Systematic study of $\alpha$ decay for odd-$A$ nuclei within a two-potential approach

TL;DR

This study systematically analyzes alpha decay half-lives of odd-A nuclei using a two-potential approach, incorporating shell effects and proton-neutron correlations to improve predictions of experimental data.

Contribution

It introduces an analytic formula for alpha preformation probabilities considering shell structure and correlations, enhancing decay half-life calculations for odd-A nuclei.

Findings

Preformation probabilities are slightly smaller for even-Z, odd-N nuclei.

Odd-even staggering influences spontaneous fission more than alpha decay.

Calculated half-lives match experimental data well.

Abstract

$\alpha$ decay is usually associated with both ground and low-lying isomeric states of heavy and superheavy nuclei, and the unpaired nucleon plays a key role on $\alpha$ decay. In this work, we systematically studied the $\alpha$ decay half-lives of odd-$A$ nuclei, including both favored and unfavored $\alpha$ decay within the two-potential approach based on the isospin dependent nuclear potential. The $\alpha$ preformation probabilities are estimated by using an analytic formula taking into account the shell structure and proton-neutron correlation, and the parameters are obtained through the $\alpha$ decay half-lives data. The results indicate that in general the $\alpha$ preformation probabilities of even-$Z$, odd-$N$ nuclei are slightly smaller than the odd-$Z$, even-$N$ ones. We found that the odd-even staggering effect may play a more important role on spontaneous fission than $\alpha$ decay. The calculated half-lives can well reproduce the experimental data.

Tables

Table 1. Table 1: The parameters of α 𝛼 \alpha preformation probaility of eq. (10) in different regions. III.1 and III.2 denote the results for even-odd and odd-even nuclei in region III, respectively.

Region	a	b	c	d	e
I	-1.8782	0.0013	0.0015	0.0049	-0.0009
II	-7.0561	0.004	0.0027	0.0248	-0.0017
III.1	51.2962	0.0044	0.0128	-0.2517	0.0172
III.2	10.6825	0.0042	0.0035	-0.0587	0.0015
IV	5.3064	-0.0085	-0.003	-0.009	0.0004

Table 2. Table 2: Calculations of α 𝛼 \alpha decay half-lives of odd- A 𝐴 A nuclei in region I within a two-potential approach.

