91	101.25	0.77	58.68	1.33
9	⁴⁰Ca(n,p)⁴⁰K	470.01 ± 50	606.43	0.78	515.21	0.91	53.14	8.84	462.20	1.02	421.60	1.11
10	⁴²Ca(n,p)⁴²K	178.25 ± 12	125.03	1.43	146.83	1.21	18.13	9.83	150.17	1.19	113.37	1.57
11	⁴⁴Ca(n,p)⁴⁴K	43.27 ± 6	28.52	1.52	28.79	1.50	3.71	11.68	49.42	0.88	32.37	1.34
12	⁴⁶Ti(n,p)⁴⁶Sc	251 ± 13	176.26	1.42	201.17	1.25	23.37	10.74	174.71	1.44	153.76	1.63
13	⁴⁸Ti(n,p)⁴⁸Sc	57.2 ± 2.7	43.59	1.31	46.54	1.23	5.97	9.58	63.47	0.90	46.13	1.24
14	⁵⁰Ti(n,p)⁵⁰Sc	11.9 ± 6	11.66	1.02	10.59	1.12	1.21	9.86	23.46	0.51	14.52	0.82
15	⁵⁰Cr(n,p)⁵⁰V	294 ± 30	242.61	1.21	262.09	1.12	28.79	10.21	199.05	1.48	206.10	1.43
16	⁵²Cr(n,p)⁵²V	80 ± 4	64.38	1.24	69.86	1.15	8.79	9.10	78.57	1.02	64.62	1.24
17	⁵⁴Cr(n,p)⁵⁴V	17.4 ± 1	18.30	0.95	17.92	0.97	2.14	8.14	31.44	0.55	21.14	0.82
18	⁵⁴Fe(n,p)⁵⁴Mn	350 ± 15	327.33	1.07	328.33	1.07	34.31	10.20	223.08	1.57	273.73	1.28
19	⁵⁶Fe(n,p)⁵⁶Mn	115 ± 6	92.46	1.24	98.96	1.16	12.10	9.50	94.48	1.22	89.29	1.29
20	⁵⁸Ni(n,p)⁵⁸Co	366 ± 19	434.19	0.84	398.72	0.92	39.88	9.18	246.75	1.48	360.86	1.01
21	⁶⁰Ni(n,p)⁶⁰Co	148 ± 8	129.68	1.14	133.82	1.11	15.82	9.35	111.03	1.33	122.02	1.21
22	⁶²Ni(n,p)⁶²Co	37 ± 4	40.91	0.90	41.84	0.88	5.13	7.21	50.41	0.73	42.72	0.87
23	⁶⁴Ni(n,p)⁶⁴Co	185 ± 10	178.22	1.04	174.26	1.06	19.89	9.30	128.07	1.44	165.20	1.12
24	⁶⁶Zn(n,p)⁶⁶Cu	73.3 ± 8	58.79	1.25	59.19	1.24	7.23	10.14	61.19	1.20	59.58	1.23
25	⁶⁸Zn(n,p)⁶⁸Cu	15.2 ± 2.5	20.30	0.75	20.07	0.76	2.37	6.40	29.53	0.51	22.13	0.69
26	⁷⁰Ge(n,p)⁷⁰Ga	74 ± 8	82.68	0.90	80.48	0.92	9.70	7.63	72.70	1.02	82.26	0.90
27	⁷²Ge(n,p)⁷²Ga	31.6 ± 1.4	29.61	1.07	29.13	1.08	3.51	9.01	36.64	0.86	31.33	1.01
28	⁷⁴Ge(n,p)⁷⁴Ga	10.1 ± 0.5	11.02	0.92	10.73	0.94	1.22	8.25	18.65	0.54	12.24	0.83
29	⁷⁶Ge(n,p)⁷⁶Ga	2.8 ± 0.4	4.25	0.66	4.07	0.69	0.43	6.54	9.61	0.29	4.90	0.57
30	⁷⁴Se(n,p)⁷⁴As	112 ± 7	114.13	0.98	105.85	1.06	12.53	8.94	84.84	1.32	112.62	0.99
31	⁷⁶Se(n,p)⁷⁶As	49 ± 3	42.27	1.16	40.66	1.20	4.93	9.94	44.44	1.10	43.90	1.12
32	⁷⁸Se(n,p)⁷⁸As	19 ± 1.1	16.22	1.17	15.80	1.20	1.86	10.23	23.48	0.81	17.52	1.08
33	⁸⁰Kr(n,p)⁸⁰Br	45 ± 6	59.20	0.76	54.90	0.82	6.65	6.77	52.88	0.85	60.97	0.74
34	⁸²Kr(n,p)⁸²Br	23 ± 5	23.37	0.98	22.39	1.03	2.69	8.56	28.89	0.80	24.83	0.93
