19Ne | $-1.88542(8)$ | [Ma82] | $-1.88542(8)$ |  | $+0.92810(37)$ | $+0.4032$ | $-1.924(25)$ | $0.039$
19F | $+2.628868(8)$ | [Li52,Ba64] | $+2.628868(8)$ |  | $+0.92810(37)$ | $-0.3972$ | $+2.695(34)$ | $0.067$
21Na | $+2.38630(10)$ | [Am65] | $+2.38630(10)$ |  | $+0.18527(25)$ | $+0.1657$ | $+2.495(36)$ | $0.109$
21Ne | $-0.661797(5)$ | [La57] | $-0.661797(5)$ |  | $+0.18527(25)$ | $-0.1881$ | $-0.763(28)$ | $0.101$
23Mg | $0.5364(3)$ | [Fu93] | $-0.5365(2)$ | 1 | $+0.14430(49)$ | $-0.2377$ | $-0.461(26)$ | $0.076$
| $-0.5366(3)$ | [Yo17] |  |  |  |  |  |
23Na | $+2.2176556(6)$ | [Wa54,Fu76] | $+2.21759(7)$ |  $\rm{unw}$222222Because the two (very precise) measurements differ by many standard deviations, the unweighted average is used. | $+0.14430(49)$ | $+0.2137$ | $+2.162(34)$ | $0.056$
| $+2.217522(2)$ | [Be74] |  |  |  |  |  |
25Al | $3.6455(12)$ | [Mi76a] | $+3.6455(12)$ |  | $+0.13750(15)$ | $+0.1459$ | $+3.700(56)$ | $0.054$
25Mg | $-0.85545(8)$ | [Al51b] | $-0.85545(8)$ |  | $+0.13750(15)$ | $-0.1678$ | $-0.880(44)$ | $0.024$
27Si | $(-)0.8652(4)$ | [Ma98] | $-0.86533(24)$ | $1$ | $+0.11851(12)$ | $-0.1692$ | $-0.701(43)$ | $0.165$
| $0.8654(3)$ | [Ma99] |  |  |  |  |  |
27Al | $+3.6415069(7)$ | [Ep68] | $+3.6415069(7)$ |  | $+0.11851(12)$ | $+0.1459$ | $+3.497(54)$ | $0.144$
29P | $1.2349(3)$ | [Su71] | $+1.23475(21)$ | $1$ | $+0.3136(11)$ | $+0.0109$ | $+1.182(15)$ | $0.052$
| $1.2346(3)$ | [Zh09] |  |  |  |  |  |
29Si | $-0.55529(3)$ | [We53] | $-0.55529(3)$ |  | $+0.3136(11)$ | $-0.0595$ | $-0.556(11)$ | $0.001$
31S | $0.48793(8)$ | [Mi76b] | $-0.48793(8)$ |  | $+0.30840(77)$ | $-0.3044$ | $-0.424(11)$ | $0.064$
31P | $+1.13160(3)$ | [Wa54] | $+1.13160(3)$ |  | $+0.30840(77)$ | $+0.2472$ | $+1.053(15)$ | $0.078$
33Cl | $+0.7523(16)$ | [Ro86] | $+0.7548(5)$ | $1.6$ | $-0.08255(76)$ | $+0.1440$ | $+0.817(32)$ | $0.062$
| $+0.7549(3)$ | [Ma04] |  |  |  |  |  |
33S | $+0.6438212(14)$ | [Dh51,Lu73] | $+0.6438212(14)$ |  | $-0.08255(76)$ | $-0.1193$ | $+0.621(25)$ | $0.023$
35Ar | $+0.6322(2)$ | [Ma03] | $+0.6322(2)$ |  | $-0.07459(51)$ | $-0.1424$ | $+0.611(25)$ | $0.021$
35Cl | $+0.8218743(4)$ | [Bl72] | $+0.8218743(4)$ |  | $-0.07459(51)$ | $+0.1681$ | $+0.831(31)$ | $0.009$
37K | $+0.20321(6)$ | [Vo71] | $+0.20321(6)$ |  | $-0.15011(35)$ | $+0.2426$ | $+0.237(34)$ | $0.034$
37Ar | $+1.145(5)$ | [Pi88] | $+1.145(5)$ |  | $-0.15011(35)$ | $-0.2027$ | $+1.121(27)$ | $0.024$
39Ca | $1.02168(12)$ | [Mi76c] | $+1.02168(12)$ |  | $-0.17135(39)$ | $0.0000$ | $+0.934(27)$ | $0.087$
39K | $+0.39150731(12)$ | [Sa74a,b] | $+0.391487(21)$ |  $\rm{unw}$232323Because the two (very precise) measurements differ by many standard deviations, the unweighted average is used. | $-0.17135(39)$ | $0.0000$ | $+0.465(35)$ | $0.074$
| $+0.3914662(3)$ | [Be50] |  |  |  |  |  |
41Sc | $+5.431(2)$ | [Mi90] | $+5.431(2)$ |  | $+0.13602(49)$ | $0.0000$ | $+5.668(78)$ | $0.237$
41Ca | $-1.594781(9)$ | [Br62] | $-1.594781(9)$ |  | $+0.13602(49)$ | $0.0000$ | $-1.800(61)$ | $0.205$
43Ti | $0.85(2)$ | [Ma93] | $-0.85(2)$ |  | $+0.1023(22)$ | $-0.1992$ | $-0.666(65)$ | $0.184$
43Sc | $+4.62(4)$ | [Co66a] | $+4.533(22)$ | $2.2$ | $+0.1023(22)$ | $+0.1723$ | $+4.562(81)$ | $0.029$
| $+4.528(10)$ | [Av11] |  |  |  |  |  |
45V |  |  |  |  | $+0.0807(25)$ | $+0.1753$ | $\it+4.230(82)$ |
45Ti | $0.095(2)$ | [Co66b] |  $(+)0.095(2)$242424The sign was obtained from shell model calculations Buck  _et al._ (2001). |  | $+0.0807(25)$ | $-0.2139$ | $-0.335(66)$ | $0.430$
47Cr |  |  |  |  | $+0.1511(43)$ | $-0.1360$ | $\it-0.651(36)$ |
47V |  |  |  |  | $+0.1511(43)$ | $+0.1050$ | $\it+2.368(43)$ |
49Mn |  |  |  |  | $+0.0916(57)$ | $+0.1222$ | $\it+3.270(80)$ |
49Cr | $0.476(3)$ | [Jo70] |  $(-)0.476(3)$252525The sign was obtained from shell model calculations Buck  _et al._ (2001). |  | $+0.0916(57)$ | $-0.1595$ | $-0.476(67)$ | $0.000$
51Fe |  |  |  |  | $+0.0989(33)$ | $-0.1073$ | $\it-0.674(52)$ |
51Mn | $3.5683(13)$ | [Jo71] | $+3.5691(26)$ | $2.1$ | $+0.0989(33)$ | $+0.0785$ | $+3.457(63)$ | $0.111$
| $+3.577(4)$ | [Ba15] |  |  |  |  |  |
53Co |  |  |  |  | $+0.0855(29)$ | $+0.1053$ | $\it+4.546(85)$ |
53Fe | $-0.65(1)$ | [Mi17] | $-0.65(1)$ |  | $+0.0855(29)$ | $-0.1431$ | $-0.645(69)$ | $0.005$
55Ni |  $(-)0.976(26)$262626The sign was obtained from shell model calculations Berryman  _et al._ (2009). | [Be09] | $(-)0.976(26)$ |  | $+0.0855(32)$ | $-0.0708$ | $-0.898(71)$ | $0.078$
55Co | $+4.822(3)$ | [Ca73] | $+4.822(3)$ |  | $+0.0855(32)$ | $+0.0397$ | $+4.776(87)$ | $0.046$
57Cu | $+2.582(7)$ | [Co10] | $+2.582(7)$ |  | $+0.1462(30)$ | $+0.0607$ | $+2.403(39)$ | $0.179$
57Ni | $-0.7975(4)$ | [Oh96] | $-0.7975(14)$ |  | $+0.1462(30)$ | $-0.1061$ | $-0.669(31)$ | $0.129$
59Zn |  |  |  |  | $+0.1263(57)$ | $-0.14(5)$ | $\it-0.51(8)$ |
59Cu | $+1.891(9)$ | [Go04] | $+1.8919(29)$ | $3.2$ | $+0.1263(57)$ | $+0.11(5)$ | $+2.20(8)$ | $0.390$
| $+1.910(4)$ | [Co10] |  |  |  |  |  |
| $+1.8910(9)$ | [Vi11] |  |  |  |  |  |
61Ga |  |  |  |  | $+0.1408(89)$ | $+0.11(5)$ | $\it+2.29(9)$ |
61Zn |  |  |  |  | $+0.1408(89)$ | $-0.14(5)$ | $\it-0.59(8)$ |
63Ga | $+1.469(5)$ | [Pr12] | $+1.469(5)$ |  | + |  |  |
67Se |  |  |  |  | $-0.067(12)$ | $-0.14(5)$ | $\it+0.94(13)$ |
67As |  |  |  |  | $-0.067(12)$ | $+0.11(5)$ | $\it+1.61(14)$ |
71Kr |  |  |  |  | $-0.078(17)$ | $-0.14(5)$ | $\it+1.04(14)$ |
71Br |  |  |  |  | $-0.078(17)$ | $+0.11(5)$ | $\it+1.50(15)$ |
75Sr |  |  |  |  | $-0.134(40)$ | $-0.14(5)$ | $\it+0.93(17)$ |
75Rb |  |  |  |  | $-0.134(40)$ | $+0.11(5)$ | $\it+0.54(19)$ |
77Sr | $-0.348(4)$ | [Li92] | $-0.348(4)$ |  | $-$ |  |  |
Table 11: References from which magnetic moment results have been rejected. The correlation between the alphabetical reference code used here and the actual reference numbers is listed in Table 13. Nucleus | Value | [Ref.] |  Reason for rejection
---|---|---|---
27Si | $-0.8554(4)$ | [Hu84] |  No correction for diamagnetism was applied. Result is strongly deviating from the two other more recent and mutually consistent results reported in [Ma89] and [Ma99].
19Ne | $-1.8846(8)$ | [Ge05] |  Uncertainty is a factor 10 or more larger than that of the most precise measurement.
35Ar | $+0.633(2)$ | [Ca65] |
35Ar | $+0.633(7)$ | [Kl96] |
37Ar | $+0.95(20)$ | [Ro65] |
39K | $+0.39147(3)$ | [Du93] |
41Ca | $-1.61(2)$ | [An82] |
| $-1.5942(7)$ | [Ar83] |
57Ni | $0.88(6)$ | [Ro75] |
59Cu | $+1.84(3)$ | [St08] |
57Cu | $2.00(5)$ | [Mi06] |  Result is much less precise and differs by more than 10 standard deviations from the newer value from [Co10] which agrees well with $\mu\simeq 2.5\mu_{N}$ as is calculated in the shell model Semon et al. (1996); Golovko  _et al._ (2004); Honma et al. (2004).

Table 12: Magnetic moments for the mirror nuclei with $A=$ 61, 67, 71 and 75, calculated with Eqs. 35-37 for the two possible signs of $\gamma_{\beta}$. The preferred values for the magnetic moments, listed in column 6 and corresponding to sgn$(\gamma_{\beta})=+$ for the $j=l+1/2$ $p_{3/2}$ subshell and sgn$(\gamma_{\beta})=-$ for the $j=l-1/2$ $f_{5/2}$ subshell, agree with the systematic of the magnetic moments for neighboring nuclei with the odd proton or neutron occupying the same shell model orbital, listed in the last column. Note that when the sign for a magnetic moment value in the last column is given, the sign has been explicitly measured. The correlation between the alphabetical reference code used here and the actual reference numbers is listed in Table 13. Isot. | $J^{\pi}$ | sub- | Magn. mom. ($\mu_{N}$) | Magn. mom. ($\mu_{N}$) | Preferred |  Reason, based on magnetic moments of neighbouring
---|---|---|---|---|---|---
|  | shell | for sgn$(\gamma_{\beta})=-$ | for sgn$(\gamma_{\beta})=+$ | value ($\mu_{N}$) |  nuclei with similar single particle configuration
61Ga | $3/2^{-}$ | $p_{3/2}$ | $+2.29(19)$ | $+0.33(7)$ | $+2.29(19)$ |  similar to the value $\mu\approx$ +1.5 to +2.6 for the 3/2- ground states of 63-77Ga [Ch10] and 57-73Cu [Vi10]
61Zn | $3/2^{-}$ |  | $-0.59(8)$ | $+0.97(8)$ | $-0.59(8)$ |  similar to the values $\mu=-0.28164(5)$ and $\mu=-0.8(2)$ for the 3/2- g.s. of 63Zn [La69], resp. the lowest-lying 3/2- state of 65Zn [We75]
67Se | $5/2^{-}$ | $f_{5/2}$ | $-0.30(13)$ | $+0.94(13)$ | $+0.94(13)$ |  similar to the values $\mu=0.735(7)$ [Ol70], $\mu=+1.018(10)$ [Mo68], $\mu=+1.124(10)$ [Bl68] and $\mu=+1.12(3)$ [Za84] for the lowest lying 5/2- states in respectively 69Ge, 71Ge, 79Kr, and 77Se (see also text)
67As | $5/2^{-}$ |  | $+3.04(14)$ | $+1.61(14)$ | $+1.61(14)$ |  similar to the values $\mu=+1.6229(16)$ [Go05], $\mu=+1.674(2)$ [He76a,b], $\mu=+1.63(10)$ [Bo63], $\mu=1.0(3)$ [Sp94] and $\mu=1.6(5)$ [Ja96] for the lowest 5/2- states in respectively 69As, 71As, 73As, 79Br, and 81Br (see also text)
71Kr | $5/2^{-}$ |  | $-0.40(14)$ | $+1.04(14)$ | $+1.04(14)$ |  similar to the values $\mu=0.735(7)$ [Ol70], $\mu=+1.018(10)$ [Mo68], $\mu=+1.124(10)$ [Bl68] and $\mu=+1.12(3)$ [Za84] for the lowest lying 5/2- states in respectively 69Ge, 71Ge, 79Kr, and 77Se
71Br | $5/2^{-}$ |  | $+3.15(15)$ | $+1.50(15)$ | $+1.50(15)$ |  similar to the values $\mu=+1.6229(16)$ [Go05], $\mu=+1.674(2)$ [He76a,b], $\mu=+1.63(10)$ [Bo63], $\mu=1.0(3)$ [Sp94] and $\mu=1.6(5)$ [Ja96] for the lowest 5/2- states in respectively 69As, 71As, 73As, 79Br, and 81Br
75Sr | $3/2^{-}$ |  | $-0.55(17)$ | $+0.93(17)$ |  $+0.93(17)$a |  (see 75Rb and text)
75Rb | $3/2^{-}$ |  | $+2.25(19)$ | $+0.54(19)$ |  $+0.54(19)$a |  similar to the value $\mu=+0.6544680(16)$ for the 3/2- g.s. state of 77Rb [Du86, Th81] (see also text)
Table 13: Reference key relating the reference codes used in Tables LABEL:Magnetic-moments to 12 and to the actual reference numbers. Table | Reference | Table | Reference | Table | Reference | Table | Reference | Table | Reference | Table | Reference
---|---|---|---|---|---|---|---|---|---|---|---
code | No. | code | No. | code | No. | code | No. | code | No. | code | No.
Al51a | Alder and Yu (1951a) | Al51b | Alder and Yu (1951b) | Am65 | Ames  _et al._ (1965) | An82 | Andl  _et al._ (1982) | Ar83 | Arnold  _et al._ (1983) | Av11 | Avgoulea  _et al._ (2011)
Ba15 | Babcock  _et al._ (2015) | Ba62 | Baldeschweiler (1962) | Ba64 | Baker  _et al._ (1964) | Be09 | Berryman  _et al._ (2009) | Be64 | Bernstein  _et al._ (1964) | Be74 | Beckmann  _et al._ (1974)
Bl68 | Bleck  _et al._ (1968) | Bl72 | Blaser  _et al._ (1972) | Bo63 | Bodenstedt  _et al._ (1963) | Br62 | Brun  _et al._ (1962) | Bu90 | Buchinger  _et al._ (1990) | Ca65 | Calaprice  _et al._ (1965)
Ca73 | Callaghan  _et al._ (1973) | Ch10 | Cheal  _et al._ (2010) | Co10 | Cocolios  _et al._ (2010) | Co63 | Commins and Feldman (1963) | Co66a | Cornwell  _et al._ (1966) | Co66b | Cornwell and McCullen (1966)
Dh51 | Dharmatti  _et al._ (1951) | Du86 | Duong et al. (1986) | Du93 | Duong  _et al._ (1993) | Ep68 | Epperlein and Lutz (1968) | Ep75 | Epperlein  _et al._ (1975) | Fu76 | Fuller (1976)
Fu93 | Fukuda  _et al._ (1993) | Ge05 | Geithner  _et al._ (2005) | Go04 | Golovko  _et al._ (2004) | Go05 | Golovko  _et al._ (2005) | He76a | Herzog  _et al._ (1976a) | He76b | Herzog  _et al._ (1976b)
Hu84 | Hugg (1984) | Ja96 | Jakob  _et al._ (1996) | Jo70 | Jonsson  _et al._ (1970) | Jo71 | Jonsson  _et al._ (1971) | Kl96 | Klein  _et al._ (1996) | La57 | LaTourrette  _et al._ (1957)
La69 | Laulainen and McDermott (1969) | Li52 | Lindstrom (1952) | Li92 | Lievens  _et al._ (1992) | Lu73 | Lutz  _et al._ (1973) | Ma02 | Matsuta  _et al._ (2002) | Ma04 | Matsuta  _et al._ (2004)
Ma82 | MacArthur  _et al._ (1982) | Ma93 | Matsuta  _et al._ (1993) | Ma98 | Matsuta  _et al._ (1998) | Ma99 | Matsuta  _et al._ (1999) | Mi06 | Minamisono  _et al._ (2006) | Mi17 | Miller  _et al._ (2017)
Mi39 | Millman  _et al._ (1939) | Mi76a | Minamisono  _et al._ (1976a) | Mi76b | Minamisono  _et al._ (1976b) | Mi76c | Minamisono  _et al._ (1976c) | Mi90 | Minamisono  _et al._ (1990) | Mi93 | Minamisono  _et al._ (1993)
Mo00 | Mohr and Taylor (2000) | Mo68 | Morgenstern  _et al._ (1968) | Ne11 | Neronov and Aleksandrov (2011) | Ne77 | Neronov and Barzakh (1977) | Oh96 | Ohtsubo  _et al._ (1996) | Ol70 | Oluwole  _et al._ (1970)
Pi88 | Pitt  _et al._ (1988) | Pr12 | Procter  _et al._ (2012) | Ro54 | Royden (1954) | Ro65 | Robertson  _et al._ (1965) | Ro75 | Rosenblum and Steyert (1975) | Ro86 | Rogers  _et al._ (1986)
Sa74a | Sahm and Schwenk (1974a) | Sa74b | Sahm and Schwenk (1974b) | Sp94 | Speidel  _et al._ (1994) | St08 | Stone  _et al._ (2008) | St14 | Stone (2014) | Su66 | Sugimoto  _et al._ (1966)
Su71 | Sugimoto  _et al._ (1971) | Ta93 | Tanigaki  _et al._ (1993) | Th81 | Thibault  _et al._ (1981) | Ul17 | Schneider  _et al._ (2017) | Vi10 | Vingerhoets  _et al._ (2010) | Vi11 | Vingerhoets  _et al._ (2011)
Vo71 | von Platen (1971) | Wa54 | Walchli (1954) | We53 | Weaver and Weaver (1953) | We75 | Wender and Cameron (1975) | Wo70 | Wolber  _et al._ (1970) | Yo17 | Yordanov  _et al._ (2017)
Za84 | Zamboni and Saxena (1984) | Zh09 | Zhou  _et al._ (2009) |  |  |  |  |  |  |  |

