-
AMD模型是量子分子动力学(QMD)的反对称化版本[23–26]。AMD模型中随时间演化的波函数能够描述核的量子效应,原子核稳定的基态是通过摩擦冷却方法构造的[27]。对于N个核子的反应系统,用斯莱特行列式(Slater determinant)表示N个高斯波包构成的波函数:
$$ \Phi (Z) = \det \left[\exp \left\{ - \nu {{({{r}_{j}}-{}^{{{Z}_{i}}}\!\!\diagup\!\!{}_{\sqrt{\nu }}\;)}^{2}} + \frac{1}{2}Z_i^2\right\} {\chi _{aj}}(j)\right], $$ (1) $ {\chi _{aj}}\left( j \right) $ 为自旋和同位旋波函数,即$ p\uparrow $ 、$ p\downarrow $ 、$ n\uparrow $ 和$ n\downarrow $ 四个状态。复变量$ Z \!=\! \left\{ {{Z_i}} \right.;i \!=\! 1,\cdots N\left. {} \right\}\ $ ,$ {Z_i} $ 表示波包质心,表示为$$ {Z_i} = \sqrt\nu {D_i} + {}^{i}\!\!\diagup\!\!{}_{2\hbar \sqrt{\nu }}\;{K_i}{\text{。}} $$ (2) AMD模型中,波包的宽度参数
$ \nu $ 被看作是恒定的常数,对于所有的波包都相同。通常选取$ \nu $ =0.16 fm–2。方程(1)给出的AMD波函数$ \Phi (Z) $ 是没有考虑多体关联效应,因此需要将有效相互作用与波函数结合起来一起使用。在AMD模型中,系统的有效哈密顿量包括动能和核子-核子相互作用势,表达式为$$ {\cal{H}} = \sum\limits_i {\frac{{p_i^2}}{{2M}}} + \sum\limits_{i < j} {{V_{ij}}}{\text{。}} $$ (3) Gogny核子-核子相互作用势由局域的两体相互作用和零程的密度相关项的和构成,表示为
$$ \begin{split} V_{\rm g0} =& \sum\limits_{i = 1,2}(W_{i}+B_{i}P_{\sigma }-H_{i}P_{\sigma }-M_{i}P_{\sigma}P_{\tau })\times \\&\exp\left[\frac{-(r-{r}')^{2}}{{\eta}_{i}^{2}}\right]+ \frac{t_{0}}{6}(W_{\rho}+B_{\rho}P_{\sigma }+\\&H_{\rho}P_{\tau}-M_{\rho }P_{\sigma }P_{\tau })\rho(r)^{\alpha }\delta (r-{r}'), \end{split} $$ (4) 其中:
$ {P_\tau} $ 为同位旋算符,$ {P_\sigma} $ 为自旋交换算符,$ \rho \left( r \right) $ 为坐标空间r处密度。根据对称能的密度依赖关系,Gogny核子-核子相互作用势有三套参数,即软(g0)[28]、硬的(g0as)、超硬(g0ass),g0为式(4)所示,g0as和g0ass[29]通过改进g0得到:$$ V_{\rm{g0as(g0ass)}} = V_{\rm{g0}}-(1-x)t_{3}[\rho (r_{1})^{\frac{1}{3}}-\rho _{0}^{\frac{1}{3}}]P_{\sigma }\delta (r_{1}-r_{2}), $$ (5) 公式中
$ t_{3} $ 和$ \rho_{0} $ =0.16 fm–3为系数,$ x $ =$ -\frac{1}{2} $ (g0as),$ x $ =$ -2 $ (g0ass),其余参数在表1给出。表 1 Gogny核子-核子相互作用势参数表
Interaction i $\eta$ W B H M $\alpha$ $t_0$/(MeV·fm–4) $\chi_0$ $W_\rho$ $B_\rho$ $H_\rho$ $M_\rho$ g0 1 0.7 $-402.4$ $-100.0$ $-496.2$ $-23.56$ 1/3 8 100 1 1 1 0 0 2 1.2 $-21.3$ $-11.77$ 37.27 $-68.81$ g0as 1 0.7 $-402.4$ $-100.0$ $-496.2$ $-23.56$ 1/3 8 100 1 1 $-0.5$ 0 0 2 1.2 $-21.3$ $-11.77$ 37.27 $ -68.81$ g0ass 1 0.7 $-402.4$ $-100.0$ $-496.2$ $-23.56$ 1/3 8 100 1 1 $-2$ 0 0 2 1.2 $-21.3 $ $-11.77$ 37.27 $-68.81$
The Influence of Gogny Nucleon-nucleon Interaction Potentials of AMD on Ground State Properties of Nuclei
-
