APPLYING THE OPTICAL MODEL TO CALCULATE THE NEUTRON CROSS SECTION

Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM II-P-1.28 APPLYING THE OPTICAL MODEL TO CALCULATE THE NEUTRON CROSS-SECTION Phan Thanh Quang, Chau Van Tao, Le Hoang Chien, Nguyen Dien Quoc Bao, Pham Minh Quan Nuclear physics Department, Faculty of Physics and Engineering physics University of Science, Viet Nam National University-HCM Email: thanhquang2392@gmail.com ABTRACT In this work, we study the optical model (OM) and apply the approach to calculate the cross sections of neutron-nucleus reactions which are very important to the study of neutron physics Some target nuclei with mass numbers from 40 to 238 and incident neutron energies varying in the range of MeV to 100 MeV are taken into account in this article The obtained results in this work are compared and reasonable to other results Also, the further calculations relating to OM are carried out in our next work Key words: optical model, fast neutron, total cross-section INTRODUCTION The optical model which is represented the two-body interaction between the projectile and the target nucleus has a significant impact on many branches of nuclear reaction physics An important feature of optical model potential (OMP) is that it can be used to reliably predict total cross-section for energies and nuclides for which no measurements exist In particular, the model has been successful in calculating the total cross-section of neutron-nucleus reactions above energy of narrow resonance region Many works have been carried out to apply this model for some peculiar nuclei such as Ref [7] or for wide mass number range like Ref [2] Some phenomenological global optical potential have been also fitted in many works such as Ref [2] and Ref [8] Most of these potential cover the incident energy from several MeV up to several tens of MeV for neutron-nucleus reactions However, this model can also be used to give average calculations in unresolved resonance region [9] Thus, in this paper, we will apply the global spherical potential which is fitted by A J Koning and J P Delaroche [2] to calculate total cross-section of neutron-nucleus reactions for two magic nuclei: Ni-56 and Sn132 in wide energy range from MeV to 100 MeV and calculate the average total cross-section for a wide mass number range from 40 to 238 in unresolved resonance region THEORY The optical model bases on an assumption that neutron can enter and freely traverses through nuclear matter, but with a finite probability of absorption less than one to form a compound nucleus Therefore, this model can be represented by a complex mean-field potential containing the real and imagine part which are related to the elastic scattering and the absorption respectively The form of the interaction potential is given by [2] ⃗𝛔 ⃗𝛔 𝐕(𝐫, 𝐄) = −𝐕𝐕 (𝐫, 𝐄) − 𝐢𝐖𝐕 (𝐫, 𝐄) − 𝐢𝐖𝐃 (𝐫, 𝐄) + 𝐕𝐒𝐎 (𝐫, 𝐄)𝓵 ⃗ + 𝐢𝐖𝐒𝐎 (𝐫, 𝐄)𝓵 ⃗ (1) where VV,SO and WV,D,SO are the real and imaginary components of the volume-central (V), surface-central (D) and spin-orbit (SO) potential respectively; ⃗ℓ and ⃗σ are the orbital angular momentum and Pauli spin operator of the incident neutron All components are separated in energy-dependent well depths, VV (E), WV (E), WD (E), VSO (E) and WSO (E), and energy-independent radial parts f(r, R i , a i ) [2] The formula of total cross section in terms of phase shifts η± ℓ can be got by studying scattering theory [7]: 𝟐𝛑 ∞ − (2) 𝛔𝐭𝐨𝐭 = 𝟐 ∑ [(𝓵 + 𝟏)𝐑𝐞(𝟏 − 𝛈+ 𝓵 ) + 𝓵𝐑𝐞(𝟏 − 𝛈𝓵 )] 𝐤 𝓵=𝟎 where the two phase shifts η± ℓ are determined separately as [4]: 𝟐𝐢𝛏 𝛈± 𝓵 =𝐞 𝐟𝓵± − 𝚫𝓵 + 𝐢𝐒𝓵 𝐟𝓵± − 𝚫𝓵 − 𝐢𝐒𝓵 | (3) 