$\alpha$ transition	$I_i^{\pi }\to I_j^{\pi }$	$Q_{\alpha }$(MeV)	$l_{\text{min}}$	$T_{1/2}^{\text{expt}}$(s)	$T_{1/2}^{\text{calc}}$(s)	$P_{\alpha }$
145Pm$\to$141Pr	5/2+$\to$5/2+	2.324	0	1.99$\times {10}^{17}$	6.47$\times {10}^{17}$	0.175
147Sm$\to$143Nd	7/2-$\to$7/2-	2.311	0	3.36$\times {10}^{18}$	6.03$\times {10}^{18}$	0.205
147Eu$\to$143Pm	5/2+$\to$5/2+	2.991	0	9.46$\times {10}^{10}$	4.02$\times {10}^{11}$	0.187
149Gd$\to$145Sm	7/2-$\to$7/2-	3.099	0	1.86$\times {10}^{11}$	1.64$\times {10}^{11}$	0.214
149Tb$\to$145Eu	1/2+$\to$5/2+	4.078	2	8.88$\times {10}^4$	5.07$\times {10}^4$	0.194
149Tbm$\to$145Eu	11/2-$\to$5/2+	4.114	3	1.13$\times {10}^6$	6.21$\times {10}^4$	0.194
151Tb$\to$147Eu	1/2(+)$\to$5/2+	3.496	2	6.67$\times {10}^8$	5.03$\times {10}^8$	0.242
151Dy$\to$147Gd	7/2(-)$\to$7/2-	4.18	0	1.92$\times {10}^4$	1.87$\times {10}^4$	0.219
151Ho$\to$147Tbm	11/2(-)$\to$11/2${}_{}{}^{-}\#$	4.644	0	1.6$\times {10}^2$	1.79$\times {10}^2$	0.197
151Hom$\to$147Tb	1/2(+)$\to$(1/2+)	4.736	0	6.13$\times {10}^1$	5.85$\times {10}^1$	0.197
153Dy$\to$149Gd	7/2(-)$\to$7/2-	3.559	0	2.45$\times {10}^8$	3.02$\times {10}^8$	0.268
153Ho$\to$149Tbm	11/2-$\to$11/2-	4.016	0	2.36$\times {10}^5$	7.29$\times {10}^5$	0.243
153Hom$\to$149Tb	1/2+$\to$1/2+	4.121	0	3.1$\times {10}^5$	1.49$\times {10}^5$	0.243
153Er$\to$149Dy	7/2(-)$\to$7/2(-)	4.802	0	7$\times {10}^1$	7.94$\times {10}^1$	0.219
153Tm$\to$149Ho	(11/2-)$\to$(11/2-)	5.248	0	1.63$\times {10}^0$	2.18$\times {10}^0$	0.195
153Tmm$\to$149Hom	(1/2+)$\to$(1/2+)	5.242	0	2.72$\times {10}^0$	2.33$\times {10}^0$	0.195
155Er$\to$151Dy	(7/2-)$\to$7/2(-)	4.118	0	1.45$\times {10}^6$	5.44$\times {10}^5$	0.265
155Tm$\to$151Ho	(11/2-)$\to$11/2(-)	4.572	0	2.43$\times {10}^3$	4.28$\times {10}^3$	0.239
155Yb$\to$151Er	(7/2-)$\to$(7/2-)	5.338	0	2.01$\times {10}^0$	2.36$\times {10}^0$	0.213
155Lu$\to$151Tm	(11/2-)$\to$(11/2-)	5.803	0	7.62$\times {10}^{-2}$	9.13$\times {10}^{-2}$	0.188
155Lum$\to$151Tmm	(1/2+)$\to$(1/2+)	5.73	0	1.82$\times {10}^{-1}$	1.81$\times {10}^{-1}$	0.188
157Yb$\to$153Er	7/2-$\to$7/2(-)	4.621	0	7.72$\times {10}^3$	7.26$\times {10}^3$	0.256
157Lum$\to$153Tm	(11/2-)$\to$(11/2-)	5.128	0	7.98$\times {10}^1$	6.56$\times {10}^1$	0.229
157Hf$\to$153Yb	7/2-$\to$7/2${}_{}{}^{-}\#$	5.885	0	1.34$\times {10}^{-1}$	1.12$\times {10}^{-1}$	0.202
159Ta$\to$155Lum	1/2+$\to$(1/2+)	5.66	0	3.06$\times {10}^0$	2.6$\times {10}^0$	0.214
159Tam$\to$155Lu	11/2-$\to$(11/2-)	5.744	0	1.02$\times {10}^0$	1.14$\times {10}^0$	0.214
159W$\to$155Hf	7/2${}_{}{}^{-}\#$$\to$7/2${}_{}{}^{-}\#$	6.445	0	1$\times {10}^{-2}$	7.58$\times {10}^{-3}$	0.187
159Rem$\to$155Ta	11/2-$\to$(11/2-)	6.965	0	2.88$\times {10}^{-4}$	4.32$\times {10}^{-4}$	0.163
161W$\to$157Hf	7/2${}_{}{}^{-}\#$$\to$7/2-	5.915	0	5.6$\times {10}^{-1}$	6.04$\times {10}^{-1}$	0.222
161Rem$\to$157Tam	11/2-$\to$11/2-	6.425	0	1.58$\times {10}^{-2}$	2.26$\times {10}^{-2}$	0.195
161Os$\to$157W	(7/2-)$\to$(7/2-)	7.065	0	6.4$\times {10}^{-4}$	4.89$\times {10}^{-4}$	0.169
163W$\to$159Hf	7/2-$\to$7/2-	5.518	0	1.88$\times {10}^1$	2.41$\times {10}^1$	0.255
163Re$\to$159Ta	1/2+$\to$1/2+	6.012	0	1.22$\times {10}^0$	6.64$\times {10}^{-1}$	0.226
163Rem$\to$159Tam	11/2-$\to$11/2-	6.068	0	3.24$\times {10}^{-1}$	3.95$\times {10}^{-1}$	0.226
163Os$\to$159W	7/2${}_{}{}^{-}\#$$\to$7/2${}_{}{}^{-}\#$	6.675	0	5.5$\times {10}^{-3}$	7.38$\times {10}^{-3}$	0.199
165Rem$\to$161Tam	(11/2-)$\to$(11/2-)	5.66	0	1.78$\times {10}^1$	1.62$\times {10}^1$	0.256
165Os$\to$161W	(7/2-)$\to$7/2${}_{}{}^{-}\#$	6.335	0	1.18$\times {10}^{-1}$	9.8$\times {10}^{-2}$	0.227
165Irm$\to$161Rem	(11/2-)$\to$11/2-	6.885	0	2.31$\times {10}^{-3}$	3.54$\times {10}^{-3}$	0.2
167Rem$\to$163Ta	1/2+$\to$1/2+	5.405	0	5.9$\times {10}^2$	1.96$\times {10}^2$	0.282
167Os$\to$163W	7/2-$\to$7/2-	5.985	0	1.65$\times {10}^0$	1.9$\times {10}^0$	0.252
167Ir$\to$163Re	1/2+$\to$1/2+	6.504	0	6.81$\times {10}^{-2}$	6.15$\times {10}^{-2}$	0.224
167Irm$\to$163Rem	11/2-$\to$11/2-	6.561	0	2.86$\times {10}^{-2}$	3.8$\times {10}^{-2}$	0.224
167Pt$\to$163Os	7/2${}_{}{}^{-}\#$$\to$7/2${}_{}{}^{-}\#$	7.155	0	8$\times {10}^{-4}$	1.15$\times {10}^{-3}$	0.197
169Rem$\to$165Ta	(1/2+,3/2+)$\to$(1/2+,3/2+)	5.189	0	7.55$\times {10}^3$	1.89$\times {10}^3$	0.301
169Os$\to$165W	(5/2-)$\to$(5/2-)	5.713	0	2.53$\times {10}^1$	2.29$\times {10}^1$	0.272
169Ir$\to$165Re	(1/2+)$\to$(1/2+)	6.141	0	7.84$\times {10}^{-1}$	1.25$\times {10}^0$	0.244
169Irm$\to$165Rem	(11/2-)$\to$(11/2-)	6.266	0	3.9$\times {10}^{-1}$	4$\times {10}^{-1}$	0.244
171Irm$\to$167Re	(11/2-)$\to$(9/2-)	6.155	2	2.72$\times {10}^0$	1.74$\times {10}^0$	0.259
171Pt$\to$167Os	7/2-$\to$7/2-	6.605	0	5.06$\times {10}^{-2}$	5.86$\times {10}^{-2}$	0.232
173Irm$\to$169Re	(11/2-)$\to$(9/2-)	5.942	2	1.83$\times {10}^1$	1.16$\times {10}^1$	0.268
173Pt$\to$169Os	(5/2-)$\to$(5/2-)	6.358	0	4.44$\times {10}^{-1}$	4.23$\times {10}^{-1}$	0.242
173Au$\to$169Ir	(1/2+)$\to$(1/2+)	6.837	0	2.91$\times {10}^{-2}$	2.38$\times {10}^{-2}$	0.217
173Hg$\to$169Pt	3/2${}_{}{}^{-}\#$$\to$(7/2-)	7.375	2	9.1$\times {10}^{-4}$	2.23$\times {10}^{-3}$	0.193
175Ir$\to$171Re	5/2${}_{}{}^{-}\#$$\to$(9/2-)	5.431	2	1.06$\times {10}^3$	2.13$\times {10}^3$	0.269
175Pt$\to$171Os	(7/2-)$\to$(5/2-)	6.178	2	3.95$\times {10}^0$	3.59$\times {10}^0$	0.246
175Au$\to$171Ir	1/2+$\to$1/2+	6.575	0	2.16$\times {10}^{-1}$	1.83$\times {10}^{-1}$	0.223
177Ir$\to$173Re	5/2-$\to$(5/2-)	5.082	0	5$\times {10}^4$	6.3$\times {10}^4$	0.263
177Pt$\to$173Os	5/2-$\to$(5/2-)	5.643	0	1.86$\times {10}^2$	3.63$\times {10}^2$	0.243
177Au$\to$173Ir	(1/2+,3/2+)$\to$(1/2+,3/2+)	6.298	0	3.65$\times {10}^0$	1.9$\times {10}^0$	0.222
177Hg$\to$173Pt	(7/2-)$\to$(5/2-)	6.735	2	1.5$\times {10}^{-1}$	2.5$\times {10}^{-1}$	0.201
177Tl$\to$173Au	(1/2+)$\to$(1/2+)	7.066	0	2.47$\times {10}^{-2}$	2.96$\times {10}^{-2}$	0.181
179Pt$\to$175Os	1/2-$\to$(5/2-)	5.412	2	8.83$\times {10}^3$	8.23$\times {10}^3$	0.233
179Au$\to$175Ir	(1/2+,3/2+)$\to$5/2${}_{}{}^{-}\#$	5.98	1	3.23$\times {10}^1$	4.48$\times {10}^1$	0.215
179Hg$\to$175Pt	7/2-$\to$(7/2-)	6.351	0	1.91$\times {10}^0$	3.55$\times {10}^0$	0.197
179Pb$\to$175Hg	(9/2-)$\to$(7/2-)	7.595	2	3.9$\times {10}^{-3}$	2.76$\times {10}^{-3}$	0.161
181Pt$\to$177Os	1/2-$\to$1/2-	5.15	0	7.03$\times {10}^4$	9.38$\times {10}^4$	0.218
181Au$\to$177Ir	(3/2-)$\to$5/2-	5.751	2	5.07$\times {10}^2$	6.73$\times {10}^2$	0.203
183Pt$\to$179Os	1/2-$\to$1/2-	4.822	0	4.06$\times {10}^6$	6.54$\times {10}^6$	0.198
183Au$\to$179Ir	5/2-$\to$(5/2)-	5.465	0	7.78$\times {10}^3$	8.07$\times {10}^3$	0.187
183Hg$\to$179Pt	1/2-$\to$1/2-	6.038	0	8.03$\times {10}^1$	6.33$\times {10}^1$	0.174
183Tl$\to$179Au	1/2(+)$\to$(1/2+,3/2+)	5.977	0	3.45$\times {10}^2$	3.99$\times {10}^2$	0.161
185Au$\to$181Ir	5/2-$\to$5/2-	5.18	0	9.81$\times {10}^4$	2.41$\times {10}^5$	0.167
185Hg$\to$181Pt	1/2-$\to$1/2-	5.773	0	8.18$\times {10}^2$	9.56$\times {10}^2$	0.157
185Pb$\to$181Hg	3/2-$\to$1/2(-#)	6.695	2	1.85$\times {10}^1$	2.74$\times {10}^0$	0.136
185Pbm$\to$181Hgm	13/2+$\to$13/2+	6.555	0	8.14$\times {10}^0$	5.13$\times {10}^0$	0.136
187Tlm$\to$183Au	(9/2-)$\to$5/2-	5.656	2	1.04$\times {10}^4$	2.26$\times {10}^4$	0.13
187Pb$\to$183Hg	3/2-$\to$1/2-	6.393	2	1.6$\times {10}^2$	4.16$\times {10}^1$	0.121
187Pbm$\to$183Hgm	13/2+$\to$13/2${}_{}{}^{+}\#$	6.208	0	1.53$\times {10}^2$	1.32$\times {10}^2$	0.121
189Pb$\to$185Hg	3/2-$\to$1/2-	5.871	2	1.26$\times {10}^4$	7.77$\times {10}^3$	0.106
191Pbm$\to$187Hgm	13/2(+)$\to$13/2+	5.404	0	6.54$\times {10}^5$	8.88$\times {10}^5$	0.089

Table 3. Table 3: Same as Table II, but for odd- A 𝐴 A nuclei in region II.