####  III.2.2 Experimental weak magnetism form factors $b$ and $b/Ac$

With the isovector magnetic moment for most of the $T=1/2$ mirror nuclei pairs
up to $A=75$ determined, we can deduce the weak magnetism $b$ form factors
using Eq. (28) and combine these with the $c$ form factors deduced from the
corrected $\mathcal{F}t^{\mathrm{mirror}}$ values (Table 6) using Eq. (II.1),
to obtain the normalized $b/Ac$ form factors ($A$ being the mass number).
Results are listed in Table 14 and shown in Figure 4. As can be seen, large
absolute values for $\mu_{f}-\mu_{i}$, and so for $b$, are observed for mirror
pairs in subshells with $j=l+1/2$ ($j$ and $l$ being, respectively, the spin
and orbital angular momentum of the subshell of the odd nucleon), i.e. in the
$p_{3/2}$ ($A$ = 11), $d_{5/2}$ ($A$ = 17-27), $s_{1/2}$ ($A$ = 29-31),
$f_{7/2}$ ($A$ = 41-55), and $2p_{3/2}$ ($A$ = 57-61) subshells. Small values
for $\mu^{\mp}$ are observed for mirror pairs in subshells with $j=l-1/2$,
i.e. the $p_{1/2}$ ($A$ = 13-15), $d_{3/2}$ ($A$ = 33-39), and $f_{5/2}$ ($A$
= 67-75) subshells. This is fully in line with the values for the difference
of the odd-proton and odd-neutron Schmidt values,
$\mu_{p}^{Sch}-\mu_{n}^{Sch}$, for the different nuclear spin values, which is
large for $j=l+1/2$ (i.e. between 4.7 and 7.7 for the spins involved here) and
relatively small for $j=l-1/2$ (i.e. ranging between -1.0 and +0.2) (see Table
15). As a consequence, the values for $b$ share the same systematic, as is
clear from Fig. 4 which shows large values for $b$ in the $d_{5/2}$,
$f_{7/2}$, and $2p_{3/2}$ subshells ($j=l+1/2$), and significantly smaller
values in the $p_{1/2}$, $d_{3/2}$, and $f_{5/2}$ subshells ($j=l-1/2$).

Table 14: Input data and CVC results for the $b$ form factors and $b/Ac$ ratios for the mirror $\beta$ transitions up to $A=39$. $\mu^{\mp}$ is the difference of the magnetic moment of the mother and daughter isotopes, where the values listed in Table LABEL:Magnetic-moments were used. If several magnetic moments values are available, the weighted average was used. Values for $b$ were calculated from Eq. (28) with $a(0)=1$. Values for the Gamow-Teller form factor $c$ are from Table 6. $\beta$ transition | $A$ | $J$ | subshell | $\mu_{f}-\mu_{i}$ |  $b$272727For $A=$ 45-53 and $A>$ 57, at least one of the mirror pair magnetic moments was not measured experimentally but calculated based on the experimental $\mathcal{F}$t value and the procedure outlined in Sec. III.2.1 Perez  _et al._ (2008). | $c$ | $(b/Ac)^{exp}$ | $(b/Ac)^{exp}$
---|---|---|---|---|---|---|---|---
|  |  |  |  [$\mu_{N}$] |  |  |  | subshell
|  |  |  |  |  |  |  |  average 282828As values and error bars sometimes vary significantly unweighted averages are given as well, in italics.
H$\rightarrow$He | 3 | 1/2 | $s_{1/2}$ | $-5.106460174(31)$ | $-26.53394540(16)$ | $-2.1053(12)$ | $4.2012(23)$ | $4.2012(23)$
C$\rightarrow$B | 11 | 3/2 | $p_{3/2}$ | $+3.6526(10)$ | $-51.871(14)$ | $-0.75442(28)$ | $6.2506(29)$ | $6.2506(29)$
N$\rightarrow$C | 13 | 1/2 | $p_{1/2}$ | $+1.02461(40)$ | $-23.0708(90)$ | $-0.55962(34)$ | $3.1712(23)$ | $2.93(21)$
O$\rightarrow$N | 15 | 1/2 |  | $-1.00269(12)$ | $26.0506(31)$ | $0.63023(46)$ | $2.7557(20)$ | $\it{2.96(29)}$
F$\rightarrow$O | 17 | 5/2 | $d_{5/2}$ | $-6.61515(30)$ | $133.0616(61)$ | $1.29555(65)$ | $6.0416(30)$ | $5.82(42)$
Ne$\rightarrow$F | 19 | 1/2 |  | $+4.51429(8)$ | $-148.5605(26)$ | $-1.60203(65)$ | $4.8807(20)$ | $\it{6.18(95)}$
Na$\rightarrow$Ne | 21 | 3/2 |  | $-3.04810(10)$ | $82.6366(27)$ | $0.71245(39)$ | $5.5233(30)$ |
Mg$\rightarrow$Na | 23 | 3/2 |  | $+2.75409(21)$ | $-81.7768(63)$ | $-0.55413(47)$ | $6.4164(55)$ |
Al$\rightarrow$Mg | 25 | 5/2 |  | $-4.5010(12)$ | $133.140(36)$ | $0.80844(42)$ | $6.5875(38)$ |
Si$\rightarrow$Al | 27 | 5/2 |  | $+4.50684(24)$ | $-143.9792(77)$ | $-0.69659(30)$ | $7.6552(33)$ |
P$\rightarrow$Si | 29 | 1/2 | $s_{1/2}$ | $-1.79004(21)$ | $89.913(11)$ | $0.53798(47)$ | $5.7631(51)$ | $5.43(21)$
S$\rightarrow$P | 31 | 1/2 |  | $+1.61953(9)$ | $-86.9584(46)$ | $-0.52939(33)$ | $5.2988(33)$ | $\it{5.53(33)}$
Cl$\rightarrow$S | 33 | 3/2 | $d_{3/2}$ | $-0.11098(50)$ | $4.728(21)$ | $-0.31416(28)$ | $-0.4561(21)$ | $0.21(59)$
Ar$\rightarrow$Cl | 35 | 3/2 |  | $+0.18967(20)$ | $-8.5704(90)$ | $0.28199(17)$ | $-0.8684(11)$ | $\it{0.5(14)}$
K$\rightarrow$Ar | 37 | 3/2 |  | $+0.9418(50)$ | $-44.99(24)$ | $-0.57789(39)$ | $2.104(11)$ |
Ca$\rightarrow$K | 39 | 3/2 |  | $-0.63019(12)$ | $31.7295(61)$ | $0.66061(50)$ | $1.2316(10)$ |
Sc$\rightarrow$Ca | 41 | 7/2 | $f_{7/2}$ | $-7.0258(20)$ | $326.626(93)$ | $1.0743(21)$ | $7.415(14)$ | $7.47(12)$
Ti$\rightarrow$Sc | 43 | 7/2 |  | $+5.383(30)$ | $-262.5(14)$ | $-0.8097(69)$ | $7.539(77)$ | $\it{8.05(97)}$
V$\rightarrow$Ti | 45 | 7/2 |  | $-4.135(82)$ | $211.0(42)$ | $0.6346(58)$ | $7.39(16)$ |
Cr$\rightarrow$V | 47 | 3/2 |  | $+3.012(57)$ | $-182.8(34)$ | $-0.5794(42)$ | $6.71(14)$ |
Mn$\rightarrow$Cr | 49 | 5/2 |  | $-3.746(80)$ | $217.2(46)$ | $0.5373(75)$ | $8.25(21)$ |
Fe$\rightarrow$Mn | 51 | 5/2 |  | $+4.243(52)$ | $-256.0(31)$ | $-0.5811(49)$ | $8.64(13)$ |
Co$\rightarrow$Fe | 53 | 7/2 |  | $-5.196(86)$ | $312.3(51)$ | $0.6730(72)$ | $8.75(17)$ |
Ni$\rightarrow$Co | 55 | 7/2 |  | $+5.798(26)$ | $-361.6(16)$ | $-0.6752(80)$ |  $9.74(12$) |
Cu$\rightarrow$Ni | 57 | 3/2 | $p_{3/2}$ | $-3.3795(71)$ | $248.69(53)$ | $0.5639(28)$ | $7.737(42)$ | $7.67(20)$
Zn$\rightarrow$Cu | 59 | 3/2 |  | $+2.402(80)$ | $-182.9(61)$ | $-0.4876(42)$ | $6.36(30)$ | $\it{6.98(70)}$
Ga$\rightarrow$Zn | 61 | 3/2 |  | $-2.87(12)$ | $226.0(95)$ | $0.5421(79)$ | $6.84(30)$ |
Se$\rightarrow$As | 67 | 5/2 | $f_{5/2}$ | $+0.67(19)$ | $-53(15)$ | $-0.3873(89)$ |  $2.05(59)$333Value for upper limit of BR; see also Table 5 and Sec. II.3.2. |
|  |  |  |  |  | $-0.2048(50)$ |  $3.9(11)$444Value for lower limit of BR; see also Table 5 and Sec. II.3.2. |
Kr$\rightarrow$Br | 71 | 5/2 |  | $0.46(21)$ | $-39(17)$ | $0.454(16)$ |  $-1.20(54)$33footnotemark: 3 |
|  |  |  |  |  | $0.1682(61)$ |  $-3.2(14)$44footnotemark: 4 |
Sr$\rightarrow$Rb | 75 | 3/2 |  | $-0.38(25)$ | $37(25)$ | $0.525(32)$ |  $0.93(63)$33footnotemark: 3 |
|  |  |  |  |  | $0.367(22)$ |  $1.34(90)$44footnotemark: 4 |
Table 15: Differences, $\mu_{p}^{Sch}-\mu_{n}^{Sch}$, of the Schmidt values for magnetic moments of odd-proton and odd-neutron nuclei with spin $j=1/2$ to $j=7/2$, for $j=l+1/2$ and $j=l-1/2$, and for both $g_{S}^{free}$ and $0.6\times g_{S}^{free}$. Large values are obtained for $j=l+1/2$ and much smaller values for $j=l-1/2$. | $\mu_{p}^{Sch}-\mu_{n}^{Sch}$
---|---
| $j=l+1/2$ | $j=l-1/2$
$j$ | $g_{S}^{free}$ | $0.6\times g_{S}^{free}$ | $g_{S}^{free}$ | $0.6\times g_{S}^{free}$
$1/2$ | $4.71$ | $2.82$ | $-0.90$ | $-0.27$
$3/2$ | $5.71$ | $3.82$ | $-1.02$ | $+0.11$
$5/2$ | $6.71$ | $4.82$ | $-0.50$ | $+0.84$
$7/2$ | $7.71$ | $5.82$ | $+0.23$ | $+1.69$
Figure 4: Weak magnetism form factor $b$ for the $T=1/2$ mirror $\beta$
transitions from the experimental/calculated magnetic moments in column 8 of
Table 14 and Eq. (28). For 67Se, 71Kr, and 75Sr the values corresponding to
the upper and lower limits of the BR values (see Tables 5 and 6) are shown.
The thin vertical lines separate the shell model subshells, the labels of
which are indicated. If not explicitly shown, error bars are smaller than the
size of the symbol.

Figure 5 shows the evolution of the Gamow-Teller form factor, $|c|$, (the sign
of $c$ is straightforwardly obtained from shell model calculations) as a
function of the mass number. While a clear local increase is visible when
crossing the doubly-magic nuclei at $A=16$, respectively $A=40$, a general
overall, slow decrease can be observed. This can be understood intuitively as
the Fermi surfaces of proton and neutron become increasingly distanced for
higher masses.

Figure 5: Absolute value of the Gamow-Teller form factor, $c$, as a function
of the mass number. A slight general downward trend with increasing mass is
apparent, with clear shell closure effects around the doubly-magic $A=16$ and
$A=40$ nuclei. For 67Se, 71Kr, and 75Sr the values corresponding to the upper
and lower limits of the BR values (see Tables 5 and 6) are shown. If not
explicitly shown, error bars are smaller than the size of the symbol.