35	⁸⁴Kr(n,p)⁸⁴Br	11 ± 2	9.52	1.16	9.31	1.18	1.07	10.24	15.91	0.69	10.33	1.07
36	⁸⁶Kr(n,p)⁸⁶Br	5 ± 1.5	3.99	1.25	3.96	1.26	0.43	11.64	8.85	0.57	4.38	1.14
37	⁸⁴Sr(n,p)⁸⁴Rb	95 ± 8	81.53	1.17	72.04	1.32	8.67	10.96	61.90	1.53	84.04	1.13
38	⁸⁶Sr(n,p)⁸⁶Rb	44 ± 4	33.05	1.33	30.73	1.43	3.73	11.80	34.85	1.26	34.87	1.26
39	⁸⁸Sr(n,p)⁸⁸Rb	17.4 ± 2	13.79	1.26	13.29	1.31	1.58	11.02	19.76	0.88	14.76	1.18
40	⁹⁰Zr(n,p)⁹⁰Y	37 ± 5	45.96	0.81	40.99	0.90	5.00	7.40	41.34	0.89	48.60	0.76
41	⁹²Zr(n,p)⁹²Y	20.23 ± 2.5	19.61	1.03	18.41	1.10	2.23	9.06	24.07	0.84	20.91	0.97
42	⁹⁴Zr(n,p)⁹⁴Y	7.5 ± 1.1	8.58	0.87	8.41	0.89	0.99	7.55	14.12	0.53	9.15	0.82
43	⁹⁶Mo(n,p)⁹⁶Nb	21.3 ± 1.1	27.44	0.78	24.81	0.86	3.05	6.99	28.83	0.74	29.40	0.72
44	⁹⁶Ru(n,p)⁹⁶Tc	146 ± 8	195.78	0.75	142.05	1.03	16.43	8.89	91.98	1.59	212.09	0.69
45	¹⁰⁴Ru(n,p)¹⁰⁴Tc	6 ± 0.7	7.99	0.75	7.83	0.77	0.95	6.30	12.88	0.47	8.46	0.71
46	¹⁰²Pd(n,p)¹⁰²Rh	93.6 ± 15	113.53	0.82	84.97	1.10	10.26	9.13	63.72	1.47	126.65	0.74
47	¹⁰⁴Pd(n,p)¹⁰⁴Rh	58 ± 15	51.45	1.13	42.08	1.38	5.23	11.09	39.66	1.46	56.98	1.02
48	¹⁰⁶Pd(n,p)¹⁰⁶Rh	22.5 ± 6	23.81	0.94	21.02	1.07	2.64	8.53	24.80	0.91	26.01	0.87
49	¹⁰⁸Pd(n,p)¹⁰⁸Rh	4 ± 1	11.24	0.36	10.64	0.38	1.33	3.01	15.60	0.26	12.04	0.33
50	¹⁰⁶Cd(n,p)¹⁰⁶Ag	130 ± 24	149.84	0.87	104.44	1.24	12.49	10.41	72.05	1.80	172.49	0.75
51	¹¹²Cd(n,p)¹¹²Ag	16.1 ± 3	15.60	1.03	14.15	1.14	1.80	8.96	18.63	0.86	17.04	0.94
52	¹¹⁴Cd(n,p)¹¹⁴Ag	8.5 ± 1.3	7.61	1.12	7.44	1.14	0.93	9.11	11.99	0.71	8.09	1.05
53	¹¹⁶Cd(n,p)¹¹⁶Ag	2.96 ± 0.3	3.78	0.78	3.97	0.75	0.48	6.18	7.75	0.38	3.89	0.76
54	¹¹⁶Sn(n,p)¹¹⁶In	14.6 ± 2	21.37	0.68	18.45	0.79	2.38	6.15	21.96	0.66	23.96	0.61
55	¹²⁰Sn(n,p)¹²⁰In	4.5 ± 0.5	5.32	0.85	5.39	0.84	0.68	6.64	9.43	0.48	5.58	0.81
56	¹²²Te(n,p)¹²²Sb	10.5 ± 1.5	14.52	0.72	12.94	0.81	1.69	6.20	16.99	0.62	16.25	0.65
57	¹²⁶Te(n,p)¹²⁶Sb	4.7 ± 0.2	3.82	1.23	4.03	1.17	0.50	9.33	7.59	0.62	3.95	1.19
58	¹²⁸Te(n,p)¹²⁸Sb	2.9 ± 0.1	2.01	1.45	2.29	1.27	0.27	10.92	5.11	0.57	1.98	1.47
59	¹³⁰Te(n,p)¹³⁰Sb	1.7 ± 0.1	1.07	1.59	1.32	1.29	0.13	12.69	3.45	0.49	1.00	1.70
60	¹²⁸Xe(n,p)¹²⁸I	27 ± 4	10.16	2.66	9.38	2.88	1.24	21.69	13.43	2.01	11.31	2.39