𝐫=𝐑 with ISBN: 978-604-82-1375-6 248 Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM ( 𝐟𝓵± = 𝐑 𝐝𝐮± 𝓵 ) 𝐝𝐫 (4) 𝐮± 𝓵 [ ]𝐫=𝐑 𝐆𝓵 (𝐫) − 𝐢𝐅𝓵 (𝐫) 𝐞 = | 𝐆𝓵 (𝐫) + 𝐢𝐅𝓵 (𝐫) 𝐫=𝐑 𝐆𝓵 𝐆𝓵′ + 𝐅𝓵 𝐅𝓵′ 𝚫𝓵 = 𝐑 | 𝐆𝓵𝟐 + 𝐅𝓵𝟐 𝐫=𝐑 𝐆𝓵 𝐅𝓵′ − 𝐅𝓵 𝐆𝓵′ 𝐒𝓵 = | 𝐆𝓵𝟐 + 𝐅𝓵𝟐 𝐫=𝐑 𝟐𝐢𝛏 (5) (6) (7) where Gℓ (r) = krnℓ (kr) and Fℓ (r) = krjℓ (kr), nℓ (kr) and jℓ (kr) are correspondingly spherical Bessel functions and spherical Neumann functions, u± ℓ are inner radial wave functions corresponding to parallel and anti-parallel ⃗ ⃗σ interactions ℓ The equation (4) shows that we need to solve inner radial wave functions to calculate the total cross-section in equation (2) In this work, we apply the first Born approximation to get the inner wave functions and the results are given as [3]: 𝛍𝐂𝓵 𝐑 ′ ± (8) ∫ 𝐫 𝐕𝐞𝐟𝐟 (𝐫′, 𝐄)𝐤𝐫′𝐣𝓵 (𝐤𝐫′) 𝐞𝐢𝐝𝐫 ′ ℏ𝟐 𝟎 ± (r′, E) which is Since the equation (8) is solved in spherical coordinates, we must use effective potential Veff the sum of optical potential and centrifugal potential The plus and minus sign denote correspondingly the parallel and anti-parallel spin-orbit interaction The function ei is given as: 𝐮𝓵 (𝐫)± = 𝐂𝓵 𝐤𝐫𝐣𝓵 (𝐤𝐫) − 𝐞𝐢 = 𝐄𝐢(𝟏, 𝐢𝐤𝐫 ′ − 𝐢𝐤𝐫) − 𝐄𝐢(𝟏, −𝐢𝐤𝐫 ′ − 𝐢𝐤𝐫) (9) ∞ e−t where Ei(1, z) = ∫z dt is exponential integral function t In the resonance region, the compound nucleus shows a series of sharp and well-spaced resonances as a function of energy At higher energies the resonances crowd together so closely that they cannot be resolved, and only the averaged or smeared-out cross-section is observed The individual resonances are characteristic of the particular compound nucleus concerned, but it is reasonable to hope that when they are averaged out, the result will be more characteristic of nuclear matter in general, and will vary smoothly from nucleus to nucleus Thus, it is necessary to develop the theory of this averaging, and see how the average quantities can be related to those experimentally observable This theory can be interpreted by the optical model Since in unresolved resonance region the energy is small enough for us to only consider neutron s-wave reactions, the total cross-section is given by [6]: 𝐈𝐦(𝛅𝟎 ) (10) 𝛔 ̃𝐓 = 𝟒𝛑 [𝐑𝟐 + ] 𝐤𝟐 Re(δ0 ) where k is the wave number, R2 = [ R ∫ r V(r, E)dr ℏ2 2m k Im(δ0 ) ] −[ k ] is the scattering length or radius and δ0 = −k [6] is the ℓ = optical model phase shift For the neutron interaction with matter, the scattering radius is given by: R = ro (A1/3 + 11/3 ), A is the mass number of the target However the contribution of the second term in right side of equation (10) at the low-energy (ℓ = 0) is extremely small compared to the first term The total cross-section in case of the angular momentum ℓ = 0, therefore, is given by: 𝛔𝐓 = 𝟒𝛑[𝐫𝐨 (𝐀𝟏/𝟑 + 𝟏)] 𝟐 (11) RESULTS We have calculated neutron total cross-section for: A wide mass number range from 40 to 238 in case of s-wave reactions in unresolved resonance region In the case of s-wave reactions, we choose the energy to get the Ref [10] cross-sections differently for different nuclei In particular, for nuclei whose mass number are below 200 we choose 0.5 MeV and for nuclei whose mass number are above 200 we choose 0.1 MeV because there are p-wave reactions for very heavy nuclei if we choose 0.5 MeV energy Two magic nuclei (Ni-56 and Sn-132) in energy range 1-100 MeV The results are given in Figure 1, Figure and Figure ISBN: 978-604-82-1375-6 249 Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM Cross-sections (barn) Average total cross-section for s-wave reactions 10 Calculations Ref [10] 50 100 150 200 250 Mass number Figure Average total neutron