This decrease persists in the ratio $b/Ac$ as is seen in Fig. 6, where again a
distinction is made between $j=l\pm 1/2$ transitions. Despite significant
variation in the absolute value of $b$ (Figure 4 and Table 14), subshell
behaviour of $b/Ac$ is reasonably uniform. The already large values in the
large $d_{5/2}$ and $f_{7/2}$ subshells are further accentuated by the
decrease of the $|c|$ values throughout these subshells (Fig. 5). The behavior
of $b/Ac$ versus $A$ for the mirror decays is thus understood in terms of the
single particle Schmidt values for the magnetic moments and the slight
decrease of the size of the Gamow-Teller form factor $c$ with increasing $A$.

Figure 6: Experimental values of $b/Ac$ for the $T=1/2$ mirror $\beta$
transitions (Table 14). The different shell model subshells are indicated. For
67Se, 71Kr, and 75Sr the values corresponding to the upper and lower limits of
the BR values are shown. See also Tables 5, 6 and 14, and text. If not
explicitly shown, error bars are smaller than the size of the symbol.

####  III.2.3 Theoretical form factor comparison

While some substructure is clearly visible in the experimental results of Fig.
6, a variation of a full order of magnitude is observed. In the face of ever-
increasing experimental precision, it is paramount that theory can meet this
challenge. Here we will compare experimental results to theoretical
predictions, using both single-particle estimates and the nuclear shell model.

#####  III.2.3.1 Single particle calculations

As noted in the previous sections, significant variation occurs in the values
for $b/Ac$ for different spin-orbit orbitals. This can be understood
intuitively from a single particle view of the transition. Using Eq. (33) this
means we are interested in the ratio of matrix elements $R=M_{L}/M_{GT}$.
Using simple harmonic oscillator wave functions this ratio is

$\displaystyle\frac{M_{L}}{M_{GT}}$ $\displaystyle=\frac{\langle
n_{f}l_{f}j_{f}|\bm{l}|n_{i}l_{i}j_{\rangle}}{\langle
n_{f}l_{f}j_{f}|\bm{\sigma}|n_{i}l_{i}j_{\rangle}}=(-1)^{j_{i}-j_{f}}$
$\displaystyle\times\frac{\left\\{\begin{array}[]{ccc}1/2&l&j_{i}\\\
1&j_{f}&l\end{array}\right\\}}{\left\\{\begin{array}[]{ccc}l&1/2&j_{i}\\\
1&j_{f}&1/2\end{array}\right\\}}\frac{\sqrt{l(l+1)(2l+1)}}{\sqrt{6}}\leavevmode\nobreak\
.$ (46)

Using the four possibilities for transitions with $j_{i,f}=l\pm 1/2$, this
results in

$\displaystyle R^{--}$ $\displaystyle=-(l+1)\qquad R^{-+}=-1/2$ (47)
$\displaystyle R^{+-}$ $\displaystyle=-1/2\qquad R^{++}=l.$ (48)

In the case of mirror decays, the signs for initial and final states are the
same. As such, the expected behaviour of $b/Ac$ is opposite for spin-orbit
partners. This split can clearly be seen in the data previously presented.
Despite this insight, there is still significant variation in the experimental
values within the same subshell (see Table 14 and Fig 6). Additional insight
can be gleaned by incorporating deformation into the single particle
potential. As discussed in the appendix, we move on to an axially deformed
Woods-Saxon potential. Thanks to mean-field results Möller  _et al._ (2016) we
have theoretical deformations available for all participating nuclei. Wave
functions are computed numerically Hayen and Severijns (2019b) and projected
on a spherical harmonic oscillator basis, for which we can use the analytical
results of Eq. (48). Performing the calculation, we find the results shown in
Fig. 7.

Figure 7: Comparison of different methods of calculating the weak magnetism
contribution $b/Ac$ to experimental data using CVC, for $Z=$6 (11C) to $Z=$23
(45V). The simple spherical harmonic oscillator results are summarized in Eq.
(48). Clearly, single particle results using the deformed Woods-Saxon
potential give excellent agreement across the board, including the outliers
33Cl and 35Ar at atomic numbers 17 and 18.

As was noted, the general trends are replicated by the simple spherical
harmonic oscillator results of Eq. (48). When taking into account nuclear
deformation through the single particle potential of Eq. (75), deviations are
introduced at an individual level. It is remarkable to note that the agreement
with experimental data now becomes excellent for all transitions considered
(Fig. 7). Of particular interest is the agreement with cases of large
deviations from traditional estimates for $A=33$ and $A=35$. In contrast to
most of the other nuclei, both initial and final states are oblate deformed
resulting in large cancellation effects. The case of 33Cl has been discussed
in more detail in Hayen and Severijns (2019b), including the dependence on the
input deformation parameters (see also Sec. III.2.4).

A final feature to note is the convergence to the simple harmonic oscillator
results as subshells are filled. This can be understood using well-known
Nilsson orbitals Nilsson (1955). In the case of prolate deformation,
projections with low spin relative to the symmetry axis are pushed down in
energy and vice versa for high spin projections. When a subshell is filled,
valence nucleons reside in low spin projections. Due to the proximity of other
low spin states, significant mixing occurs and the wave function of Eq.(76)
becomes non-trivial. All possible combinations in Eq. (48) occur, and lower
(higher) values occur for $j=l+1/2$ ($j=l-1/2$) orbitals. As the subshell is
filled, nearby equally high spin states become sparse and the resultant mixing
is strongly reduced. The results then converge to the spherical harmonic
oscillator estimates.

#####  III.2.3.2 Nuclear shell model

For the mirror nuclei up to mass 45 the $b$ and $c$ form factors, which depend
on the nuclear matrix elements $M_{GT}$ and ML, were also calculated in the
nuclear shell model. The choice of an effective interaction for these
calculations in light nuclei, whose principal configurations involve several
valence nucleons away from major shell closures, is easily made. There are
well established interactions that give excellent fits to spectra. For
$p$-shell nuclei, we use the Cohen-Kurath Cohen and Kurath (1965) interaction,
i.e. (8 - 16)POT, for $s,d$-shell nuclei the universal $s,d$-interaction USD
of Wildenthal Wildenthal (1984), and for $p,f$-shell nuclei the Kuo-Brown
$G$-matrix Kuo and Brown (1966) as modified by Poves and Zuker Poves and Zuker
(1981) and denoted KB3. Close to major shell closures the choice of a model
space and effective interaction is more problematic. Our approach is to
construct a hybrid interaction comprising the Millener-Kurath Millener and
Kurath (1975) interaction for the cross-shell matrix elements, and the
(8-16)POT, USD, or KB3 interactions for the in-shell matrix elements. The
calculations further used $g_{A}=g_{V}=1$, $g_{M}=4.706$, $g_{P}=-181.03$, and
$g_{S}=g_{T}=0$.

Results thus obtained for the form factor ratio $b/Ac$ up to $A=45$ are listed
in column 4 of Table 16. For higher masses the required truncation of the
shell model space, due to calculation power limitations, was too important to
yield reliable values. The ratio of the values for $b/Ac$ extracted from
experiment (column 3 in Table 16) and these shell model calculated values are
listed in the last column of Table 16 and displayed in Fig. 8. The average for
the ratios $(b/Ac)^{exp}/(b/Ac)^{theo}$ for the mass range up to $A=45$,
omitting the values for $A=$ 33 and 35, which both deviate significantly from
unity, is $0.96(11)$. The cases of $A=$ 33 and 35 will be discussed in more
detail in the next paragraph.

Figure 8: Ratio of experimental values (Table 6) and nuclear shell model
calculated values (Sec. III.2.3.2) for the b/Ac form factor ratio for the T =
1/2 mirror beta transitions up to A = 45. The values for $A=$ 33 and 35 have
been omitted (see text). Error bars (based only on the experimental $b/Ac$
values) are smaller than the size of the symbols.

####  III.2.4 Impulse approximation: access to $M_{GT}$ and $M_{L}$ and
comparison with shell-model calculations

Using the impulse approximation, the ratio $b/Ac_{1}$ was written in Eq. (33)
in terms of the Gamow-Teller and orbital current matrix elements, $M_{GT}$ and
$M_{L}$, respectively. The first is directly obtained from the Gamow-Teller
form factor, $c\cong g_{A}M_{GT}$ that was extracted from the experimental
${\mathcal{F}}t$ values (cf. Table 6). Using then this value for $M_{GT}$, Eq.
(33) allows extracting $M_{L}$. Note that we use here the sign convention of
Holstein, which uses a positive value for $g_{A}$, so that $M_{GT}$ and $c$
have the same sign. The resulting values for $M_{GT}^{exp}$ and $M_{L}^{exp}$
(using $g_{A}=+1.27$ for $A=$ 3 and $g_{A}=+1.00$ for all other cases (Sec.
III.1.3.2)) are listed in columns 4 and 8 of Table 17. The corresponding
shell-model calculated values are listed in columns 5 and 9.

The ratio and the difference of the experimental and shell-model calculated
values for $M_{GT}$ is shown in Fig. 9. It is seen that the shell model is
very capable in calculating $M_{GT}$, with the ratio differing 10% to 20% from
unity and the difference being typically limited to about 0.1. Taking an
unweighted average, we find the ratio of experimental to theoretical values to
be $M_{GT}^{exp}/M_{GT}^{theo}=0.97(8)$.

As a further check for the quality of the shell model calculations, Fig. 10
shows the ratio and the difference between the experimental and theoretical
values for the orbital current matrix element, $M_{L}$, for the mirror
transitions up to mass 45 (listed in Table 17). Again, reasonably good
agreement between theory and experiment is observed. A small tendency of the
shell-model calculation to slightly overestimate the value of $M_{L}$ by up to
0.5 $\it{fm}$ is seen. The average value of
$M_{L}^{exp}/M_{L}^{theo}=0.99(35)$ for $A=11-$29 and 33$-$45 is close to
unity, however. The masses 3 and 31 are left out when constructing this
average because their shell model calculations give $M_{L}=$ 0 and $M_{L}=$
-0.159, respectively. Both being close to zero, due to the dominant $s$-state
configuration in their wave function. Not omitting the case of $A=31$ slightly
modifies the ratio to $1.04(34)$. It is unclear why the experimentally deduced
$M_{L}$ value for these two transitions, i.e. -1.06 for 3He and -0.31 for 31S,
is not smaller than it is. Contributions from meson exchange which are
incorrectly estimated in the quenching of $g_{A}$ could partially explain this
discrepancy.

As can be seen from Table 14 and Figure 6, most $(b/Ac)^{exp}$ values are
positive, except for the mirror $\beta$ decays of 33Cl, 35Ar, and 71Kr, for
which slightly negative values are found. This is what one could expect from
Eq. (33), in the sign convention of Holstein Holstein (1974a). Indeed, since
the first term in Eq. (33) is equal to $g_{M}/g_{A}=4.706$ (for $g_{A}=+1$), a
negative value for $b/Ac$ can only occur if $M_{L}$ and $M_{GT}$ have opposite
signs and $|M_{L}|>4.706|M_{GT}|$ in absolute value. As can be seen from the
last two columns in Table 17, the ratio $M_{L}/M_{GT}$ is found (both from
experiment and theory) to be positive in subshells with $j=l+1/2$ (i.e. the
$s_{1/2}$, $p_{3/2}$, $d_{5/2}$, and $f_{7/2}$ subshells), leading to large
values for $b/Ac$ (Eq. (33)), and negative in subshells with $j=l-1/2$ (the
$p_{1/2}$ and $d_{3/2}$ subshells, i.e. the masses $A=$ 13-15, and $A=$
33-39), resulting in much smaller values for $b/Ac$. This is in line with the
CVC-based relation Eq. (28) and the values for $j=l\pm 1/2$ states in Table 15
(see also Sec. III.2.2), combined with small values for $c=M_{GT}$. For the
cases of 33Cl, 35Ar, and 71Kr, the values of $M_{GT}$ are indeed the smallest
among all mirror $\beta$ transitions, as is reflected by their large
$\mathcal{F}t$ values (Table II). For all three cases the ratios
$(M_{L}/M_{GT})^{exp}=(b/Ac)^{exp}-4.7$ (see Eq. 33) are $\approx-5$ thereby
over-compensating the value of 4.7 of the first term in Eq. (33) and causing
$b/Ac$ to be small and negative (cf. the last but one column in Table 6).
This, together with the fact that 33Cl, 35Ar, and 71Kr and their respective
daughter isotopes are rare cases of oblate deformation (with deformation
parameters $\beta_{2}\approx-0.23$ for 33Cl and 35Ar, and $\approx-0.36$ for
71Kr Möller  _et al._ (2016)) then poses a challenge to theoretical
calculations. However, with $M_{L}/M_{GT}$ being a ratio of matrix elements of
the same order in spherical tensor formulation, such that complex many-body
couplings drop out when neglecting core polarisation and meson exchange, one
is left with a ratio of single-particle matrix elements such that one can
expect the extreme single-particle to capture most of the required dynamic
Hayen and Severijns (2019b). The challenge then remains to pick a suitable
single-particle state based on an underlying potential. The case of 71Kr
obviously is too complex for this. For 33Cl and 35Ar Table 18 compares the
values for $b/Ac$ obtained in Ref. Hayen and Severijns (2019b) using the
spherical harmonic oscillator, the Woods-Saxon potential, and the deformed
Woods-Saxon model obtained in Ref. [119] with the here-performed shell model
calculations and the experimental results. As can be seen, only the deformed
Woods-Saxon potential can reproduce the experimental values within a factor of
about two and with the correct sign.

The good agreement with single particle estimates in a deformed potential -
particularly 33Cl and 35Ar, which turned out to be difficult for the shell
model - stresses the importance of an appropriate basis choice. While the
shell model reaches for most cases similar accuracy, it of course comes as a
result of many higher-order particle-hole excitations to effectively reproduce
nuclear deformation. Additionally, results are obtained using different
effective interactions with tuned parameters, effective charges and
renormalization of coupling constants. When the transition permits it, results
of similar accuracy can be obtained for the mirror nuclei in a much more
simple fashion in the extreme single particle approximation. Even so, while
reassuring, the possibilities for a just application of single-particle
results are slim and limited by transitions dominated by single particle
states. Inspection of $g$-factors can be a useful tool here, so that single-
particle results are an interesting cross-check when applicable.