61	¹³²Ba(n,p)¹³²Cs	15.3 ± 3.5	13.79	1.11	12.07	1.27	1.63	9.39	15.76	0.97	15.97	0.96
62	¹³⁶Ba(n,p)¹³⁶Cs	4.8 ± 0.7	3.90	1.23	4.10	1.17	0.53	9.03	7.47	0.64	4.11	1.17
63	¹³⁸Ba(n,p)¹³⁸Cs	2.18 ± 0.15	2.12	1.03	2.43	0.90	0.29	7.42	5.17	0.42	2.11	1.03
64	¹⁴⁰Ce(n,p)¹⁴⁰La	7.05 ± 0.7	5.36	1.32	5.35	1.32	0.72	9.79	8.88	0.79	5.87	1.20
65	¹⁴²Ce(n,p)¹⁴²La	4.8 ± 0.8	2.93	1.64	3.21	1.49	0.41	11.66	6.22	0.77	3.04	1.58
66	¹⁴²Nd(n,p)¹⁴²Pr	13.7 ± 1.1	13.32	1.03	11.42	1.20	1.59	8.60	14.81	0.92	16.12	0.85
67	¹⁴⁴Nd(n,p)¹⁴⁴Pr	9.8 ± 1.5	7.27	1.35	6.88	1.42	0.95	10.29	10.44	0.94	8.34	1.17
68	¹⁴⁶Nd(n,p)¹⁴⁶Pr	4.5 ± 0.7	4.02	1.12	4.19	1.07	0.56	8.02	7.39	0.61	4.36	1.03
69	¹⁴⁴Sm(n,p)¹⁴⁴Pm	19 ± 4	32.58	0.58	23.64	0.80	3.28	5.79	24.08	0.79	43.78	0.43
70	¹⁴⁸Sm(n,p)¹⁴⁸Pm	9.8 ± 0.8	9.78	1.00	8.73	1.12	1.24	7.93	12.17	0.81	11.81	0.83
71	¹⁵⁰Sm(n,p)¹⁵⁰Pm	7 ± 0.6	5.45	1.28	5.38	1.30	0.75	9.37	8.70	0.80	6.21	1.13
72	¹⁵⁸Gd(n,p)¹⁵⁸Eu	3.2 ± 1.2	2.38	1.34	2.71	1.18	0.35	9.05	5.33	0.60	2.53	1.26
73	¹⁵⁸Dy(n,p)¹⁵⁸Tb	10.6 ± 1.2	9.76	1.09	8.53	1.24	1.25	8.51	11.74	0.90	12.47	0.85
74	¹⁶⁰Dy(n,p)¹⁶⁰Tb	7 ± 1.5	5.58	1.25	5.42	1.29	0.78	9.01	8.56	0.82	6.70	1.04
75	¹⁶²Dy(n,p)¹⁶²Tb	4.3 ± 1	3.22	1.33	3.47	1.24	0.48	9.03	6.27	0.69	3.63	1.19
76	¹⁶⁴Dy(n,p)¹⁶⁴Tb	2.55 ± 0.5	1.88	1.36	2.24	1.14	0.28	8.97	4.60	0.55	1.98	1.29
77	¹⁶⁶Er(n,p)¹⁶⁶Ho	4.5 ± 0.7	4.32	1.04	4.38	1.03	0.63	7.18	7.31	0.62	5.17	0.87
78	¹⁶⁸Er(n,p)¹⁶⁸Ho	2.8 ± 0.4	2.54	1.10	2.86	0.98	0.39	7.27	5.41	0.52	2.84	0.99
79	¹⁷⁴Yb(n,p)¹⁷⁴Tm	3 ± 0.2	2.03	1.48	2.38	1.26	0.31	9.56	4.71	0.64	2.25	1.33
80	¹⁸²W(n,p)¹⁸²Ta	6.5 ± 0.5	3.59	1.81	3.73	1.74	0.55	11.91	6.36	1.02	4.59	1.42
81	¹⁸⁴W(n,p)¹⁸⁴Ta	2.61 ± 0.12	2.18	1.20	2.53	1.03	0.34	7.57	4.83	0.54	2.59	1.01
82	¹⁸⁶W(n,p)¹⁸⁶Ta	1.25 ± 0.25	1.34	0.93	1.72	0.72	0.21	5.90	3.67	0.34	1.47	0.85
83	¹⁸⁴Os(n,p)¹⁸⁴Re	9.30 ± 2	7.80	1.19	6.79	1.37	1.06	8.77	9.59	0.97	11.57	0.80
84	¹⁸⁸Os(n,p)¹⁸⁸Re	4 ± 0.8	2.89	1.38	3.14	1.27	0.45	8.84	5.58	0.72	3.70	1.08
85	¹⁹⁰Os(n,p)¹⁹⁰Re	2.2 ± 0.5	-	-	-	-	0.29	7.69	4.27	0.51	2.11	1.04
86	¹⁹²Os(n,p)¹⁹²Re	1 ± 0.03	-	-	-	-	0.18	5.70	3.28	0.30	1.21	0.82
87	¹⁹⁶Hg(n,p)¹⁹⁶Au	9.3 ± 1	-	-	-	-	0.74	12.61	7.32	1.27	7.48	1.24
88	¹⁹⁸Hg(n,p)¹⁹⁸Au	4.5 ± 0.8	-	-	-	-	0.49	9.23	5.67	0.79	4.31	1.04

