-
在Gamow-like模型中,
$ {\alpha} $ 衰变半衰期和质子放射半衰期可以用衰变常数$ \mathcal{\lambda} $ 表示[42-43]$$ T = \frac{\ln2}{\lambda}10^h, $$ (1) 其中
$ h $ 是阻碍因子,用于描述$ {\alpha} $ 衰变中奇质子和/或奇中子的影响。在原子核的质子放射性中$ h $ = 0,在原子核的$ {\alpha} $ 衰变中,对于偶-偶核,$ h $ = 0,对于奇-A核,$ h_{\rm{p}} = h_{\rm{n}} = h $ ,对于奇-奇核,$ h_{\rm{np}} = 2h $ ,$ h $ 值通过拟合$ {\alpha} $ 衰变实验数据与Gamow-like模型计算的$ {\alpha} $ 衰变理论值得到。衰变常数$ \lambda $ 可以表示为[44]$$ \lambda = \nu P, $$ (2) 式(2)中的
$ P $ 表示放射的粒子穿过势垒的概率,它通过经典WKB近似计算得出,在Gamow-like模型中具体表示为$$ P = \exp\! \Big[- \frac{2}{\hbar} \int\nolimits_{R}^{\,b} \sqrt{2\mu (V(r)-E_{\rm{k}})}\, {\rm{d}}r\Big], $$ (3) 这里
$ E_{\rm{k}} = Q{\frac{A-A'}{A}} $ 是放射粒子带走的动能。$ Q $ 是衰变能量,$ A $ 和$ A' $ 分别是母核和放射粒子的质量数。$ b $ 是外转折点,它满足条件$ V(b) = E_{\rm{k}} $ 。$\mu = \frac{M_{\rm{d}}M'} {(M_{\rm{d}} + M')}$ 是折合质量,$ M_ {{\rm{d}}} $ 和$ M' $ 是子核和放射粒子的质量。$ V(r) $ 是放射粒子-子核相互作用的总势能。一般情况下,放射粒子-子核之间的电势表示为库仑势:
$$ V_{\rm{C}}(r) = \frac{Z'Z_de^2}{r}, $$ (4) 其中
$ Z' $ 和$ Z_{\rm{d}} $ 是放射粒子和子核的质子数。在考虑静电屏蔽效应后,我们使用Hulthen势来代替这种屏蔽效应的库仑势,它表示为$$ V_{\rm{h}}(r) = \frac{a Z_{\rm{d}} Z' e^2}{e^{ar}-1}, $$ (5) 其中
$ a $ 是静电屏蔽参数。总的放射粒子-子核相互作用势
$ V(r) $ 由下式给出:$$ V(r) = \left\{ \begin{array}{rcl} -V_0,& & {0 \leqslant r \leqslant R,}\\ V_{\rm{h}}(r)+V_{\rm{l}}(r), & & {r \geqslant R,} \end{array} \right. $$ (6) 其中
$ V_0 $ 是方势阱的深度。$ V_{\rm{h}}(r) $ 和$ V_{\rm{l}}(r) $ 分别表示Hulthen势和离心势。球形方阱半径$ R $ 等于子核和放射粒子的半径之和,表示为$$ R = r_0\left({A_{\rm{d}}}^\frac{1}{3}+{A'}^\frac{1}{3}\right), $$ (7) 其中
$ A_{\rm{d}} $ 是子核的质量数,半径常数$ r_{0} $ 是可调参数。对于偶偶核基态到基态的
$ {\alpha} $ 衰变,角动量为零,不用考虑离心势的影响,但对于非偶偶核的$ {\alpha} $ 衰变以及原子核的质子放射性的一维问题,$ l(l +1)\rightarrow(l + 1/2)^2 $ 是必要的修正[45],所以本工作中的离心势$ V_{\rm{l}}(r) $ 的表达式为$$ V_{\rm{l}}(r) = \frac{\hbar^2\left(l+\frac{1}{2}\right)^2}{2{\mu}r^2}, $$ (8) 其中
$ l $ 是放射粒子带走的轨道角动量。根据角动量守恒定律和选择定则,可以确定放射粒子带走的最小角动量$ l_{\text {min}} $ :$$ {l_{\min }} = \left\{ {\begin{array}{*{20}{l}} {{\Delta _j},}&{{\rm{even}}\;{\Delta _j},{\pi _{\rm{p}}}{\rm{ = }}{\pi _{\rm{d}}},}\\ {{\Delta _j} + 1,}&{{\rm{even}}\;{\Delta _j},{\pi _{\rm{p}}} \ne {\pi _{\rm{d}}},}\\ {{\Delta _j},}&{{\rm{odd}}\;{\Delta _j},{\pi _{\rm{p}}} \ne {\pi _{\rm{d}}},}\\ {{\Delta _j} + 1,}&{{\rm{odd}}\;{\Delta _j},{\pi _{\rm{p}}}{\rm{ = }}{\pi _{\rm{d}}},} \end{array}} \right. $$ (9) 其中
$ {\Delta}_j = |j_{\rm{p}}-j_{\rm{d}}| $ 。$ j_{\rm{p}} $ ,$ {\pi}_{\rm{p}} $ ,$ j_{\rm{d}} $ ,$ {\pi}_{\rm{d}} $ 分别表示母核和子核的自旋和宇称。$ \nu $ 表示放射粒子的碰撞频率,它可以用振荡频率$ \omega $ 来计算,并表示为[46]$$ \nu = \omega/2\pi = \dfrac{\left(2n_{\rm{r}}+l+\frac{3}{2}\right)\hbar}{2\pi \mu {R_{\rm{n}}}^2} = \frac{\left(G+\frac{3}{2}\right)\hbar}{1.2\pi \mu{R_{0}}^2}, $$ (10) 其中
$ R_{\rm{n}} = \sqrt{3/5}R_{0} $ 是原子核均方根(rms)半径,$ R_0 = 1.28A^{1/3}-0.76+0.8A^{-1/3} $ 是母核的半径。$ G = 2n_{\rm{r}}+l $ 是主要量子数,其中$ n_{\rm{r}} $ 和$ l $ 分别是径向量子数和角量子数。$ G $ 的值可以通过下式获得[47]:$$ G = 2n_{\rm{r}}+l = \left\{ \begin{array}{rcl} 18, & & {N \leqslant 82,}\\ 20, & & {82 < N \leqslant 126,}\\ 22, & & {N > 126\text{。}} \end{array} \right. $$ (11) -
首先,我们根据