$\alpha$ transition	$I_i^{\pi }\to I_j^{\pi }$	$Q_{\alpha }$(MeV)	$l_{\text{min}}$	$T_{1/2}^{\text{expt}}$(s)	$T_{1/2}^{\text{calc}}$(s)	$P_{\alpha }$
179Pb$\to$175Hg	(9/2-)$\to$(7/2-)	7.595	2	3.9$\times {10}^{-3}$	1.27$\times {10}^{-2}$	0.034
185Pb$\to$181Hg	3/2-$\to$1/2(-#)	6.695	2	1.85$\times {10}^1$	5.62$\times {10}^0$	0.065
185Pbm$\to$181Hgm	13/2+$\to$13/2+	6.555	0	8.14$\times {10}^0$	1.05$\times {10}^1$	0.065
185Bim$\to$181Tl	1/2+$\to$1/2+	8.218	0	5.8$\times {10}^{-4}$	7.62$\times {10}^{-5}$	0.087
187Pb$\to$183Hg	3/2-$\to$1/2-	6.393	2	1.6$\times {10}^2$	6.82$\times {10}^1$	0.072
187Pbm$\to$183Hgm	13/2+$\to$13/2${}_{}{}^{+}\#$	6.208	0	1.53$\times {10}^2$	2.16$\times {10}^2$	0.072
187Bim$\to$183Tl	1/2${}_{}{}^{+}\#$$\to$1/2(+)	7.887	0	3.7$\times {10}^{-4}$	5.27$\times {10}^{-4}$	0.099
187Po$\to$183Pb	(1/2-,5/2-)$\to$3/2-	7.976	2	1.4$\times {10}^{-3}$	9.91$\times {10}^{-4}$	0.133
189Pb$\to$185Hg	3/2-$\to$1/2-	5.871	2	1.26$\times {10}^4$	1.04$\times {10}^4$	0.077
189Bim$\to$185Tl	(1/2+)$\to$1/2${}_{}{}^{+}\#$	7.452	0	9.8$\times {10}^{-3}$	9.55$\times {10}^{-3}$	0.108
189Po$\to$185Pb	(5/2-)$\to$3/2-	7.694	2	3.8$\times {10}^{-3}$	5.73$\times {10}^{-3}$	0.147
191Pbm$\to$187Hgm	13/2(+)$\to$13/2+	5.404	0	6.54$\times {10}^5$	9.93$\times {10}^5$	0.079
191Bim$\to$187Tl	(1/2+)$\to$(1/2+)	7.018	0	1.82$\times {10}^{-1}$	2.42$\times {10}^{-1}$	0.111
191Atm$\to$187Bi	(7/2-)$\to$9/2${}_{}{}^{-}\#$	7.88	2	2.2$\times {10}^{-3}$	2.52$\times {10}^{-3}$	0.209
193Bim$\to$189Tl	(1/2+)$\to$(1/2+)	6.613	0	3.81$\times {10}^0$	7.05$\times {10}^0$	0.11
193Atm$\to$189Bi	7/2${}_{}{}^{-}\#$$\to$(9/2-)	7.581	2	2.1$\times {10}^{-2}$	1.92$\times {10}^{-2}$	0.213
193Rn$\to$189Po	3/2${}_{}{}^{-}\#$$\to$(5/2-)	8.04	2	1.15$\times {10}^{-3}$	1.4$\times {10}^{-3}$	0.287
195Bim$\to$191Tl	(1/2+)$\to$(1/2+)	6.232	0	2.64$\times {10}^2$	2.42$\times {10}^2$	0.103
195Po$\to$191Pb	(3/2-)$\to$(3/2-)	6.755	0	4.94$\times {10}^0$	3.91$\times {10}^0$	0.147
195Pom$\to$191Pbm	(13/2+)$\to$13/2(+)	6.84	0	2.13$\times {10}^0$	1.87$\times {10}^0$	0.147
195Rn$\to$191Po	(3/2-)$\to$(3/2-)	7.694	0	7$\times {10}^{-3}$	8.4$\times {10}^{-3}$	0.283
195Rnm$\to$191Pom	(13/2+)$\to$(13/2+)	7.714	0	6$\times {10}^{-3}$	7.21$\times {10}^{-3}$	0.283
197Po$\to$193Pb	(3/2-)$\to$(3/2-)	6.405	0	1.22$\times {10}^2$	9.54$\times {10}^1$	0.134
197At$\to$193Bi	(9/2-)$\to$(9/2-)	7.108	0	4.04$\times {10}^{-1}$	3.88$\times {10}^{-1}$	0.19
197Rn$\to$193Po	(3/2-)$\to$(3/2-)	7.415	0	5.4$\times {10}^{-2}$	6.6$\times {10}^{-2}$	0.265
199Po$\to$195Pb	3/2${}_{}{}^{-}\#$$\to$3/2${}_{}{}^{-}\#$	6.074	0	4.38$\times {10}^3$	2.67$\times {10}^3$	0.116
199At$\to$195Bi	(9/2-)$\to$(9/2-)	6.778	0	7.89$\times {10}^0$	6.63$\times {10}^0$	0.167
199Fr$\to$195At	1/2${}_{}{}^{+}\#$$\to$1/2+	7.811	0	1.6$\times {10}^{-2}$	6.55$\times {10}^{-3}$	0.328
201Po$\to$197Pb	3/2-$\to$3/2-	5.799	0	8.28$\times {10}^4$	5.63$\times {10}^4$	0.095
201At$\to$197Bi	(9/2-)$\to$(9/2-)	6.473	0	1.2$\times {10}^2$	1.18$\times {10}^2$	0.14
201Fr$\to$197At	(9/2-)$\to$(9/2-)	7.515	0	6.2$\times {10}^{-2}$	6.25$\times {10}^{-2}$	0.283
203Po$\to$199Pb	5/2-$\to$3/2-	5.496	2	2$\times {10}^6$	4.02$\times {10}^6$	0.075
203At$\to$199Bi	9/2-$\to$9/2-	6.209	0	1.43$\times {10}^3$	1.77$\times {10}^3$	0.112
203Rn$\to$199Po	3/2${}_{}{}^{-}\#$$\to$3/2${}_{}{}^{-}\#$	6.63	0	6.67$\times {10}^1$	6.16$\times {10}^1$	0.163
203Fr$\to$199At	9/2${}_{}{}^{-}\#$$\to$(9/2-)	7.274	0	5.5$\times {10}^{-1}$	4.61$\times {10}^{-1}$	0.233
203Ra$\to$199Rn	(3/2-)$\to$(3/2-)	7.745	0	3.6$\times {10}^{-2}$	2.28$\times {10}^{-2}$	0.326
205Po$\to$201Pb	5/2-$\to$5/2-	5.325	0	1.57$\times {10}^7$	2.21$\times {10}^7$	0.056
205At$\to$201Bi	9/2-$\to$9/2-	6.02	0	2.03$\times {10}^4$	1.51$\times {10}^4$	0.085
205Rn$\to$201Po	5/2-$\to$3/2-	6.39	2	6.9$\times {10}^2$	1.29$\times {10}^3$	0.126
205Fr$\to$201At	(9/2-)$\to$(9/2-)	7.054	0	3.82$\times {10}^0$	3.3$\times {10}^0$	0.183
207Po$\to$203Pb	5/2-$\to$5/2-	5.216	0	9.94$\times {10}^7$	1.15$\times {10}^8$	0.04
207At$\to$203Bi	9/2-$\to$9/2-	5.872	0	6.52$\times {10}^4$	9.38$\times {10}^4$	0.061
207Rn$\to$203Po	5/2-$\to$5/2-	6.251	0	2.64$\times {10}^3$	3.52$\times {10}^3$	0.093
207Fr$\to$203At	9/2-$\to$9/2-	6.894	0	1.56$\times {10}^1$	1.62$\times {10}^1$	0.137
207Ra$\to$203Rn	5/2${}_{}{}^{-}\#$$\to$3/2${}_{}{}^{-}\#$	7.274	2	1.6$\times {10}^0$	2.17$\times {10}^0$	0.198
207Ac$\to$203Fr	9/2${}_{}{}^{-}\#$$\to$9/2${}_{}{}^{-}\#$	7.849	0	3.1$\times {10}^{-2}$	2.64$\times {10}^{-2}$	0.279
209Bi$\to$205Tl	9/2-$\to$1/2+	3.137	5	6.28$\times {10}^{26}$	2.26$\times {10}^{26}$	0.017
209At$\to$205Bi	9/2-$\to$9/2-	5.757	0	4.75$\times {10}^5$	4.51$\times {10}^5$	0.042
209Rn$\to$205Po	5/2-$\to$5/2-	6.155	0	1.01$\times {10}^4$	1.23$\times {10}^4$	0.065
209Fr$\to$205At	9/2-$\to$9/2-	6.777	0	5.62$\times {10}^1$	5.83$\times {10}^1$	0.097
209Ra$\to$205Rn	5/2-$\to$5/2-	7.135	0	5.23$\times {10}^0$	4.92$\times {10}^0$	0.143
211At$\to$207Bi	9/2-$\to$9/2-	5.982	0	6.21$\times {10}^4$	5.19$\times {10}^4$	0.028
211Fr$\to$207At	9/2-$\to$9/2-	6.662	0	2.14$\times {10}^2$	2.27$\times {10}^2$	0.066
211Ra$\to$207Rn	5/2(-)$\to$5/2-	7.042	0	1.42$\times {10}^1$	1.43$\times {10}^1$	0.098
211Ac$\to$207Fr	9/2${}_{}{}^{-}\#$$\to$9/2-	7.619	0	2.13$\times {10}^{-1}$	2.41$\times {10}^{-1}$	0.144
213Fr$\to$209At	9/2-$\to$9/2-	6.904	0	3.5$\times {10}^1$	3.56$\times {10}^1$	0.043
213Ra$\to$209Rn	1/2-$\to$5/2-	6.862	2	2.05$\times {10}^2$	1.72$\times {10}^2$	0.065
213Pa$\to$209Ac	9/2${}_{}{}^{-}\#$$\to$(9/2-)	8.395	0	7$\times {10}^{-3}$	3.81$\times {10}^{-3}$	0.197
215Ac$\to$211Fr	9/2-$\to$9/2-	7.746	0	1.7$\times {10}^{-1}$	1.78$\times {10}^{-1}$	0.061
215Th$\to$211Ra	(1/2-)$\to$5/2(-)	7.665	2	1.2$\times {10}^0$	1.03$\times {10}^0$	0.09
215Pa$\to$211Ac	9/2${}_{}{}^{-}\#$$\to$9/2${}_{}{}^{-}\#$	8.245	0	1.4$\times {10}^{-2}$	1.49$\times {10}^{-2}$	0.13
217Pa$\to$213Ac	9/2${}_{}{}^{-}\#$$\to$9/2${}_{}{}^{-}\#$	8.485	0	3.48$\times {10}^{-3}$	4.21$\times {10}^{-3}$	0.081

Table 4. Table 4: Same as Table II, but for odd- A 𝐴 A nuclei in region III.