Table 5 Measured and calculated data for reaction cross-sections at 14.7 MeV (for odd-A nuclides)

		(n, p) Cross-Section (mb) at 14.7 MeV
		Experimental	Eder et al.	Ratio	Kasugai et al.	Ratio	Korovin et al.	Ratio	Dόczi et al.	Ratio	Present	Ratio
1	¹¹B(n,p)¹¹Be	4.2 ± 0.2	-	-	-	-	0.03	151.41	25.03	0.17	-	-
2	¹⁹F(n,p)¹⁹O	18.6 ± 0.88	29.27	0.64	17.61	1.06	3.00	6.20	88.83	0.21	-	-
3	²³Na(n,p)²³Ne	47.0 ± 2	49.50	0.95	44.55	1.06	7.16	6.56	122.90	0.38	-	-
4	²⁵Mg(n,p)²⁵Na	88.04 ± 13	62.51	1.41	63.38	1.39	9.72	9.06	139.70	0.63	-	-
5	²⁷Al(n,p)²⁷Mg	70 ± 2	77.76	0.90	85.60	0.82	12.49	5.61	156.23	0.45	-	-
6	²⁹Si(n,p)²⁹Al	135.33 ± 15	95.51	1.42	110.90	1.22	15.39	8.79	172.48	0.78	105.21	1.29
7	³¹P(n,p)³¹Si	91.85 ± 3.48	116.03	0.79	138.97	0.66	18.38	5.00	188.40	0.49	103.20	0.89
8	³³S(n,p)³³P	134 ± 22	139.63	0.96	169.44	0.79	21.42	6.26	204.00	0.66	101.15	1.32
9	³⁵Cl(n,p)³⁵S	110 ± 10	166.62	0.66	201.96	0.54	24.48	4.49	219.28	0.50	99.08	1.11
10	³⁷Cl(n,p)³⁷S	25.4 ± 3	30.87	0.82	30.49	0.83	4.13	6.15	58.92	0.43	75.48	0.34
11	³⁹K(n,p)³⁹Ar	314 ± 14	232.26	1.35	271.84	1.16	30.59	10.27	248.89	1.26	94.92	3.31
12	⁴¹K(n,p)⁴¹Ar	51.36 ± 3	48.05	1.07	52.04	0.99	6.94	7.40	76.67	0.67	73.17	0.70
13	⁴³Ca(n,p)⁴³K	99.83 ± 13	59.01	1.69	65.50	1.52	8.59	11.62	85.95	1.16	71.96	1.39
14	⁴⁵Sc(n,p)⁴⁵Ca	57 ± 8	71.84	0.79	80.78	0.71	10.40	5.48	95.45	0.60	70.73	0.81
15	⁴⁷Ti(n,p)⁴⁷Sc	136 ± 9	86.75	1.57	97.85	1.39	12.33	11.03	105.12	1.29	69.49	1.96
16	⁵¹V(n,p)⁵¹Ti	33.3 ± 1.7	27.69	1.20	28.19	1.18	3.51	9.49	43.91	0.76	54.11	0.62
17	⁵³Cr(n,p)⁵³V	42 ± 2.1	34.05	1.23	35.31	1.19	4.41	9.52	49.60	0.85	53.32	0.79
18	⁵⁵Mn(n,p)⁵⁵Cr	45 ± 10	41.53	1.08	43.51	1.03	5.43	8.28	55.54	0.81	52.51	0.86