Table 16: Comparison of experimental and theoretical values for the ratio $(b/Ac)$ for the mirror $\beta$ transitions up to 45V. Theoretical values were calculated using the shell model. Values for $(b/Ac)^{exp}$ are obtained from the experimental ${\mathcal{F}}t$ values and the magnetic moments of the mirror nuclei pairs. See text for more details. $\beta$ decay | $A$ | $(b/Ac)^{exp}$ | $(b/Ac)^{theo}$ | $\frac{(b/Ac)^{exp}}{(b/Ac)^{theo}}$
---|---|---|---|---
H $\rightarrow$ He | 3 | $4.2012(23)$ | $4.706$ | $0.893$
C $\rightarrow$ B | 11 | $6.2506(29)$ | $5.759$ | $1.085$
N $\rightarrow$ C | 13 | $3.1712(23)$ | $3.479$ | $0.912$
O $\rightarrow$ N | 15 | $2.7556(20)$ | $2.753$ | $1.001$
F $\rightarrow$ O | 17 | $6.0416(30)$ | $6.682$ | $0.904$
Ne $\rightarrow$ F | 19 | $4.8807(20)$ | $5.134$ | $0.951$
Na $\rightarrow$ Ne | 21 | $5.5233(30)$ | $6.005$ | $0.920$
Mg $\rightarrow$ Na | 23 | $6.4164(55)$ | $6.004$ | $1.069$
Al $\rightarrow$ Mg | 25 | $6.5875(38)$ | $6.858$ | $0.961$
Si $\rightarrow$ Al | 27 | $7.6552(33)$ | $7.320$ | $1.046$
P $\rightarrow$ Si | 29 | $5.7631(51)$ | $5.790$ | $0.995$
S $\rightarrow$ P | 31 | $5.2988(33)$ | $5.030$ | $1.053$
Cl $\rightarrow$ S | 33 | $-0.4561(21)$ | $-0.206$ | $2.219$
Ar $\rightarrow$ Cl | 35 | $-0.8684(11)$ | $0.154$ | $-5.633$
K $\rightarrow$ Ar | 37 | $2.104(11)$ | $2.437$ | $0.863$
Ca $\rightarrow$ K | 39 | $1.2316(10)$ | $1.863$ | $0.661$
Sc $\rightarrow$ Ca | 41 | $7.415(14)$ | $7.669$ | $0.967$
Ti $\rightarrow$ Sc | 43 | $7.539(77)$ | $6.520$ | $1.156$
V $\rightarrow$ Ti | 45 | $7.39(16)$ | $8.121$ | $0.910$
Table 17: Comparison of experimental and theoretical values for the $M_{GT}$ and $M_{L}$ matrix elements (in $fm$ units) for the mirror $\beta$ transitions up to 45V. Theoretical values were calculated using the shell model (see text for details). Values for $M_{GT}^{exp}$ are obtained from the Gamow-Teller form factors, $c$, listed in Table 6. Values for $M_{L}^{exp}$ were calculated from Eq. (33) using the $(b/Ac)^{exp}$ values listed in Table 16. In both cases $g_{A}=1.27$ was used for $A=$ 3, and $g_{A}=1.00$ for all other cases (see Sec. III.1.3.2). $\beta$ decay | $A$ | shell | $M_{GT}^{exp}$ | $M_{GT}^{theo}$ | $\frac{M_{GT}^{exp}}{M_{GT}^{theo}}$ | $M_{GT}^{(exp-theo)}$ | $M_{L}^{exp}$ | $M_{L}^{theo}$ | $\frac{M_{L}^{exp}}{M_{L}^{theo}}$ | $M_{L}^{(exp-theo)}$ | $\frac{M_{L}^{theo}}{M_{GT}^{theo}}$ | $\frac{M_{L}^{exp}}{M_{GT}^{exp}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
H $\rightarrow$ He | 3 | $s_{1/2}$ | $-1.6577(9)$ | $-1.706$ | $0.972$ | $+0.048$ | $-1.0436(48)$ | $0$ | $-$ | $-1.044$ | $0.000$ | $+0.627$
C $\rightarrow$ B | 11 | $p_{3/2}$ | $-0.75442(28)$ | $-0.789$ | $0.956$ | $+0.035$ | $-1.1653(22)$ | $-0.831$ | $1.402$ | $-0.334$ | $+1.053$ | $+1.545$
N $\rightarrow$ C | 13 | $p_{1/2}$ | $-0.55962(34)$ | $-0.568$ | $0.985$ | $+0.008$ | $+0.8589(14)$ | $+0.697$ | $1.232$ | $+0.162$ | $-1.227$ | $-1.535$
O $\rightarrow$ N | 15 |  | $+0.63023(46)$ | $+0.576$ | $1.094$ | $+0.054$ | $-1.2291(15)$ | $-1.125$ | $1.093$ | $-0.104$ | $-1.953$ | $-1.950$
F $\rightarrow$ O | 17 | $d_{5/2}$ |  $+1.29555(65$) | $+1.182$ | $1.096$ | $+0.114$ | $+1.7303(40)$ | $+2.336$ | $0.741$ | $-0.606$ | $+1.976$ | $+1.336$
Ne $\rightarrow$ F | 19 |  | $-1.60203(65)$ | $-1.676$ | $0.956$ | $+0.074$ | $-0.2799(32)$ | $-0.717$ | $0.390$ | $+0.437$ | $+0.428$ | $+0.175$
Na $\rightarrow$ Ne | 21 |  | $+0.71245(39)$ | $+0.726$ | $0.981$ | $-0.014$ | $+0.5823(22)$ | $+0.943$ | $0.617$ | $-0.361$ | $+1.299$ | $+0.817$
Mg $\rightarrow$ Na | 23 |  | $-0.55413(47)$ | $-0.588$ | $0.942$ | $+0.034$ | $-0.9478(32)$ | $-0.763$ | $1.242$ | $-0.185$ | $+1.298$ | $+1.710$
Al $\rightarrow$ Mg | 25 |  | $+0.80844(42)$ | $+0.781$ | $1.035$ | $+0.027$ | $+1.5211(32)$ | $+1.681$ | $0.905$ | $-0.160$ | $+2.152$ | $+1.881$
Si $\rightarrow$ Al | 27 |  | $-0.69659(30)$ | $-0.769$ | $0.906$ | $+0.072$ | $-2.0544(25)$ | $-2.010$ | $1.022$ | $-0.044$ | $+2.614$ | $+2.949$
P $\rightarrow$ Si | 29 | $s_{1/2}$ | $+0.53798(47)$ | $+0.513$ | $1.049$ | $+0.025$ | $+0.5687(28)$ | $+0.556$ | $1.023$ | $+0.013$ | $+1.084$ | $+1.057$
S $\rightarrow$ P | 31 |  | $-0.52939(33)$ | $-0.490$ | $1.080$ | $-0.039$ | $-0.3138(18)$ | $-0.159$ | $1.974$ | $-0.155$ | $+0.324$ | $+0.593$
Cl $\rightarrow$ S | 33 | $d_{3/2}$ | $-0.31416(28)$ | $-0.328$ | $0.958$ | $+0.014$ | $+1.6217(16)$ | $+1.611$ | $1.007$ | $+0.011$ | $-4.912$ | $-5.162$
Ar $\rightarrow$ Cl | 35 |  | $+0.28199(17)$ | $+0.328$ | $0.860$ | $-0.046$ | $-1.5719(10)$ | $-1.493$ | $1.053$ | $-0.079$ | $-4.552$ | $-5.574$
K $\rightarrow$ Ar | 37 |  | $-0.57789(39)$ | $-0.624$ | $0.926$ | $+0.046$ | $+1.5037(64)$ | $+1.416$ | $1.062$ | $+0.088$ | $-2.269$ | $-2.602$
Ca $\rightarrow$ K | 39 |  | $+0.66061(50)$ | $+0.764$ | $0.865$ | $-0.103$ | $-2.2952(19)$ | $-2.172$ | $1.057$ | $-0.123$ | $-2.843$ | $-3.474$
Sc $\rightarrow$ Ca | 41 | $f_{7/2}$ | $+1.0743(21)$ | $+1.116$ | $0.963$ | $-0.042$ | $+2.910(16)$ | $+3.307$ | $0.880$ | $-0.397$ | $+2.963$ | $+2.709$
Ti $\rightarrow$ Sc | 43 |  | $-0.8097(69)$ | $-0.989$ | $0.819$ | $+0.179$ | $-2.294(65)$ | $-1.794$ | $1.279$ | $-0.500$ | $+1.814$ | $+2.833$
V $\rightarrow$ Ti | 45 |  | $+0.6346(58)$ | $+0.619$ | $1.025$ | $+0.016$ | $+1.70(10)$ | $+2.114$ | $0.806$ | $-0.411$ | $+3.415$ | $+2.684$
---
Figure 9: Ratio (top) and difference (bottom) of experimental and nuclear
shell model (Sec. III.2.3.2) calculated values of the $M_{GT}$ matrix elements
for the $T=1/2$ mirror $\beta$ transitions with masses $A=3$ to 45. See also
Table 17. Error bars (based only on the experimental $M_{GT}$ values) are
smaller than the size of the symbols.
---
Figure 10: Ratio (top) and difference (bottom) of experimental and nuclear shell model (Sec. III.2.3.2) calculated values of the $M_{L}$ matrix elements for the $T=1/2$ mirror $\beta$ transitions with masses $A=3$ to 45. The ratio for $A=3$ is not shown as $M_{L}^{theo}=0$ in this case. See also Table 17. If not explicitly shown, error bars (based on the experimental $M_{L}$ values only) are smaller than the size of the symbols. Table 18: Extreme single-particle values of $b/Ac$ for the mirror $\beta$ decays of 33Cl and 35Ar calculated with three different possibilities for the nuclear potential, i.e. spherical harmonic oscillator (SHO), spherical Woods–Saxon (WS) and deformed Woods–Saxon (DWS), and within the nuclear shell model (NSM) using the universal $s,d$-interaction USD of Wildenthal Wildenthal (1984). The experimental values are listed in the last column. decay | SHO | WS | DWS | NSM | exp.
---|---|---|---|---|---
33Cl | $2.46$ | $2.46$ | $-1.13$ | $-0.21$ | $-0.456$
35Ar | $2.46$ | $2.46$ | $-0.94$ | $+0.15$ | $-0.868$

####  III.2.5 Discussion

Good agreement between theoretical and experimental values for the Gamow-
Teller matrix elements, $M_{GT}$, the orbital current matrix elements,
$M_{L}$, and the form factor ratio $b/Ac$ for the mirror $\beta$ transitions
considered here is obtained. Differences reveal no significant trend in $A$ or
in the shell model states occupied, except for a small downward slope towards
higher $A$ values related to the required truncation of the shell-model space.
The observed deviation in $b/Ac$ for the cases of 33Cl and 35Ar was understood
in terms of the smallness of these $b/Ac$ values, due to the large ratio of
$M_{L}/M_{GT}$ in the $d_{3/2}$ subshell ($j=l-1/2$), and the large oblate
deformation of these isotopes.

The good correspondence between experimental and theoretical values for the
$M_{L}$ matrix elements, with the exception of the case of 31S with a strong
$s$-state configuration in the wave function, gives confidence in the
reliability of the shell-model calculations performed and thus perhaps also in
the calculations of the matrix elements determining yet other, although
usually less important form factors (e.g. the tensor form factor, $d$, and the
induced pseudoscalar form factor, $h$, Holstein (1974a)). We therefore list in
Table 19 the full set of form factors from the shell model calculations
discussed above (Sec. III.2.3.2). These can be used in the analysis of
experiments determining the $\beta-\nu$ correlation or the $\beta$ asymmetry
parameter for the mirror $\beta$ transitions, that are being planned at
different laboratories (e.g. Brodeur  _et al._ (2016b); Liénard et al. (2015);
Fenker  _et al._ (2018)). Additional information on the effect of recoil
terms, radiative corrections and specific experimental conditions on such
measurements are also discussed in Refs. Hayen (2021); Vanlangendonck  _et
al._ (2021).

Table 19: Form factors calculated in the shell model for the $T=1/2$ mirror $\beta$ transitions up to mass $A=45$. Details on the calculations and the interactions used can be found in Sec. III.2.3.2. Note that the first-class part of the $d$ form factor, i.e. $d\approx Ag_{A}M_{\sigma L}$ (see Table 8), is zero because these are transitions within an isospin multiplet. Note that the form factor $j_{2}$ is in absolute value smaller than $8\times 10^{-2}$ and so negligible for all transitions listed here. $\beta$ decay | Shell model | $c=c_{1}$ | $c_{2}$ | $b/Ac$ | $f/Ac$ | $g/A^{2}c$ | $j_{1}/A^{2}c$ | $j_{3}/A^{2}c$ | $h/A^{2}c$
---|---|---|---|---|---|---|---|---|---
| interaction |  |  |  |  |  |  |  |
3He | MSDI3 | $-1.706$ | $-0.971$ | $4.709$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ | $+181.2$
11C | CK816POT | $-0.789$ | $-1.092$ | $5.761$ | $0.064$ | $-66.83$ | $-70.39$ | $+185.4$ | $+114.2$
13N | CK816POT | $-0.568$ | $-1.133$ | $3.480$ | $0.0$ | $0.0$ | $-232.3$ | $0.0$ | $-39.38$
15O | MK | $+0.576$ | $+1.267$ | $2.755$ | $0.0$ | $0.0$ | $-277.0$ | $0.0$ | $-81.79$
17F | MK | $+1.182$ | $+2.319$ | $6.684$ | $-0.188$ | $+127.9$ | $-57.38$ | $+172.1$ | $+126.5$
19Ne | USD | $-1.676$ | $-3.044$ | $5.134$ | $0.0$ | $0.0$ | $-1.311$ | $0.0$ | $+180.2$
21Na | USD | $+0.726$ | $+1.397$ | $6.008$ | $0.098$ | $-49.66$ | $-19.12$ | $-206.8$ | $+163.0$
23Mg | USD | $-0.588$ | $-1.219$ | $6.004$ | $-0.059$ | $+25.49$ | $-49.19$ | $-244.3$ | $+134.4$
25Al | USD | $+0.781$ | $+1.767$ | $6.858$ | $-0.161$ | $+65.56$ | $-91.98$ | $+41.79$ | $+93.83$
27Si | USD | $-0.769$ | $-1.871$ | $7.321$ | $0.419$ | $-149.3$ | $-130.2$ | $+130.2$ | $+57.44$
29P | USD | $+0.513$ | $+1.293$ | $5.794$ | $0.0$ | $0.0$ | $-143.9$ | $0.0$ | $+44.73$
31S | USD | $-0.490$ | $-1.260$ | $5.030$ | $0.0$ | $0.0$ | $-149.3$ | $0.0$ | $+39.29$
33Cl | USD | $-0.328$ | $-1.133$ | $-0.212$ | $0.898$ | $-271.3$ | $-393.9$ | $+166.3$ | $-191.8$
35Ar | USD | $+0.328$ | $+1.143$ | $0.148$ | $-0.092$ | $+25.88$ | $-390.7$ | $+161.0$ | $-190.6$
37K | USD | $-0.624$ | $-1.873$ | $2.434$ | $0.147$ | $-40.03$ | $-244.7$ | $+49.63$ | $-50.92$
39Ca | MK | $+0.764$ | $+2.336$ | $1.859$ | $-0.856$ | $+217.7$ | $-247.0$ | $+67.30$ | $-53.35$
41Sc | MK | $+1.116$ | $+3.591$ | $7.667$ | $-0.848$ | $+217.5$ | $-109.3$ | $+253.7$ | $+77.29$
43Ti | KB3 | $-0.989$ | $-3.095$ | $6.523$ | $0.437$ | $-105.5$ | $-71.64$ | $+141.1$ | $+113.2$
45V | KB3 | $+0.619$ | $+2.218$ | $8.120$ | $-0.689$ | $+159.6$ | $-149.2$ | $+308.7$ | $+392.5$

###  III.3 Isospin $T=$ 1, 3/2, and 2 multiplet decays

We have also investigated weak magnetism for $\beta$ decays from states in
isospin $T=1,T=3/2$, and $T=2$ isobar multiplets for which experimental input
data required to calculate the $b$ and $c$ form factors are available. Weak
magnetism studies for $\beta$ decays of isobaric $T=1$ triplet states have
been performed before Calaprice and Holstein (1976); Huber (2011). Here we
update and significantly extend these works, yielding results for as many as
68 (previously only 14 Calaprice and Holstein (1976); Huber (2011)) $\beta$
transitions from isospin multiplet states and the corresponding analog
$\gamma$ transitions.

Figure 11 displays the generic decay scheme for the $\beta$ and $\gamma$
decays of isobaric analog states in a $T=1$ triplet to a common $T^{\prime}=0$
final state. For the $\Delta T=1,\Delta T=\pm 1$ transitions considered here
the weak magnetism form factor for the $\beta$ transitions is now given by Eq.
(30). The decay width, $\Gamma$, of the $\gamma$-decaying analog state is
related to its half-life $t_{1/2}$ (in seconds) by

$\Gamma=\frac{4.56238\times 10^{-16}}{t_{1/2}}\leavevmode\nobreak\
\rm{eV}\leavevmode\nobreak\ .$ (49)

When several $\gamma$ rays are deexciting the analog state the partial decay
width for each transition is obtained by multiplying $\Gamma$ with the
transition’s fractional intensity. In tables, sometimes the $B(M1)$ transition
strength for the specific $\gamma$ ray is listed. The $\Gamma_{M1}$ width is
then obtained as

$\Gamma_{M1}=20.723\times 10^{-3}\leavevmode\nobreak\
E_{\gamma}^{3}\leavevmode\nobreak\ B(M1)\leavevmode\nobreak\
\rm{eV}\leavevmode\nobreak\ ,$ (50)

with $E_{\gamma}$ in MeV and $B(M1)$ in Weiskopff units.

