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热核系统的驱动力不仅仅是裸势的负梯度,它还应该包含热力学修正[12, 34]。因此,在Langevin方程中,熵被用于描述裂变系统的动力学演化。我们用下面的一维Langevin方程进行动力学轨道计算:
$$ \begin{split} \frac{{\rm d} q}{{\rm d} t} = \frac{p}{m}, \;\;\;\;\frac{{\rm d} p}{{\rm d} t} = K -\beta p + \sqrt{m\beta T}\varGamma(t){\text{。}}\end{split} $$ (1) 其中:t是时间变量;q是无量纲形变坐标,表示未来裂变碎片质心间距离的一半(以复合核半径为单位);p是其共轭动量;m是惯性质量参数;
$ \beta $ 和T分别表示耗散强度和温度。$ \varGamma(t) $ 表示涨落力,满足$ <\varGamma(t)> $ =0和$ <\varGamma(t)\varGamma(t')> $ =2$ \delta(t-t') $ 。方程(1)中的驱动力K通过在恒定激发能下的熵
$ S(q) $ 对q的导数来获得:$$ K = T\frac{{\rm d} S}{{\rm d} q}\; {\text{。}} $$ (2) 熵
$ S(q) $ 可由Fermi-gas模型[12]得到:$$ S(q) = 2\sqrt{a(q)\big[E^*-V(q)-E_{\mathrm{coll}}\big]}, $$ (3) 其中
$ E^* $ 为裂变系统的总激发能,$ E_{\mathrm{coll}} $ 为系统集体自由度的动能。有限力程液滴势能$ V(q) $ [35]包含了形变依赖的表面能、库仑能和转动能项。构建熵
$ S(q) $ 时,使用了Ignatyuk等[36]的方法来计算形变相关的能级密度参数:$$ a(q) = 0.073 A + 0.095 A^{2/3} B_{\rm s}(q), $$ (4) 式中:A为原子核的质量数;
$ B_{\rm s}(q) $ 是描述原子核表面形状的无量纲参数[12]。轻粒子的衰变宽度用Blann的参数化公式[37]计算:
$$ \begin{split} \varGamma_{\rm \nu} =& (2s_{\nu}+1)\frac{m_{\nu}}{\pi^2 \hbar^2 \rho_{\rm c}(E^*_{\mathrm{intr}})}\times \\ & \int\nolimits_{0}^{E^*_{\mathrm{intr}}-B_{\nu}} {\rm d}\varepsilon_{\nu} \rho_{\rm R}(E^*_{\mathrm{intr}}-B_{\nu}-\varepsilon_{\nu})\varepsilon_{\nu}\sigma_{\mathrm{inv}}(\varepsilon_{\nu}),\end{split}$$ (5) 这里,
$ s_{\nu} $ 是被发射粒子$ \nu $ (=n, p,$ ^{2}{\rm H} $ ,$ ^{3}{\rm H} $ ,$ ^{3}{\rm He} $ ,$ \alpha $ )的自旋,$ m_{\nu} $ 是其相对于蒸发余核的约化质量。系统内部的激发能是$ E^*_{\mathrm{intr}}\big[ = E^*-V(q)-E_{\mathrm{intr}}-E_{\mathrm{evap}}(t)\big] $ ,其中$ E_{\mathrm{evap}}(t) $ 代表在时间t之前所有被蒸发粒子所带走的能量。$ \rho_{\rm c}(E^*_{\mathrm{intr}}) $ 和$ \rho_{\rm R}(E^*_{\mathrm{intr}}-B_{\nu}-\varepsilon_{\nu}) $ 是母核和子核的能级密度[38]。$ B_{\nu} $ 和$ \varepsilon_{\nu} $ 分别表示被发射粒子$ {\nu} $ 的结合能和动能。$ \sigma_{\mathrm{inv}}(\varepsilon_{\nu}) $ 为逆反应截面[37]。本文采用离散化的方式[12-13, 33]处理轻粒子发射。首先计算每个时间步长
$ \tau $ 时的轻粒子衰变宽度;然后在时间步长$ \tau $ 内判断一个随机数$ \zeta $ ($ 0\leqslant\zeta\leqslant1 $ )是否小于$ \tau/\tau_{\mathrm{dec}} $ ;如是,则粒子被允许发射,并根据$ \varGamma_{\nu}/\varGamma_{\mathrm{part}} $ 权重通过Monte Carlo方法挑选出被发射粒子的种类。这里,$ \tau_{\mathrm{dec}} \!=\! \hbar/\varGamma_{\mathrm{part}} $ ,$ \varGamma_{\mathrm{part}} $ 代表所有轻粒子衰变宽度的和。粒子发射会引起衰变系统角动量的损失。本工作中假定如下[12-13]:发射一个中子、一个质子、一个$ \alpha $ 粒子和一个$ \gamma $ 光子分别带走角动量1$ \hbar $ 、1$ \hbar $ 、2$ \hbar $ 和1$ \hbar $ 。在粒子发射以后,Langevin方程中的内能$ E^*_{\mathrm{intr}} $ 、势能$ V(q) $ 和熵将会被重新计算并继续进行动力学方程的演化。当动力学轨道从基态越过鞍点到达断点,它就被视作一次裂变事件。断前粒子多重性是通过计算在裂变轨道中蒸发的粒子数获得的。进而,通过记录粒子发射时的形变坐标q并比较其与鞍点位置的大小来判断该粒子是来自鞍点内还是来自鞍点外,从而得到鞍点前、后的粒子多重性。
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摘要: 基于考虑了粒子发射的随机Langevin模型,计算了重裂变核240Am在 鞍点后发射的中子、质子和
$ \alpha $ 粒子多重性作为鞍点后摩擦强度($ \beta $ )的函数。结果表明在高激发能($ E^* $ )和高角动量($ \ell $ )条件下,这些轻粒子发射对摩擦的敏感性变强。进而,比较了在(高$ E^* $ ,低$ \ell $ )和(低$ E^* $ ,高$ \ell $ )这两个不同初始条件下,240Am核在鞍点后蒸发的粒子随$ \beta $ 的演化。发现前者不但能增强核摩擦对粒子发射的影响,也显著提高了带电粒子对$ \beta $ 的敏感性。在实验方面,我们建议可以用中能重离子碰撞的方式产生高激发的重裂变系统,来更精确地用粒子发射(尤其是轻带电粒子)来探测鞍点后的摩擦强度。Abstract: The stochastic Langevin model is applied to calculate the evolution of postsaddle emitted neutrons, protons and$ \alpha $ particles of heavy 240Am nuclei with the postsaddle friction strength($ \beta $ ). A significantly enhanced sensitivity of these particles to$ \beta $ has been observed at a high energy($ E^* $ ) and a large angular momentum($ \ell $ ). Moreover, the postsaddle emission as a function of$ \beta $ is compared under two contrasting initial conditions of (high$ E^* $ , low$ \ell $ ) and (low$ E^* $ , high$ \ell $ ) for the populated 240Am system. It is shown that for the former type of conditions, the influence of friction on particle evaporation is amplified and the sensitivity of light charged particles to friction is significantly increased. These results suggest that in experiment, to precisely probe the postsaddle friction through the measurement of particle multiplicity, in particular light charged-particle multiplicities, it is best to choose intermediate-energy heavy-ion collisions to produce highly excited heavy fissioning nuclei. -
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