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在研究丰中子Cr 同位素的奇特结构之前,先简要介绍一下RMFPC-CMR-BCS的理论框架[68,72]。点耦合顶点的基本模块是一般类型的两费米子项
$(\bar \psi {o_\tau }\varGamma \psi ), {o_\tau } \in \{ 1,{\tau _i}\} ,\;\varGamma \in \{ 1,{\gamma _\mu },{\gamma _5},{\gamma _5}{\gamma _\mu },{\sigma _{\mu v}}\}$ ,其中,$ \psi $ 为核子的Dirac旋量场,$ {\tau _i} $ 是同位旋泡利矩阵,$\varGamma$ 为4×4的Dirac矩阵之一。RMFPC理论是从拉格朗日密度出发$$ L = {L^{{\text{free}}}} + {L^{{\text{4f}}}} + {L^{{\text{em}}}} + {L^{{\text{der}}}} + {L^{{\text{hot}}}}, $$ (1) 其中:
$ {L^{{\text{free}}}} $ 、$ {L^{{\text{4f}}}} $ 、$ {L^{{\text{em}}}} $ 、$ {L^{{\text{der}}}} $ 和$ {L^{{\text{hot}}}} $ 分别为自由核子项、四费米子点耦合项、质子之间电磁相互作用项、密度和流微商项以及高阶耦合项的拉氏量密度,具体如下:$$ {L^{{\text{free}}}} = \bar \psi ({\rm{i}}{\gamma _\mu }{\partial ^\mu } - m)\psi , $$ $$ \begin{split} {L^{{\text{4f}}}} =& - \frac{1}{2}{\alpha^{} _S}{(\bar \psi \psi )^2} - \frac{1}{2}{\alpha ^{}_V}(\bar \psi {\gamma _\mu }\psi )(\bar \psi {\gamma ^\mu }\psi ) - \\& \frac{1}{2}{\alpha^{} _{{TS} }}(\bar \psi {\tau }\psi )(\bar \psi {\tau }\psi ) - \frac{1}{2}{\alpha^{} _{TV}}(\bar \psi {\tau }{\gamma^{} _\mu }\psi )(\bar \psi {\tau }{\gamma ^\mu }\psi ), \end{split} $$ $$ {L^{{\text{em}}}} = - \frac{1}{4}\big[{F^{\mu v}}{F_{\mu v}} + 2e(1 - {\tau _3})(\bar \psi {\gamma ^\mu }\psi ){A_\mu }\big], $$ $$ \begin{split} {L^{{\text{der}}}} =& - \frac{1}{2}{\delta _S}{\partial _v}(\bar \psi \psi ){\partial ^v}(\bar \psi \psi ) - \frac{1}{2}{\delta _{\text{V}}}{\partial _v}(\bar \psi {\gamma _\mu }{\psi} ){\partial ^v}(\bar \psi {\gamma ^\mu }\psi ) - \\& \frac{1}{2}{\delta _{{TS} }}{\partial _v}(\bar \psi {{{{\boldsymbol{\tau}}}} }\psi ){\partial ^v}(\bar \psi {{\boldsymbol{\tau}} }\psi ) - \frac{1}{2}{\delta _{{TV} }}{\partial _v}(\bar \psi {{\boldsymbol{\tau}} }{\gamma _\mu }\psi ){\partial ^v}(\bar \psi {{\boldsymbol{\tau}} }{\gamma ^\mu }\psi ), \end{split}$$ $$ {L^{{\text{hot}}}} = - \frac{1}{3}{\beta _S}{(\bar \psi \psi )^3} - \frac{1}{4}{\gamma _S}{(\bar \psi \psi )^4} - \frac{1}{4}{\gamma _V}{\big[(\bar \psi {\gamma _\mu }\psi )(\bar \psi {\gamma ^\mu }\psi )\big]^2}, $$ 式(1)所有符号的含义与文献[67]中相同。由拉格朗日密度出发,经过一系列的计算可以得到核子的狄拉克方程
$$ \big[ {{{{\boldsymbol{\alpha}} }} \boldsymbol\cdot {{{\boldsymbol{p}}}} + \beta (m + S) + V} \big]\psi = \varepsilon \psi , $$ (2) 这里,
$S(r) = {{\varSigma} _s}$ 和$V(r) = {{\varSigma} ^\mu } + {\boldsymbol{\tau }} \boldsymbol\cdot {\boldsymbol\varSigma} _{Tv}^\mu$ 分别为标量势和矢量势,$$ \left\{ \begin{gathered} {\varSigma}_s = {\alpha^{} _s}{\rho^{} _s} + {\beta ^{}_s}\rho _s^2 + {\gamma^{} _s}\rho _s^3 + {\delta^{} _s}\vartriangle {\rho^{} _s} \\ {\varSigma}^\mu = {\alpha ^{}_v}j_v^\mu + {\gamma ^{}_v}{(j_v^\mu )^3} + {\delta ^{}_v}\vartriangle j_v^\mu + e{A^\mu } \\ {\boldsymbol\varSigma} _{Tv}^\mu = {\alpha ^{}_{Tv}}\boldsymbol j_{Tv}^\mu + {\delta^{} _{Tv}}\vartriangle \boldsymbol j_{Tv}^\mu \\ \end{gathered} \right. , $$ (3) 其中:
${{\varSigma} _s}$ 是同位旋标量-标量自能;${{\varSigma} ^\mu }$ 是同位旋标量-矢量自能;${\boldsymbol{\varSigma}} _{Tv}^\mu$ 是同位旋矢量-矢量自能。方程(2)中束缚态的解可以用常规的方法得到。为了得到物理共振态的非束缚解,我们将式(2)转化到动量表象下$$ \int {{\rm d}{{\boldsymbol{k}}^{'}}\left\langle {{\boldsymbol{k}}\left| H \right|{{\boldsymbol{k}}^{'}}} \right\rangle } \psi \left( {{{\boldsymbol{k}}^{'}}} \right) = \varepsilon \psi \left( {\boldsymbol{k}} \right)。 $$ (4) 这里,
