Luận án tiến sĩ Vật lý: Skyrme Hartree-Fock theory for elastic scattering and proton radiative capture reactions of CNO nucleosynthesis cycle

Pierre Descouvemont từ Dai hoc Tự do Brussels ULB vì đã cung cấp cho tôi chương trình tính toán đơn giản đầu tiên về phan ứng bắt hạt phát gamma trong quá trình tôi học Thạc sĩ, điều này

VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY

ĐẠI HỌC QUỐC GIA TP HCM

TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN

VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY

UNIVERSITY OF SCIENCE

NGUYEN LE ANH

SKYRME HARTREE-FOCK THEORY FOR ELASTIC SCATTERING AND PROTON RADIATIVE CAPTURE

REACTIONS OF CNO NUCLEOSYNTHESIS CYCLE

Speciality: Theoretical and Mathematical Physics

Code: 9440103

Reviewer 1: Assoc Prof Tran Viet Nhan HaoReviewer 2: Assoc Prof Nguyen Quang HungReviewer 3: Assoc Prof Ho Manh Dung

Independent reviewer 1: Not applicable Independent reviewer 2: Not applicable

SUPERVISORS

1 Dr Bui Minh Loc

2 Dr Nguyen Huu Nha

Ho Chi Minh City - 2024

DẠI HỌC QUỐC GIA TP HCM

TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN

NGUYÊN LÊ ANH

LÝ THUYET SKYRME HARTREE-FOCK CHO TAN XA DAN HOI VA PHAN UNG BAT PROTON PHAT GAMMA

CUA CHU TRINH TONG HOP HAT NHAN CNO

Ngành: Vật lý lý thuyết và vật lý toán

Mã số ngành: 9440103 Phản biện 1: PGS TS Trần Viết Nhân Hao

Phản biện 2: PGS TS Nguyễn Quang Hưng

Declaration of Authorship

I declare that this thesis titled “Skyrme Hartree-Fock theory for elastic tering and proton radiative capture reactions of CNO nucleosynthesis cycle” andthe work presented in it are my own No part of this thesis has been submittedfor a degree at any other university I confirm that this work was done wholly

scat-and faithfully while in cscat-andidature for the Doctor of Philosophy degree at the

University of Science, VNU-HCM under the supervision of Dr Bui Minh Loc

and Dr Nguyen Huu Nha.

Supervisors PhD Student

4.7

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+ 1/A4)/YTYVUUL „y ;

Ho Chi Minh City, May 2024

I am sincerely thankful to my supervisors, Dr Bui Minh Loc and Dr Nguyen Huu Nha, for instilling in me the fundamental importance of attaining a thor-

ough, simplified, and meticulous understanding of research while being attentive

to every intricate detail and subtlety I thank Professor Naftali Auerbach fromTel Aviv University (TAU) and Professor Vladimir Zelevinsky from Michigan State University (MSU) for their invaluable support throughout this research.

A special mention of gratitude goes to Professor Pierre Descouvemont from

Uni-versité Libre de Bruxelles (ULB) for providing the first simple computer program

of radiative capture during my Master’s studies, which laid the foundation for

my research trajectory.

I want to express my heartfelt gratitude to my colleagues at the Department

of Physics, Ho Chi Minh City University of Education (HCMUE), where I am

currently employed, for their invaluable assistance and unwavering support in

teaching, fostering an academic environment that has significantly contributed

to my professional growth at HCMUE Furthermore, I am deeply grateful to allthose affiliated with the Department of Theoretical Physics, Faculty of Physics

and Engineering Physics, Ho Chi Minh City University of Science (HCMUS),

VNU-HCM, for their guidance and contributions, which have played a crucial role in shaping my academic journey over the past three years.

il

I am incredibly honored to receive the 2023 Young Research Award from

the Vietnamese Theoretical Physics Society (VTPS) This recognition is a

sig-nificant encouragement for my scientific career I am indebted to the Master,

PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), code

VINIF.2023.TS.003, for their support during my third year as a PhD student I

would also like to acknowledge the assistance provided by the VNU-HCM

Post-graduate Scholarship Program during my second year as a PhD student in 2022.

Additionally, I am grateful for the Vallet scholarship in 2021

Lastly, I wish to convey my most profound appreciation to my family for

their unwavering and unconditional support throughout my academic pursuits.

Specifically, my gratitude knows no bounds for my mother, whose unwaveringfaith in my abilities has been an inexhaustible source of strength Her continuousencouragement and belief in my potential have been a guiding beacon, motivat-ing me to surpass my limitations and strive for excellence in every endeavor.Without her love and support, completing this thesis would not have been pos-sible I will forever be grateful for her immeasurable sacrifices and steadfast

dedication.

11

Lời cảm ơn

Thành phố Hồ Chí Minh, tháng 5 năm 2024

Tôi muốn thể hiện lòng biết ơn sâu sắc đến hai người thầy hướng dẫn, TS Bùi Minh Lộc và TS Nguyễn Hữu Nhã, vì đã truyền đạt cho tôi những hiểu biết

sâu sắc, tính cẩn thận và sự liêm chính trong nghiên cứu khoa học Tôi cũng

muốn bày tỏ lòng biết ơn đặc biệt đến GS Naftali Auerbach từ Dai hoc Tel Aviv (TAU) va GS Vladimir Zelevinsky từ Dai học Michigan (MSU) về những

