Relativistic Hartree-Fock model of nuclear single-particle resonances based on real stabilization method

Yang Wei; Ding Shi-Yuan; Sun Bao-Yuan

doi:10.7498/aps.73.20231632

Abstract

With the development of radioactive ion beam devices along with associated nuclear experimental detection technologies, the research areas in atomic nuclei have been further expanded, illustrating many new aspects of nuclear excitation as well as the physics of exotic nuclei far from the β-stability line. For weakly bound nuclei, the Fermi surface may lie near the continuum, which facilitates the easy scattering of valence nucleons into the continuum to occupy the resonance state. These continuum effects are of crucial importance in explaining the unusual structure of unstable nuclei. In this work, with the real stabilization method in coordinate space, nuclear structure model for single-particle resonances is developed within the framework of the relativistic Hartree-Fock (RHF) theory. In order to extract potential single-particle resonance structures, we study the evolution of single-particle states with box size in the continuum. To avoid the instability of nuclear binding energy, the pairing correlations are not taken into account in the calculation. As an important motivation, the roles of Fock terms in determining the energy, widths and spin-orbit splitting are discussed for low-lying neutron resonance states of $^{120}$Sn. By comparing with the relativistic mean field (RMF) model, it is found that the inclusion of exchange terms in the RHF model changes the in-medium balance of nuclear interactions and the equilibrium of nuclear dynamics, which in turn affects the description of the single-particle effective potential. For several neutron resonance states in $^{120}$Sn with finite resonant width, RHF model predicts lower resonant energy and smaller widths than RMF. For the single-particle states around the continuum threshold, the featured signals of resonance can depend sensitively on the effective interactions. In addition, for the spin-partner states $\nu {\mathrm{i}}_{13/2}$ and $\nu {\mathrm{i}}_{11/2}$ in resonance states, the effect of Fock terms on their spin-orbit splitting is analyzed. In comparison with the bound states, the wave functions of resonant spin-partner states can differ remarkably from each other, changing the effective potential and single-particle energies correspondingly. Thus, additional components in the single-particle effective potential may also contribute to the spin-orbit splitting of resonance states, aside from the spin-orbit interaction. In order to elucidate the mechanism of Fock term in single-particle resonance physics, in the subsequent study more numerical techniques that have been recently developed will be incorporated into the RHF methodology.

Keywords:

Authors and contacts

1.
MOE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China
2.
School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China

Corresponding author: Sun Bao-Yuan, sunby@lzu.edu.cn

Funds: Project supported by the Fundamental Research Fund for the Central Universities, China (Grant Nos. lzujbky-2022-sp02, lzujbky-2023-stlt01) and the National Natural Science Foundation of China (Grant No. 11875152).

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References

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Cited By

图 1 $ ^{120}$Sn中子共振态的能量随坐标空间截断$ R_\mathrm{max} $的变化. 对$ \nu {\mathrm{i}}_{13/2}, \nu {\mathrm{i}}_{11/2} $和$ \nu {\mathrm{j}}_{15/2} $态, 主量子数n从左到右由$ n=1 $开始依次增加. 图中实线表示RHF有效相互作用 PKO3, 虚线表示RMF有效相互作用PKDD

Figure 1. Dependence of single-particle energies on the coordinate space cutoff $ R_\mathrm{max} $ for neutron resonance states of $ ^{120}$Sn. For $ \nu {\mathrm{i}}_{13/2},\; \nu {\mathrm{i}}_{11/2} $ and $ \nu {\mathrm{j}}_{15/2} $, the principal quantum numbers start from $ n=1 $ and increase from left to right. The solid lines in the figure is represented as the RHF effective interaction PKO3 and RMF’s PKDD with dashed lines are selected for comparison.

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图 2 不同主量子数n下¹²⁰Sn中子$ \nu {\mathrm{j}}_{15/2} $共振态的径向波函数, 有效相互作用选取为PKO3

Figure 2. Radial wave functions of $ \nu {\mathrm{j}}_{15/2} $ neutron resonance state in ¹²⁰Sn with different principal quantum numbers n, the effective interaction is selected as PKO3.