$ {\alpha} $ 衰变以及质子放射的一些实验数据拟合出静电屏蔽参数$ a $ 和半径常数$ r_0 $ 。半衰期、自旋和宇称的实验数据取自最新核数据表NUBASE2016[41],衰变能实验数据取自最新的原子质量表AME2016[48-49]。对于$ {\alpha} $ 衰变,在拟合过程中,首先使用169个偶-偶核的实验数据拟合得到$ a^{\alpha} $ 和$ r^{\alpha}_0 $ 的值,然后再使用132个奇$ N $ 偶$ Z $ 核、94个偶$ N $ 奇$ Z $ 核和66个奇-奇核拟合得到$ h $ 的值,且$ h_{\rm{n}} = h_{\rm{p}} = \frac{1}{2}h_{\rm{np}} = h $ 。对于质子放射,使用41个质子放射的实验数据拟合得到相应的$ a^{\rm{p}} $ 和$ r^{\rm{p}}_0 $ 。拟合的结果如下:$$ \ r^{\alpha}_0 = 1.14\; \text{fm}, \; a^{\alpha} = 7.7\times10^{-4},\; \; h = 0.342\text{。} $$ (12) $$ \ r^{\rm{p}}_0 = 1.14\; \text{fm}, \; a^{\rm{p}} = 5.6\times10^{-4}\text{。} $$ (13) 为了直观地给出Hulthen势对外转折点的影响,在图1中,我们描绘了不同的动能
$ E_{\rm{k}} $ 值对应于纯库仑势和Hulthen势的外转折点的值之差,其中$ b_{\rm{c}} $ 和$ b_{\rm{h}} $ 分别代表使用纯库仑势和Hulthen势计算的外转折点的值。从该图可以发现,子核的动能越小,质子数越大,纯库仑势和Hulthen势的经典转折点外转折点的值之差就越大。图 1 (在线彩图)对于原子核的
${\alpha}$ 衰变,通过$V(r) = E_{\rm{k}}$ 获得的纯库仑势转折点$(b_{\rm{c}})$ 和Hulthen势转折点($b_{\rm{h}}$ )之差对于质子发射,由于衰变的质子数据只有41个,所以每一个核使用Hulthen势与库仑势计算的转折点的差值都可以被直接描绘。在图2中,库仑势对应的转折点势
$ R_{\rm{out}}^{\rm{C}} $ ,Hulthen势对应的转折点是$ R_{\rm{out}}^{\rm{H}} $ 。从图2中可以看出,质子放射性的屏蔽效应相当大。静电排斥的屏蔽将该半径缩短了百分之几。此外,在库仑势的情况下,$ R^{\rm{C}}_{\rm{out}} $ 是$ Z_{\rm{d}}/Q_{\rm{p}} $ 的解析函数,因此势垒的差值取决于$ Z_{\rm{d}}/Q_{\rm{p}} $ ,它在图2中随$ Z_{\rm{d}}/Q_{\rm{p}} $ 增加。使用改进的Gamow-like模型,我们系统地计算了偶-偶核,奇-A核,奇-奇核
$ {\alpha} $ 衰变半衰期以及41个原子核的质子放射半衰期。对于原子核的$ {\alpha} $ 衰变,我们也基于最新的数据重新拟合了原Gamow-like模型,并计算得到了原Gamow-like模型对应的参数$ r^{\alpha'}_0 = 1.2\; \text{fm}, h' = 0.5135 $ ,计算的详细结果如图3~6所示。图3给出了169个偶-偶核的$ {\alpha} $ 衰变半衰期的实验数据以及使用Gamow-like和改进的Gamow-like计算的$ {\alpha} $ 衰变半衰期的理论值。X轴表示原子核质量数,Y轴表示对数形式的$ {\alpha} $ 衰变半衰期。$ \text{lg}T_{1/2}^{\text{exp}} $ 代表$ {\alpha} $ 衰变实验数据的对数形式,$ \text{lg}T_{1/2}^{\text{cal1}} $ 代表使用改进的Gamow-like计算的$ {\alpha} $ 衰变半衰期的对数形式,$ \text{lg}T_{1/2}^{\text{cal2}} $ 代表使用Gamow-like计算的$ {\alpha} $ 衰变半衰期的对数形式。图4~6分别代表了奇数$ N $ 偶数$ Z $ 原子核,偶数$ N $ 奇数$ Z $ 原子核和奇-奇核的计算结果。图 2 (在线彩图)对于原子核的质子发射,通过
$V(r) = E_{\rm{k}}$ 获得的库仑势转折点($R_{\rm{out}}^{\rm{C}}$ )和Hulthen势转折点($R_{\rm{out}}^{\rm{H}}$ )之差从图3~6可以看出,相比于原Gamow-like模型,我们的方法与实验数据符合得更好。为了更直观地比较两种模型的差异,我们在表1中给出了
$ {\alpha} $ 衰变理论半衰期和实验数据的标准差$ \sigma = \sqrt{\sum ({\text{lg}{T^{\text{expt}}_{1/2}}}-{\text{lg}{T^{\text{cal}}_{1/2}}})^2/n} $ 。在表1的第一列中,$ {\pi_\text{z}} $ 和$ {\pi_{\rm{n}}} $ 分别表示质子和中子的奇偶性,第二列是相应的原子核总数,第三列与第四列分别表示了改进的Gamow-like模型和Gamow-like模型对应的$ h $ 值。第五列和第六列分别是改进的Gamow-like模型和Gamow-like模型与实验数据的均方根偏差,其中$ {\text{lg}{T^{\text{cal1}}_{1/2}}} $ 和$ {\text{lg}{T^{\text{cal2}}_{1/2}}} $ 与$ {\text{lg} {T^{\text{expt}}_{1/2}}} $ 之间的标准差用$ \sigma_1 $ 和$ \sigma_2 $ 分别表示。通过表1可以看出,对于Gamow-like模型,考虑库仑势的屏蔽效应和离心势可以更好地再现$ \alpha $ 实验数据。表 1 改进的Gamow-like模型与Gamow-like模型的均方根偏差
${\pi_\text{z}}-{\pi_\text{n}}$ n $h$ $h'$ $\sigma_1$ $\sigma_2$ e-e 169 – – 0.347 0.471 e-o 132 0.342 0.5135 0.680 0.821 o-e 94 0.342 0.5135 0.597 0.673 o-o 66 0.684 1.027 0.747 0.890 对于41个质子发射核,我们在表2详细地列出了使用Gamow-like和改进的Gamow-like模型计算得到的计算结果。从表2可以计算得出我们的模型和Gamow-like模型与实验数据的均方根差分别为0.468和0.559,同样地可以看出我们的工作更加符合实验数据。
表 2 使用Gamow-like模型与改进的Gamow-like模型计算的质子放射半衰期