19	⁵⁷Fe(n,p)⁵⁷Mn	59 ± 4	50.30	1.17	52.83	1.12	6.57	8.98	61.72	0.96	51.70	1.14
20	⁵⁹Co(n,p)⁵⁹Fe	51 ± 1	60.51	0.84	63.30	0.81	7.82	6.52	68.12	0.75	50.88	1.00
21	⁶¹Ni(n,p)⁶¹Co	64 ± 4	72.36	0.88	74.96	0.85	9.18	6.97	74.71	0.86	50.06	1.28
22	⁶³Cu(n,p)⁶³Ni	70 ± 13	86.03	0.81	87.82	0.80	10.65	6.58	81.49	0.86	49.23	1.42
23	⁶⁵Cu(n,p)⁶⁵Ni	20.93 ± 4	28.45	0.74	28.55	0.73	3.45	6.07	38.18	0.55	40.06	0.52
24	⁶⁷Zn(n,p)⁶⁷Cu	37.9 ± 6	34.35	1.10	34.39	1.10	4.17	9.09	42.45	0.89	39.49	0.96
25	⁶⁹Ga(n,p)⁶⁹Zn	64 ± 5	41.24	1.55	40.99	1.56	4.98	12.85	46.91	1.36	38.92	1.64
26	⁷¹Ga(n,p)⁷¹Zn	20.5 ± 1.5	14.81	1.38	14.50	1.41	1.68	12.19	23.27	0.88	32.00	0.64
27	⁷³Ge(n,p)⁷³Ga	19.4 ± 1.1	17.98	1.08	17.63	1.10	2.07	9.35	26.11	0.74	31.60	0.61
28	⁷⁵As(n,p)⁷⁵Ge	18.7 ± 2.1	21.71	0.86	21.21	0.88	2.52	7.41	29.11	0.64	31.19	0.60
29	⁸¹Br(n,p)⁸¹As	21 ± 5	12.33	1.70	12.01	1.75	1.40	15.02	19.21	1.09	25.28	0.83
30	⁸⁹Y(n,p)⁸⁹Sr	22.2 ± 4.2	25.28	0.88	23.59	0.94	2.86	7.76	28.81	0.77	24.07	0.92
31	⁹¹Zr(n,p)⁹¹Y	29 ± 4	29.92	0.97	27.42	1.06	3.34	8.67	31.52	0.92	23.76	1.22
32	⁹⁵Mo(n,p)⁹⁵Nb	41.3 ± 2	41.43	1.00	36.33	1.14	4.46	9.26	37.31	1.11	23.14	1.78
33	⁹⁷Mo(n,p)⁹⁷Nb	14.6 ± 0.8	18.28	0.80	17.01	0.86	2.08	7.02	22.32	0.65	19.55	0.75
34	¹⁰¹Ru(n,p)¹⁰¹Tc	21.2 ± 1.2	25.44	0.83	22.72	0.93	2.82	7.52	26.63	0.80	19.08	1.11
35	¹⁰³Rh(n,p)¹⁰³Ru	17 ± 3	29.85	0.57	26.05	0.65	3.25	5.24	28.94	0.59	18.85	0.90
36	¹⁰⁵Pd(n,p)¹⁰⁵Rh	38 ± 2.9	34.91	1.09	29.70	1.28	3.71	10.23	31.34	1.21	18.61	2.04
37	¹¹¹Cd(n,p)¹¹¹Ag	23.25 ± 2.1	22.48	1.03	19.62	1.19	2.49	9.33	23.27	1.00	15.49	1.50
38	¹¹⁵In(n,p)¹¹⁵Cd	13.26 ± 2.95	12.79	1.04	11.78	1.13	1.51	8.80	16.31	0.81	13.12	1.01