trao đổi chuyên môn quý giá trong suốt quá trình nghiên cứu Một lời cảm ơn

đặc biệt cũng được gửi đến GS Pierre Descouvemont từ Dai hoc Tự do Brussels (ULB) vì đã cung cấp cho tôi chương trình tính toán đơn giản đầu tiên về phan ứng bắt hạt phát gamma trong quá trình tôi học Thạc sĩ, điều này đã đặt nền

tảng cho hành trình nghiên cứu của tôi sau này

Tôi muốn bày tỏ lòng biết ơn chân thành đến các đồng nghiệp tại Khoa Vật

lý, Trường Dai học Sư phạm Thanh phố Hồ Chí Minh (HCMUE), nơi tôi hiện

đang làm việc, vì những hỗ trợ và ủng hộ trong việc giảng dạy, xây dựng một

môi trường học thuật đóng góp đáng kể vào sự phát triển chuyên môn của tôi

tại HCMUE Hơn nữa, tôi rất biết ơn các thầy tại Bộ môn Vật lý lý thuyết,

Khoa Vật lý và Vật lý kỹ thuật, Trường Đại học Khoa học Tự nhiên (HCMUS),

ĐHQG-HCM, vì đã giảng day cho tôi nhiều kiến thức chuyên sâu, đóng vai trò

quan trọng trong hành trình học thuật của tôi trong suốt thời gian học Thạc sĩ

và Tiến sĩ.

lv

Tôi vô cùng vinh dự khi được nhận Giải thưởng Nghiên cứu Trẻ năm 2023 của

Hội Vật lý lý thuyết Việt Nam (VTPS) Sự ghi nhận này là sự khích lệ đáng kế cho sự nghiệp khoa học của tôi Tôi rất biết ơn Chương trình Hoc bổng Thạc sĩ, Tiên sĩ của Quỹ Đổi mới sáng tạo Vingroup (VINIF), mã VINIF.2023.TS.003,

vì sự hỗ trợ trong năm thứ ba làm nghiên cứu sinh của tôi Tôi cũng muốn ghi

nhận sự hỗ trợ được cung cấp bởi Chương trình Học bổng Sau đại học

ĐHQG-HCM trong năm thứ hai làm nghiên cứu sinh của tôi vào năm 2022 Hơn nữa,

tôi cũng biết ơn Học bổng Vallet vào năm 2021.

Cuối cùng, tôi muốn bày tỏ lòng biết ơn sâu sắc nhất đến gia đình của tôi

vì sự ủng hộ không ngừng và vô điều kiện trong suốt quá trình theo đuổi học

vấn của tôi Đặc biệt, lòng biết ơn không giới hạn đối với mẹ, người đã luôn tin

tưởng vào khả năng của tôi Sự khích lệ và niềm tin vào tiềm năng của tôi là

ngọn đèn chỉ dẫn, thúc đẩy tôi vượt qua những khó khăn va phan đấu không

ngừng trong mọi công việc Nếu không có tình yêu và sự hỗ trợ của mẹ, việc

hoàn thành luận án này không thể thành hiện thực Tôi sẽ mãi biết ơn vì sự hi

sinh của mẹ

1 Radiative capture reactions in stars 9

1.1 Radiative capture reactions 200000004 9 1.2 Potential model for radiative capture reaction 17

121 Phenomenological potential 17

1.2.2 Example for proton-deuteron radiative capture 20

2 Consistent microscopic approach for radiative capture reactions 23

vi

2.1 Skyrme Hartree-Fock calculation for bound state

2.2 Skyrme Hartree-Fock calculation for scattering state

2.3 Bound-to-continuum potential model for radiative capture reaction 38

3 Results and discussions 41

3.1 Low-energy elastic scattering and s-state resonances 42

3.1.1 Simple test cases: °O(p,p) and !C(p,p) 42

3.1.2 !°Be(p,p) and the missing resonance statein 4B 46

3.1.3 !*!15O(p,p) and effect of spin-spin potential 48

3.2 Proton radiative capture processes 2.000 ol

3.2.2 BC(p,y)MN 2 ee 57

3.2.3 MN(p,7)PO 2 aI 61

3.2.5 Applications to !2C(n,+)!12C and O(n, yO 2 69

Conclusions and future perspectives 75

Appendix C Link between Skyrme Hartree-Fock and relativistic

mean-field models 113

Appendix D Tables of numerical values for relevant nuclear

reac-tion rates 117

vill

Name of PhD Student: Nguyen Le Anh

Academic year: 2020-2023 (Extended to 12/2024 due to COVID-19 pandemic)

Supervisor: Dr Bui Minh Loc and Dr Nguyen Huu Nha

At: VNUHCM - University of Science

1 Summary

The thesis focuses on using the framework of Skyrme Hartree-Fock in

con-tinuum to analyze low-energy proton-induced reactions, particularly radiative capture reactions at energies relevant to nuclear astrophysics These reactions

are of great importance in nuclear astrophysics The study uses the continuum potential model to examine cross sections The work includes the

bound-to-determination of both single-particle bound and scattering states to calculate

electromagnetic transition matrix elements The low-energy behavior of theastrophysical S factor is extracted and compared with previous studies Theresearch unequivocally illustrates that the Skyrme Hartree-Fock framework is

a highly effective tool for examining radiative capture reactions at stellar

ener-gies In addition, the approach is shown to accurately reproduce the excitationfunctions of proton elastic scattering at the energies near the proton-emissionthreshold, demonstrating the applicability of the Skyrme Hartree-Fock in con-

tinuum approach.

1X

2 Novelty of thesis

The thesis introduces a microscopic and consistent approach for both energy proton elastic scattering and radiative capture reactions of the CNOnucleosynthesis cycle The proposed framework can offer a practical approach

low-to explain nuclear scattering at the energies near the prolow-ton-emission threshold

with minimal experimental input.