Nucleus $Q_{\rm p}$ $l_{\rm{min}}$ lg$T^{\rm{expt}}_{1/2} $ lg${T_{1/2}^{\rm{calc1}}} $ lg${T_{1/2}^{\rm{calc2}}} $ $^{105}\mathrm{Sb}$ 0.491 2 2.086 1.906 1.768 $^{109}\mathrm{I}$ 0.821 2 –3.897 –4.320 –4.153 $^{112}\mathrm{Cs}$ 0.821 2 –3.310 –3.585 –3.432 $^{113}\mathrm{Cs}$ 0.972 2 –4.752 –5.662 –5.438 $^{121}\mathrm{Pr}$ 0.891 2 –1.921 –3.236 –3.069 $^{130}\mathrm{Eu}$ 1.031 2 –3.000 –3.830 –3.616 $^{131}\mathrm{Eu}$ 0.951 2 –1.703 –2.763 –2.586 $^{135}\mathrm{Tb}$ 1.181 3 –2.996 –4.152 –3.849 $^{140}\mathrm{Ho}$ 1.092 3 –2.222 –2.549 –2.287 $^{141}\mathrm{Ho^m}$ 1.251 0 –5.137 –5.972 –5.738 $^{145}\mathrm{Tm}$ 1.741 5 –5.499 –5.595 –5.077 $^{146}\mathrm{Tm}$ 0.891 0 –0.810 –0.604 –0.550 $^{146}\mathrm{Tm^m}$ 1.201 5 –1.125 –1.030 –0.629 $^{147}\mathrm{Tm}$ 1.059 5 0.573 0.707 1.05 $^{147}\mathrm{Tm^m}$ 1.12 2 –3.444 –3.117 –2.890 $^{150}\mathrm{Lu}$ 1.271 5 –1.201 –1.261 –0.919 $^{150}\mathrm{Lu^m}$ 1.291 2 –4.398 –4.433 –4.226 $^{151}\mathrm{Lu}$ 1.243 5 –0.916 –0.972 –0.638 $^{151}\mathrm{Lu^m}$ 1.291 2 –4.783 –4.442 –4.235 $^{155}\mathrm{Ta}$ 1.451 5 –2.495 –2.524 –2.139 $^{156}\mathrm{Ta}$ 1.021 2 –0.828 –0.520 –0.438 $^{156}\mathrm{Ta^m}$ 1.111 5 0.924 1.167 1.437 $^{157}\mathrm{Ta}$ 0.941 0 –0.529 –0.057 –0.068 $^{159}\mathrm{Re^m}$ 1.816 5 –4.666 –4.874 –4.425 $^{160}\mathrm{Re}$ 1.271 0 –3.164 –3.889 –3.742 $^{161}\mathrm{Re}$ 1.201 0 –3.357 –3.094 –2.974 $^{161}\mathrm{Re^m}$ 1.321 5 –0.680 –0.786 –0.443 $^{164}\mathrm{Ir}$ 1.844 5 –3.959 –4.661 –4.210 $^{165}\mathrm{Ir^m}$ 1.721 5 –3.430 –3.819 –3.388 $^{166}\mathrm{Ir}$ 1.161 2 –0.842 –1.228 –1.094 $^{166}\mathrm{Ir^m}$ 1.331 5 –0.091 –0.387 –0.049 $^{167}\mathrm{Ir}$ 1.071 0 –1.128 –0.763 –0.716 $^{167}\mathrm{Ir^m}$ 1.246 5 0.778 0.559 0.865 $^{170}\mathrm{Au}$ 1.471 2 –3.487 –4.070 –3.832 $^{170}\mathrm{Au^m}$ 1.751 5 –2.975 –3.621 –3.188 $^{171}\mathrm{Au}$ 1.448 0 –4.652 –4.607 –4.415 $^{171}\mathrm{Au^m}$ 1.702 5 –2.587 –3.267 –2.842 $^{176}\mathrm{Tl}$ 1.261 0 –2.208 –2.053 –1.932 $^{177}\mathrm{Tl}$ 1.155 0 –1.178 –0.698 –0.627 $^{177}\mathrm{Tl^m}$ 1.962 5 –3.459 –4.674 –4.210 $^{185}\mathrm{Bi^m}$ 1.607 0 –4.192 –5.050 –4.822 如今超重核的合成和研究已成为核物理领域的热门话题。作为应用,现在我们将该模型推广到预测
$ Z = 120 $ 的原子核和它们的$ \alpha $ 衰变链上的一些未合成核的$ \alpha $ 衰变半衰期,以及16个原子核的放射性半衰期,这些原子核的质子放射性已经被观测到,但尚未准确定量。对于
$ Z = 120 $ 的偶偶超重核从文献中[50]可以获得这些原子核的$ \alpha $ 衰变链,即$^{296}120\to^{292}\text{Og}\to^{288}\text{Lv}\to ^{284}\text{Fl}\!\to ^{280}\!\text{Cn}\!\to^{276}\! \text{Ds}\!\to^{272}\!\text{Hs}\!\to^{268}\!\text{Sg}$ ,$^{298}120\to ^{294}\text{Og}\to ^{290}\text{Lv}\!\to^{286}\!\text{Fl}\!\to^{282}\!\text{Cn}\!\to^{278} \!\text{Ds}\!\to^{274}\!\text{Hs}$ ,$^{300}120\to^{296}\text{Og}\to ^{292}\text{Lv}\to^{288}\text{Fl}\to^{284}\text{Cn}$ ,$ ^{302}120\to\!^{298}{\text{Og}}\!\to^{294}\!\text{Lv}\!\to^{290}\!\text{Fl} $ ,$ ^{304}120\!\to^{300}\!\text{Og}\!\to^{296}\!\text{Lv}\!\to^{294}\!\text{Fl} $ ,$^{306}120\!\to^{302}\!\text{Og}\!\to ^{298}\!\text{Lv}$ ,$ ^{308}120\to^{304}\text{Og}\to^{300}\text{Lv} $ 。在我们先前对超重核的研究中[51-52],$ {\alpha} $ 衰变能是计算$ {\alpha} $ 衰减半衰期的关键。同时,Sobiczewski[53]发现,WS3+[54]计算的$ {\alpha} $ 衰变能可较好地重现$ {\alpha} $ 衰变能的实验数据。在本工作中,我们使用来自WS3+的$ {\alpha} $ 衰变能来计算质子数为$ Z = 120 $ 的偶-偶核素及其$ \alpha $ 衰变链的核的$ {\alpha} $ 衰变半衰期。衰变链有五个已知的原子核的$ \alpha $ 实验半衰期,即$ ^{294} {\rm{Og}}$ ,$ ^{290} {\rm{Lv}}$ ,$ ^{286} {\rm{Fl}}$ ,$ ^{292} {\rm{Lv}}$ 和$ ^{288} {\rm{Fl}}$ ,这些核的数据来自 NUBASE2016[41]。作为对比,我们还分别使用库仑势和亲和势模型(CPPM-Bass73)[55],Viola-Seaborg-Sobiczewski (VSS)经验公式[56],通用曲线(UNIV)[44],Royer公式[57],通用衰变定律(UDL)[58]以及Ni-Ren-Dong-Xu(NRDX)经验公式[59]系统地计算了质子数