Figure 11: Generic scheme showing the $\beta$ and $\gamma$ decays from the
isobaric analog states in a $T=1$ triplet to a common $T^{\prime}=0$ daughter
state. The weak magnetism form factor of the $\beta$ transitions can be
obtained from the energy, $E_{\gamma}$, and the M1-decay width, $\Gamma_{M1}$,
of the corresponding isovector transition from the isobaric $T=1$ analog
state.

Using Eqs. (II.1) and (II.1), the partial half-life, $t$, for a general mixed
Fermi/Gamow-Teller (F/GT) transition Eq. (II.1) can be rewritten as

$\displaystyle t=\frac{2\leavevmode\nobreak\ \mathcal{F}t^{0^{+}\rightarrow
0^{+}}(1+\Delta_{R}^{V})}{(1+\delta_{R}^{\prime})\leavevmode\nobreak\
\left[f_{V}\leavevmode\nobreak\ a^{2}+f_{A}\leavevmode\nobreak\
c^{2}\right]}.$ (51)

All $\beta$ transitions considered here are either $J^{\pi}\rightarrow
J^{\pi}$, $\Delta T=1$ transitions or $J^{\pi}\rightarrow J^{\pi}\pm 1$,
$\Delta T=1$ transitions. The latter are of pure Gamow-Teller type while in
the former isospin-symmetry breaking due to the electromagnetic interaction
may induce a small isospin-forbidden Fermi component into the otherwise pure
Gamow-Teller decay. In the latter case the Fermi matrix element in the form
factor $a=g_{V}M_{F}$ can be written as $M_{F}=\alpha\sqrt{2T}$ with $\alpha$
the amplitude of the admixture of the analog state to the final state of the
$\beta$ decay into the initial state Raman  _et al._ (1975). An extensive
survey of isospin-forbidden $J^{\pi}\rightarrow J^{\pi}$ Fermi transtions
Raman  _et al._ (1975) has shown $\alpha$ to be of the order of $10^{-3}$ and
lower (see also Schuurmans  _et al._ (2000); Severijns  _et al._ (2005); Soti
_et al._ (2014)). At the level of precision we are dealing with here (which is
determined by the uncertainty on the values of $\Gamma_{M1}$ or $B_{M1}$ used
to extract the weak magnetism form factor, $b$), the term $a^{2}$ in Eq. (51)
can be neglected with respect to $c^{2}$. One then has, for all transitions
considered here:

$\displaystyle f_{A}t=\frac{2\leavevmode\nobreak\
\mathcal{F}t^{0^{+}\rightarrow
0^{+}}(1+\Delta_{R}^{V})}{(1+\delta_{R}^{\prime})\leavevmode\nobreak\ c^{2}}.$
(52)

The inner radiative correction, $\Delta_{R}^{V}$, has recently been subject of
new, more complete calculations Czarnecki  _et al._ (2019); Seng  _et al._
(2020); Hayen (2021). We use here the value $\Delta_{R}^{V}=0.02473(27)$ of
Ref.Hayen (2021), which is in agreement with the value of Seng  _et al._
(2020) (both calculations being also slightly more encompassing than Ref.
Czarnecki  _et al._ (2019)) and used a slightly more conservative approach in
determining the uncertainty. The values for $\delta_{R^{\prime}}$ are listed
in the last column of Tables 20 and 21.

Note that Eq. (52) only holds for ’normal’ allowed $\beta$ transitions. For
strongly hindered transitions, with large log $ft$ values (i.e. typically
larger than about 6.7) the beta spectrum is unlikely to have an allowed shape,
and the connection between log$ft$ and $c$ is in principle lost because the
electromagnetic interaction is very much amplified in this case Calaprice and
Holstein (1976). For similar reasons Huber Huber (2011) recently suggested
that a detailed study of the breakdown of the impulse approximation in nuclei
with large log $ft$ transitions would be useful.

Combining the weak magnetism form factor, $b$, obtained from the energy and
decay width of the analog M1 $\gamma$ transition via Eq. (30), and the Gamow-
Teller form factor, $c$, obtained from the $\beta$ decay $ft$ value via Eq.
(52), the $b/Ac$ value for the $\beta$ transitions in $T\geq 1$ isospin
multiplets can be obtained. The new analysis presented here significantly
extends the available data set. Thereafter, experimental values for the
$M_{L}$ matrix elements will again be deduced from these experimental $b/Ac$
ratios and Eq. (33). Finally, the experimental results can again be compared
to those from shell model calculations that have been performed for several
$\beta$ transitions from the $T=1$ multiplets, as will be discussed below.

####  III.3.1 Experimental values for Gamow-Teller and weak magnetism form
factors

Searching the Brookhaven National Nuclear Data Base
(https://www.nndc.bnl.gov/ensdf/) files, input data were found for 52 pairs of
$\beta$ and $\gamma$ transitions from analog states in isospin $T=1$ isobaric
triplets with masses ranging from $A$ = 6 to 42, for 15 pairs of transitions
from analog states in $T=3/2$ quartets with $A$ = 9 to 23 and $A=$ 53, and a
single pair of transitions from analog states in the $T=2$ multiplet with $A$
= 32.

Detailed information and input data related to the $\beta$ and $\gamma$
transitions considered here, as well as the results obtained in the analysis,
are listed in the Tables 20 to 25 that are discussed below. To clearly
indicate which transitions are considered here, Figs. 12 to 16 show the
partial decay scheme for all pairs of analog $\beta$ and $\gamma$ transitions
from states in the $T=1$ triplets and the $T=2$ multiplet, indicating the
log$ft$ value for the $\beta$ transitions and the energy and the $\Gamma_{M1}$
or $B(M1)$ value (Eq. (50)) for the $\gamma$ transitions. Figs. 17 and 18
provide the same information for the pairs of analog $\beta$ and $\gamma$
transitions from states in the $T=3/2$ multiplets with mass $A$ = 9 to 23 and
$A=$ 53.

Figure 12: Partial decay schemes (not to scale) showing the $\beta$
transitions and their analog $\gamma$ transitions in $T=1$ multiplets with
$A=$ 6 to 14 for which the weak magnetism form factor is determined, with
their respective log$ft$ value ($\beta$ transitions), or the transition energy
and $\Gamma_{M1}$ or $B(M1)$ value ($\gamma$ transitions). For each ground
state and excited state the spin-parity, isospin, and energy (in keV) is
given. The excited state labels (e.g. LiIA or C1) are also used in the first
column of Tables 20 to 25 for easy reference. Figure 13: Partial decay schemes
(not to scale) showing the $\beta$ transitions and their analog $\gamma$
transitions in $T=1$ multiplets with $A=$ 16 to 22 for which the weak
magnetism form factor is determined, with their respective log$ft$ value
($\beta$ transitions), or the transition energy and $\Gamma_{M1}$ or $B(M1)$
value ($\gamma$ transitions). For each ground state and excited state the
spin-parity, isospin, and energy (in keV) is given. The excited state labels
(e.g. LiIA or C1) are also used in the first column of Tables 20 to 25 for
easy reference. Figure 14: Partial decay schemes (not to scale) showing the
$\beta$ transitions and their analog $\gamma$ transitions in $T=1$ multiplets
with $A=$ 24 to 28 for which the weak magnetism form factor is determined,
with their respective log$ft$ value ($\beta$ transitions), or the transition
energy and $\Gamma_{M1}$ or $B(M1)$ value ($\gamma$ transitions). For each
ground state and excited state the spin-parity, isospin, and energy (in keV)
is given. The excited state labels (e.g. LiIA or C1) are also used in the
first column of Tables 20 to 25 for easy reference. Figure 15: Partial decay
schemes (not to scale) showing the $\beta$ transitions and their analog
$\gamma$ transitions in $T=1$ multiplets with $A=$ 30 to 36 for which the weak
magnetism form factor is determined, with their respective log$ft$ value
($\beta$ transitions), or the transition energy and $\Gamma_{M1}$ or $B(M1)$
value ($\gamma$ transitions). For each ground state and excited state the
spin-parity, isospin, and energy (in keV) is given. The excited state labels
(e.g. LiIA or C1) are also used in the first column of Tables 20 to 25 for
easy reference. Figure 16: Partial decay schemes (not to scale) showing the
$\beta$ transitions and their analog $\gamma$ transitions in $T=1$ multiplets
with $A=$ 38 to 42, and a single $T=2$ multiplet with $A=$ 32 for which the
weak magnetism form factor is determined, with their respective log$ft$ value
($\beta$ transitions), or the transition energy and $\Gamma_{M1}$ or $B(M1)$
value ($\gamma$ transitions). For each ground state and excited state the
spin-parity, isospin, and energy (in keV) is given. The excited state labels
(e.g. LiIA or C1) are also used in the first column of Tables 20 to 25 for
easy reference. Figure 17: Partial decay schemes (not to scale) showing the
$\beta$ transitions and their analog $\gamma$ transitions in $T=3/2$
multiplets with $A=$ 9 to 13 for which the weak magnetism form factor is
determined, with their respective log$ft$ value ($\beta$ transitions), or the
transition energy and $\Gamma_{M1}$ or $B(M1)$ value ($\gamma$ transitions).
For each ground state and excited state the spin-parity, isospin, and energy
(in keV) is given. The excited state labels (e.g. LiIA or C1) are also used in
the first column of Tables 20 to 25 for easy reference. Figure 18: Partial
decay schemes (not to scale) showing the $\beta$ transitions and their analog
$\gamma$ transitions in $T=3/2$ multiplets with $A=$ 15 to 23, and 53 for
which the weak magnetism form factor is determined, with their respective
log$ft$ value ($\beta$ transitions), or the transition energy and
$\Gamma_{M1}$ or $B(M1)$ value ($\gamma$ transitions). For each ground state
and excited state the spin-parity, isospin, and energy (in keV) is given. The
excited state labels (e.g. LiIA or C1) are also used in the first column of
Tables 20 to 25 for easy reference.

Tables 20 and 21 provide, for each $\beta$ transition considered here, the
input data leading to the $ft$ value, as well as the transition-dependent
radiative correction, $\delta^{\prime}_{R}$, required to extract the Gamow-
Teller form factor, $c$, from this $ft$ value via Eq. (52). Columns 4 and 5
thus list, respectively, the energy of the final state of the $\beta$
transition, $E_{level}(J_{f})$, and the $\beta$ decay transition energy,
$Q_{\beta^{-}}$ or $Q_{EC}$ (obtained by combining $E_{level}(J_{f})$ with the
ground state-to-ground state $Q$ values listed in Wang  _et al._ (2021)). The
resulting $f_{A}$ value is listed in column 6 and was calculated from Eq. (4)
using the beta spectrum generator code described in Ref. Hayen and Severijns
(2019b). The latter is based on the recent high-precision analytical
description of the allowed $\beta$ spectrum shape Hayen  _et al._ (2018).

Columns 7 to 10 list the half-life, $t_{1/2}$, of the $\beta$-decaying states,
the branching ratio, $BR$ and the electron-capture fraction, $P_{EC}$, of the
respective $\beta$ transitions, and the partial half-life, $t$, resulting from
these input values (Eq. (5)). Most half-lives and all branching ratio values
were obtained from the Brookhaven National Nuclear Data Base
(https://www.nndc.bnl.gov/ensdf/). For isotopes with a superallowed pure Fermi
transition, i.e. 10C, 14O, 18Ne, 22Mg, 26Si, 30S, 34Ar, 38Ca, and 42Ti, the
half-lives were taken from the detailed analysis presented in Ref. Hardy and
Towner (2020). For 20F, with varying values being reported in the literature,
the weighted average of the values listed in Table V of Hughes  _et al._
(2018) was used with the error bar increased by a factor
$\sqrt{(}\chi^{2}/\nu)$. The electron-capture fractions were calculated using
the same procedure as described in Sec. II.2.2 for the mirror $\beta$
transitions.

For all 68 pairs of analog $\beta$ and $\gamma$ transitions considered, the
log$ft$ values are given in column 4 in Tables 22 to 25. These values result
from the values for $f_{A}$ and $t$ listed in Tables 20 and 21. The $\beta$
transitions are identified by the information in columns 1 to 3. Column 5 to 8
list for the analog $\gamma$ transitions the energies of the initial state,
the transition energies, the M1 decay widths and the resulting weak magnetism
values, $b_{\gamma}$, respectively. Combining the latter with the Gamow-Teller
form factors, $|c|^{exp}$, obtained from the log$ft$ values using Eq. (52) and
listed in column 9, the $|b/Ac|^{exp}$ values listed in column 10 are obtained
for the different $\beta$ transitions.

From Eqs. (30) and (52) it follows that the signs of $b$ and $c$ cannot be
determined experimentally. However, as the ratio $b/Ac$ was found to be
positive (in the convention of Holstein Holstein (1974a), where the sign of
$g_{A}$ is taken to be positive) for nearly all mirror $\beta$ transitions
(see Table 14 and Sec. III.2.4), it was chosen to be positive for the
transitions considered here as well. Shell model calculations for a subset of
the transitions considered here indeed showed the ratio $b/Ac$ to be positive
when using $g_{A}=+1$ (see Sec. III.3.2, and Sec. III.2.4). As it turns out,
only one value out of 68 is found to be close to zero (i.e. $b/Ac=0.189(42)$
for the transition 36K$\rightarrow^{36}$Ar1 (Table 24)) and could thus
possibly have a negative sign as was seen for the transitions of 33Cl, 35Ar,
and 71Kr (Table 14 and Sec. III.2.4)).

In the mass $A=14$ triplet the $\beta$ decays of the ground states of 14C and
14O to the ground state of 14N yield values for $b/Ac$ that are very large
(i.e. $|b/Ac|^{exp}=$ 324(7) and 35.6(8), respectively) compared to all other
$T$ = 1 decays listed in Tables 22 to 24. These are clear cases of strongly
hindered transitions (with log $ft$ values of 9.225(2) and 7.309(7)) for which
the beta spectrum is unlikely to have an allowed shape and the connection
between log $ft$ and $c$ (Eq. (52)) is lost (see also Calaprice and Holstein
(1976)). When nevertheless trying to use Eq. (52) for such cases very small
values for $c$ are obtained, often leading to large values for the ratio
$b/Ac$. However, other hindered transitions, i.e. 22Na (with
log$ft=7.4227(6)$; Table 23) and 32Si (log$ft=8.23(55)$; Table 25), do not
lead to similarly large $b/Ac$ values, but rather yield values that are in
line with the non-hindered transitions. The reason for this is not clear.

The $|b/Ac|$ values for the beta transitions from the $T=1$ and $T=3/2$
states, except for the transitions from the $A=14$ triplet, are shown
graphically in Fig. 20. As can be seen, for almost all $\beta$ transitions the
$|b/Ac|$ values range between 0 and 10, except for the transitions from the
$A=24$ triplet consisting of the isomeric states 24mNa and 24mAl and the
excited state at 9967 keV in 24Mg (Fig. 14). Even though the three $\beta$
transitions in this triplet have low log$ft$ values (i.e. between 5.79(13) and
5.936(88)), they show large $|b/Ac|$ values ranging from 15.8(21) to 20.8(30).
Other $\beta$ transitions with similar or even larger log$ft$ values, such as
e.g. the $\beta$ transitions from 9C ($T=3/2$) or from the ground states of
24Na and 24Al $(T=1)$ (with log$ft$ values from 5.876(45) to 6.585(40)), and
even the strongly hindered transitions from 22Na (log$ft=7.4227(6)$) and 32Si
(log$ft=8.23(55)$), all show values for $|b/Ac|$ ranging from 2.2(6) to
7.3(11). The large values of $b/Ac$ for the $A=24$ triplet of the isomeric
states 24mNa and 24mAl and the excited state at 9967 keV in 24Mg are, however,
not so much due to the small value for the Gamow-Teller form factor, $c$, but
mainly to the large values of the M1 $\gamma$-decay width, $\Gamma_{M1}$ of
the analog $\gamma$ transitions, i.e. $\Gamma_{M1}$ = 4.3(9) eV and 1.65(39)
eV (Table 23). The value of $|b/Ac|$ = 17.5(32) for the transition from 24mAl
to the ground state of 24Mg was also addressed in shell model calculations (to
be discussed in the next section), with the theoretical result being in
agreement with experiment within error bars.