$\alpha$ transition	$I_i^{\pi }\to I_j^{\pi }$	$Q_{\alpha }$(MeV)	$l_{\text{min}}$	$T_{1/2}^{\text{expt}}$(s)	$T_{1/2}^{\text{calc}}$(s)	$P_{\alpha }$
209Bi$\to$205Tl	9/2-$\to$1/2+	3.137	5	6.28$\times {10}^{26}$	9.64$\times {10}^{25}$	0.039
211Bi$\to$207Tl	9/2-$\to$1/2+	6.75	5	1.28$\times {10}^2$	3.96$\times {10}^1$	0.045
211Po$\to$207Pb	9/2+$\to$1/2-	7.594	5	5.16$\times {10}^{-1}$	6.88$\times {10}^{-2}$	0.082
211At$\to$207Bi	9/2-$\to$9/2-	5.982	0	6.21$\times {10}^4$	2.22$\times {10}^4$	0.065
213Bi$\to$209Tl	9/2-$\to$(1/2+)	5.983	5	1.31$\times {10}^5$	5.2$\times {10}^4$	0.047
213Po$\to$209Pb	9/2+$\to$9/2+	8.536	0	3.72$\times {10}^{-6}$	5.36$\times {10}^{-6}$	0.11
213At$\to$209Bi	9/2-$\to$9/2-	9.254	0	1.25$\times {10}^{-7}$	3.08$\times {10}^{-7}$	0.075
213Rn$\to$209Po	9/2${}_{}{}^{+}\#$$\to$1/2-	8.243	5	1.95$\times {10}^{-2}$	5.85$\times {10}^{-3}$	0.06
213Fr$\to$209At	9/2-$\to$9/2-	6.904	0	3.5$\times {10}^1$	1.53$\times {10}^1$	0.1
215Po$\to$211Pb	9/2+$\to$9/2+	7.526	0	1.78$\times {10}^{-3}$	3.81$\times {10}^{-3}$	0.116
215At$\to$211Bi	9/2-$\to$9/2-	8.178	0	1$\times {10}^{-4}$	1.53$\times {10}^{-4}$	0.08
215Rn$\to$211Po	9/2+$\to$9/2+	8.839	0	2.3$\times {10}^{-6}$	5.31$\times {10}^{-6}$	0.094
215Fr$\to$211At	9/2-$\to$9/2-	9.54	0	8.6$\times {10}^{-8}$	2.04$\times {10}^{-7}$	0.116
215Ra$\to$211Rn	9/2${}_{}{}^{+}\#$$\to$1/2-	8.864	5	1.67$\times {10}^{-3}$	8.93$\times {10}^{-4}$	0.04
215Ac$\to$211Fr	9/2-$\to$9/2-	7.746	0	1.7$\times {10}^{-1}$	7.76$\times {10}^{-2}$	0.141
217Po$\to$213Pb	(9/2+)$\to$(9/2+)	6.662	0	1.59$\times {10}^0$	4.68$\times {10}^0$	0.098
217At$\to$213Bi	9/2-$\to$9/2-	7.201	0	3.23$\times {10}^{-2}$	1.63$\times {10}^{-1}$	0.081
217Rn$\to$213Po	9/2+$\to$9/2+	7.887	0	5.4$\times {10}^{-4}$	1.73$\times {10}^{-3}$	0.117
217Ra$\to$213Rn	(9/2+)$\to$9/2${}_{}{}^{+}\#$	9.161	0	1.63$\times {10}^{-6}$	5.2$\times {10}^{-6}$	0.075
217Ac$\to$213Fr	9/2-$\to$9/2-	9.832	0	6.9$\times {10}^{-8}$	1.42$\times {10}^{-7}$	0.166
217Th$\to$213Ra	9/2${}_{}{}^{+}\#$$\to$1/2-	9.435	5	2.47$\times {10}^{-4}$	2.49$\times {10}^{-4}$	0.025
217Pa$\to$213Ac	9/2${}_{}{}^{-}\#$$\to$9/2${}_{}{}^{-}\#$	8.485	0	3.48$\times {10}^{-3}$	1.86$\times {10}^{-3}$	0.185
217Pam$\to$213Ac	29/2${}_{}{}^{+}\#$$\to$9/2${}_{}{}^{-}\#$	10.345	11	1.48$\times {10}^{-3}$	3.08$\times {10}^{-3}$	0.185
219Rn$\to$215Po	5/2+$\to$9/2+	6.946	2	3.96$\times {10}^0$	4.24$\times {10}^0$	0.115
219Fr$\to$215At	9/2-$\to$9/2-	7.449	0	2$\times {10}^{-2}$	9.62$\times {10}^{-2}$	0.129
219Ra$\to$215Rn	(7/2)+$\to$9/2+	8.138	2	1$\times {10}^{-2}$	3.32$\times {10}^{-3}$	0.109
219Ac$\to$215Fr	9/2-$\to$9/2-	8.827	0	1.18$\times {10}^{-5}$	3.22$\times {10}^{-5}$	0.184
219Th$\to$215Ra	9/2${}_{}{}^{+}\#$$\to$9/2${}_{}{}^{+}\#$	9.511	0	1.05$\times {10}^{-6}$	4.85$\times {10}^{-6}$	0.055
219Pa$\to$215Ac	9/2-$\to$9/2-	10.084	0	5.3$\times {10}^{-8}$	1.31$\times {10}^{-7}$	0.221
219U$\to$215Th	9/2${}_{}{}^{+}\#$$\to$(1/2-)	9.943	5	5.5$\times {10}^{-5}$	1.24$\times {10}^{-4}$	0.015
221Rn$\to$217Po	7/2+$\to$(9/2+)	6.163	2	7.01$\times {10}^3$	8.13$\times {10}^3$	0.09
221Fr$\to$217At	5/2-$\to$9/2-	6.458	2	2.87$\times {10}^2$	9.14$\times {10}^2$	0.124
221Ra$\to$217Rn	5/2+$\to$9/2+	6.88	2	2.8$\times {10}^1$	5.07$\times {10}^1$	0.125
221Ac$\to$217Fr	9/2${}_{}{}^{-}\#$$\to$9/2-	7.78	0	5.2$\times {10}^{-2}$	3.3$\times {10}^{-2}$	0.191
221Th$\to$217Ra	(7/2+)$\to$(9/2+)	8.625	2	1.68$\times {10}^{-3}$	8.46$\times {10}^{-4}$	0.093
221Pa$\to$217Ac	9/2-$\to$9/2-	9.251	0	5.9$\times {10}^{-6}$	9.67$\times {10}^{-6}$	0.248
223Rn$\to$219Po	7/2(-#)$\to$9/2${}_{}{}^{+}\#$	5.284	1	3.65$\times {10}^8$	2.35$\times {10}^8$	0.055
223Fr$\to$219At	3/2(-)$\to$5/2${}_{}{}^{-}\#$	5.562	2	2.2$\times {10}^7$	1.71$\times {10}^7$	0.111
223Ra$\to$219Rn	3/2+$\to$5/2+	5.979	2	9.88$\times {10}^5$	4.12$\times {10}^5$	0.114
223Ac$\to$219Fr	(5/2-)$\to$9/2-	6.783	2	1.27$\times {10}^2$	2.16$\times {10}^2$	0.185
223Th$\to$219Ra	(5/2)+$\to$(7/2)+	7.566	2	6$\times {10}^{-1}$	1.08$\times {10}^0$	0.126
223Pa$\to$219Ac	9/2${}_{}{}^{-}\#$$\to$9/2-	8.325	0	5.1$\times {10}^{-3}$	2.9$\times {10}^{-3}$	0.261
225Ac$\to$221Fr	3/2${}_{}{}^{-}\#$$\to$5/2-	5.935	2	8.57$\times {10}^5$	1.33$\times {10}^6$	0.169
225Th$\to$221Ra	(3/2+)$\to$5/2+	6.922	2	5.83$\times {10}^2$	2.16$\times {10}^2$	0.134
225U$\to$221Th	5/2${}_{}{}^{+}\#$$\to$(7/2+)	8.015	2	6.1$\times {10}^{-2}$	2.34$\times {10}^{-1}$	0.116
225Np$\to$221Pa	9/2${}_{}{}^{-}\#$$\to$9/2-	8.785	0	3$\times {10}^{-3}$	5.85$\times {10}^{-4}$	0.33
227Ac$\to$223Fr	3/2-$\to$3/2(-)	5.042	0	4.98$\times {10}^{10}$	8.28$\times {10}^{10}$	0.144
227Th$\to$223Ra	1/2+$\to$3/2+	6.147	2	1.61$\times {10}^6$	5.3$\times {10}^5$	0.113
227Pa$\to$223Ac	(5/2-)$\to$(5/2-)	6.58	0	2.7$\times {10}^3$	4.77$\times {10}^3$	0.237
227U$\to$223Th	(3/2+)$\to$(5/2)+	7.211	2	6.6$\times {10}^1$	1.09$\times {10}^2$	0.145