摘要: 在反对称化的分子动力学(AMD)模型中,分别采用g0、g0as和g0ass三套Gogny相互作用参数研究了4He、6Li、12C、20Ne、40Ca和60Ni的结合能和均方根半径在自身平均场中随时间的涨落。系统地研究了原子序数从1到18的同位素的结合能,并与现有实验数据进行比较,发现g0的结果更接近实验值。这为利用AMD研究低能区的熔合和中能区的多重碎裂等核反应机制提供了基础。Abstract: In antisymmetrized molecular dynamics(AMD) model, the binding energies and root mean square (RMS) radii of 4He, 6Li, 12C, 20Ne, 40Ca and 60Ni in their own mean field were studied by three sets of Gogny interactions (g0, g0as and g0ass). The binding energies of isotopes with atomic numbers from 1 to 18 were studied systematically. Comparing with the experimental data, it is found that the result of g0 is the best. These investigations would provide clues to the study of the nuclear reaction mechanism such as fusion in low energy region and multi fragmentation in medium energy region by AMD.
-
Key words:
- AMD /
- binding energy /
- root mean square
-
图 1 (在线彩图) 原子核结合能在自身平均场中随时间的演化,黑色虚线为实验值,实验值取自文献[30]
图 2 (在线彩图) 原子核方均根半径在自身平均场中随时间的演化,黑色虚线为实验值,实验值取自文献[31]
图 3 (在线彩图) AMD模型计算的
$ Z\! =\! 1\!\thicksim\!9$ 原子核同位素的结合能理论值与实验值的比较,实验值取自文献[30]图 4 (在线彩图) AMD模型计算的
$ Z\!= \!10\!\thicksim\!18$ 原子核同位素的结合能理论值与实验值的比较,实验值取自文献[30]表 1 Gogny核子-核子相互作用势参数表
Interaction i $\eta$ W B H M $\alpha$ $t_0$ /(MeV·fm–4)$\chi_0$ $W_\rho$ $B_\rho$ $H_\rho$ $M_\rho$ g0 1 0.7 $-402.4$ $-100.0$ $-496.2$ $-23.56$ 1/3 8 100 1 1 1 0 0 2 1.2 $-21.3$ $-11.77$ 37.27 $-68.81$ g0as 1 0.7 $-402.4$ $-100.0$ $-496.2$ $-23.56$ 1/3 8 100 1 1 $-0.5$ 0 0 2 1.2 $-21.3$ $-11.77$ 37.27 $ -68.81$ g0ass 1 0.7 $-402.4$ $-100.0$ $-496.2$ $-23.56$ 1/3 8 100 1 1 $-2$ 0 0 2 1.2 $-21.3 $ $-11.77$ 37.27 $-68.81$ -
[1] BECK C, SOUZA F A, ROWLEY N, et al. Phys Rev C, 2003, 67: 054602. doi: 10.1103/PhysRevC.67.054602 [2] KEELEY N, RAABE R, ALAMANOS N, et al. Prog in Part and Nucl Phys, 2007, 59: 579. doi: 10.1016/j.ppnp.2007.02.002 [3] CANTO L F, GOMES P R S, HUSSEIN M S, et al. Phys Rep, 2006, 424: 1. doi: 10.1016/j.physrep.2005.10.006 [4] XU J. Progress in Particle and Nuclear Physics, 2019, 106: 312. doi: 10.1016/j.ppnp.2019.02.009 [5] MA C W, ZHANG Y L, WANG S S, et al. Chin Phys Lett, 2015, 32: 072501. doi: 10.1088/0256-307X/32/7/072501 [6] BERTSCH G F, GUPTA S D. Phys Rep, 1988, 160: 189. doi: 10.1016/0370-1573(88)90170-6 [7] BONASERA A, GULMINELLI F, MOLITORIS J. Phys Rep, 1994, 1: 243. [8] AICHELIN J, STÖCKER H. Phys Lett B, 1986, 176: 14. doi: 10.1016/0370-2693(86)90916-0 [9] AICHELIN J. Phys Rep, 1991, 202: 233. doi: 10.1016/0370-1573(91)90094-3 [10] 王闪闪, 曹喜光, 张同林, 等. 