3 Applications/ Applicability /Perspective

In this thesis, a computer program has been meticulously developed to

per-form comprehensive calculations of radiative capture reactions, accounting forboth £1 and M1 transitions The program uses the framework of SkyrmeHartree-Fock, developed in the 1970s, enabling the reproduction of highly ac-

curate and intricate nuclear reaction calculations It is important to highlight

my contribution to this program, as the landscape of available tools for

radia-tive capture reactions is relaradia-tively sparse Existing programs are often vast andcumbersome, or the omission of M1 transition calculations Our goal is to offer

an invaluable resource for researchers seeking highly accurate calculations of the

cross sections in radiative capture reactions.

Our theoretical calculation of radiative capture cross sections plays a

cru-cial role in understanding stellar nucleosynthesis, nuclear astrophysics, and the

synthesis of elements in various astrophysical environments Furthermore, thethesis contributes significantly to the enrichment of nuclear data evaluations

Building upon the foundations laid out in this study, several further works can

be carried out Firstly, radiative capture reactions involving exotic and unstable

nuclei will be investigated using the same Skyrme Hartree-Fock framework Thisextension can shed light on the role of nuclear structure effects in astrophysical

scenarios involving rare isotopes Additionally, integrating our precise cross

sec-tion predicsec-tions into astrophysical network calculasec-tions represents a pivotal step

toward enhancing our understanding of nucleosynthesis across a range of physical environments Furthermore, the continuous development of advancedcomputational techniques and the incorporation of high-performance comput-ing resources may enable even more intricate and comprehensive simulations,pushing the boundaries of our knowledge in nuclear astrophysics Overall, this

astro-research not only addresses critical questions in the field but also paves the way

for continued innovation and discovery in the realms of nuclear physics and

as-trophysical modeling.

x1

TRANG THONG TIN LUẬN AN

Tên đề tài luận án: Lý thuyết Skyrme Hartree-Fock cho tán xa đàn hồi và phan

ứng bắt proton phát gamma của chu trình tổng hợp hạt nhân CNO

Ngành: Vật lý lý thuyết và vật lý toán

Mã số ngành: 9440103

Họ tên nghiên cứu sinh: Nguyễn Lê Anh

Khóa đào tạo: 2020-2023 (Gia hạn do COVID: 12/2024)

Người hướng dẫn khoa học: TS Bùi Minh Lộc và TS Nguyễn Hữu Nhã

Co sở đào tạo: Trường Dai hoc Khoa học Tự nhiên, DHQG.HCM

1 TÓM TẮT NỘI DUNG LUẬN ÁN

Luận ấn tập trung vào việc sử dụng phương pháp Skyrme Hartree-Fock cho

vùng liên tục để phân tích các phản ứng bắt proton tại năng lượng thấp, đặc

biệt là phản ứng bắt hạt phát gamma ở các năng lượng liên quan đến vật lý

thiên văn hạt nhân Những phan ứng nay có vai trò đặc biệt quan trọng trong

vật lý thiên văn hạt nhân Nghiên cứu này liên quan đến việc sử dụng mô hình

thé cho vùng liên tục để tính toán tiết diện phản ứng Các tính toán các trạng thái liên kết và trạng thái tán xạ đơn hạt được xác định để tính toán các phần

tử ma trận chuyển dịch điện từ Hệ số thiên văn S tai năng lượng thấp được

trích xuất và so sánh với các nghiên cứu trước đây Luận án chứng minh rằng

phương pháp Skyrme Hartree-Fock là một công cụ hữu ích để khảo sát các phản

ứng bắt hạt phát gamma ở năng lượng trên các ngôi sao Ngoài ra, phương

pháp nay cũng được chứng minh là có thé mô tả chính xác các hàm kích thích

của tán xạ đàn hồi proton ở mức năng lượng gần ngưỡng phát proton, chứng td

khả năng ứng dụng của phương pháp Skyrme Hartree-Fock trong cách tiếp cận

xii

vùng liên tục.

2 NHỮNG KẾT QUẢ MỚI CỦA LUẬN ÁN

Luận án giới thiệu một cách tiếp cận vi mô và nhất quán để nghiên cứu cả

phản ứng tán xạ đàn hồi proton và phản ứng bắt proton phát gamma thuộc chu

trình CNO tại năng lượng thấp Phương pháp này được đề xuất có thể sử dụng

để giải thích các quá trình tán xạ hạt nhân ở mức năng lượng gần ngưỡng phát

proton với ít sự phụ thuộc vào thực nghiệm.

3 CAC UNG DUNG/ KHẢ NANG UNG DUNG TRONG

THUC TIEN HAY NHUNG VAN DE CON BO NGO CAN TIẾP TỤC NGHIÊN CỨU

Trong quá trình thực hiện luận ấn, chúng tôi đã phat triển chương trình tính

toán chi tiết để thực hiện các tính toán về tiết diện của các phản ứng bắt hạt phát gamma, bao gồm các chuyển dịch lưỡng cực điện từ El và M1 Chương trình sử dụng phương pháp Skyrme Hartree-Fock được phát triển vào những

năm 1970, cho phép mô tả các tiết diện phản ứng hạt nhân phức tạp và có độ chính xác cao Đây được xem là đóng góp thực tiễn quan trọng vì số lượng các công cụ có sẵn hiện nay dành cho phân tích phản ứng bắt hạt phát gamma

tương đối ít Các chương trình hiện tại thường rất nặng và cồng kềnh hoặc thiếu

các tính toán chuyển dịch M1 Mục tiêu của chúng tôi là cung cấp một nguồn

tài nguyên có giá trị cho các nhà nghiên cứu đang tìm kiếm những tính toán có

độ chính xác cao về tiết diện của các phản ứng bắt hạt phát gamma.