$ Z = 120 $ 的偶-偶及其$ {\alpha} $ 衰变链上的核的$ {\alpha} $ 衰变半衰期。计算的结果分别列在了表3和表4中,其中能量的单位为MeV,使用库仑势和亲和势模型、VSS经验公式、UNIV、Royer公式、UDL、NRDX经验公式以及改进的Gamow-like模型得到地对数形式的预测结果分别用$ {\text{lg}{T^{\text{CPPM}}_{1/2}}} $ 、$ {\text{lg}{T^{\text{VSS}}_{1/2}}} $ 、$ {\text{lg}{T^{\text{UNIV}}_{1/2}}} $ 、$ {\text{lg}{T^{\text{Royer}}_{1/2}}} $ 、$ {\text{lg}{T^{\text{UDL}}_{1/2}}} $ 、$ {\text{lg}{T^{\text{NRDX}}_{1/2}}} $ 以及$ {\text{lg}{T^{\text{Gamow}}_{1/2}}} $ 表示,$ \text{lg}{T^{\text{expt}}_{1/2}} $ 则代表了部分已知原子核的$ \alpha $ 衰变半衰期。为了更直观地展示预测结果,我们将不同理论计算得到的理论$ {\alpha} $ 衰变半衰期以及部分已知的实验值描绘在图7~11中。在这些图中,衰变链以质子数$ Z = 120 $ 的原子核开始,每个衰变链末端的原子核的衰变模式为自发裂变,其余原子核的衰变模式均为$ {\alpha} $ 衰变。X轴表示相应的$ {\alpha} $ 衰变链中的衰变母核的质量数,Y轴表示$ {\alpha} $ 衰变半衰期的对数。表 3 库仑势和亲和势模型、VSS 经验公式、UNIV以及Royer公式预测的质子数
$Z = 120$ 的偶-偶及其${\alpha}$ 衰变链上的核的${\alpha}$ 衰变半衰期Nucleus $Q_{\alpha}$ ${\text{lg}{T^{\text{CPPM}}_{1/2}}}$ ${\text{lg}{T^{\text{VSS}}_{1/2}}}$ ${\text{lg}{T^{\text{UNIV}}_{1/2}}}$ ${\text{lg}{T^{\text{Royer}}_{1/2}}}$ $\begin{array}{c}^{296}120\to^{292}\text{Og}\to^{288}\text{Lv}\to^{284}\text{Fl} \to^{280}\text{Cn}\to^{276}\text{Ds}\to^{272}\text{Hs}\to^{268}\text{Sg}\end{array}$ $^{296}120$ 13.187 –6.189 –5.613 –5.884 –5.774 $^{292}{\rm{Og}}$ 12.015 –4.264 –3.662 –4.044 –3.842 $^{288}{\rm{Lv}}$ 11.105 –2.698 –2.082 –2.527 –2.275 $^{284}{\rm{Fl}}$ 10.666 –2.202 –1.568 –2.018 –1.767 $^{280}{\rm{Cn}}$ 10.911 –3.471 –2.797 –3.183 –2.999 $^{276}{\rm{Ds}}$ 10.976 –4.259 –3.555 –3.891 –3.76 $^{272}{\rm{Hs}}$ 9.54 –1.077 –0.406 –0.823 –0.603 $\begin{array}{c}^{298}120\to^{294}\text{Og}\to^{290}\text{Lv}\to^{286}\text{Fl} \to^{282}\text{Cn}\to^{278}\text{Ds}\to^{274}\text{Hs}\end{array}$ $^{298}120$ 12.9 –5.643 –5.032 –5.371 –5.231 $^{294}{\rm{Og}}$ 11.835 –3.889 –3.254 –3.688 –3.471 $^{290}{\rm{Lv}}$ 11.005 –2.482 –1.832 –2.319 –2.062 $^{286}{\rm{Fl}}$ 10.365 –1.431 –0.771 –1.28 –1.008 $^{282}{\rm{Cn}}$ 10.106 –1.375 –0.695 –1.186 –0.934 $^{278}{\rm{Ds}}$ 10.31 –2.601 –1.882 –2.315 –2.122 $^{300}120\to^{296}\text{Og}\to^{292}\text{Lv}\to^{288}\text{Fl}\to^{284}\text{Cn}$ $^{300}120$ 13.287 –6.461 –5.811 –6.13 –6.045 $^{296}{\rm{Og}}$ 11.561 –3.279 –2.612 –3.109 –2.867 $^{292}{\rm{Lv}}$ 10.775 –1.922 –1.243 –1.784 –1.51 $^{288}{\rm{Fl}}$ 10.065 –0.624 0.06 –0.506 –0.214 $^{302}120\to^{298}{\text{Og}}\to^{294}\text{Lv}\to^{290}\text{Fl}$ $^{302}120$ 12.878 –5.671 –4.986 –5.391 –5.259 $^{298}{\rm{Og}}$ 12.118 –4.607 –3.893 –4.358 –4.182 $^{294}{\rm{Lv}}$ 10.451 –1.083 –0.379 –0.981 –0.683 $^{304}120\to^{300}\text{Og}\to^{296}\text{Lv}\to^{294}\text{Fl}$ $^{304}120$ 12.745 –5.43 –4.71 –5.162 –5.019 $^{300}{\rm{Og}}$ 11.905 –4.162 –3.414 –3.935 –3.741 $^{296}{\rm{Lv}}$ 10.777 –2.002 –1.248 –1.853 –1.588 $^{306}120\to^{302}\text{Og}\to^{298}\text{Lv}$ $^{306}120$ 13.823 –7.59 –6.836 –7.169 –7.175 $^{302}{\rm{Og}}$ 11.995 –4.404 –3.618 –4.16 –3.98 $^{308}120\to^{304}\text{Og}\to^{300}\text{Lv}$ $^{308}120$ 13.036 –6.102 –5.309 –5.784 –5.689 $^{304}{\rm{Og}}$ 13.104 –6.789 –5.96 –6.389 –6.354 表 4 UDL、NRDX经验公式以及改进的Gamow-like模型预测的质子数
