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为了研究晕核的单粒子共振态,我们首先将哈密顿量表示为
$$H = T + V,$$ (1) 这里动量算符
$T = {{{{{p}}^2}} / {2m}}$ ,其中,${{p}} = \hbar {{k}}$ (${{k}}$ 是波矢),根据文献[25],相互作用势V由中心势、形变势以及自旋轨道耦合势三部分组成:$$\begin{split} &{V_{{\rm{cent}}}}(r) = {V_0}f(r), \\ & {V_{{\rm{def}}}}(r) = - {\beta _2}k(r){Y_{20}}(\vartheta ,\varphi ), \\ & {V_{sl}}(r) = - 4{V_0}{\Lambda ^2}\frac{1}{r}\frac{{{\rm{d}}f(r)}}{{{\rm{d}}r}}(s \cdot l), \\ \end{split} $$ (2) 上式中
$\Lambda $ 是核子的约化康普顿波长${\hbar / {{m_r}}}c$ ,$${\text{且}}\qquad\qquad\qquad\begin{array}{l} f(r) = \dfrac{1}{{1 + {{\rm e}^{\frac{{r - R}}{a}}}}}, \\ k(r) = {V_0}r\dfrac{{{\rm{d}}f(r)}}{{{\rm{d}}r}}{\text{。}} \\ \end{array} \qquad\qquad\qquad\qquad$$ 和文献[26]类似,参数
$a\!=\!0.67\;{\rm{fm}}$ ,$R\!=\!{r_0}{A^{{1 / 3}}}$ ,${r_0}\!=\! $ $1.27\;{\rm{fm}}$ 。为了获得共振态,我们将薛定谔方程转到动量空间,可以同时得到束缚态、共振态和连续谱:
$$\int {\rm{d}} {{k}}'\left\langle {{{k}}\left| H \right|{{k}}'} \right\rangle \psi ({{k}}') = E\psi ({{k}}),$$ (3) 对于轴对称形变核,宇称
$\pi $ 以及总角动量的第三个分量是个好量子数,动量波函数$\psi ({{k}})$ 用径向和角向部分来表达,故$$\psi ({{k}}) = {\psi _{{m_j}}}({{k}}) = \sum\limits_{lj} {{f^{lj}}(k)} {\phi _{lj{m_j}}}({\Omega _k}),$$ (4) 这里
${f^{lj}}(k)$ 是波函数的径向分量。波函数的角向部分表示为$${\phi _{lj{m_j}}}({\Omega _k}) = \sum\limits_{{m_s}} {\left\langle {\left. {lm\frac{1}{2}{m_s}} \right|} \right.} \left. {j{m_j}} \right\rangle {Y_{lm}}({\Omega _k}){\chi _{{m_s}}},$$ (5) 将上述的波函数代入薛定谔方程,我们可以得到:
$$\begin{split} & \frac{{{\hbar ^2}k_a^2}}{{2M}}{f^{lj}}({k_a}) + \sum\limits_b {{w_b}} k_b^2{V_s}(l,j,{k_a},{k_b}){f^{lj}}({k_b}) - \\ & {\beta _2}\sum\limits_{l'j'} {\sum\limits_b {{w_b}} k_b^2{V_d}(l,j,l',j',{k_a},{k_b}){f^{l'j'}}({k_b})} = E{f^{lj}}({k_a}), \\ \end{split} $$ (6) 式(6)中的矩阵是不对称的,为了简便计算,我们变为对称矩阵
$$\begin{split} &\frac{{{\hbar ^2}k_a^2}}{{2M}}{F^{lj}}({k_a}) + \sum\limits_b {\sqrt {{w_a}{w_b}} {k_a}{k_b}} {V_s}(l,j,{k_a},{k_b}){F^{lj}}({k_b}) - \\ &{\beta _2}\sum\limits_b {\sum\limits_{l'j'} {\sqrt {{w_a}{w_b}} {k_a}{k_b}} } {V_d}(l,j,l',j',{m_j},{k_a},{k_b}){F^{l'j'}}({k_b}) =\\ &E{F^{lj}}({k_a}) {\text{。}} \\[-12pt] \end{split} $$ (7) 至此,通过求解对称矩阵的本征解就可以解出薛定谔方程,获得束缚态和共振态的能量,为了获得坐标空间波函数,做如下变换:
$$\psi ({ r}) = {\psi _{{m_j}}}({ r}) = \frac{1}{{{{(2\pi )}^{3/2}}}}\displaystyle\int {{\rm{d}}{{k}}{{\rm e}^{{\rm{i}}{{k}} \cdot {{r}}}}} {\psi _{{m_j}}}({{k}}){\text{。}}$$ (8) 假设
${\psi _{{m_j}}}({{k}})$ 具有如下形式$${\psi _{{m_j}}}( { r} ) = \sum\limits_{lj} {{f^{lj}}} (r){\phi _{lj{m_j}}}({\Omega _r}),$$ (9) 则得到波函数的径向部分
$${f^{lj}}(r) = {i^l}\sqrt {\frac{2}{p}} \sum\limits_{a = 1}^N {\sqrt {{w_a}} } {k_a}{j_l}({k_a}r){F^{lj}}({k_a}),$$ (10) 和径向密度分布
$$ {\rho _{{m_j}}}(r) = \sum\limits_{lj} {{f^{lj*}}} (r){f^{lj}}(r)\text{。} $$ (11)
Study on Halo Phenomenon in Exotic Nuclei by Complex Momentum Representation Method
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摘要: 晕现象的研究使人们对核结构有了新的认识。连续谱,尤其是连续谱中的共振态在其中扮演着重要角色。复动量表象(CMR)方法不仅能够统一描述束缚态、共振态和连续谱,而且能够很好地描述窄共振和宽共振。本文介绍了CMR方法对原子核共振态的研究。给出了31Ne和19C等核的束缚态和共振态的单粒子能量随形变参数β2的变化规律,分析了19C和31Ne中单中子晕形成的物理机制和在中子数N=20附近能级反转的原因,并预测了比37Mg重的核中的单中子晕现象,这一预测结果对在实验中寻找较重的晕核具有一定的参考价值。这些研究表明CMR 方法不仅适用于描述稳定核,也适用于描述具有弥散物质分布的奇特核。Abstract: The study of halo phenomenon gives us a new understanding of nuclear structure, in which the continuum, especially the resonance in the continuum, plays an important role. The complex momentum representation (CMR) method can not only describe the bound state, resonant state and continuous spectrum uniformly, but also describe the narrow and wide resonance well. In this paper, the CMR method is introduced for the study of nuclear resonance. The single particle energy of bound state and resonance state of 31Ne and 19C with deformation parameter β2 is given. The physical mechanism of halo formation in 19C and 31Ne and the reason of energy level inversion near the neutron number N=20 are analyzed. The halo phenomenon in nuclei heavier than 37Mg is predicted. The result of this prediction is helpful to find heavier halo nuclei in experiments. These studies show that the CMR method is suitable for describing not only stable nuclei, but also exotic nuclei with diffuse material distribution.
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图 1 (在线彩图)计算得到的随四极形变系数β2变化的单粒子能量,本图取自文献[22]
图 2 (在线彩图)单粒子态1/2[310]的主要组分的占比与形变参数的变化关系,本图取自文献[22]
图 3 (在线彩图)单粒子态3/2[321]的主要组分的占比与形变参数的变化关系,本图取自文献[22]
图 4 (在线彩图)单粒态1/2 [110]和1/2 [310]的径向密度随
$r$ 的变化曲线,本图取自文献[22] -
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