Các tính toán lý thuyết của chúng tôi về tiết diện của các phan ứng bắt hạt

phát gamma đóng một vai trò quan trọng trong việc tìm hiểu quá trình tổng hợp hạt nhân trên các ngôi sao, vật lý thiên văn hạt nhân và sự tổng hợp các

nguyên tố trong các môi trường vật lý thiên văn khác nhau Hơn nữa, nghiên

xii

cứu của chúng tôi góp phần đáng kể vào việc làm phong phú thêm cho dit liệu

hạt nhân.

Dựa trên những gì đã đạt được trong nghiên cứu này, một số nghiên cứu tiếp

theo có thể được thực hiện Dầu tiên, các phản ứng bắt hạt phát gamma liên quan đến hạt nhân lạ và không ổn định sẽ được nghiên cứu bằng cách sử dụng phương pháp Skyrme Hartree-Fock Phần mở rộng này có thé làm sáng tỏ vai

trò của các hiệu ứng cấu trúc hạt nhân trong các kịch bản vật lý thiên văn liên quan đến các đồng vị lạ Ngoài ra, việc tích hợp các tiết diện tính toán của

chúng tôi vào các tính toán mạng lưới vật lý thiên văn thể hiện một bước đi quan trọng nhằm nâng cao hiểu biết của chúng ta về quá trình tổng hợp hạt nhân trong các môi trường vật lý thiên văn Hơn nữa, sự phát triển liên tục của

các kỹ thuật tính toán tiên tiến và việc kết hợp các tài nguyên máy tính hiệu

suất cao có thể cho phép các tính toán mô phỏng phức tạp và toàn diện hơn

nữa, mở rộng ranh giới kiến thức của chúng ta về vật lý thiên văn hạt nhân Nhìn chung, nghiên cứu này không chỉ giải quyết các câu hỏi quan trọng trong

lĩnh vực này mà còn mở đường cho sự đổi mới và khám phá liên tục trong lĩnh

vực vật lý hạt nhân và mô hình vật lý thiên văn.

XIV

Schematic diagram of nuclear radiative capture reaction and

di-rect mechanism 2 c2

Gamow peak illustrated in the case of !2C(p,+)!1NÑ at 7 = 109 K .

Astrophysical S factor of proton-deuteron radiative capture The

experimental data are from Refs |40-4.4]

Effective mass m*/m for protons (q = 1) and neutrons (q¢ = 0) in

the closed-shell nuclei with SLy4 force

Neutron and proton distributions of !©O, 4°Ca, 48Ca, Zr, !132S%n, and ?°8Pb predicted using the HF method with the SLy4 force Calculated elastic cross section for n + !8O as a function of the

neutron energy in center-of-mass frame Force SkM* is

repre-sented by a dashed line, while SLy4 is shown as a solid line

Computed phase shifts ổ¿; of dạ; and ƒ;/¿ for n + '°O using SLy4

XV

32

36

Selected low-lying states and proton-decay thresholds (all in MeV)

in 1N, E, HB, and 151%Ƒ, The proton-decay channels from the

JTM states caused by s-wave resonances are shown in dashed arrows 43

Elastic scattering excitation functions of !2C(p,p)!2C at 6 = 170°

with and without adjustment The experimental data are

ob-tained from Ref |69| Ặ 000.0002 0000.

Elastic scattering excitation functions of !°O(p, p)!®O at scattering

angle Ø = 170° within the energy range of 1-2.5 MeV The Skyrme

HF calculations are performed with adjustments of +5% compared

to the original calculation without any adjustment The available

experimental measurements are from Ref [72] .

Evidence of the s-wave resonance occurring in the excitation

func-tion for °Be(p,p)!°Be at a scattering angle of 6 = 180° within

Skyrme HF theory The solid line represents the best fit of the

resonance located at Eyes, = 182 keV The data points in the recent

measurement are from Ref [75] 0004Analysis of the s-state resonance at 1.27 MeV in the excitation

function of “O(p,p)O The differential cross sections calculated

with NM, = 1.02 are in agreement with the experimental data

ob-tained from Refs [90,91] 02.000.0.

The excitation function of !5O(ø,p)!5O below 1000 keV exhibiting

two s-state resonances For the calculation of these resonances,

N,(J = 0) = 0.99 and N,(J = 1) = 0.97 were adopted The mental data are from Ref [90|

experi-Xvl

3.7 The level schemes of tN, !4N, 5O, and !’F near the proton

thresh-old (dashed lines) The solid arrows indicate the considered #1

transitions to bound states The dashed arrows represent other

potential £1 transitions Irrelevant states have been omitted for

the sake of clarity [29) 2 2 ee ee

3.8 Theoretical results for the reaction !2C(p,+)!#NÑ in comparison

with the experimental data obtained from Refs [97-99, 101].

3.9 Reaction rate of !2C(p, y)!8N in comparison with NACRE II

com-pilation [14], 2 ee

3.10 The astrophysical S factor of the !3C(p,+)!14NÑ reaction for the

transition to the ground state The experimental data used in

this analysis are from Refs [105-107]

3.11 The astrophysical Š factor of the !C(p,y)!#Ñ* reaction for the

transitions to the excited states at 2.31 MeV (a) and 3.95 MeV(b) The experimental data are taken from Ref [107] .