$H = {\boldsymbol{\alpha }} \boldsymbol\cdot {\boldsymbol{p}} + \beta (m + S) + V$ ,$\psi \left( {\boldsymbol k} \right)$ 和$ \varepsilon $ 分别为动量波函数和单粒子能量。为了求解方程(4),波函数可以用分离变量法得到$$ \psi ({\boldsymbol{k}}) = \left( {\begin{array}{*{20}{c}} {f(k){\phi _{lj{m_j}}}({\Omega _k})} \\ {g(k){\phi _{\tilde lj{m_j}}}({\Omega _k})} \end{array}} \right)。 $$ (5) 其中:
$ f\left( k \right) $ 是关于l和j的径向分量;${\phi _{lj{m_j}}}\left( {{\Omega _k}} \right) = \sum\limits_{{m_s}} {\left\langle {{{\rm lm}\frac{1}{2}{m_s}}} \mathrel{\left | {\vphantom {{lm\frac{1}{2}{m_s}} {j\Omega }}} \right. } {{j\Omega }} \right\rangle } {Y_{lm}}\left( {{\Omega _k}} \right){\chi _{{m_s}}}$ 是角向部分;$ {Y_{lm}}\left( {{\Omega _k}} \right) $ 和$ {\chi _{{m_s}}} $ 分别是球谐函数和自旋波函数。将式(5)带入到式(4)中进行一系列的计算,式(4)的Dirac方程可以写成$$ \begin{gathered} mf(k) - kg(k) + \int {k{'^2}} {{\rm{d}}}k'{V_ + }(k,k')f(k') = \varepsilon f(k), \\ - kf(k) - mg(k) + \int {k{'^2}} {{\rm{d}}}k'{V_ - }(k,k')g(k') = \varepsilon g(k)。 \\ \end{gathered} $$ (6) 这里
$$ \begin{gathered} {V_ + }(k,k') = \frac{2}{\pi }\int {{r^2}{{\rm{d}}}r\left[ {V(r) + S(r)} \right]{j_l}} (k'r){j_l}(kr), \\ {V_ - }(k,k') = \frac{2}{\pi }\int {{r^2}{{\rm{d}}}r\left[ {V(r) - S(r)} \right]{j_{\tilde l}}} (k'r){j_{\tilde l}}(kr)。 \\ \end{gathered} $$ (7) 在复动量空间中求解式(6),不仅可以得到束缚态,而且也可以得到共振态,在束缚态和共振态的基础上,用BCS近似处理对关联,推导细节见文献[68, 72]。
Study on the Exotic Structures of the Neutron-rich Cr Isotopes by the Complex Momentum Representation Method
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摘要: 奇特核的研究是核物理中最有趣的前沿课题之一。中等质量核中是否存在晕现象,目前的研究相对较少。复动量表象(CMR)方法可以用来探索原子核中的奇特结构。连续谱阈值附近的共振态在奇特现象的形成中起着重要的作用。利用相对论点耦合框架下的复动量表象(RMFPC-CMR)方法研究了丰中子Cr 同位素中的奇特结构,得到的非常接近于零的双中子分离能和迅速增大的中子均方根半径表明在靠近中子滴线的Cr 同位素中存在晕结构。从获得的单粒子能级、费米面附近价核子占据几率、中子和质子密度分布以及各能级对原子核密度的贡献可以发现
$ 3{s_{1/2}} $ 和$ 2{d_{3/2}} $ 能级的占据有利于中子晕的形成。研究发现,低角动量弱束缚能级对异常增大的半径和弥散密度分布有显著的贡献,这导致靠近中子滴线的76-82Cr 是中子晕核。这一预测结果对在实验中探索中等质量区的晕核具有一定的参考价值。Abstract: The study of exotic nuclei is one of the most interesting frontier topics in nuclear physics. There are relatively few studies on the existence of halo in medium-mass nuclei. The complex momentum representation(CMR) method can be used to explore exotic structures in nuclei. The resonant states near the continuum threshold play an important role in the formation of exotic phenomena. Therefore, the relativistic point coupled and the complex momentum representation (RMFPC-CMR) method is used to explore the exotic structures for the neutron-rich Cr isotopes. The two-neutron separation energies have been calculated to be very close to zero, and the mean square (rms) radii of neutron also have increased sharply, implying the presence of halo structures in the Cr isotopes near the neutron drip line. The single-particle levels, the occupation probabilities of valence nucleons on the levels near the Fermi surface, the neutron and proton density distributions, and the contribution of every level to the nucleus density are obtained, it can be found that the occupations of the levels$ 3{s_{1/2}} $ and$ 2{d_{3/2}} $ are beneficial to the formation of neutron halo. It is found that the unusual increases of rms radii and diffuse distributions of neutron densities come mainly from the contributions of the weakly bound levels with lower orbital angular momentum, which leads to the neutron halos of 76-82Cr near the neutron drip line. This prediction has a certain reference value for exploring the halo nuclei in the medium mass region in experiments. -
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