227Np$\to$223Pa	5/2${}_{}{}^{-}\#$$\to$9/2${}_{}{}^{-}\#$	7.815	2	5.1$\times {10}^{-1}$	8.89$\times {10}^{-1}$	0.33
229Th$\to$225Ra	5/2+$\to$1/2+	5.168	2	2.5$\times {10}^{11}$	1.51$\times {10}^{11}$	0.075
229Pa$\to$225Ac	(5/2+)$\to$3/2${}_{}{}^{-}\#$	5.834	1	2.7$\times {10}^7$	2.1$\times {10}^7$	0.206
229U$\to$225Th	(3/2+)$\to$(3/2+)	6.475	0	1.74$\times {10}^4$	6.22$\times {10}^4$	0.144
229Np$\to$225Pa	5/2${}_{}{}^{+}\#$$\to$5/2${}_{}{}^{-}\#$	7.015	1	3.53$\times {10}^2$	5.25$\times {10}^2$	0.309
229Pu$\to$225U	3/2${}_{}{}^{+}\#$$\to$5/2${}_{}{}^{+}\#$	7.587	2	1.82$\times {10}^2$	3.05$\times {10}^1$	0.145
231Th$\to$227Ra	5/2+$\to$3/2+	4.213	2	2.3$\times {10}^{17}$	2.27$\times {10}^{18}$	0.04
231Pa$\to$227Ac	3/2-$\to$3/2-	5.15	0	1.03$\times {10}^{12}$	1.63$\times {10}^{11}$	0.167
231U$\to$227Th	(5/2)(+#)$\to$1/2+	5.577	2	9.07$\times {10}^9$	4.25$\times {10}^9$	0.112
231Np$\to$227Pa	(5/2)(+#)$\to$(5/2-)	6.373	1	1.46$\times {10}^5$	3.21$\times {10}^5$	0.271
231Pu$\to$227U	3/2${}_{}{}^{+}\#$$\to$(3/2+)	6.839	0	3.97$\times {10}^3$	1.11$\times {10}^4$	0.168
233Pu$\to$229U	5/2${}_{}{}^{+}\#$$\to$(3/2+)	6.414	2	1.04$\times {10}^6$	1.46$\times {10}^6$	0.153
233Am$\to$229Np	5/2${}_{}{}^{-}\#$$\to$5/2${}_{}{}^{+}\#$	7.055	1	4.27$\times {10}^3$	2.25$\times {10}^3$	0.331
233Cm$\to$229Pu	3/2${}_{}{}^{+}\#$$\to$3/2${}_{}{}^{+}\#$	7.475	0	1.35$\times {10}^2$	2.19$\times {10}^2$	0.182
235U$\to$231Th	7/2-$\to$5/2+	4.678	1	2.22$\times {10}^{16}$	5.25$\times {10}^{15}$	0.034
235Np$\to$231Pa	5/2+$\to$3/2-	5.194	1	1.32$\times {10}^{12}$	1.09$\times {10}^{12}$	0.172
235Pu$\to$231U	(5/2+)$\to$(5/2)(+#)	5.951	0	5.42$\times {10}^7$	2.05$\times {10}^8$	0.111
235Am$\to$231Np	5/2${}_{}{}^{-}\#$$\to$(5/2)(+#)	6.575	1	1.55$\times {10}^5$	2.81$\times {10}^5$	0.276
237Np$\to$233Pa	5/2+$\to$3/2-	4.959	1	6.76$\times {10}^{13}$	4.92$\times {10}^{13}$	0.125
237Pu$\to$233U	7/2-$\to$5/2+	5.748	1	9.39$\times {10}^{10}$	4.84$\times {10}^9$	0.063
237Am$\to$233Np	5/2(-)$\to$5/2${}_{}{}^{+}\#$	6.195	1	1.77$\times {10}^7$	2.07$\times {10}^7$	0.216
237Cm$\to$233Pu	5/2${}_{}{}^{+}\#$$\to$5/2${}_{}{}^{+}\#$	6.775	0	6.67$\times {10}^4$	1.36$\times {10}^5$	0.164
239Np$\to$235Pa	5/2+$\to$(3/2-)	4.599	1	4.07$\times {10}^{16}$	2.7$\times {10}^{16}$	0.085
239Am$\to$235Np	(5/2)-$\to$5/2+	5.922	1	4.28$\times {10}^8$	6.54$\times {10}^8$	0.159
239Cm$\to$235Pu	(7/2-)$\to$(5/2+)	6.541	1	1.45$\times {10}^8$	2.51$\times {10}^6$	0.11
241Pu$\to$237U	5/2+$\to$1/2+	5.14	2	1.84$\times {10}^{13}$	1.65$\times {10}^{14}$	0.01
241Am$\to$237Np	5/2-$\to$5/2+	5.638	1	1.36$\times {10}^{10}$	3.22$\times {10}^{10}$	0.109
243Am$\to$239Np	5/2-$\to$5/2+	5.439	1	2.32$\times {10}^{11}$	6.8$\times {10}^{11}$	0.07
243Cm$\to$239Pu	5/2+$\to$1/2+	6.169	2	9.18$\times {10}^8$	8.49$\times {10}^8$	0.024
243Bk$\to$239Am	3/2${}_{}{}^{-}\#$$\to$(5/2)-	6.874	2	1.08$\times {10}^7$	2.38$\times {10}^5$	0.13
243Es$\to$239Bk	(7/2+)$\to$(7/2+)	8.075	0	3.54$\times {10}^1$	1.58$\times {10}^1$	0.204
243Fm$\to$239Cf	7/2${}_{}{}^{-}\#$$\to$5/2${}_{}{}^{+}\#$	8.685	1	2.54$\times {10}^{-1}$	3.86$\times {10}^{-1}$	0.259
245Cm$\to$241Pu	7/2+$\to$5/2+	5.623	2	2.66$\times {10}^{11}$	2.19$\times {10}^{12}$	0.008
245Bk$\to$241Am	3/2-$\to$5/2-	6.454	2	3.56$\times {10}^8$	2.68$\times {10}^7$	0.085
247Cm$\to$243Pu	9/2-$\to$7/2+	5.353	1	4.92$\times {10}^{14}$	2.3$\times {10}^{14}$	0.002
247Bk$\to$243Am	(3/2-)$\to$5/2-	5.89	2	4.35$\times {10}^{10}$	3.08$\times {10}^{10}$	0.052
247Cf$\to$243Cm	7/2${}_{}{}^{+}\#$$\to$5/2+	6.495	2	3.2$\times {10}^7$	1.97$\times {10}^8$	0.02
247Fm$\to$243Cf	(7/2+)$\to$1/2${}_{}{}^{+}\#$	8.255	4	6.2$\times {10}^1$	8.72$\times {10}^1$	0.107
247Fmm$\to$243Cf	(1/2+)$\to$1/2${}_{}{}^{+}\#$	8.305	0	5.1$\times {10}^0$	1.07$\times {10}^1$	0.107
247Md$\to$243Es	(7/2-)$\to$(7/2+)	8.765	1	1.19$\times {10}^0$	7.78$\times {10}^{-1}$	0.147
247Mdm$\to$243Es	(1/2-)$\to$(7/2+)	9.025	3	3.16$\times {10}^{-1}$	3$\times {10}^{-1}$	0.147
249Bk$\to$245Am	7/2+$\to$(5/2)+	5.524	2	1.97$\times {10}^{12}$	6.35$\times {10}^{12}$	0.03
249Cf$\to$245Cm	9/2-$\to$7/2+	6.296	1	1.11$\times {10}^{10}$	3.85$\times {10}^9$	0.006
251Es$\to$247Bk	(3/2-)$\to$(3/2-)	6.598	0	2.38$\times {10}^7$	5.49$\times {10}^7$	0.035
251Fm$\to$247Cf	(9/2-)$\to$7/2${}_{}{}^{+}\#$	7.425	1	1.06$\times {10}^6$	1.01$\times {10}^5$	0.017
251Md$\to$247Es	(7/2-)$\to$(7/2+)	7.964	1	2.53$\times {10}^3$	7.03$\times {10}^2$	0.062
251No$\to$247Fm	(7/2+)$\to$(7/2+)	8.755	0	9.64$\times {10}^{-1}$	1.99$\times {10}^0$	0.106
253Md$\to$249Es	(7/2-)$\to$7/2+	7.575	1	1.2$\times {10}^5$	3.05$\times {10}^4$	0.037
253Lr$\to$249Md	(7/2-)$\to$(7/2-)	8.925	0	7.02$\times {10}^{-1}$	2.33$\times {10}^0$	0.061
255Lrm$\to$251Md	(7/2-)$\to$(7/2-)	8.594	0	2.54$\times {10}^0$	3.88$\times {10}^1$	0.036
255Rfm$\to$251No	5/2${}_{}{}^{+}\#$$\to$(7/2+)	8.975	2	1$\times {10}^0$	3.44$\times {10}^0$	0.105
257Db$\to$253Lr	(9/2+)$\to$(7/2-)	9.205	1	2.45$\times {10}^0$	3.46$\times {10}^0$	0.033