39	¹¹⁷Sn(n,p)¹¹⁷In	11.7 ± 0.6	15.01	0.78	13.50	0.87	1.74	6.72	17.75	0.66	12.98	0.90
40	¹¹⁹Sn(n,p)¹¹⁹In	7.1 ± 0.8	7.49	0.95	7.29	0.97	0.93	7.63	11.63	0.61	11.16	0.64
41	¹²⁷I(n,p)¹²⁷Ie	11.7 ± 0.8	6.25	1.87	6.17	1.89	0.80	14.58	10.14	1.15	9.35	1.25
42	¹³¹Xe(n,p)¹³¹I	6.1 ± 0.6	3.86	1.58	4.06	1.50	0.52	11.79	7.52	0.81	8.02	0.76
43	¹³³Cs(n,p)¹³³Xe	10.5 ± 2	4.54	2.31	4.67	2.25	0.61	17.30	8.23	1.28	7.95	1.32
44	¹³⁹La(n,p)¹³⁹Ba	4.5 ± 1.1	3.37	1.33	3.62	1.24	0.47	9.64	6.80	0.66	6.79	0.66
45	¹⁴¹Pr(n,p)¹⁴¹Ce	11.5 ± 0.9	8.46	1.36	7.85	1.47	1.08	10.63	11.51	1.00	7.66	1.50
46	¹⁴³Nd(n,p)¹⁴³Pr	11.5 ± 2.3	9.83	1.17	8.86	1.30	1.23	9.33	12.43	0.92	7.59	1.52
47	¹⁴⁵Nd(n,p)¹⁴⁵Pr	7.25 ± 1.6	5.40	1.34	5.36	1.35	0.73	9.89	8.78	0.83	6.62	1.10
48	¹⁵³Eu(n,p)¹⁵³Sm	4.2 ± 0.4	4.75	0.88	4.79	0.88	0.66	6.32	7.98	0.53	5.64	0.75
49	¹⁵⁷Gd(n,p)¹⁵⁷Eu	5.4 ± 1.1	3.14	1.72	3.41	1.59	0.46	11.75	6.25	0.86	4.90	1.10
50	¹⁵⁹Tb(n,p)¹⁵⁹Gd	5.1 ± 0.4	3.65	1.40	3.85	1.33	0.53	9.61	6.77	0.75	4.86	1.05
51	¹⁶⁷Er(n,p)¹⁶⁷Ho	3.4 ± 0.3	3.31	1.03	3.54	0.96	0.49	6.90	6.29	0.54	4.18	0.81
52	¹⁷⁵Lu(n,p)¹⁷⁵Y	4 ± 0.7	3.03	1.32	3.28	1.22	0.46	8.67	5.88	0.68	3.60	1.11
53	¹⁸¹Ta(n,p)¹⁸¹Hf	2.94 ± 0.18	2.43	1.21	2.75	1.07	0.38	7.75	5.14	0.57	3.14	0.94
54	¹⁸⁷Re(n,p)¹⁸⁷W	3.73 ± 0.28	1.97	1.89	2.33	1.60	0.31	11.88	4.54	0.82	2.74	1.36
55	¹⁷⁹Au(n,p)¹⁷⁹Pt	2 ± 0.5	197.04	0.01	79.19	0.03	11.11	0.18	50.83	0.04	7.03	0.28
56	¹⁹⁹Hg(n,p)¹⁹⁹Au	2.5 ± 0.5	-	-	-	-	0.39	6.35	4.99	0.50	2.35	1.06
57	²⁰³Tl(n,p)²⁰³Hg	4.2 ± 0.8	-	-	-	-	0.29	14.52	4.16	1.01	2.08	2.02
58	²⁰⁵Tl(n,p)²⁰⁵Hg	1.9 ± 0.2	-	-	-	-	0.18	10.44	3.25	0.58	1.86	1.02
59	²⁰⁹Bi(n,p)²⁰⁹Pb	0.8 ± 0.3	-	-	-	-	0.24	3.28	3.75	0.21	-	-