3.12 Reaction rate of !3C(p, y)'4N in comparison with NACRE II

com-pilation (14) 2 HH kg va

3.13 The partial-wave analysis of the '4N(p, y)!°O reaction for the

tran-sitions to the ground state (1/27) The experimental data aretaken from Refs [II1,116,117]

3.14 The calculated astrophysical S$ factor (in black) of the 4N(p, y)!’O*

reaction for the transitions to the excited state (3/27) at 6.18MeV The measured data are taken from Ref [111] .

3.15 The astrophysical $ factor of the '4N(p,7)!°O* reaction for the

transitions to the excited state (3/27) at 6.79 MeV The data are taken from Ref [111] 0.000.000.0000 000 000.0.

The total astrophysical 9 factor of the O(p,7)!"F reaction for

both transitions to the ground state (5/2T) and the first excited state (1/2) at 495 keV The data points are measured fromRefs [28, 120,121,123,124] The data source is taken from Ref [125] 68

Result in Ref [33] for cross sections of the C(n,7)8C reaction

including transitions to the ground state (a) and the excited states (b), (c), and (d) The data points are from Refs [133,134] 71

The calculated cross section from Ref [33] of the O(n, 7)!7O

reaction including transitions to the ground state (5/2+) and the

excited states (1/2T) in O The measured data are from Ref [136] 72

The reaction rates of the proton radiative capture in CNO cycle 117

XVill

List of Tables

2.1 Parameters of the Skyrme interactions 26

2.2 The ground-state calculations of root-mean-square radii (in fm)

and total binding energies per nucleon (in MeV) using interactionsSkM*, SLy4, and SAMI The experimental charge radii r¿ aretaken from Ref [62] The experimental values of £/A are obtained from Refs [45,46] 002.000.000.002 0 000 312.3 The single-particle energies (in MeV) for protons (g = 1) and

neutrons (q = 0) obtained using SLy4 force The spacings refer to

the gaps relevant to magic numb@rs 33

3.1 The main configurations and s.p transitions in the calculation of

resonances MN, and MN are the scaling factors for the scatteringand bound potentials, respectively The resonance energies Eyes,for scattering states and proton separation energies e„ for boundstates are in MeV The HF s.p energies eyp corresponding to

Ny = 1 are in MeV The Sp is the spectroscopic factor 54

3.2 Calculated $(0) values (in units of keV b) of 4N(p,y)!°O in

com-parison with the values from SCH87 [111], ANGO1 [112], IMB0ð [113],RUNOS [114], and MUK03 [Hö| 62

D.1 Reaction rates in cm? mole~! s~! from 0.017) to 0.25Ty 118

XIX

D.2 Reaction rates in cm? mole~! s~! from 0.267 to 0.507ạ.

D.3 Reaction rates in cm? mole~! s~! from 0.5179 to 0.7579.

D.4 Reaction rates in cm® mole! s~! from 0.767g to 1.007.

XX

AGB Asymptotic Giant Branch

BBN Big Bang nucleosynthesis c.m center of mass

At first sight, the atomic nucleus is a two-superfluid quantum many-bodysystem consisting of a finite number of protons and neutrons that have similarmasses, are different in charge, and are known as nucleons The properties ofatomic nuclei can be, therefore, described by quantum many-body techniques

in theoretical physics such as second quantization, canonical transformation,Green’s functions, and the independent-pair approximation These are the same

techniques that are applied to other many-body systems, such as atoms and

molecules However, two main properties distinguish nuclear physics from otherquantum many-body systems

The first is the nature of the nuclear force While the theoretical framework

of quantum chromodynamics (QCD) provides the foundation for understanding

the nuclear force, obtaining precise quantitative predictions for the force directly

from QCD remains a significant challenge Nowadays, although ab initio

calcu-lations have been strongly developed [1], phenomenological models have often

been used in practice Phenomenological models incorporate experimental data and observations to parameterize the nuclear force, allowing for more straight-

forward calculations and predictions In addition to the nuclear strong force,there is always the presence of the nuclear weak force and the Coulomb forcebetween protons that also affect nuclear structure studies

Second, simply describing atomic nuclei as a two-superfluid quantum body system does not capture the full complexity and variety of behaviors ofatomic nuclei For example, while there are some similarities between superfluid

many-behavior and certain aspects of nuclear physics, such as the pairing of protonsand neutrons, it is important to note that the behavior of particles within theatomic nucleus is more complex and diverse than that of a traditional superfluid.While both protons and neutrons can exhibit superfluid behavior individually

in certain contexts, it is not accurate to consider the nucleus as two separate

superfluids In general, nuclear many-body calculations are incredibly complex.

Regarding approximations in nuclear physics, there are two pillars: the consistent mean-field theory and the configuration interaction nuclear shell model,

self-or the shell model fself-or shself-ort The main difference between these two methods lies in their underlying approximations and the type of information they provide.

The self-consistent mean-field theory views the nucleus as a system of nucleonsinteracting through an average potential This approach provides informationabout the overall properties of the nucleus, such as its energy, size, and shape.The self-consistent mean-field theory is particularly useful for describing groundstates, low-lying excited states, and nuclear giant resonances The interactingshell model presents the many-body Schrédinger equation as a matrix eigen-

value problem The shell model is particularly useful for understanding the

spectroscopy of nuclei and their responses to external probes It is emphasized

that in both approaches, the effective nucleon-nucleon (NN) interaction is

es-sential and has to be carefully specified at first sight The calculation method

in the thesis is based on the mean-field theory with the effective interactionbeing the Skyrme force named after the British physicist Tony Skyrme, who

introduced it in the 1950s [2] The application of Skyrme force was introduced

to the nuclear physics community in the 1970s [3] A brief discussion of widely

used effective NN interactions was provided in nuclear textbooks, including the

notable Ref [4].