Table 5. Table 5: Same as Table II, but for odd- A 𝐴 A nuclei in region IV.

$\alpha$ transition	$I_i^{\pi }\to I_j^{\pi }$	$Q_{\alpha }$(MeV)	$l_{\text{min}}$	$T_{1/2}^{\text{expt}}$(s)	$T_{1/2}^{\text{calc}}$(s)	$P_{\alpha }$
253Es$\to$249Bk	7/2+$\to$7/2+	6.739	0	1.77$\times {10}^6$	4.22$\times {10}^6$	0.087
255Cf$\to$251Cm	(7/2+)$\to$(1/2+)	5.736	4	2.55$\times {10}^{12}$	1.4$\times {10}^{12}$	0.064
255Fm$\to$251Cf	7/2+$\to$1/2+	7.24	4	7.23$\times {10}^4$	6.21$\times {10}^5$	0.059
255Md$\to$251Es	(7/2-)$\to$(3/2-)	7.905	2	2.03$\times {10}^4$	1.29$\times {10}^3$	0.061
257Fm$\to$253Cf	(9/2+)$\to$(7/2+)	6.864	2	8.68$\times {10}^6$	9.71$\times {10}^6$	0.042
257Md$\to$253Es	(7/2-)$\to$7/2+	7.558	1	1.32$\times {10}^5$	2.42$\times {10}^4$	0.042
257Rfm$\to$253No	(11/2-)$\to$(9/2-)	9.156	2	4.3$\times {10}^0$	1.5$\times {10}^0$	0.059
259Sgm$\to$255Rfm	(9/2-)$\to$5/2${}_{}{}^{+}\#$	9.885	2	3.11$\times {10}^{-1}$	6.25$\times {10}^{-2}$	0.061
261Rf$\to$257No	3/2${}_{}{}^{+}\#$$\to$(7/2+)	8.645	2	8.15$\times {10}^0$	9.93$\times {10}^1$	0.029
261Sg$\to$257Rf	(3/2+)$\to$(1/2+)	9.713	2	1.87$\times {10}^{-1}$	2.45$\times {10}^{-1}$	0.042
263Rf$\to$259No	3/2${}_{}{}^{+}\#$$\to$9/2${}_{}{}^{+}\#$	8.255	4	2.2$\times {10}^3$	7.93$\times {10}^3$	0.022
263Sg$\to$259Rf	7/2${}_{}{}^{+}\#$$\to$7/2${}_{}{}^{+}\#$	9.405	0	1.08$\times {10}^0$	1.42$\times {10}^0$	0.03
265Sgm$\to$261Rf	3/2${}_{}{}^{+}\#$$\to$3/2${}_{}{}^{+}\#$	9.125	0	2.46$\times {10}^1$	1.16$\times {10}^1$	0.023
265Hs$\to$261Sg	3/2${}_{}{}^{+}\#$$\to$(3/2+)	10.47	0	1.96$\times {10}^{-3}$	7.5$\times {10}^{-3}$	0.037
267Hs$\to$263Sg	5/2${}_{}{}^{+}\#$$\to$7/2${}_{}{}^{+}\#$	10.035	2	6.88$\times {10}^{-2}$	1.93$\times {10}^{-1}$	0.028
267Ds$\to$263Hs	9/2${}_{}{}^{+}\#$$\to$7/2${}_{}{}^{+}\#$	11.775	2	1$\times {10}^{-5}$	3.51$\times {10}^{-5}$	0.053
269Ds$\to$265Hs	9/2${}_{}{}^{+}\#$$\to$3/2${}_{}{}^{+}\#$	11.514	4	2.3$\times {10}^{-4}$	4.46$\times {10}^{-4}$	0.041
271Ds$\to$267Hs	13/2${}_{}{}^{-}\#$$\to$5/2${}_{}{}^{+}\#$	10.875	5	9$\times {10}^{-2}$	3.21$\times {10}^{-2}$	0.033
271Dsm$\to$267Hs	9/2${}_{}{}^{+}\#$$\to$5/2${}_{}{}^{+}\#$	10.945	2	1.7$\times {10}^{-3}$	3.41$\times {10}^{-3}$	0.033
273Ds$\to$269Hs	13/2${}_{}{}^{-}\#$$\to$9/2${}_{}{}^{+}\#$	11.365	3	2.4$\times {10}^{-4}$	6.22$\times {10}^{-4}$	0.028
273Dsm$\to$269Hs	3/2${}_{}{}^{+}\#$$\to$9/2${}_{}{}^{+}\#$	11.565	4	1.2$\times {10}^{-1}$	4.12$\times {10}^{-4}$	0.028
277Ds$\to$273Hs	11/2${}_{}{}^{+}\#$$\to$3/2${}_{}{}^{+}\#$	10.835	4	2.2$\times {10}^{-2}$	1.9$\times {10}^{-2}$	0.025
281Ds$\to$277Hs	3/2${}_{}{}^{+}\#$$\to$3/2${}_{}{}^{+}\#$	9.325	0	9.33$\times {10}^1$	3.95$\times {10}^1$	0.027
281Cn$\to$277Ds	3/2${}_{}{}^{+}\#$$\to$11/2${}_{}{}^{+}\#$	10.465	4	3.7$\times {10}^{-1}$	3.73$\times {10}^{-1}$	0.043
285Cn$\to$281Ds	5/2${}_{}{}^{+}\#$$\to$3/2${}_{}{}^{+}\#$	9.315	2	3.2$\times {10}^1$	1.68$\times {10}^2$	0.053
289Fl$\to$285Cn	5/2${}_{}{}^{+}\#$$\to$5/2${}_{}{}^{+}\#$	9.965	0	2.4$\times {10}^0$	2.25$\times {10}^0$	0.131

Equations

$$V_{\text{N}}(r)=-V_0\frac{1+cosh(R/a)}{cosh(r/a)+cosh(R/a)},$$

$$R=1.28A^{1/3}-0.76+0.8A^{-1/3}.$$

$$V_{\text{C}}(r)=\{\begin{array}{r}\frac{Z_{\text{d}}{Z}_{\alpha }{e}^2}{2R}[3-(\frac{r}{R}{)}^2],r<R,\\ \frac{Z_{\text{d}}{Z}_{\alpha }{e}^2}{r},r>R,\end{array}$$

$$V_{\text{l}}(r)=\frac{l(l+1){\hslash }^2}{2\mu r^2},$$

$$T_{1/2}=\frac{\hslash ln2}{\Gamma }=\frac{ln2}{\lambda },$$

$$\Gamma =\frac{P_{\alpha }FP}{4\mu }.$$

$$F{\int }_{r_1}^{r_2}\frac{dr}{2k(r)}=1,$$

$$P=exp(-2{\int }_{r_2}^{r_3}k(r)dr).$$

$$P_{\alpha }=P_0\frac{T_{1/2}^{\text{calc}}}{T_{1/2}^{\text{expt}}},$$

$$lo{g}_{10}{P}_{\alpha }=$$

Full text

I Introduction

The odd-even staggering effect in nuclear physics is ubiquitous, for example binding energy, abundance of isotopes, excited energy spectrum, moment of inertia, $\alpha$ preformation probability and so on Koura et al. (2005); Delion et al. (2016); Xu et al. (1999). It is primarily caused by pairing correlations that is similar to the electronic superconductivity and the unpaired nucleon blocking effect, while comparable contribution comes from the deformed mean field Satuła et al. (1998). Generally, it is believed that nuclei with the even number of protons and neutrons are more stable than the neighboring odd-$A$ and doubly-odd nuclei. However, for some heavy and superheavy nuclei dominated by $\alpha$ decay and fission, this understanding is biased and the effect of unpaired nucleon on the nuclear stability cannot be neglected NND . An explicit example is 255Rf with a half-life of 2.3 second, while the neighboring even-even nuclei are more instable, e.g., spontaneous fission for 254,256Rf isotopes take place within 23 microsecond and 6.4 millisecond at average, respectively. Within the Gamow picture, the process of $\alpha$ decay can be qualitatively described as an $\alpha$ cluster preforms in the surface of the decaying nucleus followed by a quantum tunneling of the potential barrier Gamow (1928); Yarman et al. (2016). Among the above two processes, $\alpha$ particle preformation contains more nuclear structure information Qi (2016); Zhang and Royer (2008); Xu and Ren (2007); Xu et al. (2016). And the penertration depends on the $\alpha$ decay energy and interaction between the $\alpha$ particle and daughter nucleus without obvious odd-even staggers, while the $\alpha$ preformation probability shows odd-even staggering effect Xu and Ren (2005).