a region of low A. The present results are closer to the experimental results at a high A. But in general, we can see a good agreement between the present results with the results of other authors. In Figure 12 for the even-A nuclides, we can see the big difference between the author Korovin et al. and the present results, as we can notice that the present results are very close to the experimental results compared to the author’s results.

As given in Table 5 the comparisons between the authors’ results and the present results for odd-A nuclides are given and also plotted in Figure 13 for the odd-A nuclides, and we notice that the authors’ results Eder et al and Dόczi et al. approach the experimental values at the middle A and a Scattering occurs in the low and high A. In other hand the author Kasugai et al. approaches the experimental values at the low A and it scatters as the mass number A increases. Conversely, our results are approximately equal to the experimental value at high A.

As shown in Figure 14, the present result is clearly closer to the experimental values compared to the results of the author Korovin et al. for the odd-A nuclides.

5. Conclusions

The current work’s objective is to develop a semi-empirical equation for computing the reaction’s cross-section at 14.7 MeV neutron energy. The findings are as follows:

· On the basis of the liquid drop model, we were able to derive a semi-empirical equation that contained the four constants ( a 0 = ln c 3 ( 1 − V p ε p ) , coulomb constant, symmetry, coulomb barrier) and the calculated (n, p) reaction cross sections based on these constants were consistent with the experimental values for both even-A and odd-A nuclides.

· The calculated (n, p) reaction cross sections using the present empirical formula for both odd and even A nuclides were compared with that obtained by other authors and it was found that there is a good agreement between the present results and that obtained by other authors.

· Also, in the present work we study the odd even effect on the (n, p) reaction cross sections at 14.7 MeV neutron energy and the present systematics shows clearly the presence of odd-even effect.

· In addition, more simple empirical formulae were obtained in this work to calculate the (n, p) reaction cross sections at 14.7 MeV neutron energy depending on odd-even character.

We suggest more studies for other interactions of neutrons, such as (n, α), (n, 2n) and (n, t) at 14.7 MeV neutron energy.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Alsuhaibani, S.A. and Osman, K.T. (2022) The Systematics Study of (n, p) Reaction Cross-Sections at 14.7 MeV Neutron Energy. World Journal of Nuclear Science and Technology, 12, 113-132. https://doi.org/10.4236/wjnst.2022.124010

The Variables that Were Used in This Work

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