$Z = 120$ 的偶-偶及其${\alpha}$ 衰变链上的核的${\alpha}$ 衰变半衰期Nucleus $Q_{\alpha}$ ${\text{lg}{T^{\text{UDL}}_{1/2}}}$ ${\text{lg}{T^{\text{NRDX}}_{1/2}}}$ ${\text{lg}{T^{\text{Gamow}}_{1/2}}}$ $\text{lg}{T^{\text{expt}}_{1/2}}$ $\begin{array}{c}^{296}120\to^{292}\text{Og}\to^{288}\text{Lv}\to^{284}\text{Fl} \to^{280}\text{Cn}\to^{276}\text{Ds}\to^{272}\text{Hs}\to^{268}\text{Sg}\end{array}$ $^{296}120$ 13.187 –6.189 –5.613 –5.884 –5.774 $^{292}{\rm{Og}}$ 12.015 –4.264 –3.662 –4.044 –3.842 $^{288}{\rm{Lv}}$ 11.105 –2.698 –2.082 –2.527 –2.275 $^{284}{\rm{Fl}}$ 10.666 –2.202 –1.568 –2.018 –1.767 $^{280}{\rm{Cn}}$ 10.911 –3.471 –2.797 –3.183 –2.999 $^{276}{\rm{Ds}}$ 10.976 –4.259 –3.555 –3.891 –3.76 $^{272}{\rm{Hs}}$ 9.54 –1.077 –0.406 –0.823 –0.603 $\begin{array}{c}^{298}120\to^{294}\text{Og}\to^{290}\text{Lv}\to^{286}\text{Fl} \to^{282}\text{Cn}\to^{278}\text{Ds}\to^{274}\text{Hs}\end{array}$ $^{298}120$ 12.9 –5.258 –4.826 –5.148 – $^{294}{\rm{Og}}$ 11.835 –3.408 –3.115 –3.474 –2.939 $^{290}{\rm{Lv}}$ 11.005 –1.932 –1.748 –2.12 –2.097 $^{286}{\rm{Fl}}$ 10.365 –0.833 –0.731 –1.097 –0.456 $^{282}{\rm{Cn}}$ 10.106 –0.776 –0.677 –1.014 – $^{278}{\rm{Ds}}$ 10.31 –2.057 –1.862 –2.15 – $^{300}120\to^{296}\text{Og}\to^{292}\text{Lv}\to^{288}\text{Fl}\to^{284}\text{Cn}$ $^{300}120$ 13.287 –6.116 –5.591 –5.907 – $^{296}{\rm{Og}}$ 11.561 –2.759 –2.484 –2.895 – $^{292}{\rm{Lv}}$ 10.775 –1.338 –1.168 –1.587 –1.602 $^{288}{\rm{Fl}}$ 10.065 0.017 0.087 –0.325 –0.125 $^{302}120\to^{298}{\text{Og}}\to^{294}\text{Lv}\to^{290}\text{Fl}$ $^{302}120$ 12.878 –5.273 –4.781 –5.166 – $^{298}{\rm{Og}}$ 12.118 –4.148 –3.741 –4.144 – $^{294}{\rm{Lv}}$ 10.451 –0.453 –0.319 –0.785 – $^{304}120\to^{300}\text{Og}\to^{296}\text{Lv}\to^{294}\text{Fl}$ $^{304}120$ 12.745 –5.011 –4.509 –4.937 – $^{300}{\rm{Og}}$ 11.905 –3.672 –3.27 –3.719 – $^{296}{\rm{Lv}}$ 10.777 –1.406 –1.172 –1.654 – $^{306}120\to^{302}\text{Og}\to^{298}\text{Lv}$ $^{306}120$ 13.823 –7.296 –6.595 –6.949 – $^{302}{\rm{Og}}$ 11.995 –3.92 –3.47 –3.944 – $^{308}120\to^{304}\text{Og}\to^{300}\text{Lv}$ $^{308}120$ 13.036 –5.709 –5.096 –5.559 – $^{304}{\rm{Og}}$ 13.104 –6.434 –5.769 –6.178 – 从这些图中我们可以清楚地看到,由于模型依赖,同一个原子核的不同模型对
$ {\alpha} $ 衰变半衰期的理论计算是不同的,但是所有理论上计算出的$ {\alpha} $ 衰变半衰期曲线都具有相同的趋势。我们的模型预测结果在这些结果中靠近中间位置。对于16个尚未测出半衰期的质子放射的质子放射半衰期预测,我们选择质子的放射衰变定律(UDLP)[31]与改进的Gamow-like模型作比较,预测结果列在表5中。表 5 使用改进的Gamow-like模型和UDLP模型对16个母核的质子放射半衰期地的预测
Nucleus $Q_{\rm p}$ $l_{\rm{min}}$ lg$T^{\rm{UDLP}}_{1/2} $ lg${T_{1/2}^{\rm{This-Work}}} $ $^{108}\mathrm{I}$ 0.601 2 0.164 –0.161 $^{111}\mathrm{Cs}$ 1.811 2 –10.405 –11.521 $^{117}\mathrm{La}$ 0.821 2 –2.322 –2.728 $^{127}\mathrm{Pm}$ 0.911 2 –2.372 –2.694 $^{137}\mathrm{Tb}$ 0.831 5 2.907 3.274 $^{141}\mathrm{Ho}$ 1.181 3 –3.132 –3.304 $^{144}\mathrm{Tm}$ 1.711 5 –4.609 –4.873 $^{146}\mathrm{Tm^n}$ 1.131 5 –0.037 0.166 $^{159}\mathrm{Re}$ 1.591 0 –6.121 –6.611 $^{165}\mathrm{Ir}$ 1.541 0 –5.341 –5.728 $^{169}\mathrm{Ir^m}$ 0.765 5 7.727 8.616 $^{169}\mathrm{Au}$ 1.931 0 –7.483 –7.986 $^{172}\mathrm{Au}$ 0.861 2 3.873 4.262 $^{172}\mathrm{Au^m}$ 0.611 2 9.926 10.603 $^{185}\mathrm{Bi}$ 1.523 5 –0.881 –0.525 $^{185}\mathrm{Bi^m}$ 1.703 6 –1.044 –0.747 由于我们前面的工作发现41个质子放射的