$\alpha$ decay is a common and important decay mode for heavy and superheavy nuclei, which provides many unique nuclear structure information especially for low lying isomeric and ground states of heavy and superheavy nuclei Oganessian and Utyonkov (2015); Hofmann et al. (2016); Heßberger et al. (2016); Ren and Ren (2012); Wang et al. (2010); Yang et al. (2015). And the $\alpha$ decay and $\alpha$ clustering are very old subjects but still attracting much attention in both theortical and experimental research Bethe and Bacher (1936); Röpke et al. (1998). The nuclear cluster structure appears in the regions of either light or heavy nuclei Delion et al. (2016); Tohsaki et al. (2001); Varga et al. (1992); Astier et al. (2010). Theoretically, microscopic mechanism on how $\alpha$ cluster forms in the surface of heavy nucleus is still an open question, meanwhile some conclusions on the $\alpha$ preformation probability have been obtained to some extent Qi (2016); Zhang and Royer (2008); Seif et al. (2015). Notably, there is a striking similarity between the tendency of the neutron pairing gaps and the $\alpha$ preformation probability Qi (2016). The calculations with an effective isospin-dependent contact interaction show that the di-neutron cluster structure in middle and heavy nuclei is related to the strength of pairing correlations in dilute nuclear matter Isayev (2008). Also, a quartet correlations may be of importance for very low density Röpke et al. (1998); Elhatisari et al. (2016). Phenomenologically, by analyzing the data of $\alpha$ decay, it is founded that the $\alpha$ preformation probability takes local minimum for nuclei with nucleons number near to the magic numbers Zhang and Royer (2008); Xu and Ren (2007). Moreover, other factors, e.g., the valence proton-neutron correlation, the $\alpha$ particle angular momentum effect and the deformation of nuclei, also play a role on the $\alpha$ preformation probability Kaneko and Hasegawa (2003); Sun et al. (2016a); Seif et al. (2015). However, because the precise size of decaying nucleus is unknown, by adjusting the radius parameters and keeping the $\alpha$ preformation probability constant, good reproduces of experimental $\alpha$ decay width can also be obtained Zdeb et al. (2013). This problem is actually related to the properties of dilute nuclear matter and the definition of nuclear radius. For nuclei with the finite number of protons and neutrons, the active valence nucleon, especially the odd nucleon, plays an important role on the bulk and collective properties, such as the $\alpha$ preformation probability. In order to investigate the odd-even staggering effect in $\alpha$ decay, we systematically calculate the $\alpha$ decay half-lives for odd-$A$ nuclei.

Recently we have preformed systematic calculation on $\alpha$ decay half-lives and $\alpha$ preformation probability for even-even nuclei based on the isospin dependent nuclear potential Sun et al. (2016b). In this paper, our previous work Sun et al. (2016b) is extended to calculate the $\alpha$ decay half-lives of odd-$A$ nuclei. The odd-even staggering effect of $\alpha$ decay half-lives is discussed in detail. The unpaired nucleon plays a key role on the $\alpha$ preformation probability, which resulting in the complexity of $\alpha$ decay especially for unfavored $\alpha$ decay. This paper is organized as follows. Sec. II briefly shows the theoretical framework of the two-potential approach and the analytic expression for the $\alpha$ preformation probability. In Sec. III we show the systematic results of $\alpha$ decay half-lives for odd-$A$ nuclei and discuss the odd-even staggering effect. A summary is given in Sec. IV.

II Theoretical framework

In the two-potential approach Gurvitz and Kalbermann (1987), the total potential between the $\alpha$ particle and the daughter nucleus contains the nuclear and Coloumb potential besides the centrifugal potential for unfavored $\alpha$ decay. The nuclear potential $V_{\text{N}}(r)$ is determined within a two-body model, which is different from the quasi-molecular shape path proposed by the Generalized Liquid Drop Model Guo et al. (2015). In the two-body model, the $\alpha$ particle is assumed to preformed at the surface of decaying nucleus, and the strong attractive nuclear interaction can be roughly approximated by a square well potential Zdeb et al. (2013). A better description can be obtained by the double-folding integral of M3Y nucleon-nucleon force between the $\alpha$ particle and residual nucleus Xu and Ren (2005). In this work we choose a type of cosh parametrized form for the nuclear potential, which is also successful in describing the rotatinally spaced energies, enhanced E2 transition strengths and so on Buck et al. (1992); Buck and Merchant (1989). The nuclear potential $V_{\text{N}}(r)$ can be expressed as

[TABLE]

where $V_{0}$ and $a$ are the depth and diffuseness for the nuclear potential, respectively. In previous work, we have obtained a set of isospin dependent parameters, which is $a=0.5958$ fm and $V_{0}=192.42+31.059\frac{N-Z}{A}$ MeV Sun et al. (2016b), where $N$, $Z$ and $A$ are the neutron, proton and mass numbers of the daughter nucleus, respectively. Besides, $R$ and $r$ denote the nuclear sharp radius of the parent nucleus and the distance between the $\alpha$ particle and daughter nucleus, respectively. $R$ is empirically given by Royer (2000)

[TABLE]

The Coloumb potential $V_{\text{C}}(r)$ is obtained under the assumption of a uniformly charged sphere, which can be written as

[TABLE]

where $Z_{\text{d}}$ and $Z_{\alpha}$ are proton numbers of the daughter nucleus and $\alpha$ particle, respectively. In addition, unfavored $\alpha$ decay has $l\neq 0$, where $l$ is the orbital angular momentum taken away by the $\alpha$ particle. In this $\alpha$ decay mode, the centrifugal potential $V_{\text{l}}(r)$ plays a role on the total potential, which is given by

[TABLE]

where $\mu$ denotes the reduced mass of the two-body model between $\alpha$ particle and the daughter nucleus in center of mass system.

The $\alpha$ decay half-life $T_{1/2}$ can be calculated by the WKB approxination. It can be written as

[TABLE]

where the $\alpha$ decay constant $\lambda$ or decay width $\Gamma$ depends on the $\alpha$ particle preformation probability $P_{\alpha}$, the penetration probability $P$ and the normalized factor $F$. Within the two-potential approach framework Gurvitz and Kalbermann (1987), the decay width can be expressed as

[TABLE]

The normalized factor $F$ represents the collision probability or assault frequency. It satifies the expression

[TABLE]

where $r_{1}$, $r_{2}$ and the following $r_{3}$ denote the classical turning points, which satisfy the conditions $V(r_{1})=V(r_{2})=V(r_{3})=Q$. The middle turning point $r_{2}$ is slightly larger than the nuclear sharp radius $R$ of the parent nucleus. $k(r)=\sqrt{\frac{2\mu}{\hbar^{2}}\mid Q_{\alpha}-V(r)\mid}$ is the wave number. $V(r)$ and $Q$ are $\alpha$-core potential and $\alpha$ decay energy, respectively. The penetration probability $P$ can be calculated using

[TABLE]

At last, we focus on the most unclear concept, the $\alpha$ particle preformation in the decaying parent nucleus. Due to the absence of effictive method to calculate the $\alpha$ preformation probability, which may be obtained by

[TABLE]

where $T_{1/2}^{\text{calc}}$ is calculated by eq. (5) with $P_{\alpha}=1$. According to the density-dependent cluster model Xu and Ren (2005), $P_{0}$ is the average $\alpha$ preformation factor for a certain kind of $\alpha$ decay, such as $P_{0}=0.7$ for even-even nuclei, $P_{0}=0.35$ for odd-$A$ nuclei, and $P_{0}=0.18$ for doubly-odd nuclei. And it takes as $P_{0}=0.35$ in this work. The variation of $\alpha$ preformation probability can be estimated by the analytic formula Zhang and Royer (2008); Guo et al. (2015); Sun et al. (2016b), which can be described by the following expression

[TABLE]

where $Z$, $N$ and $A$ are the proton, neutron and mass numbers of parent nucleus. $Z_{1}$ and $Z_{2}$ ($N_{1}$ and $N_{2}$) are the proton (neutron) magic numbers around $Z$ ($N$). $a$, $b$, $c$, $d$ and $e$ are the adjustable parameters. The first and fourth terms describe the magnitude and the trend of the $\alpha$ preformation probability, the second and third terms show a parabolic dependence of $log_{10}P_{\alpha}$ taking into account the valence protons (neutrons) and holes, the last term relates to the correlation of valence neutron-proton Guo et al. (2015).