原子核物理评论, 2015, 32: 24. doi: 10.11804/NuclPhysRev.32.01.024 WANG S S, CAO X G, ZHANG T L, et al. Nuclear Physics Review, 2015, 32: 24. (in Chinese) doi: 10.11804/NuclPhysRev.32.01.024 [11] MARUYAMA T, NⅡTA K, IWAMOTO A. Phys Rev C, 1996, 53: 297. doi: 10.1103/PhysRevC.53.297 [12] PAPA M, MARUYAMA T, BONASERA A, et al. Phys Rev C, 2001, 64: 024612. doi: 10.1103/PhysRevC.64.024612 [13] ZHANG S, WANG J C, HUANG M R, et al. Chines Physics C, 2019, 43: 064102. doi: 10.1088/1674-1137/43/6/064102 [14] ZHANG S, BONASERA A, HUANG M, et al. Phys Rev C, 2019, 99: 044605. doi: 10.1103/PhysRevC.99.044605 [15] ONO A, HORIUCHI H. Phys Rev Lett, 1992, 68: 2898. doi: 10.1103/PhysRevLett.68.2898 [16] FELDMEIER H. Nucl Phys A, 1990, 515: 147. doi: 10.1016/0375-9474(90)90328-J [17] SUN X Y, FANG D Q, MA Y G, et al. Phys Lett B, 2010, 682: 396. doi: 10.1016/j.physletb.2009.11.031 [18] YONG G C, GAO Y, ZUO W, et al. Phys Rev C, 2011, 84: 034609. doi: 10.1103/PhysRevC.84.034609 [19] DAI Z T, FANG D Q, MA Y G, et al. Phys Rev C, 2014, 89: 014613. doi: 10.1103/PhysRevC.89.014613 [20] WEI G F, LI B A, XU J, et al. Phys Rev C, 2014, 90: 014610. doi: 10.1103/PhysRevC.90.014610 [21] WEI G F. Phys Rev C, 2015, 91: 014616. doi: 10.1103/PhysRevC.91.014616 [22] ONO A, HORIUCHI H. Phys Rev C, 1999, 59: 853. doi: 10.1103/PhysRevC.59.853 [23] ONO A, HORIUCHI H, MARUYAMA T, et al. Phys Rev C, 1993, 48: 2946. doi: 10.1103/PhysRevC.48.2946 [24] ONO A, HORIUCHI H. Phys Rev C, 1996, 53: 2341. doi: 10.1103/PhysRevC.53.2341 [25] LIN W, LIU X, WADA R, et al. Phys Rev C, 2016, 94: 064609. doi: 10.1103/PhysRevC.94.064609 [26] WADA R. Phys Rev C, 2000, 62: 034601. doi: 10.1103/PhysRevC.62.034601 [27] ONO A, HORIUCHI H, MARUYAMA T, et al. Progress of Theoretical Physics, 1992, 87: 1185. doi: 10.1143/ptp/87.5.1185 [28] DECHARGÉ J, GOGNY D. Phys Rev C, 1980, 21: 1568. doi: 10.1103/PhysRevC.21.1568 [29] ONO A, DANIELEWICZ P, FRIEDMAN W A, et al. Phys Rev C, 2003, 68: 051601. doi: 10.1103/PhysRevC.68.051601 [30] WANG M, AUDI G, KONDEV F G, et al. Chin Phys C, 2017, 41: 030003. doi: 10.1088/1674-1137/41/3/030003 [31] ANGELI I, MARINOVA K P. Atomic Data and Nuclear Data Tables, 2013, 99: 69. doi: 10.1016/j.adt.2011.12.006