The Skyrme force is an effective, phenomenological, and mean-field imation of the nuclear force It assumes that the nucleons interact through

approx-a density-dependent, locapprox-al, approx-and effective potentiapprox-al The theory does not tapprox-ake

into account the explicit exchange of mesons (such as pions) as in other, more

traditional models of the nuclear force The Skyrme force has been used in avariety of nuclear physics applications and it is particularly successful in semi-

classical calculations and mean-field theories (see [5] for a review) Over the

years, various parametrizations of the Skyrme interaction were proposed to fit

experimental data and reproduce nuclear properties, such as binding energies,

radii, and excitation spectra, across a wide range of nuclei The earliest Skyrme

force was introduced in Ref [6], while the most recent was the SAMi interaction [7] A list of over 200 Skyrme forces can be found in Ref [8].

With a successful microscopic description, along with numerous applications

in everyday life, nuclear physics has a significant impact on the studies of the

universe Nuclear science, astronomy, and astrophysics are inextricably

inter-twined Nowadays, an interdisciplinary physics was formed and well-known asnuclear astrophysics, which involves close collaboration among researchers invarious subfields of both nuclear physics and astrophysics Indeed, astrophysics

has long relied upon nuclear physics, both within the scope of experiments and

theoretical models Notably, the field involves treating a diversity of atomicnuclei, each situated within highly distinctive environments that significantlyinfluence their static properties, the range of their transformative processes, andthe associated probabilities of these transformations Tackling the challenges

within nuclear astrophysics highly demands close collaboration between physicists and nuclear physicists.

astro-The question of the origin of elements holds a pivotal place in science Howdid the universe transform from a composition primarily consisting of hydrogen

and helium, with only slight traces of lithium, to a domain characterized by the

remarkable chemical diversity of elements? These elements are considered the

foundational constituents for both planets and life itself Nuclear astrophysics

aims to unravel the origin of the chemical elements and isotopes, along with therole of nuclear energy generation, in cosmic sources such as stars, supernovae,

novae, and violent binary-star collisions.

Nuclear reactions occurring in the early universe and within stars are crucial driving forces for the evolution of our universe The basic concept that the

energy generation within the Sun and other stars results from the outputs of

nuclear reactions was initially proposed in Ref [9] These reactions take place via the mechanisms known as the “proton-proton (pp) chain” and the “Carbon-Nitrogen-Oxygen (CNO) cycle” For example, the energy origin of the Sun was

attributed to reactions of the pp chain culminating in the fusion of four protons

into a nucleus of He, often referred to as “burning” On the other hand, the CNO

cycle (see Fig 1) is the primary nuclear fusion process that powers stars alongside

the pp chain The CNO cycle is dominant in massive stars with higher coretemperatures and pressures than our Sun Both in the pp chain and the CNOcycle, radiative capture reactions hold a significant role These reactions involvethe fusion of an atomic nucleus with one or more nucleons or nuclei, accompanied

by the emission of electromagnetic radiation Furthermore, radiative capture reactions are crucial in explosive phenomena such as novae, X-ray bursts, and

supernovae [10] They contribute significantly to the understanding of these

astrophysical events and their impact on stellar evolution and element synthesis

The precise determination of radiative-capture cross sections is a fundamental

problem in astrophysics [11-14] Radiative capture reactions with the positive

Q value make knowledge of reaction rates essential for determining reactionpathways and energy release The electromagnetic processes are much slowerthan those due to the strong interactions This assigns them the role of rate-limiting steps in several reaction pathways and cycles As a result, they often

govern the flow of reactions and the pace of nucleosynthesis For the study of radiative capture reaction at astrophysical energies, one needs to describe the

Figure 1 The diagram of the CNO bi-cycle for hydrogen burning.

state of the nuclear system before and after the capture process It requires both

experimental measurement and theoretical calculation.

In experiments, measuring low-energy radiative capture reactions is

challeng-ing due to their extremely small cross sections [15,16] This arises because the

thermonuclear energies relevant to astrophysical phenomena are considerably

lower than the Coulomb and centrifugal barriers At these energies, reactionrates frequently prove too small for practical measurement through accelerator

experiments Theoretical insights are necessary to estimate relevant stellar

en-ergies Extrapolation of experimental radiative capture reaction cross sectionsdownward from accessible to astrophysical energies relies on theoretical models

[17-19]

In theories, the radiative capture process is mainly associated with the

elec-tromagnetic transitions from the scattering states to the bound states Thisprocess is more complicated than usual transitions in bound spectra This com-plexity arises due to the continuum state being not square-integrable and energy-dependent The continuum state may involve resonances at energy levels in the

structure of the final nucleus To microscopically describe this process, it iscrucial to rely on solutions derived from the nuclear many-body problem for the

involved states within the nucleon-nucleus or nucleus-nucleus system It is thus

essential to have a comprehensive physical understanding of the mechanism ofthe radiative capture reactions