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图 3 不同CDF有效相互作用下, $ ^{120} $Sn中子$ \nu {\mathrm{f}}_{5/2}, \nu {\mathrm{i}}_{13/2}, $$ \nu {\mathrm{i}}_{11/2} $和$ \nu {\mathrm{j}}_{15/2} $共振态的共振能量和宽度. 图中黑色、蓝色和绿色标记分别表示RHF, DDRMF和NLRMF有效相互作用的结果

Figure 3. Single-particle energies and widths of neutron $ \nu {\mathrm{f}}_{5/2}, \nu {\mathrm{i}}_{13/2}, \nu {\mathrm{i}}_{11/2} $ and $ \nu {\mathrm{j}}_{15/2} $ resonances in $ ^{120}\mathrm{Sn} $ with different CDF effective interactions. The black, blue and olive marks in the figure indicate the results with the effective interactions of RHF, DDRMF and NLRMF, respectively

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图 4 (a)不同CDF有效相互作用下$ ^{120} $Sn中子$ \nu {\mathrm{j}}_{15/2} $共振态的径向大分量波函数; (b)对中子$ \nu {\mathrm{j}}_{15/2} $共振态, PKO3与PKDD有效相互作用下类薛定谔方程中单粒子有效势和单粒子能级, 其中实线是总的势场, 虚线为直接项贡献, 阴影区域为交换项产生的影响

Figure 4. (a) Radial wave functions of large component of $ ^{120} $Sn neutron $ \nu {\mathrm{j}}_{15/2} $ resonance state with different CDF effective interactions; (b) for neutron $ \nu {\mathrm{j}}_{15/2} $ resonance state, effective potentials in Schrödinger-like equation and corresponding single-particle energies with PKO3 and PKDD effective interactions. Solid lines are the total potentials, dotted line is the contribution from the direct terms and shaded area comes from the exchange terms of RHF model.

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图 5 ¹²⁰Sn核中共振态$ (\nu {\mathrm{i}}_{13/2}, \nu {\mathrm{i}}_{11/2}) $和束缚态$ (\nu 1 {\mathrm{g}}_{9/2}, $$ \nu 1 {\mathrm{g}}_{7/2}) $自旋伙伴态对应的径向大分量波函数, 选取RHF有效相互作用PKO3计算得到

Figure 5. Radial wave functions of large component of resonance states $ (\nu {\mathrm{i}}_{13/2}, \nu {\mathrm{i}}_{11/2}) $ and bound states $ (\nu 1 {\mathrm{g}}_{9/2}, $$ \nu 1 {\mathrm{g}}_{7/2}) $, given by the RHF effective interaction PKO3.

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表 1 根据$^{120}$Sn 中子$\nu {\mathrm{j}}_{15/2}$共振态在不同主量子数n下的$E {\text{-}} R_\mathrm{max}$曲线得到对应的拐点大小$\bar{R}_\mathrm{max}$, 共振能量$E_{\gamma}$以及宽度 Γ. 以RHF有效相互作用PKO3结果为例

Table 1. Resonant energies $E_{\gamma}$, widths Γ and inflection points $\bar{R}_\mathrm{max}$ derived from $E {\text{-}} R_\mathrm{max}$ curves with different principal quantum numbers n for the neutron resonance state $\nu {\mathrm{j}}_{15/2}$ of $^{120}$Sn, illustrated by the RHF effective interaction PKO3.

n	$\bar{R}_\mathrm{max}$/fm	$E_{\gamma}$/MeV	Γ/MeV
1	12.000	12.457	0.518
2	17.731	12.377	0.653
3	22.396	12.368	0.703
4	26.825	12.365	0.738
5	31.145	12.360	0.755
6	35.393	12.358	0.767

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表 2 采取不同的CDF有效相互作用给出的¹²⁰Sn中子共振态的能量和宽度, 表中“\”表示无法提取相关信息. 所有单位均为MeV

Table 2. Single-particle energies and widths of neutron resonances in ¹²⁰Sn with different CDF effective interactions, “\” in table means that the relevant information could not be calculated. All units are in MeV

	$\nu 3 {\mathrm{p}}_{1/2}$		$\nu 1 {\mathrm{h}}_{9/2}$		$\nu {\mathrm{f}}_{5/2}$		$\nu {\mathrm{i}}_{13/2}$		$\nu {\mathrm{i}}_{11/2}$		$\nu {\mathrm{j}}_{15/2}$
	$E_{\gamma}$	Γ	$E_{\gamma}$	Γ	$E_{\gamma}$	Γ	$E_{\gamma}$	Γ	$E_{\gamma}$	Γ	$E_{\gamma}$	Γ
PKO1	–0.071	\	0.262	$\sim$0.000	0.675	0.028	2.802	0.001	9.763	1.152	11.963	0.705
PKO2	–0.096	\	0.491	$\sim$0.000	1.150	0.127	2.516	0.001	10.171	1.161	11.882	0.586
PKO3	0.028	0.013	0.312	$\sim$0.000	0.834	0.049	3.084	0.002	9.963	1.206	12.358	0.767
DD-LZ1	–0.326	\	1.437	$6\times 10^{-4}$	0.268	0.001	4.221	0.016	10.370	1.895	13.277	1.387
PKDD	\	\	1.054	$1\times 10^{-4}$	1.173	0.153	3.874	0.009	10.737	1.953	13.313	1.279
DD-ME2	–0.057	\	0.949	$6\times 10^{-5}$	0.787	0.047	4.038	0.012	10.541	1.874	13.329	1.366
NL3	–0.015	\	\	\	0.673	0.029	3.263	0.004	9.559	1.205	12.561	0.973
PK1	0.046	0.034	0.250	$\sim$0.000	0.870	0.063	3.468	0.005	9.808	1.274	12.875	1.036
PK1(RMF-GF)	0.050	0.033	0.251	$8\times 10^{-8}$	0.871	0.065	3.469	0.005	9.854	1.283	12.893	1.065