cross-section for s-wave reactions in mass number range of 40-238 DISCUSSION The calculation data for s-wave reactions in unresolved resonance region has a good agreement with the Ref [10] data for mass number range of about 40-80, about 130 - 200 and about 220 - 238, but it underestimates for nuclei which mass number region from 80 to 130 and overestimates in mass number region from 200 to 220 The most important reason for these bad agreements is that the choice of radius parameter ro has no sense We have chosen ro = 1.35 (fm), and then, the radius is R = ro (A1/3 + 1) = 1.35(A1/3 + 1) Thus, the calculation gives a simple average values In the other hand, we choose Pb-206, Pb-207, Pb-208 and Bi-209 as representatives to calculate the data in mass number region around 210 and they all have resonance region up to several hundreds of keV, thus, the calculation data deviates strongly from the Ref [10] for these mass number In general, the deviations of data of the two magic nuclei in the energy range of 1-20 MeV are about or higher than 10% These bad agreements appear due to the lack of interference effect between the wave which has traversed through the nuclear matter and the wave that has gone around [5] In total cross-section data, the interference causes oscillations which are called Ramsauer resonances [5] In fact, the interference changes the amplitude of out-going wave that goes to the detectors due to the difference between phases of the two waves, and hence, it changes the number of neutrons which are in the incident beam go to the detectors Since we can control the incident beam, the number of neutrons that are removed from the beam must be impacted on by the interference Then, there are oscillations in the total cross-section data Because we have not involved this effect, the calculation data is only the average values of the Ref [10] data Total neutron cross-section for (n+ Ni-56) Total cross-section (barn) Cal Ni-56 Ref [10] Ni-56 0 20 40 60 80 100 120 Incident neutron energy (MeV) Figure Total neutron cross-section for (n+Ni-56) ISBN: 978-604-82-1375-6 250 Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM Total neutron cross-section for (n+Sn-132) Total cross-section (barn) Cal Sn-132 Ref [10] Sn-132 0 20 40 60 80 100 120 Incident neutron energy (MeV) Figure Total neutron cross-section for (n+Sn-132) The interference happens in all neutron energy regions, but there are some good agreement data in our calculation Indeed, the calculation data of Sn-132 in region 40-100 MeV (except for the data at 65 MeV) is good agreement with the Ref [10] data although there is the first maximum of Ramsauer resonances in this region [5] This can be explained by the wave properties of neutron As the incident energy increases, the wave properties of neutron become less clearly, hence, the amplitude of first maximum is smaller than higher maxima It is also the reason why calculation data of Ni-56 in the region around its first maximum (about 50-60 MeV) does not agree well with the Ref [10] Another reason for Ni-56 in energy region around 50 MeV is the surface absorption WD is not globalized as well as the other OMP parameters [2] In very high energy region which is near 100 MeV, the relativistic effect becomes stronger However, it affects light nuclei more strongly than heavy nuclei This statement can be proved by the following equation: vC vC -βc E+E0,n -1= = (12) βc βc E0,A where vC , βc are correspondingly the velocity of central mass system in classical and relativistic cases, E0,n , E0,A are correspondingly the rest energy of neutron and target nucleus