Fig. 20 does not show evidence for the single particle-related structure that
was noticed for the $\beta$ transitions of the $T=1/2$ mirror nuclei (Fig. 6).
This is not surprising as for most of the $T=1$, 3/2 and 2 transitions the
final states are in fact excited states that usually have a more complicated
structure. Nevertheless, some lower values of $b/Ac$ are clearly present in
the $p_{1/2}$ (around $A=$ 13 to 15) and $d_{3/2}$ (region from $A=$ 33 to 40)
subshells with $j=l-1/2$ (see Sec. III.2.2).

The ratios $|b/Ac|^{exp}$ for the $\beta$ transitions from $T=1$ triplet
states listed in Tables 22 to 24, except for the strongly hindered transitions
from the ground states of 14O and 14C to the ground state of 14N, are plotted
in Fig. 20. The unweighted average (an unweighted average is used because
individual error bars differ a lot) is $|b/Ac|^{exp}=4.9\pm 3.5$, with the
error bar indicating one standard deviation. Similarly, for the transitions
from $T=3/2$ states listed in Table 25 and also plotted in Fig. 20, an
unweighted average value $|b/Ac|^{exp}=6.3\pm 2.4$ is found.

Table 20: Input data to the log$f_{A}t$ values and values for the radiative correction $\delta^{\prime}_{R}$ (last column) for the $\beta$ transitions from the $A=$ 6 to $A=$ 26, $T=1$ triplet states (see Fig. 12 to 14) for which $|b/Ac|^{exp}$ can be obtained from the $\Gamma_{M1}$ transition strength of the corresponding analog $\gamma$ transitions. $Q_{\beta^{-}/EC}$ values were obtained from the ground state-to-ground state $Q$ values from the 2016 Atomic Mass Evaluation Wang  _et al._ (2017) corrected for the energy of the final state, $E_{level}(J_{f})$, as listed in the Brookhaven National Nuclear Data Base ENSDF , from which also most values for the half-life, $t_{1/2}$ (see text), and all branching ratios, $BR$, were obtained. Values for the phase space factor $f_{A}$ were calculated with the beta spectrum generator code described in Ref. Hayen and Severijns (2019b) (see text for more details). Element labels $X$ without superscripts denote nuclear ground states. Superspcripts ”1” to ”4” indicate excited states (see Fig. 12 to 14). For the energies of the 24mAl and 24mNa isomers the values 472.2074(8) keV and 425.8(1) keV were used ENSDF . Transition | $A$ | $J_{i}\xrightarrow{\beta}J_{f}$ | $E_{level}(J_{f})$ | $Q_{\beta^{-}/EC}$ | $f_{A}$ | $t_{1/2}$ | $BR$ | $P_{EC}$ | $t$ | $\delta^{\prime}_{R}$
---|---|---|---|---|---|---|---|---|---|---
|  |  | (keV) | (keV) |  | (s) | (%) | (%) | (s) | (%)
He $\xrightarrow{\beta^{-}}$ Li | 6 |  0+ $\rightarrow$ 1+ | 0 | 3505.215(53) | 997.795(68) | 0.8067(1) | 100 | - | 0.8067(10) | 1.203
C $\xrightarrow{\beta^{+}}$ B1 | 10 |  0+ $\rightarrow$ 1+ | 718.380(11) | 2929.682(70) | 56.6306(90) | 19.3016(24) | 98.53(2) | 0.0285 | 19.583(13) | 1.454
B $\xrightarrow{\beta^{-}}$ C | 12 |  1+ $\rightarrow$ 0+ | 0 | 13369.4(13) | 557180(260) | 0.02020(2) | 98.216(28) | - | 0.020567(21) | 0.740
N $\xrightarrow{\beta^{+}}$ C |  |  1+ $\rightarrow$ 0+ | 0 | 17338.1(10) | 1128063(335) | 0.011000(16) | 96.17(5) | 0.0000817 | 0.011438(18) | 0.700
B $\xrightarrow{\beta{-}}$ C1 | 12 |  1+ $\rightarrow$ 2+ | 4439.82(21) | 8929.6(13) | 81196(57) | 0.02020(2) | 1.182(19) | - | 1.709(28) | 0.911
N $\xrightarrow{\beta^{+}}$ C1 |  |  1+ $\rightarrow$ 2+ | 4439.82(21) | 12898.3(60) | 243533(591) | 0.011000(16) | 1.898(32) | 0.000209 | 0.5796(98) | 0.814
B $\xrightarrow{\beta^{-}}$ C2 | 12 |  1+ $\rightarrow$ 0+ | 7654.07(19) | 5715.3(13) | 10016(11) | 0.02020(2) | 0.54(2) | - | 3.74(14) | 1.093
N $\xrightarrow{\beta^{+}}$ C2 |  |  1+ $\rightarrow$ 0+ | 7654.07(19) | 9684.0(60) | 53912(177) | 0.011000(16) | 1.41(3) | 0.000533 | 0.780(17) | 0.934
N $\xrightarrow{\beta^{+}}$ C3 | 12 |  1+ $\rightarrow$ 1+ | 12710(6) | 4628.1(61) | 925.5(99) | 0.011000(16) | 0.120(3) | 0.00708 | 9.17(23) | 1.265
O $\xrightarrow{\beta^{+}}$ N1 | 14 |  0+ $\rightarrow$ 1+ | 3948.10(20) | 1196.26(20) | 0.003800(24) | 70.619(11) | 0.054(2) | 176 |  360(13)$\times 10^{3}$ | 2.000
C $\xrightarrow{\beta^{-}}$ N | 14 |  0+ $\rightarrow$ 1+ | 0 | 156.476(4) | 0.00933138(82) | 5700(30) y | 100 | - |  179.88(95)$\times 10^{9}$ | 2.033
O $\xrightarrow{\beta^{+}}$ N |  |  0+ $\rightarrow$ 1+ | 0 | 5144.364(25) | 1760.522(49) | 70.619(11) | 0.61(1) | 0.00701 | 11576(190) | 1.268
N $\xrightarrow{\beta^{-}}$ O1 | 16 |  2- $\rightarrow$ 3- | 6129.89(4) | 4291.0(13) | 2847.3(39) | 7.13(2) | 66.2(6) | - | 10.77(10) | 1.267
N $\xrightarrow{\beta^{-}}$ O2 | 16 |  2- $\rightarrow$ 1- | 7116.85(14) | 3304.1(23) | 876.9(27) | 7.13(2) | 4.8(4) | - | 149(12) | 1.371
N $\xrightarrow{\beta^{-}}$ O3 | 16 |  2- $\rightarrow$ 2- | 8871.9(5) | 1549.0(24) | 33.94(22) | 7.13(2) | 1.06(7) | - | 673(44) | 1.643
Ne $\xrightarrow{\beta^{+}}$ F | 18 |  0+ $\rightarrow$ 1+ | 0 | 4444.50(59) | 689.42(54) | 1.66422(47) | 92.11(21) | 0.0270 | 1.8157(96) | 1.368
F $\xrightarrow{\beta^{+}}$ O |  |  1+ $\rightarrow$ 0+ | 0 | 1655.93(46) | 0.5564(16) | 6586.2(30) | 100 | 3.33 | 6805.6(31) | 1.851
Ne $\xrightarrow{\beta^{+}}$ F1 | 18 |  0+ $\rightarrow$ 1+ | 1700.81(18) | 2743.69(62) | 33.411(54) | 1.66422(47) | 0.188(6) | 0.212 | 891(29) | 1.611
F $\xrightarrow{\beta^{-}}$ Ne1 | 20 |  2+ $\rightarrow$ 2+ | 1633.674(15) | 5390.795(34) | 8626.32(27) |  11.096(36)292929Weighted average of the values listed in Table V of Hughes  _et al._ (2018). | 99.9913(8) | - | 11.097(36) | 1.213
Na $\xrightarrow{\beta^{+}}$ Ne1 |  |  2+ $\rightarrow$ 2+ | 1633.674(15) | 12258.7(11) | 175906(83) | 0.4479(23) | 79.3(11) | 0.00109 | 0.5648(84) | 0.929
Na $\xrightarrow{\beta^{+}}$ Ne2 | 20 |  2+ $\rightarrow$ 2+ | 7421.9(12) | 6470.5(16) | 5675.1(97) | 0.4479(23) | 16.4(13) | 0.00943 | 2.73(22) | 1.208
Na $\xrightarrow{\beta^{+}}$ Ne3 | 20 |  2+ $\rightarrow$ 2+ | 7833.4(15) | 6059.0(19) | 3933.2(87) | 0.4479(23) | 0.67(6) | 0.0119 | 66.9(60) | 1.238
Mg $\xrightarrow{\beta^{+}}$ Na1 | 22 |  0+ $\rightarrow$ 1+ | 583.05(10) | 4198.36(18) | 473.65(14) | 3.87445(69) | 41.33(20) | 0.0626 | 9.383(45) | 1.449
Na $\xrightarrow{\beta^{+}}$ Ne1 | 22 |  3+ $\rightarrow$ 2+ | 1274.537(7) | 1568.78(15) | 0.29144(32) |  82108(69)$\times 10^{3}$ | 99.944(14) | 10.8 |  91016(78)$\times 10^{3}$ | 1.950
Na $\xrightarrow{\beta^{-}}$ Mg2 | 24 |  4+ $\rightarrow$ 4+ | 4122.889(12) | 1392.79(24) | 24.726(17) | 53989(43) | 99.855(5) | - | 54067(43) | 1.815
Al $\xrightarrow{\beta^{+}}$ Mg2 |  |  4+ $\rightarrow$ 4+ | 4122.889(12) | 9761.88(23) | 50529.1(63) | 2.053(4) | 7.7(10) | 0.00413 | 26.7(35) | 1.054
Al $\xrightarrow{\beta^{+}}$ Mg3 | 24 |  4+ $\rightarrow$ 3+ | 5235.12(4) | 8649.65(23) | 26246.4(38) | 2.053(4) | 1.40(13) | 0.00624 | 147(14) | 1.105
Al $\xrightarrow{\beta^{+}}$ Mg4 | 24 |  4+ $\rightarrow$ 4+ | 6010.84(4) | 7873.93(23) | 15951.0(26) | 2.053(4) | 1.2(1) | 0.00851 | 171(14) | 1.157
Al $\xrightarrow{\beta^{+}}$ Mg5 | 24 |  4+ $\rightarrow$ 4+ | 8439.36(4) | 5445.41(23) | 2081.61(50) | 2.053(4) | 50.0(20) | 0.0312 | 4.11(16) | 1.340
mNa $\xrightarrow{\beta^{-}}$ Mg | 24 |  1+ $\rightarrow$ 0+ | 0 | 5987.884(21) | 21350.49(33) | 0.02020(7) | 0.05(1) | - | 40.4(81) | 1.177
mAl $\xrightarrow{\beta^{+}}$ Mg |  |  1+ $\rightarrow$ 0+ | 0 | 14310.6(25) | 466680(410) | 0.1313(25) | 10(3) | 0.000961 | 1.31(39) | 0.941
mAl$\xrightarrow{\beta^{+}}$Mg1 | 24 |  1+ $\rightarrow$ 2+ | 1368.672(5) | 12941.83(23) | 279859(25) | 0.1313(25) | 4.4(5) | 0.00131 | 2.98(34) | 0.968
Si $\xrightarrow{\beta^{+}}$ Al1 | 26 |  0+ $\rightarrow$ 1+ | 1057.739(12) | 4011.397(86) | 348.764(45) | 2.2453(7) | 21.9(4) | 0.128 | 10.27(19) | 1.529
Si $\xrightarrow{\beta^{+}}$ Al2 | 26 |  0+ $\rightarrow$ 1+ | 1850.62(3) | 3218.52(9) | 87.844(17) | 2.2453(7) | 2.73(7) | 0.328 | 82.5(21) | 1.646
Si $\xrightarrow{\beta^{+}}$ Al3 | 26 |  0+ $\rightarrow$ 1+ | 2071.64(4) | 2997.50(9) | 55.087(12) | 2.2453(7) | 0.290(11) | 0.454 | 778(30) | 1.685
Si $\xrightarrow{\beta^{+}}$ Al4 | 26 |  0+ $\rightarrow$ 1+ | 2740.03(3) | 2329.11(9) | 9.4555(29) | 2.2453(7) | 0.0618(25) | 1.60 | 3691(149) | 1.826
Table 21: Input data to the log$f_{A}t$ values and values for the radiative correction $\delta^{\prime}_{R}$ (last column) for the $\beta$ transitions from the $A=$ 28 to $A=$ 53, $T=$ 1, 3/2 and 2 multiplet states (see Fig. 14 to 18) for which $|b/Ac|^{exp}$ can be obtained from the $\Gamma_{M1}$ transition strength of the corresponding analog $\gamma$ transitions. $Q_{\beta^{-}/EC}$ values were obtained from the ground state-to-ground state $Q$ values from the 2020 Atomic Mass Evaluation Wang  _et al._ (2021) corrected for the energy of the final state, $E_{level}(J_{f})$, as listed in the Brookhaven National Nuclear Data Base ENSDF , from which most values for the half-life, $t_{1/2}$ (see text), and all branching ratios, $BR$, were obtained. Values for the phase space factor $f_{A}$ were calculated with the beta spectrum generator code described in Ref. Hayen and Severijns (2019b). See text for more details. Element labels $X$ without superscripts denote nuclear ground states. Superscripts ”1” to ”4” indicate excited states. For the 53Ni$\rightarrow^{53}$Co transition the half-life is from Dossat  _et al._ (2007) and the branching ratio from Su  _et al._ (2016). Transition | $A$ | $J_{i}\xrightarrow{\beta}J_{f}$ | $E_{level}(J_{f})$ | $Q_{\beta^{-}/EC}$ | $f_{A}$ | $t_{1/2}$ | $BR$ | $P_{EC}$ | $t$ | $\delta^{\prime}_{R}$
---|---|---|---|---|---|---|---|---|---|---
|  |  | (keV) | (keV) |  | (s) | (%) | (%) | (s) | (%)
Al $\xrightarrow{\beta^{-}}$ Si1 | 28 |  3+ $\rightarrow$ 2+ | 1779.030(11) | 2663.048(48) | 396.2(35) | 134.70(12) | 99.99(1) | - | 134.71(12) | 1.640
P $\xrightarrow{\beta^{+}}$ Si1 |  |  3+ $\rightarrow$ 2+ | 1779.030(11) | 12566.1(11) | 180803(1554) | 0.2703(5) | 69.1(7) | 0.00302 | 0.3912(40) | 0.944
P $\xrightarrow{\beta^{+}}$ Si2 | 28 |  3+ $\rightarrow$ 3+ | 6276.20(7) | 8065.9(11) | 17215(331) | 0.2730(5) | 7.6(4) | 0.0129 | 3.56(19) | 1.177
S $\xrightarrow{\beta^{+}}$ P | 30 |  0+ $\rightarrow$ 1+ | 0 | 6141.60(20) | 3804(104) | 1.17977(77) | 21.3(5) | 0.0415 | 5.53(13) | 1.344
P$\xrightarrow{\beta^{+}}$ Si |  |  1+ $\rightarrow$ 0+ | 0 | 4232.106(61) | 459(12) | 149.88(24) | 99.939(3) | 0.132 | 150.17(24) | 1.522
Cl $\xrightarrow{\beta^{+}}$ S1 | 32 |  1+ $\rightarrow$ 2+ | 2230.57(15) | 10450.26(58) | 67076(20) | 0.298(1) | 60(4) | 0.00855 | 0.497(33) | 1.086
Cl $\xrightarrow{\beta^{+}}$ S2 | 32 |  1+ $\rightarrow$ 0+ | 3778.4(10) | 8902.4(11) | 28917(26) | 0.298(1) | 2.6(8) | 0.0144 | 11.5(35) | 1.178
Ar $\xrightarrow{\beta^{+}}$ Cl1 | 34 |  0+ $\rightarrow$ 1+ | 461.00(4) | 5600.793(75) | 2189(84) | 0.84644(35) | 0.91(10) | 0.0874 | 93(10) | 1.439
Ar $\xrightarrow{\beta^{+}}$ Cl2 | 34 |  0+ $\rightarrow$ 1+ | 665.56(4) | 5396.233(75) | 1774(15) | 0.84644(35) | 2.49(11) | 0.103 | 33.9(15) | 1.460
Ar $\xrightarrow{\beta^{+}}$ Cl3 | 34 |  0+ $\rightarrow$ 1+ | 2589.5(14) | 3482.3(14) | 132.3(13) | 0.84644(35) | 0.86(5) | 0.585 | 98.7(57) | 1.711
Ar $\xrightarrow{\beta^{+}}$ Cl4 | 34 |  0+ $\rightarrow$ 1+ | 3129.2(10) | 2932.5(10) | 43.59(60) | 0.84644(35) | 1.30(7) | 1.23 | 65.7(35) | 1.812
K $\xrightarrow{\beta^{+}}$ Ar1 | 36 |  1+ $\rightarrow$ 2+ | 1970.38(5) | 10883.98(33) | 79044(77) | 0.342(2) | 44(4) | 0.0114 | 0.777(71) | 1.084