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表 3 ¹²⁰Sn核中$\nu {\rm i}$共振态和$\nu 1 {\rm g}$束缚态的自旋-轨道劈裂$\Delta \varepsilon$, 以及利用类薛定谔方程(9)得到各部分的贡献. 所有单位均为 MeV

Table 3. Spin-orbit splitting of resonance states $\nu {\rm i}$ and bound spin partners $\nu 1 {\rm g}$ in ¹²⁰Sn, as well as their contributions from various components according to the Schrodinger-like Eq. (9). All units are in MeV

	PKO3		PKDD
	$l=4$	$l=6$	$l=4$	$l=6$
$G''$	–0.856	–1.093	–0.994	–0.434
$\varSigma_+$	0.297	23.228	0.319	22.183
$V_{{\mathrm{CB}}}$	0.473	–16.907	0.452	–20.589
$V^{\mathrm{D}}$	4.362	4.086	7.069	5.703
$V^{\mathrm{E}}$	1.800	–2.436	0.000	0.000
$\Delta \varepsilon$	6.074	6.878	6.846	6.863

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map

[1]	Meng J, Ring P 1998 Phys. Rev. Lett. 80 460 Google Scholar
[2]	Dobaczewski J, Nazarewicz W, Werner T R, Berger J F, Chinn C R, Dechargé J 1996 Phys. Rev. C 53 2809 Google Scholar
[3]	Meng J, Ring P 1996 Phys. Rev. Lett. 77 3963 Google Scholar
[4]	Pöschl W, Vretenar D, Lalazissis G A, Ring P 1997 Phys. Rev. Lett. 79 3841 Google Scholar
[5]	Meng J, Toki H, Zhou S G, Zhang S Q, Long W H, Geng L S 2006 Prog. Part. Nucl. Phys. 57 470 Google Scholar
[6]	Liu W, Lou J L, Ye Y L, Pang D Y 2020 Nucl. Sci. Tech. 31 20 Google Scholar
[7]	Li W J, Ma Y G, Zhang G Q, et al 2019 Nucl. Sci. Tech. 30 180 Google Scholar
[8]	Khumalo T C, Pellegri L, Wiedeking M, et al. 2023 J. Phys.: Conf. Ser. 2586 012065 Google Scholar
[9]	Curutchet P, Vertse T, Liotta R J 1989 Phys. Rev. C 39 1020 Google Scholar
[10]	Cao L G, Ma Z Y 2002 Phys. Rev. C 66 024311 Google Scholar
[11]	苟秉聪 1993 42 223 Google Scholar Gou B C 1993 Acta Phys. Sin. 42 223 Google Scholar
[12]	孙言, 胡峰, 桑萃萃, 梅茂飞, 刘冬冬, 苟秉聪 2019 68 163101 Google Scholar Sun Y, Hu F, Sang C C, Mei M F, Liu D D, Gou B C 2019 Acta Phys. Sin. 68 163101 Google Scholar
[13]	Wigner E P, Eisenbud L 1947 Phys. Rev. 72 29 Google Scholar
[14]	Hale G M, Brown R E, Jarmie N 1987 Phys. Rev. Lett. 59 763 Google Scholar
[15]	Humblet J, Filippone B W, Koonin S E 1991 Phys. Rev. C 44 2530 Google Scholar
[16]	Taylor J R 1972 Scattering Theory: The Quantum Theory on Nonrelativistic Collisions (New York: John Wiley & Son) p240
[17]	Kukulin V I, Krasnopl’sky V M, Horácek J 1989 Theory of Resonances: Principles and Applications (Dordrecht: Kluwer Academic) p219
[18]	Yang S C, Meng J, Zhou S G 2001 Chin. Phys. Lett. 18 196 Google Scholar
[19]	Tanaka N, Suzuki Y, Varga K, Lovas R G 1999 Phys. Rev. C 59 1391 Google Scholar
[20]	Cattapan G, Maglione E 2000 Phys. Rev. C 61 067301 Google Scholar
[21]	Gyarmati B, Kruppa A T 1986 Phys. Rev. C 34 95 Google Scholar
[22]	Kruppa A T, Heenen P H, Flocard H, Liotta R J 1997 Chin. Phys. Lett. 79 2217 Google Scholar
[23]	Arai K 2006 Phys. Rev. C 74 064311 Google Scholar