III Numerical results and discussion

In this work, we systematically study and analyse the $\alpha$ decay half-lives, including 270 odd-$A$ nuclei, i.e., even $Z$, odd $N$ (even-odd) and odd $Z$, even $N$ (odd-even) nuclei. The experimental data of $\alpha$ decay energy and half-lives are calculated based on the NUBASE2012 Audi et al. (2012). We have previously preformed the systematic study for even-even nuclei with isospin dependent nuclear potential and a description of the $\alpha$ preformation probability Sun et al. (2016b). Compared to the even-even nuclei, the unpaired proton or neutron in odd-$A$ nuclei results in complex single particle level structure and even unfavored $\alpha$ decay. For unfavored transition, the additional centrifugal potential will increases the height of total potential barrier and decreases the penetrate probability. Besides, it is awkward that the odd nucleon may hinder the cluster behaviour of nucleon pairs by the Pauli blocking effect Xu et al. (1999). Because of these difficulties, calculations on $\alpha$ decay half-lives for odd-$A$ nuclei deserve further study and improvement. There is no doubt that the systematic study will help to design experiments and test the validity of new measured data in future.

At first, we discuss the odd-even staggering effect of $\alpha$ decay half-lives. The measured half-lives of ground states of Cf, Fm, No and Rf isotopes are plotted in Fig. 1. Using yellow diamonds, green squares, blue asterisks and peach stars denote the main decay modes, i.e., $\alpha$ decay, spontaneous fission (SF), electron capture (EC) and $\beta$- decay, respectively. From the picture we can clearly see that from Cf to Rf isotopes the average half-life drops off with the increasing of proton number. Moreover, the odd-$A$ nuclei of Cf isotopes tend to transform through EC and $\beta$- decay because their $\alpha$ decay half-lives are longer than the corresponding $\beta$ decay ones Ren and Ren (2014). As shown in Fig. 1, the half-lives of 249,251Cf isotopes are longer than the even-even isotopes, while the half-lives of EC nuclei 239,241,243,245,247Cf and $\beta$- nuclei 253,255Cf isotopes are as much as or even shorter than their neighboring ones. Additionally, the odd-$A$ nuclei 243,257Fm, 259No and 255,257,259,261Rf isotopes prefer $\alpha$ decay but not SF as the main decay mode, which are not similar to their neighboring nuclei, indicating that the odd nucleon blocking effect seems play a more important role on SF than on $\alpha$ decay.

The $\alpha$ decay energy is one of the most important inputs for calculation of the half-life without obvious odd-even staggering effect. $\alpha$ decay was first successfully explained by Gamow Gamow (1928) and independently by Gurney and Condon Gurney and Condon (1928) as the penetration through the Coulomb barrier, leading to the $Q_{\alpha}^{-1/2}$ dependence of the Geiger-Nuttall law. Besides, the $\alpha$ preformation probability also depend linearly on $Q_{\alpha}^{-1/2}$, especially for nuclei with neutron number below $N$=126 Qi et al. (2014). The $\alpha$ decay energies which are obtained by the atomic mass difference between the ground states of the parent and daughter nuclei, including Rn, Th, Pu, Cf, No, Sg and Ds isotopes, are shown in Fig. 2. It can be seen that $\alpha$ decay energy approximates several or over ten MeVs, and decreases with increasing neutron number on an isotope chain without obvious odd-even staggering effect. The shell closure will abruptly increases $\alpha$ decay energy, which takes local maximum value at the place of magic number plus two additional nucleons as shown by the dotted lines at $N$=128, 154 and 164. It should be special mentioned the strong dependence of $\alpha$ decay half-life on decay energy, and the decay energy is relative large resulting in the unstability of superheavy nuclei. In view of the general downward trend for $\alpha$ decay energy with increasing of neutron number, new synthesised more neutron-rich superheavy nuclei would survive longer.

Now that the $\alpha$ decay energy has no obvious odd-even differences, the odd-even staggering effect of $\alpha$ decay half-lives maybe comes from the $\alpha$ particle preformation probability mostly. According to the method of treating even-even nuclei, we extract the $\alpha$ particle preformation probabilities from ratios of the calculated $\alpha$ decay half-lives within the two-potential approach to the experimental data by eq. (9). Then the trend of the extracted $\alpha$ preformation probabilities can be described by eq. (10). Using the esitimated probabilities, the calculated half-lives can well reproduce the experimental data Sun et al. (2016b). For odd-$A$ nuclei, the unpaired nucleon is easy to be scattered to the higher single particle level. It indicates that the $\alpha$ decay daughter nuclei have uncertain configuration, resulting in unfavored $\alpha$ decay is quite common. In the following text, we will systematically study the $\alpha$ decay half-lives for odd-$A$ nuclei.

The whole $\alpha$ decay nuclei is divided into 4 regions according to the major shell closures Zhang and Royer (2008); Sun et al. (2016b). In the chart of nuclides for $\alpha$ decay, there are 4 regions, region I: 50$<Z<$82 and 82$<N<$126, region II: 82$<Z<$126 and 82$<N<$126, region III: 82$<Z<$126 and 126$<N<$152, region IV: 82$<Z<$126 and 152$<N<$184. Here $Z$=50, 82 and $N$=82, 126 are the well-known spherical shell closures, while $N$=152 is the deformed neutron shell given by many experimental data, and $Z$=126, $N$=184 are the hypothetical spherical closed shells predicted by different theoritical models. The parameters of $\alpha$ preformation probaility of eq. (10) and the calculated $\alpha$ decay half-lives in different regions are listed in Tables from I to V, respectively. The nuclei in region III are separately treated for the cases of even-odd and odd-even nuclei as shown in Table I.

The blocking effects coming from the unpaired proton and neutron are different, which could not be ignored especially for the nuclei in region III. The comparison between in a unified way and separately treating the $\alpha$ preformation probability of even-odd and odd-even nuclei in region III is shown in Fig. 3. We can see that in the case of separately treating, in Fig. 3(b), the deviations between the calculated half-livse and experimental data are smaller than those in a unified way in Fig. 3(a). And the estimated $\alpha$ preformation probability of even-odd and odd-even nuclei in region III are denoted by blue triangles and red circles in Fig. 4, respectively. In general, the $\alpha$ preformation probabilities of even-odd nuclei are more smaller than the odd-even ones, which is consistent with the results within the Viola-Seaborg formula Dong and Ren (2005).

From Table II to V, we show the experimental $\alpha$ decay half-lives data and numerical results within the two-potential approach of odd-$A$ nuclei. For all the tables, the first column denotes the $\alpha$ transition from the parent to daughter nuclei, including ground and isomeric states. The next four columns express the spin-parity transition, $\alpha$ decay energy, the minimal angular momentum quantum number carried out by the emitted $\alpha$ particle and the experimental $\alpha$ decay half-lives, respectively. The sixth and seventh columns denote the calculated $\alpha$ decay half-lives and $\alpha$ preformation probabilities, respectively. To clearly show the results, we plot the deviations between the calculated half-lives and experimental data in Fig. 5. The green squares and magenta triangles denote the favored and unfavored $\alpha$ decay, respectively. For nuclei with neutron number below $N$=126 in regions I and II, the results can better reproduce the experimental half-lives than those in regions III and IV where the unfavored $\alpha$ decay with nonzero angular momentum transfer is quite common. And in the case of unfavored $\alpha$ decay, the $\alpha$ preformation probability seems to be smaller and uncertain, which is worthy to further study in the future.

IV Summary

In summary, we systematically study the $\alpha$ decay half-lives for odd-$A$ nuclei within the two-potential approach. The $\alpha$ preformation probabilities are extracted from the measured $\alpha$ decay half-lives, and estimated with the analytic formula taking into account the shell effect and proton-neutron correlation. The odd-even staggering effect of half-lives is investigated. We found that electron capture and $\beta$- decay are insensitive to unpaired nucleon, and the odd nucleon blocking effect play a more important role on spontaneous fission than $\alpha$ decay. The fact that $\alpha$ decay energy without obvious odd-even effect results in the $\alpha$ preformation probability are directly related to the unpaired nucleon. The difference of blocking effect between odd proton and neutron can not be ignored, and the $\alpha$ preformation probabilities of even-$Z$, odd-$N$ nuclei are smaller than the odd-$Z$, even-$N$ ones. In general, the numerical results of $\alpha$ decay half-lives can well reproduce the experimental data for odd-$A$ nuclei just as the case of even-even nuclei, and the unfavored $\alpha$ decay deserves further study.

Acknowledgements.

This work is supported in part by the National Natural Science Foundation of China (Grant No. 11205083), the construct program of the key discipline in hunan province, the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 15A159), the Natural Science Foundation of Hunan Province, China (Grant No. 2015JJ3103, 2015JJ2123), the Innovation Group of Nuclear and Particle Physics in USC, Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX2015B398).