$ \sigma = 0.468 $ ,因此预测的质子放射半衰期与实验的误差可能会在2.94的范围内。此外,为了更直观地与UDLP模型比较,我们在图12中绘制了使用我们的模型和UDLP模型预测的质子放射半衰期与母核质子数的关系,图中UDLP和Calc分别代表了使用UDLP模型与我们的模型预测的质子放射半衰期。结果表明,UDLP的预测结果与我们的模型是一致的。
Study of α Decay and Proton Radioactivity Half-lives Based on Improved Gamow-like Model
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摘要: 通过引入离心势和静电屏蔽效应对Gamow-like模型进行了改进,并将其用于α衰变和质子放射性研究,发现改进的Gamow-like模型能更好地符合实验数据。另外,还利用改进的Gamow-like模型预言了16个丰质子核的质子放射性的半衰期以及7个
$Z=120$ 超重核素($^{296-308}120$ )α衰变链上原子核的α衰变的半衰期,为将来在大科学装置上合成和鉴别这些新核素提供重要的理论参考。-
关键词:
- α衰变 /
- 质子放射 /
- Gamow-like模型 /
- 静电屏蔽
Abstract: In this paper, the Gamow-like model is improved by introducing centrifugal potential and electrostatic shielding, and it is used in the study of α decay and proton radioactivity. It is found that our calculations can well reproduce the experimental data. In addition, the modified Gamow-like model is used to predict the proton radioactivity half-lives of 116 proton-rich nuclei and α decay half-lives of seven even-even nuclei with$Z=120$ ($^{296-308}120$ ) and some nuclei on their α decay chains. It will provide important theoretical references for the synthesis and identification of these new nuclides on large scientific devices in the future.-
Key words:
- α decay /
- proton radioactivity /
- Gamow-like model /
- electrostatic shielding
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表 1 改进的Gamow-like模型与Gamow-like模型的均方根偏差
${\pi_\text{z}}-{\pi_\text{n}}$ n $h$ $h'$ $\sigma_1$ $\sigma_2$ e-e 169 – – 0.347 0.471 e-o 132 0.342 0.5135 0.680 0.821 o-e 94 0.342 0.5135 0.597 0.673 o-o 66 0.684 1.027 0.747 0.890 表 2 使用Gamow-like模型与改进的Gamow-like模型计算的质子放射半衰期
Nucleus $Q_{\rm p}$ $l_{\rm{min}}$ lg $T^{\rm{expt}}_{1/2} $ lg ${T_{1/2}^{\rm{calc1}}} $ lg ${T_{1/2}^{\rm{calc2}}} $ $^{105}\mathrm{Sb}$ 0.491 2 2.086 1.906 1.768 $^{109}\mathrm{I}$ 0.821 2 –3.897 –4.320 –4.153 $^{112}\mathrm{Cs}$ 0.821 2 –3.310 –3.585 –3.432 $^{113}\mathrm{Cs}$ 0.972 2 –4.752 –5.662 –5.438 $^{121}\mathrm{Pr}$ 0.891 2 –1.921 –3.236 –3.069 $^{130}\mathrm{Eu}$ 1.031 2 –3.000 –3.830 –3.616 $^{131}\mathrm{Eu}$ 0.951 2 –1.703 –2.763 –2.586 $^{135}\mathrm{Tb}$ 1.181 3 –2.996 –4.152 –3.849 $^{140}\mathrm{Ho}$ 1.092 3 –2.222 –2.549 –2.287 $^{141}\mathrm{Ho^m}$ 1.251 0 –5.137 –5.972 –5.738 $^{145}\mathrm{Tm}$ 1.741 5 –5.499 –5.595 –5.077 $^{146}\mathrm{Tm}$ 0.891 0 –0.810 –0.604 –0.550 $^{146}\mathrm{Tm^m}$ 1.201 5 –1.125 –1.030 –0.629 $^{147}\mathrm{Tm}$ 1.059 5 0.573 0.707 1.05 $^{147}\mathrm{Tm^m}$ 1.12 2 –3.444 –3.117 –2.890 $^{150}\mathrm{Lu}$ 1.271 5 –1.201 –1.261 –0.919 $^{150}\mathrm{Lu^m}$ 1.291 2 –4.398 –4.433 –4.226 $^{151}\mathrm{Lu}$ 1.243 5 –0.916 –0.972 –0.638 $^{151}\mathrm{Lu^m}$ 1.291 2 –4.783 –4.442 –4.235 $^{155}\mathrm{Ta}$ 1.451 5 –2.495 –2.524 –2.139 $^{156}\mathrm{Ta}$ 1.021 2 –0.828 –0.520 –0.438 $^{156}\mathrm{Ta^m}$ 1.111 5 0.924 1.167 1.437 $^{157}\mathrm{Ta}$ 0.941 0 –0.529 –0.057 –0.068 $^{159}\mathrm{Re^m}$ 