Due to equation (12), as the mass of target nucleus decreases, the ratio becomes larger than unitary Then, the classical calculation deviates larger from the relativistic one Since the relativistic effect affects light nuclei more strongly than heavy nuclei, the calculation data near 100 MeV of Ni-56 is bad, on the contrary the data of Sn-132 is good compared to Reference data [10] Another problem that we must concern is that the assumption of optical model is no longer true in high energy region which is about several tens of MeV In fact, as the incident energy increases, the direct nuclear reactions become stronger and the compound nuclear reactions can be ignored However, the calculation data for Sn-132 in energy region 40-100 MeV deviates little from the Ref [10] It is due to that we have used the phenomenological potential which is fitted in with the experimental data CONCLUSION The calculation data for s-wave reactions is good in light mass region, but it deviates strongly from the Ref [10] in heavy mass region due to the choice of radius parameter The calculation data of 1-100 MeV incident neutron is good agreement for Sn-132 in region 40-100 MeV Since we have not involved the Ramsauer effect and the relativistic effect, the data of Ni-56 in region 1-100 MeV is bad agreement with the Ref [10] The Ramsauer effect also explains why our data of Sn-132 in low energy region deviates strongly from the Ref [10] ÁP DỤNG MẪU QUANG HỌC TÍNH TOÁN TIẾT DIỆN PHẢN ỨNG NEUTRON Phan Thanh Quang, Châu Văn Tạo, Lê Hoàng Chiến, Nguyễn Điền Quốc Bảo, Phạm Minh Quân Khoa Vật lý - Vật lý Kỹ thuật, Trường ĐH KHTN, ĐHQG-HCM TÓM TẮT Trong báo này, nghiên cứu mẫu quang học áp dụng để tính toán tiết diện phản ứng neutron với hạt nhân Những tiết diện đóng vai trò quan trọng lĩnh vực ISBN: 978-604-82-1375-6 251 Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM nghiên cứu vật lý neutron Đối với báo cáo này, sử sử dụng mẫu quang học để tính tiết diện phản ứng neutron với hạt nhân bia có khối lượng nguyên tử nằm khoảng từ 40 tới 238 mức lượng tới từ MeV tới 100 MeV đồng thời so sánh, biện luận kết thu với số liệu tham khảo khác Những tính toán có liên quan đến việc sử dụng mẫu quang học trình bày công trình tới REFERENCE [1] Chau Van Tao, Vat Ly Hat Nhan Dai Cuong, Publishing house of National University, Ho Chi Minh city, (2013) [2] A J Koning and J P Delaroche, Local and global nucleon optical models from keV to 200 MeV, Nuclear Physics A, 713 (2003) 231–310 [3] David J Griffiths, Introduction to Quantum Mechanics, Upper Saddle River, New Jersey 07458, (2005) [4] John M Blatt and Victor F Weisskof, Theoretical nuclear physics, John Wiley & Sons, Inc., USA, (1952) [5] J M Peterson, Neutron Giant Resonances- Nuclear Ramsauer Effect, Physical Review, Vol 125 (3) (1962) p.955-963 [6] P E Hodgson, The optical model of elastic scattering, Oxford at the Clarendon Press, (1963) [7] Randall S Caswell, Nuclear Optical Model Analysis of Neutron Elastic Scattering for Cadium, Journal of Research of the National Bureaur of Stanrdards, Vol 66A(5) (1962) 389-400 [8] Satoshi Kunieda, Satoshi Chiba, Keiichi Shibata, Akira Ichihara and Efrem Sh Sukhovitskii, Coupledchannels Optical Model Analyses of Nucleon-induced Reactions for Medium and Heavy Nuclei in the Energy Region from keV to 200MeV, Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol 44(6) (2007) p 838–852 [9] S F Mughabghab, Atlas of Neutron Resonances, National Nuclear Data Center Brookhaven National Laboratory Upton, USA, (2006) [10] Library TENDL-2012 , https://www-nds.iaea.org/exfor/endf.htm ISBN: 978-604-82-1375-6 252