K $\xrightarrow{\beta^{+}}$ Ar2 | 36 |  1+ $\rightarrow$ 0+ | 4440.11(19) | 8374.25(38) | 19589(137) | 0.342(2) | 8.4(10) | 0.0263 | 4.07(49) | 1.234
Ca $\xrightarrow{\beta^{+}}$ K1 | 38 |  0+ $\rightarrow$ 1+ | 458.53(16) | 6283.73(17) | 4019(134) | 0.44370(25) | 2.84(6) | 0.0856 | 15.64(33) | 1.421
Ca $\xrightarrow{\beta^{+}}$ K2 | 38 |  0+ $\rightarrow$ 1+ | 1697.65(25) | 5044.61(26) | 1166.0(92) | 0.44370(25) | 19.48(13) | 0.195 | 2.282(15) | 1.549
Ti $\xrightarrow{\beta^{+}}$ Sc1 | 42 |  0+ $\rightarrow$ 1+ | 611.051(6) | 6405.60(22) | 4318(152) | 0.20833(80) | 55.9(36) | 0.114 | 0.374(24) | 1.452
Li $\xrightarrow{\beta^{-}}$ Be | 9 |  3/2- $\rightarrow$ 3/2- | 0 | 13606.45(20) | 582680(41) | 0.1783(4) | 49.2(9) | - | 0.3624(67) | 0.676
Li $\xrightarrow{\beta^{-}}$ Be1 | 9 |  3/2- $\rightarrow$ 5/2- | 2429.4(13) | 11177.1(13) | 227139(179) | 0.1783(4) | 29.7(30) | - | 0.600(61) | 0.759
C $\xrightarrow{\beta^{+}}$ B | 9 |  3/2- $\rightarrow$ 3/2- | 0 | 16494.5(23) | 895467(644) | 0.1265(9) | 54.1(15) | 0.0000571 | 0.2338(67) | 0.735
C $\xrightarrow{\beta^{+}}$ B1 | 9 |  3/2- $\rightarrow$ 5/2- | 2345(11) | 14150(11) | 405910(2317) | 0.1265(9) | 30.4(58) | 0.0000927 | 0.416(79) | 0.796
C $\xrightarrow{\beta^{+}}$ B2 | 9 |  3/2- $\rightarrow$ 1/2- | 2780(16) | 13715(16) | 344584(2957) | 0.1265(9) | 5.8(6) | 0.000103 | 2.18(23) | 0.802
B $\xrightarrow{\beta^{-}}$ C | 13 |  3/2- $\rightarrow$ 1/2- | 0 | 13436.9(10) | 569259(203) | 0.01736(16) | 92.1(18) | - | 0.01885(41) | 0.744
B $\xrightarrow{\beta^{-}}$ C1 | 13 |  3/2- $\rightarrow$ 3/2- | 3684.507(19) | 9752.4(10) | 123890(60) | 0.01736(16) | 7.6(8) | - | 0.228(24) | 0.863
O $\xrightarrow{\beta^{+}}$ N | 13 |  3/2- $\rightarrow$ 1/2- | 0 | 17770.0(95) | 1253437(3444) | 0.00858(5) | 89.2(22) | 0.000118 | 0.00962(24) | 0.693
O $\xrightarrow{\beta^{+}}$ N1 | 13 |  3/2- $\rightarrow$ 3/2- | 3502(2) | 14268(10) | 406858(1509) | 0.00858(5) | 9.8(20) | 0.000234 | 0.088(18) | 0.809
C $\xrightarrow{\beta^{-}}$ N1 | 15 |  1/2+ $\rightarrow$ 1/2+ | 5298.822(14) | 4472.89(80) | 3325.0(27) | 2.449(5) | 63.2(8) | - | 3.875(50) | 1.213
O $\xrightarrow{\beta^{-}}$ F1 | 19 |  5/2+ $\rightarrow$ 3/2+ | 1554.038(9) | 3266.3(26) | 863.9(31) | 26.88(5) | 54.4(12) | - | 49.4(11) | 1.399
O $\xrightarrow{\beta^{-}}$ F2 | 19 |  5/2+ $\rightarrow$ 7/2+ | 4377.700(42) | 442.6(26) | 0.2684(58) | 26.88(5) | 0.0984(30) | - | 27317(834) | 1.991
Ne $\xrightarrow{\beta^{-}}$ Na | 23 |  5/2+ $\rightarrow$ 3/2+ | 0 | 4375.80(10) | 3431.23(36) | 37.24(12) | 66.9(13) | - | 55.7(11) | 1.331
Ne $\xrightarrow{\beta^{-}}$ Na1 | 23 |  5/2+ $\rightarrow$ 5/2+ | 439.990(9) | 3935.82(10) | 2115.26(24) | 37.24(12) | 32.0(13) | - | 116.4(47) | 1.379
Ni $\xrightarrow{\beta^{+}}$ Co1 | 53 |  7/2- $\rightarrow$ 9/2- | 1327.0(9) | 11702(25) | 101177(1135) | 0.0552(7) | 17(8) | 0.0389 | 0.32(15) | 1.201
Si $\xrightarrow{\beta^{-}}$ P | 32 |  0+ $\rightarrow$ 1+ | 0 | 227.19(30) | 0.03376(86) |  4828(600)$\times 10^{6}$ | 100 | 4.09 |  5.03(62)$\times 10^{3}$ | 2.329
Table 22: Data for the $\beta$ transitions between $A=$ 6 to 18, $T=1$ triplet states and their corresponding analog $\gamma$ transitions (see Fig. 12 and 13), leading to the form factor ratio $|b/Ac|^{exp}$. Log$(f_{A}t)$ values are from Table 20. Level energies and $E_{\gamma}$ and $\Gamma_{M1}$ values are from Ref. ENSDF . Element labels $X$ in column 1 without superscripts denote nuclear ground states, while superscripts ”1” to ”3” indicate excited states, and ”IA” indicates the analogue state to the $\beta$ decaying state(s) (see Fig. 12 and 13). For the $|b/Ac|^{exp}$ ratios flagged with a dagger (†) the value of $|b|^{exp}$ is divided (multiplied) by $\sqrt{3}$ because the gamma transition that yields $b$ is from a $T=1$ ($T=0$) to a $T=0$ ($T=1$) state while the $\beta$ transition that yields $|c|^{exp}$ is in the reverse direction (see Eq. (30)). Note that the sign of $M_{L}$ is relative to the sign of $M_{GT}$ since $M_{L}$ is extracted using the absolute value of the $c$ form factor. $T=1$ |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
$\xrightarrow{\beta,\gamma}$ |  |  |  |  level of $\gamma$ |  Eγ | $\Gamma_{M1}$ |  |  |  |
$T=0$ | $A$ | $J_{i}(T_{i})\rightarrow J_{f}(T_{f})$ | $\log ft$ | (keV) | (keV) | (eV) | $|b|_{\gamma}^{exp}$ | $|c|^{exp}$ | $|b/Ac|^{exp}$ | $M_{L}^{exp}$
He $\xrightarrow{\beta^{-}}$ Li | 6 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 2.905753(61) |  |  |  |  | 2.7802(19) | 4.088(42) | $-1.72(12)$
LiIA $\xrightarrow{\gamma}$ Li |  |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ |  | 3562.88(10) | 3561.75(10) | 8.19(17) | 68.2(7) |  |  |
C $\xrightarrow{\beta^{+}}$ B1 | 10 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.04520(13) |  |  |  |  | 2.3649(19) | 3.34(72) | $-3.2(17)$
BIA $\xrightarrow{\gamma}$ B1 |  |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ |  | 1740.05(4) | 1021.646(14) | 0.094(40) | 79(17) |  |  |
B $\xrightarrow{\beta^{-}}$ C | 12 |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ | 4.05917(49) |  |  |  |  | 0.73852(55) | 3.825(45) | $-0.650(33)$
N $\xrightarrow{\beta^{+}}$ C |  |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ | 4.11069(68) |  |  |  |  | 0.69613(64) | 4.058(48) | $-0.451(33)$
CIA $\xrightarrow{\gamma}$ C |  |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ |  | 15110(3) | 15100(3) | 38.5(8) | 33.9(4) |  |  |
B $\xrightarrow{\beta^{-}}$ C1 | 12 |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 5.1423(70) |  |  |  |  | 0.2121(17) | 3.54(24) | $-0.248(50)$
N $\xrightarrow{\beta^{+}}$ C1 |  |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 5.1497(74) |  |  |  |  | 0.2104(18) | 3.57(24) | $-0.240(50)$
CIA $\xrightarrow{\gamma}$ C1 |  |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ |  | 15110(3) | 10666(3) | 0.96(13) | 9.0(6) |  |  |
B $\xrightarrow{\beta^{-}}$ C2 | 12 |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ | 4.574(16) |  |  |  |  | 0.4077(76) | 3.35(23) | $-0.552(96)$
N $\xrightarrow{\beta^{+}}$ C2 |  |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ | 4.6239(94) |  |  |  |  | 0.3851(42) | 3.55(24) | $-0.446(93)$
CIA $\xrightarrow{\gamma}$ C2 |  |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ |  | 15110(3) | 7453(3) | 1.09(14) | 16.4(11) |  |  |
N $\xrightarrow{\beta^{+}}$ C3 | 12 |  1${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.929(12) |  |  |  |  | 0.856(12) | 6.44(93) | 1.49(80)
CIA $\xrightarrow{\gamma}$ C3 |  |  1${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ |  | 15110(3) | 2400(7) | 0.59(17) | 66.2(95) |  |  |
O $\xrightarrow{\beta^{+}}$ N1 | 14 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.137(16) |  |  |  |  | 2.123(40) |  3.15(70)† | $-3.3(15)$
N1 $\xrightarrow{\gamma}$ NIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 3948.10(20) | 1635.2(2) | 0.091(40) | 54(12) |  |  |
C $\xrightarrow{\beta^{-}}$ N | 14 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 9.2249(23) |  |  |  |  | 0.0019174(54) | 324.1(71) | 0.612(14)
O $\xrightarrow{\beta^{+}}$ N |  |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 7.3093(71) |  |  |  |  | 0.01747(14) | 35.58(83) | 0.539(15)
NIA $\xrightarrow{\gamma}$ N |  |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ |  | 2312.798(11) | 2312.593(11) | 0.0067(3) | 8.70(19) |  |  |
N $\xrightarrow{\beta^{-}}$ O1 | 16 |  2${}^{-}(1)$ $\rightarrow$ 3${}^{-}(0)$ | 4.4867(42) |  |  |  |  | 0.4503(22) | 4.46(25) | $-0.11(11)$
OIA $\xrightarrow{\gamma}$ O1 |  |  2${}^{-}(1)$ $\rightarrow$ 3${}^{-}(0)$ |  | 12968.6(4) | 6837.1(4) | 1.8(2) | 32.1(18) |  |  |
N $\xrightarrow{\beta^{-}}$ O2 | 16 |  2${}^{-}(1)$ $\rightarrow$ 1${}^{-}(0)$ | 5.115(36) |  |  |  |  | 0.2184(91) | 3.95(41) | $-0.165(89)$
OIA $\xrightarrow{\gamma}$ O2 |  |  2${}^{-}(1)$ $\rightarrow$ 1${}^{-}(0)$ |  | 12968.6(4) | 5850.7(5) | 0.21(4) | 13.8(13) |  |  |
N $\xrightarrow{\beta^{-}}$ O3 | 16 |  2${}^{-}(1)$ $\rightarrow$ 2${}^{-}(0)$ | 4.359(29) |  |  |  |  | 0.521(17) | 7.57(56) | 1.49(30)
OIA $\xrightarrow{\gamma}$ O3 |  |  2${}^{-}(1)$ $\rightarrow$ 2${}^{-}(0)$ |  | 12968.6(4) | 4096.1(7) | 1.5(2) | 63.1(42) |  |  |
Ne $\xrightarrow{\beta^{+}}$ F | 18 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.0955(11) |  |  |  |  | 2.2328(32) | 5.72(55) | 2.3(12)
F $\xrightarrow{\beta^{+}}$ O |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ | 3.5783(13) |  |  |  |  | 1.2778(22) |  5.77(55)† | 1.36(71)
FIA $\xrightarrow{\gamma}$ F |  |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ |  | 1041.55(8) | 1041.55(8) | 0.26(5) | 230(22) |  |  |
Ne $\xrightarrow{\beta^{+}}$ F1 | 18 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 4.472(14) |  |  |  |  | 0.4572(73) |  4.13(10)† | $-0.262(44)$
F1 $\xrightarrow{\gamma}$ FIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 1700.81(18) | 659.25(20) | 0.000478(16) | 19.64(33) |  |  |
Table 23: Data for the $\beta$ transitions between $A=$ 20 to 26, $T=1$ triplet states and their corresponding analog $\gamma$ transitions (see Fig. 13 and 14), leading to the form factor ratio $|b/Ac|^{exp}$. Log$(f_{A}t)$ values are from Table 20. Level energies and $E_{\gamma}$ and $\Gamma_{M1}$ values are from Ref. ENSDF . The $\Gamma_{M1}$ value for the transition from 22Na $\rightarrow^{22}$Ne1 is from Triambak  _et al._ (2017a, b). Element labels $X$ in column 1 without superscripts denote nuclear ground states, while superscripts ”1” to ”4” indicate excited states, and ”IA” indicates the analogue state to the $\beta$ decaying state(s) (see Fig. 13 and 14). For the $|b/Ac|^{exp}$ ratios flagged with a dagger (†) the value of $|b|^{exp}$ is divided (multiplied) by $\sqrt{3}$ because the gamma transition that yields $b$ is from a $T=1$ ($T=0$) to a $T=0$ ($T=1$) state while the $\beta$ transition that yields $|c|^{exp}$ is in the reverse direction (see Eq. (30)). Note that the sign of $M_{L}$ is relative to the sign of $M_{GT}$ since $M_{L}$ is extracted using the absolute value of the $c$ form factor. $T=1$ |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
$\xrightarrow{\beta,\gamma}$ |  |  |  |  level of $\gamma$ |  Eγ | $\Gamma_{M1}$ |  |  |  |
$T=0$ | $A$ | $J_{i}(T_{i})\rightarrow J_{f}(T_{f})$ | $\log ft$ | (keV) | (keV) | (eV) | $|b|_{\gamma}^{exp}$ | $|c|^{exp}$ | $|b/Ac|^{exp}$ | $M_{L}^{exp}$
F $\xrightarrow{\beta^{-}}$ Ne1 | 20 |  2${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 4.9810(14) |  |  |  |  | 0.25493(45) | 8.26(41) | 0.91(11)
Na $\xrightarrow{\beta^{+}}$ Ne1 |  |  2${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 4.9972(64) |  |  |  |  | 0.2506(19) | 8.40(42) | 0.93(11)
Ne${}^{IA}\xrightarrow{\gamma}$Ne1 |  |  2${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ |  | 10273.2(19) | 8638(3) | 4.0(4) | 42.1(21) |  |  |
Na $\xrightarrow{\beta^{+}}$ Ne2 | 20 |  2${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 4.190(35) |  |  |  |  | 0.634(25) | 4.88(37) | 0.11(23)
Ne${}^{IA}\xrightarrow{\gamma}$Ne2 |  |  2${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ |  | 10273.2(19) | 2852(4) | 0.31(4) | 61.8(40) |  |  |