The basic ingredient of the calculation of cross sections contains the overlapbetween a scattering wave function describing the collision of two nuclei and thebound-state wave function of the nucleus created by the capture process Thereare different theoretical models which have been developed including R-matrix

theory [20,21], potential model [22], and Gamow shell model [23-25] Among

them, the potential model has been demonstrated as a convenient and powerfultool for examining radiative capture reactions, particularly those that involve

light nuclei [22,26] In the present thesis, the radiative-capture reactions are

studied in the potential model at very low energies, including energy zero Themost straightforward approach to the potential model is using the phenomeno-

logical potentials for scattering and bound states, which are assumed to have

simple forms such as Woods-Saxon (WS), harmonic oscillator, or even a squaredwell [22,27,28] Modern nuclear physics nowadays studies not only stable nuclei

but also exotic nuclei that have unique or unusual properties compared to

sta-ble or commonly encountered nuclei Exotic nuclei often exist for only a short

lifetime or under specific conditions The need for a microscopic and consistent

calculation has naturally arisen

In this thesis, we introduced a new microscopic and consistent approach to

the potential model for (p,y) reactions Despite being a method with a 50-year

history, the Skyrme Hartree-Fock (HF) formalism employed in nuclear structure

studies proves remarkably effective in the investigation of (p,+) reactions Our

result reproduced excellent experimental data and provided a benchmark for extrapolation to zero energy Not only did we successfully reproduce the exper-

V (MeV) ,

continuum

Scattering statesThreshold l[F===zzzzz7

Bound statesdiscrete

Figure 2 Schematic illustration of bound-to-continuum approach for very energy nuclear reactions

low-imental data, but we also uncovered significant nuclear structure information using the bound-to-continuum approach The results presented in the thesis are

drawn from Refs [29-31] The organizational structure of the thesis is outlined

below, highlighting the key contributions to the field

To begin, Chapter 1 provides an introduction to nuclear radiative capture

reactions in stars Our focus centers on determining electric dipole (E1) sitions, which prevail over magnetic dipole (/1) or higher-order transitions in

tran-(p, y) reactions The coupling representation for the potential model is detailed

in Refs [29,30] Specifically, in Ref [29], we conducted a comprehensive analysis

of E1 transitions for (p,) reactions of the CNO cycle The examined reactions

are (p,y) on 122C, !N and !°O in Ref [29] The keV-nucleon radiative capture

on 12C and !°O were revisited in Ref [30].

Moving forward, in Chapter 2, we introduce an approach named the

bound-to-continuum potential model for radiative capture reaction (see Fig 2) The

approach is based on the Skyrme HF calculation for nuclear bound and scattering

state The single-particle (s.p.) states, which are the outputs of the Skyrme

HF calculation, are used to calculate the astrophysical S$ factor of the nucleonradiative capture reactions within the potential model The standard formalism

is detailed in Ref [30], where we developed a comprehensive Skyrme HF model for both bound and scattering states in (p, 7) reactions Additionally, in Ref [32],

we successfully reproduced the scattering state through elastic scattering on

2C and oxygen isotopes (14°O) It is crucial to highlight the significance of

resonances in proton elastic scattering on 12C and '*°O in relation to the 2s; /2

resonance near the threshold.

Finally, Chapter 3 presents our results based on the mean-field approach,

providing a consistent microscopic description for the nucleon radiative capture

reaction at stellar energy The success of our approach is illustrated by theexcellent results of the proton radiative capture reaction in the first CNO cycle

[29] The neutron radiative capture reactions at keV energy, which is moredifficult in experimental measurement, are also well described [33] All resonanceand non-resonant electromagnetic transitions are described consistently [30] In

addition, our calculation also solves the 20-year question about the near

proton-emission threshold resonance in !!B [31].

nu-in pure nuclear physics, especially nu-in nuclear astrophysics In this chapter, we

present the general methodology for calculating rates of radiative capture

reac-tions, proceeding through electric dipole transition processes We analyze the

radiative capture processes using a simplified two-body approximation for the

system composed of the particle and the target.

1.1 Radiative capture reactions

A radiative capture process involves an electromagnetic transition from ascattering state to a bound state, as illustrated in Fig 1.1 Consider a radiative

capture reaction in which an incoming particle (a) is captured by the target (A)

a+A>+B+y or A(a,+)B, (1.1)

Target Projectile Daughter nucleus + tay

where excited final/daughter nucleus B is formed and the + ray is immediately

radiated The nuclei a and A with charges Z¡e and Z2e with e being elementarycharge and masses A; and Ag, respectively, scatter in a reaction at a relativevelocity v, the energy corresponding to their relative motion is determined as

[34]

5“ am” (1.2)

where ñ is reduced Planck constant, is the reduced mass of the nuclei, and k

is wave number We will begin by considering the capturing charged particles

where Z¡Z2 #0.

In the case of charged particles, the Coulomb repulsion causes a barrier, whilereactions occur below the maximum Coulomb barrier energy The approximatemagnitude of the penetration probability through the Coulomb barrier is deter-

mined by the Gamow factor e~?"”, where the Sommerfeld parameter representing

10

the s-wave (¢ = 0) barrier penetration reads as [34]

_ Z\ Zoe _ En

)= = \j DR (1.3)

The alternate formulation of 7 can be expressed through the nuclear Rydberg

energy Ey and the fine structure constant a = e2/(ñe) [34]

As the energy approaches zero (EF — 0), the Sommerfeld parameter behaves

proportionally to 1/VE As the energy # (or k) decreases, the Gamow factor

e-?TM shows a rapid drop Therefore, it cannot be expanded around zero energy

in the case of charged particles

As the energy approaches zero, both reaction cross sections and radiativecapture cross sections z(#) exhibit a significant decrease similar to the Gamowfactor It is also not good to perform an expansion around zero energy It is

customary to use the energy-dependent astrophysical S factor defined as [35]

S(E) = EeTM"o(E), (1.6)

which shows a significantly slower rate of change at low energy, especially when

not affected by resonances Therefore, S(£) can be expanded around E = 0 as

11

S(E) = S(0) + 5(0)E+ 35(0)E? fee, (1.7)

This property is important in investigating the behavior of the astrophysical Sfactor at low energy

In stellar models, a significant quantity of interest is the reaction rate (cv), which represents the average value of the product of the cross section and the

relative velocity The investigation of stellar evolution requires a wide range

of reaction rates for various reactions [35] Considering the star as a gas in

equilibrium, the energy distribution is determined by the Maxwell-Boltzmannfunction Consequently, the reaction rate is influenced by the temperature T of

the star [34].