1.816 5 –4.666 –4.874 –4.425 $^{160}\mathrm{Re}$ 1.271 0 –3.164 –3.889 –3.742 $^{161}\mathrm{Re}$ 1.201 0 –3.357 –3.094 –2.974 $^{161}\mathrm{Re^m}$ 1.321 5 –0.680 –0.786 –0.443 $^{164}\mathrm{Ir}$ 1.844 5 –3.959 –4.661 –4.210 $^{165}\mathrm{Ir^m}$ 1.721 5 –3.430 –3.819 –3.388 $^{166}\mathrm{Ir}$ 1.161 2 –0.842 –1.228 –1.094 $^{166}\mathrm{Ir^m}$ 1.331 5 –0.091 –0.387 –0.049 $^{167}\mathrm{Ir}$ 1.071 0 –1.128 –0.763 –0.716 $^{167}\mathrm{Ir^m}$ 1.246 5 0.778 0.559 0.865 $^{170}\mathrm{Au}$ 1.471 2 –3.487 –4.070 –3.832 $^{170}\mathrm{Au^m}$ 1.751 5 –2.975 –3.621 –3.188 $^{171}\mathrm{Au}$ 1.448 0 –4.652 –4.607 –4.415 $^{171}\mathrm{Au^m}$ 1.702 5 –2.587 –3.267 –2.842 $^{176}\mathrm{Tl}$ 1.261 0 –2.208 –2.053 –1.932 $^{177}\mathrm{Tl}$ 1.155 0 –1.178 –0.698 –0.627 $^{177}\mathrm{Tl^m}$ 1.962 5 –3.459 –4.674 –4.210 $^{185}\mathrm{Bi^m}$ 1.607 0 –4.192 –5.050 –4.822 表 3 库仑势和亲和势模型、VSS 经验公式、UNIV以及Royer公式预测的质子数
$Z = 120$ 的偶-偶及其${\alpha}$ 衰变链上的核的${\alpha}$ 衰变半衰期Nucleus $Q_{\alpha}$ ${\text{lg}{T^{\text{CPPM}}_{1/2}}}$ ${\text{lg}{T^{\text{VSS}}_{1/2}}}$ ${\text{lg}{T^{\text{UNIV}}_{1/2}}}$ ${\text{lg}{T^{\text{Royer}}_{1/2}}}$ $\begin{array}{c}^{296}120\to^{292}\text{Og}\to^{288}\text{Lv}\to^{284}\text{Fl} \to^{280}\text{Cn}\to^{276}\text{Ds}\to^{272}\text{Hs}\to^{268}\text{Sg}\end{array}$ $^{296}120$ 13.187 –6.189 –5.613 –5.884 –5.774 $^{292}{\rm{Og}}$ 12.015 –4.264 –3.662 –4.044 –3.842 $^{288}{\rm{Lv}}$ 11.105 –2.698 –2.082 –2.527 –2.275 $^{284}{\rm{Fl}}$ 10.666 –2.202 –1.568 –2.018 –1.767 $^{280}{\rm{Cn}}$ 10.911 –3.471 –2.797 –3.183 –2.999 $^{276}{\rm{Ds}}$ 10.976 –4.259 –3.555 –3.891 –3.76 $^{272}{\rm{Hs}}$ 9.54 –1.077 –0.406 –0.823 –0.603 $\begin{array}{c}^{298}120\to^{294}\text{Og}\to^{290}\text{Lv}\to^{286}\text{Fl} \to^{282}\text{Cn}\to^{278}\text{Ds}\to^{274}\text{Hs}\end{array}$ $^{298}120$ 12.9 –5.643 –5.032 –5.371 –5.231 $^{294}{\rm{Og}}$ 11.835 –3.889 –3.254 –3.688 –3.471 $^{290}{\rm{Lv}}$ 11.005 –2.482 –1.832 –2.319 –2.062 $^{286}{\rm{Fl}}$ 10.365 –1.431 –0.771 –1.28 –1.008 $^{282}{\rm{Cn}}$ 10.106 –1.375 –0.695 –1.186 –0.934 $^{278}{\rm{Ds}}$ 10.31 –2.601 –1.882 –2.315 –2.122 $^{300}120\to^{296}\text{Og}\to^{292}\text{Lv}\to^{288}\text{Fl}\to^{284}\text{Cn}$ $^{300}120$ 13.287 –6.461 –5.811 –6.13 –6.045 $^{296}{\rm{Og}}$ 11.561 –3.279 –2.612 –3.109 –2.867 $^{292}{\rm{Lv}}$ 10.775 –1.922 –1.243 –1.784 –1.51 $^{288}{\rm{Fl}}$ 10.065 –0.624 0.06 –0.506 –0.214 $^{302}120\to^{298}{\text{Og}}\to^{294}\text{Lv}\to^{290}\text{Fl}$ $^{302}120$ 12.878 –5.671 –4.986 –5.391 –5.259 $^{298}{\rm{Og}}$ 12.118 –4.607 –3.893 –4.358 –4.182 $^{294}{\rm{Lv}}$ 10.451 –1.083 –0.379 –0.981 –0.683 $^{304}120\to^{300}\text{Og}\to^{296}\text{Lv}\to^{294}\text{Fl}$ $^{304}120$ 12.745 –5.43 –4.71 –5.162 –5.019 $^{300}{\rm{Og}}$ 11.905 –4.162 –3.414 –3.935 –3.741 $^{296}{\rm{Lv}}$ 10.777 –2.002 –1.248 –1.853 –1.588 $^{306}120\to^{302}\text{Og}\to^{298}\text{Lv}$ $^{306}120$ 13.823 –7.59 –6.836 –7.169 –7.175 $^{302}{\rm{Og}}$ 11.995 –4.404 –3.618 –4.16 –3.98 $^{308}120\to^{304}\text{Og}\to^{300}\text{Lv}$ $^{308}120$ 13.036 –6.102 –5.309 –5.784 –5.689 $^{304}{\rm{Og}}$ 13.104 –6.789 –5.96 –6.389 –6.354 表 4 UDL、NRDX经验公式以及改进的Gamow-like模型预测的质子数