[24]	Guo J Y, Yu M, Wang J, Yao B M, Jiao P 2010 Comput. Phys. Commun. 181 550 Google Scholar
[25]	Maier C H, Cederbaum L S, Domcke W 1980 J. Phys. B 13 L119 Google Scholar
[26]	Taylor H S, Hazi A U 1970 Phys. Rev. A 1 1109 Google Scholar
[27]	Mandelshtam V A, Ravuri T R, Taylor H S 1993 Phys. Rev. Lett. 70 1932 Google Scholar
[28]	Mandelshtam V A, Taylor H S, Ryaboy V, Moiseyev N 1994 Phys. Rev. A 50 2764 Google Scholar
[29]	Serot B D, Walecka J D 1986 Adv. Nucl. Phys. 16 1
[30]	Reinhard P G 1989 Rep. Prog. Phys. 52 439 Google Scholar
[31]	Ring P 1996 Prog. Part. Nucl. Phys. 37 193 Google Scholar
[32]	Vretenar D, Afanasjev A, Lalazissis G A, Ring P 2005 Phys. Rep. 409 101 Google Scholar
[33]	Zhang S S, Meng J, Zhou S G, Hillhouse G C 2004 Phys. Rev. C 70 034308 Google Scholar
[34]	Guo J Y, Fang X Z, Jiao P, Wang J, Yao B M 2010 Phys. Rev. C 82 034318 Google Scholar
[35]	刘野, 陈寿万, 郭建友 2012 61 112101 Google Scholar Liu Y, Chen S W, Guo J Y 2012 Acta Phys. Sin. 61 112101 Google Scholar
[36]	Zhang L, Zhou S G, Meng J, Zhao E G 2008 Phys. Rev. C 77 014312 Google Scholar
[37]	张力, 周善贵, 孟杰, 赵恩广 2007 56 3839 Google Scholar Zhang L, Zhou S G, Meng J, Zhao E G 2007 Acta Phys. Sin. 56 3839 Google Scholar
[38]	Lu B N, Zhao E G, Zhou S G 2012 Phys. Rev. Lett. 109 072501 Google Scholar
[39]	Lu B N, Zhao E G, Zhou S G 2013 Phys. Rev. C 88 024323 Google Scholar
[40]	Li Z P, Meng J, Zhang Y, Zhou S G, Savushkin L N 2010 Phys. Rev. C 81 034311 Google Scholar
[41]	Sun T T, Zhang S Q, Zhang Y, Hu J N, Meng J 2014 Phys. Rev. C 90 054321 Google Scholar
[42]	Sun T T, Qian L, Chen C, Ring P, Li Z P 2020 Phys. Rev. C 101 014321 Google Scholar
[43]	Chen C, Li Z P, Li Y X, Sun T T 2020 Chin. Phys. C 44 084105 Google Scholar
[44]	Sun T T, Li Z P, Ring P 2023 Phys. Lett. B 847 138320 Google Scholar
[45]	Li N, Shi M, Guo J Y, Niu Z M, Liang H Z 2016 Phys. Rev. Lett. 117 062502 Google Scholar
[46]	Zhang Y, Qu X Y 2020 Phys. Rev. C 102 054312 Google Scholar
[47]	Long W H, Giai N V, Meng J 2006 Phys. Lett. B 640 150 Google Scholar
[48]	Geng J, Long W H 2022 Phys. Rev. C 105 034329 Google Scholar
[49]	Long W H, Geng J, Liu J, Wang Z H 2022 Commun. Theor. Phys. 74 097301 Google Scholar
[50]	Long W H, Ring P, Giai N V, Meng J 2010 Phys. Rev. C 81 024308 Google Scholar
[51]	Jiang L J, Yang S, Sun B Y, Long W H, Gu H Q 2015 Phys. Rev. C 91 034326 Google Scholar
[52]	Zong Y Y, Sun B Y 2018 Chin. Phys. C 42 024101 Google Scholar
[53]	Wang Z H, Naito T, Liang H Z, Long W H 2021 Chin. Phys. C 45 064103 Google Scholar
[54]	Long W H, Sagawa H, Meng J, Giai N V 2008 Europhys. Lett. 82 12001 Google Scholar
[55]	Wang L J, Dong J M, Long W H 2013 Phys. Rev. C 87 047301 Google Scholar
[56]	Liang H Z, Zhao P W, Meng J 2012 Phys. Rev. C 85 064302 Google Scholar
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