When considering models of the chemical evolution of certain astrophysicalsystems, vital information is provided by the reaction rate per pair of particles.The astrophysical reaction rates per particle pair at a specific temperature T

can be calculated with [11, 12,35]

8 1 °° E

(ov) = Vàmmm | S(E) exp -= — 2mi(P) dE, (1.8)

where kg is the Boltzmann constant Notice that the astrophysical $(£) factor

varies slowly at low energy The exponential function in the integrand is known

as the Gamow window function, which consists of two rapidly varying factors

The Gamow window function can be expressed as [11, 12]

g(E) = exp (-sn) exp (—277) = exp (ấn) exp +» 2] ; (1.9)

which can be seen in Fig 1.2 as an illustrative example for !2C(p,+)!13Ñ reaction

at the temperature of 109 K The Gamow peak, often referred to as the most

12

Probability (arb, uniits)

0.01 0.1 1

Figure 1.2 Gamow peak illustrated in the case of !?C(p,+)!13NÑ at 7 = 10° K.

efficient energy, can be found at the energy given by [11, 12]

Additionally, it is worth noting that the position of the peak in the integrand,

as stated in (1.8), may also have a slight dependence on $(F£) The Boltzmann

constant relates the average kinetic energy of particles The energy of 1 MeV is

equivalent to the temperature of 1.16 x 10!° K Therefore, it is convenient to be

expressed in MeV as a function of temperature in billions of degrees, denoted as

Ty = T/109 K [11, 12, 14]

Eụ © 0.122(Z?Z30)!/3T?/” Mev, (1.11)

This energy can be compared to the energy of the Coulomb barrier (Ez)

To determine the energy of the Coulomb barrier, a model can be used where

13

the target nuclei and the projectile are assumed to be in contact, such that

the distance rc is taken as the sum of the radii of the projectile and the targetnucleus Within this model, the energy of the Coulomb barrier can be calculated

as [34]

e 122 _„ 0.9621242 MeV © 0 Ep= `

where the Coulomb radius zc + 1.4 x (At? 4.43%) (in fm) is used For Ty < 1, the

capture with the energy of Ep occurs significantly below the Coulomb barrier

Ep.

The reaction rate in (1.8) depends on the coefficient of g(£o) and the value of

S(£o) In general, the reaction rate is influenced by the behavior of $(£) within

the Gamow-peak region The main goal is to calculate the astrophysical S factor

for radiative capture reactions at extremely low energies To accomplish this, it

is necessary to calculate the precise cross section at low-energy regions.

The general radiative capture cross section ø(#) in Eq (1.6) can be provided

by the literature of NACRE collaboration [13,14] When describing nuclear

reac-tions of astrophysical significance, several theoretical models focus on describingthe behavior of cross sections at low energies, such as the phenomenological R-

matrix method and potential models The potential models rely on certain

phys-ical parameters, such as nucleus-nucleus or nucleon-nucleon interactions, whichcan be determined based solely on experimental observations The cross sec-tion is determined by calculating the matrix elements of the transitions between

two states using perturbation theory, wherein the electromagnetic operators are sandwiched within the long-wavelength approximation In the present thesis,

we focus on the formalism for the radiative capture of a nucleon, particularly

an incident nucleon such as a proton or a neutron captured by a target nucleuswith the mass number A and charge number Z

14

The scattering (initial) state is denoted as |[ƒ @ (£¿ ® s)js|JsMs), while thebound (final) state is |[J @ (4 ® s)j»|JpMy) The system is assumed as the core (target nucleus) with one additional nucleon captured into a s.p state Within

the potential model, the internal structure of the interacting nuclei is essentially

neglected The intrinsic spins of the core (J) and incident nucleon (s = 1/2)

are therefore kept unchanged in the calculation The total relative angular

momentum of the system is ji = ¿+ 8 with 6 being the relative orbital angular momentum (i = s,b) The channel spin is a result of coupling J; = 1+ j; The

radiative capture cross section for the transition to a certain bound state is

therefore written as [14, 22, 28]

41 (4m 1

Onlejode (E) ( 3 = 2 — 3hø 15) xxTJBF+T), 2 Meal (1.13)sJs2s;

where the y-ray wave number is defined using energy conservation [22]

— E+Q-Ex

as a function of bombarding energy E The Q value of the nuclear reaction

and the excitation energy E, of the daughter nucleus can be determined byexperimental measurements The state of the daughter nucleus in the bound

state has the energy E above the elastic threshold In Eq (1.13), the reduced

matrix elements Mg, with Q = E or 0 = M present the electric or magnetic

transitions of a specific photo-emission multipolarity À (A = 1 for dipole sitions, \ = 2 for quadrupole transitions, ) In the case of radiative capture,

tran-the energy of tran-the emitted photons is typically smaller than 10 MeV, allowing

us to use the long-wavelength approximation It is emphasized that #2 and M1

transitions typically contribute less to the transition probabilities compared toE1 transitions Several works have indicated that the contribution of M1 can

be considered negligible in the low-energy region, which plays a crucial role in

15