$Z = 120$ 的偶-偶及其${\alpha}$ 衰变链上的核的${\alpha}$ 衰变半衰期Nucleus $Q_{\alpha}$ ${\text{lg}{T^{\text{UDL}}_{1/2}}}$ ${\text{lg}{T^{\text{NRDX}}_{1/2}}}$ ${\text{lg}{T^{\text{Gamow}}_{1/2}}}$ $\text{lg}{T^{\text{expt}}_{1/2}}$ $\begin{array}{c}^{296}120\to^{292}\text{Og}\to^{288}\text{Lv}\to^{284}\text{Fl} \to^{280}\text{Cn}\to^{276}\text{Ds}\to^{272}\text{Hs}\to^{268}\text{Sg}\end{array}$ $^{296}120$ 13.187 –6.189 –5.613 –5.884 –5.774 $^{292}{\rm{Og}}$ 12.015 –4.264 –3.662 –4.044 –3.842 $^{288}{\rm{Lv}}$ 11.105 –2.698 –2.082 –2.527 –2.275 $^{284}{\rm{Fl}}$ 10.666 –2.202 –1.568 –2.018 –1.767 $^{280}{\rm{Cn}}$ 10.911 –3.471 –2.797 –3.183 –2.999 $^{276}{\rm{Ds}}$ 10.976 –4.259 –3.555 –3.891 –3.76 $^{272}{\rm{Hs}}$ 9.54 –1.077 –0.406 –0.823 –0.603 $\begin{array}{c}^{298}120\to^{294}\text{Og}\to^{290}\text{Lv}\to^{286}\text{Fl} \to^{282}\text{Cn}\to^{278}\text{Ds}\to^{274}\text{Hs}\end{array}$ $^{298}120$ 12.9 –5.258 –4.826 –5.148 – $^{294}{\rm{Og}}$ 11.835 –3.408 –3.115 –3.474 –2.939 $^{290}{\rm{Lv}}$ 11.005 –1.932 –1.748 –2.12 –2.097 $^{286}{\rm{Fl}}$ 10.365 –0.833 –0.731 –1.097 –0.456 $^{282}{\rm{Cn}}$ 10.106 –0.776 –0.677 –1.014 – $^{278}{\rm{Ds}}$ 10.31 –2.057 –1.862 –2.15 – $^{300}120\to^{296}\text{Og}\to^{292}\text{Lv}\to^{288}\text{Fl}\to^{284}\text{Cn}$ $^{300}120$ 13.287 –6.116 –5.591 –5.907 – $^{296}{\rm{Og}}$ 11.561 –2.759 –2.484 –2.895 – $^{292}{\rm{Lv}}$ 10.775 –1.338 –1.168 –1.587 –1.602 $^{288}{\rm{Fl}}$ 10.065 0.017 0.087 –0.325 –0.125 $^{302}120\to^{298}{\text{Og}}\to^{294}\text{Lv}\to^{290}\text{Fl}$ $^{302}120$ 12.878 –5.273 –4.781 –5.166 – $^{298}{\rm{Og}}$ 12.118 –4.148 –3.741 –4.144 – $^{294}{\rm{Lv}}$ 10.451 –0.453 –0.319 –0.785 – $^{304}120\to^{300}\text{Og}\to^{296}\text{Lv}\to^{294}\text{Fl}$ $^{304}120$ 12.745 –5.011 –4.509 –4.937 – $^{300}{\rm{Og}}$ 11.905 –3.672 –3.27 –3.719 – $^{296}{\rm{Lv}}$ 10.777 –1.406 –1.172 –1.654 – $^{306}120\to^{302}\text{Og}\to^{298}\text{Lv}$ $^{306}120$ 13.823 –7.296 –6.595 –6.949 – $^{302}{\rm{Og}}$ 11.995 –3.92 –3.47 –3.944 – $^{308}120\to^{304}\text{Og}\to^{300}\text{Lv}$ $^{308}120$ 13.036 –5.709 –5.096 –5.559 – $^{304}{\rm{Og}}$ 13.104 –6.434 –5.769 –6.178 – 表 5 使用改进的Gamow-like模型和UDLP模型对16个母核的质子放射半衰期地的预测
Nucleus $Q_{\rm p}$ $l_{\rm{min}}$ lg $T^{\rm{UDLP}}_{1/2} $ lg ${T_{1/2}^{\rm{This-Work}}} $ $^{108}\mathrm{I}$ 0.601 2 0.164 –0.161 $^{111}\mathrm{Cs}$ 1.811 2 –10.405 –11.521 $^{117}\mathrm{La}$ 0.821 2 –2.322 –2.728 $^{127}\mathrm{Pm}$ 0.911 2 –2.372 –2.694 $^{137}\mathrm{Tb}$ 0.831 5 2.907 3.274 $^{141}\mathrm{Ho}$ 1.181 3 –3.132 –3.304 $^{144}\mathrm{Tm}$ 1.711 5 –4.609 –4.873 $^{146}\mathrm{Tm^n}$ 1.131 5 –0.037 0.166 $^{159}\mathrm{Re}$ 1.591 0 –6.121 –6.611 $^{165}\mathrm{Ir}$ 1.541 0 –5.341 –5.728 $^{169}\mathrm{Ir^m}$ 0.765 5 7.727 8.616 $^{169}\mathrm{Au}$ 1.931 0 –7.483 –7.986 $^{172}\mathrm{Au}$ 0.861 2 3.873 4.262 $^{172}\mathrm{Au^m}$ 0.611 2 9.926 10.603 $^{185}\mathrm{Bi}$ 1.523 5 –0.881 –0.525 $^{185}\mathrm{Bi^m}$ 1.703 6 –1.044 –0.747 -
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