Na $\xrightarrow{\beta^{+}}$ Ne3 | 20 |  2${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 5.420(39) |  |  |  |  | 0.1538(69) | 4.03(58) | $-0.104(90)$
Ne${}^{IA}\xrightarrow{\gamma}$Ne3 |  |  2${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ |  | 10273.2(19) | 2440.4(33) | 0.0078(21) | 12.4(17) |  |  |
Mg $\xrightarrow{\beta^{+}}$ Na1 | 22 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.6477(21) |  |  |  |  | 1.1819(30) | 5.419(93) | 0.84(11)
Na${}^{IA(Mg)}\xrightarrow{\gamma}$Na1 |  |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ |  | 657.00(14) | 73.9(1) | 0.0000232(8) | 140.9(24) |  |  |
Na $\xrightarrow{\beta^{+}}$ Ne1 | 22 |  3${}^{+}(0)$ $\rightarrow$ 2${}^{+}(1)$ | 7.42366(60) |  |  |  |  | 0.015259(19) |  7.5(12)† | 0.043(19)
Na${}^{IA(Ne)}\xrightarrow{\gamma}$Na |  |  2${}^{+}(1)$ $\rightarrow$ 3${}^{+}(0)$ |  | 1951.8(3) | 1951.8(3) | 0.000269(88) | 2.99(49) |  |  |
Na $\xrightarrow{\beta^{-}}$ Mg2 | 24 |  4${}^{+}(1)$ $\rightarrow$ 4${}^{+}(0)$ | 6.12609(46) |  |  |  |  | 0.068014(74) | 4.8(12) | 0.005(83)
Al $\xrightarrow{\beta^{+}}$ Mg2 |  |  4${}^{+}(1)$ $\rightarrow$ 4${}^{+}(0)$ | 6.129(57) |  |  |  |  | 0.0680(44) | 4.8(13) | 0.005(86)
Mg${}^{IA1}\xrightarrow{\gamma}$Mg2 |  |  4${}^{+}(1)$ $\rightarrow$ 4${}^{+}(0)$ |  | 9516.28(4) | 5392.68(9) | 0.023(12) | 7.8(20) |  |  |
Al $\xrightarrow{\beta^{+}}$ Mg3 | 24 |  4${}^{+}(1)$ $\rightarrow$ 3${}^{+}(0)$ | 6.585(40) |  |  |  |  | 0.0402(19) | 2.16(56) | $-0.102(23)$
Mg${}^{IA1}\xrightarrow{\gamma}$Mg3 |  |  4${}^{+}(1)$ $\rightarrow$ 3${}^{+}(0)$ |  | 9516.28(4) | 4280.62(13) | 0.00083(42) | 2.09(53) |  |  |
Al $\xrightarrow{\beta^{+}}$ Mg4 | 24 |  4${}^{+}(1)$ $\rightarrow$ 4${}^{+}(0)$ | 6.436(36) |  |  |  |  | 0.0478(20) | 4.28(11) | $-0.021(51)$
Mg${}^{IA1}\xrightarrow{\gamma}$Mg4 |  |  4${}^{+}(1)$ $\rightarrow$ 4${}^{+}(0)$ |  | 9516.28(4) | 3505.61(9) | 0.0025(12) | 4.9(12) |  |  |
Al $\xrightarrow{\beta^{+}}$ Mg5 | 24 |  4${}^{+}(1)$ $\rightarrow$ 4${}^{+}(0)$ | 3.932(17) |  |  |  |  | 0.852(17) | 3.86(93) | $-0.72(79)$
Mg${}^{IA1}\xrightarrow{\gamma}$Mg5 |  |  4${}^{+}(1)$ $\rightarrow$ 4${}^{+}(0)$ |  | 9516.28(4) | 1076.86(4) | 0.019(9) | 79(19) |  |  |
Na(m) $\xrightarrow{\beta^{-}}$ Mg | 24 |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ | 5.936(88) |  |  |  |  | 0.0849(85) | 20.7(30) | 1.36(29)
Al(m) $\xrightarrow{\beta^{+}}$ Mg |  |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ | 5.79(13) |  |  |  |  | 0.101(15) | 17.5(32) | 1.29(38)
MgIA2 $\xrightarrow{\gamma}$ Mg |  |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ |  | 9967.19(22) | 9963.0(15) | 4.3(9) | 42.3(44) |  |  |
Al(m)$\xrightarrow{\beta^{+}}$Mg1 | 24 |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 5.921(50) |  |  |  |  | 0.0860(50) | 15.8(21) | 0.96(19)
Mg${}^{IA2}\xrightarrow{\gamma}$Mg1 |  |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ |  | 9967.19(22) | 8595.1(15) | 1.65(39) | 32.7(39) |  |  |
Si $\xrightarrow{\beta^{+}}$ Al1 | 26 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.5539(79) |  |  |  |  | 1.316(12) |  6.18(61)† | 1.93(80)
Al${}^{1}\xrightarrow{\gamma}$AlIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 1057.739(12) | 829.3(4) | 0.0177(35) | 122(12) |  |  |
Si $\xrightarrow{\beta^{+}}$ Al2 | 26 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.861(11) |  |  |  |  | 0.924(12) |  2.88(13)† | $-1.69(13)$
Al${}^{2}\xrightarrow{\gamma}$AlIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 1850.62(3) | 1622(7) | 0.0141(13) | 39.9(18) |  |  |
Si $\xrightarrow{\beta^{+}}$ Al3 | 26 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 4.632(16) |  |  |  |  | 0.3802(72) |  1.63(16)† | $-1.170(66)$
Al${}^{3}\xrightarrow{\gamma}$AlIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 2071.64(4) | 1842.8(7) | 0.00112(22) | 9.29(91) |  |  |
Si $\xrightarrow{\beta^{+}}$ Al4 | 26 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 4.543(18) |  |  |  |  | 0.4209(85) |  3.31(20)† | $-0.588(86)$
Al${}^{4}\xrightarrow{\gamma}$AlIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 2740.03(3) | 2511.59(10) | 0.0144(16) | 20.9(12) |  |  |
Table 24: Data for the $\beta$ transitions between $A=$ 28 to 42, $T=1$ triplet states and their corresponding analog $\gamma$ transitions (see figs. 14 to 16), leading to the form factor ratio $|b/Ac|^{exp}$. Log$(f_{A}t)$ values are from Table 21. Level energies and $E_{\gamma}$ and $\Gamma_{M1}$ values are from Ref. ENSDF . Element labels $X$ in column 1 without superscripts denote nuclear ground states, while superscripts ”1” to ”4” indicate excited states, and ”IA” indicates the analogue state to the $\beta$ decaying state(s) (see figs. 14 to 16). For the $|b/Ac|^{exp}$ ratios flagged with a dagger (†) the value of $|b|^{exp}$ is divided (multiplied) by $\sqrt{3}$ because the gamma transition that yields $b$ is from a $T=1$ ($T=0$) to a $T=0$ ($T=1$) state while the $\beta$ transition that yields $|c|^{exp}$ is in the reverse direction (see Eq. (30)). Note that the sign of $M_{L}$ is relative to the sign of $M_{GT}$ since $M_{L}$ is extracted using the absolute value of the $c$ form factor. $T=1$ |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
$\xrightarrow{\beta,\gamma}$ |  |  |  |  level of $\gamma$ |  Eγ | $\Gamma_{M1}$ |  |  |  |
$T=0$ | $A$ | $J_{i}(T_{i})\rightarrow J_{f}(T_{f})$ | $\log ft$ | (keV) | (keV) | (eV) | $|b|_{\gamma}^{exp}$ | $|c|^{exp}$ | $|b/Ac|^{exp}$ | $M_{L}^{exp}$
Al $\xrightarrow{\beta^{-}}$ Si1 | 28 |  3${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 4.7273(38) |  |  |  |  | 0.3407(15) | 1.74(38) | $-1.01(13)$
P $\xrightarrow{\beta^{+}}$ Si1 |  |  3${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 4.8496(58) |  |  |  |  | 0.2970(20) | 2.00(43) | $-0.80(13)$
Si${}^{IA}\xrightarrow{\gamma}$Si1 |  |  3${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ |  | 9315.92(10) | 7535.7(4) | 0.21(9) | 16.6(36) |  |  |
P $\xrightarrow{\beta^{+}}$ Si2 | 28 |  3${}^{+}91)$ $\rightarrow$ 3${}^{+}(0)$ | 4.787(24) |  |  |  |  | 0.3188(89) | 4.47(85) | $-0.08(27)$
Si${}^{IA}\xrightarrow{\gamma}$Si2 |  |  3${}^{+}(1)$ $\rightarrow$ 3${}^{+}(0)$ |  | 9315.92(10) | 3039.16(17) | 0.08(3) | 39.9(75) |  |  |
S $\xrightarrow{\beta^{+}}$ P | 30 |  0${}^{+}(1)\rightarrow$1${}^{+}(0)$ | 4.324(16) |  |  |  |  | 0.5429(98) | 6.08(34) | 0.75(18)
P$\xrightarrow{\beta^{+}}$ Si |  |  1${}^{+}(0)\rightarrow$0${}^{+}(1)$ | 4.839(12) |  |  |  |  | 0.2998(41) |  6.36(34)† | 0.50(10)
PIA $\xrightarrow{\gamma}$ P |  |  0${}^{+}(1)\rightarrow$1${}^{+}(0)$ |  | 677.01(3) | 677.01(3) | 0.0048(5) | 99.1(52) |  |  |
Cl $\xrightarrow{\beta^{+}}$ S1 | 32 |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 4.523(29) |  |  |  |  | 0.432(14) | 3.08(52) | $-0.70(23)$
S${}^{IA}\xrightarrow{\gamma}$S1 |  |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ |  | 7001.4(4) | 4770.5(3) | 0.27(9) | 42.6(71) |  |  |
Cl $\xrightarrow{\beta^{+}}$ S2 | 32 |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ | 5.52(14) |  |  |  |  | 0.137(21) | 5.5(19) | 0.11(25)
S${}^{IA}\xrightarrow{\gamma}$S2 |  |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ |  | 7001.4(4) | 3223.4(10) | 0.027(16) | 24.3(72) |  |  |
Ar $\xrightarrow{\beta^{+}}$ Cl1 | 34 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 5.309(51) |  |  |  |  | 0.175(10) |  7.91(58)† | 0.56(11)
Cl${}^{1}\xrightarrow{\gamma}$ClIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 461.00(4) | 461.00(4) | 0.000087(8) | 27.1(12) |  |  |
Ar $\xrightarrow{\beta^{+}}$ Cl2 | 34 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 4.781(20) |  |  |  |  | 0.3206(72) |  1.72(11)† | $-0.957(41)$
Cl${}^{2}\xrightarrow{\gamma}$ClIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 665.56(4) | 665.55(5) | 0.000042(5) | 10.84(65) |  |  |
Ar $\xrightarrow{\beta^{+}}$ Cl3 | 34 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 4.117(26) |  |  |  |  | 0.688(20) |  1.89(26)† | $-1.94(19)$
Cl${}^{3}\xrightarrow{\gamma}$ClIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 2580.4(2) | 2579.4(14) | 0.0135(36) | 25.5(34) |  |  |
Ar $\xrightarrow{\beta^{+}}$ Cl4 | 34 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.458(24) |  |  |  |  | 1.467(41) |  2.78(94)† | $-2.8(14)$
Cl${}^{4}\xrightarrow{\gamma}$ClIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 3129.13(12) | 3129(10) | 0.24(16) | 80(27) |  |  |
K $\xrightarrow{\beta^{+}}$ Ar1 | 36 |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ | 4.778(40) |  |  |  |  | 0.318(15) | 0.189(42) | $-1.438(67)$
Ar${}^{IA}\xrightarrow{\gamma}$Ar1 |  |  1${}^{+}(1)$ $\rightarrow$ 2${}^{+}(0)$ |  | 6612.12(20) | 4641.0(5) | 0.00051(22) | 2.17(47) |  |  |
K $\xrightarrow{\beta^{+}}$ Ar2 | 36 |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ | 4.902(52) |  |  |  |  | 0.279(17) | 1.37(31) | $-0.93(10)$
Ar${}^{IA}\xrightarrow{\gamma}$Ar2 |  |  1${}^{+}(1)$ $\rightarrow$ 0${}^{+}(0)$ |  | 6612.12(20) | 2170.29(20) | 0.0021(9) | 13.8(30) |  |  |
Ca $\xrightarrow{\beta^{+}}$ K1 | 38 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 4.798(17) |  |  |  |  | 0.3143(62) |  6.26(32)† | 0.49(10)
K${}^{1}\xrightarrow{\gamma}$KIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 458.4(4) | 327.9(2) | 0.000064(6) | 43.2(20) |  |  |
Ca $\xrightarrow{\beta^{+}}$ K2 | 38 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.4251(45) |  |  |  |  | 1.5262(80) |  1.49(15)† | $-4.91(23)$
K ${}^{2}\xrightarrow{\gamma}$ KIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 1697.84(12) | 1567.39(12) | 0.0093(19) | 49.9(51) |  |  |
Ti $\xrightarrow{\beta^{+}}$ Sc1 | 42 |  0${}^{+}(1)$ $\rightarrow$ 1${}^{+}(0)$ | 3.207(32) |  |  |  |  | 1.963(72) |  6.2(14)† | 3.0(27)
Sc${}^{1}\xrightarrow{\gamma}$ScIA |  |  1${}^{+}(0)$ $\rightarrow$ 0${}^{+}(1)$ |  | 611.051(6) | 611.046(6) | 0.016(7) | 297(65) |  |  |
Table 25: Data for the $\beta$ transitions between $A=$ 9 to 53, $T=3/2$ quartet states and their corresponding analog $\gamma$ transitions (see Fig. 17 and 18), leading to the form factor ratio $|b/Ac|^{exp}$. At the bottom of the table the single $\beta$ transition from a $T=2$ quintet state and its analog $\gamma$ transition are included as well. Log$(f_{A}t)$ values are from Table 21. Level energies and $E_{\gamma}$ and $\Gamma_{M1}$ values are from Ref. ENSDF . The data for the transition 53Ni $\rightarrow^{53}$Co1 are from Su  _et al._ (2016). The $\Gamma_{M1}$ value for this case is from theory Su  _et al._ (2016), with a 20% error being assumed. Element labels $X$ in column 1 without superscripts denote nuclear ground states, while superscripts ”1” to ”2” indicate excited states, and ”IA” indicates the analogue state to the $\beta$ decaying state(s) (see Fig. 17 and 18). Note that the sign of $M_{L}$ is relative to the sign of $M_{GT}$ since $M_{L}$ is extracted using the absolute value of the $c$ form factor. $T=3/2$ |  |  |  |  level of $\gamma$ |  Eγ | $\Gamma_{M1}$ |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
$decays$ | $A$ | $J_{i}(T_{i})\rightarrow J_{f}(T_{f})$ | $\log ft$ | (keV) | (keV) | (eV) | $|b|_{\gamma}^{exp}$ | $|c|^{exp}$ | $|b/Ac|^{exp}$ | $M_{L}^{exp}$
Li $\xrightarrow{\beta^{-}}$ Be | 9 |  3/2${}^{-}(3/2)$ $\rightarrow$ 3/2${}^{-}(1/2)$ | 5.3246(80) |  |  |  |  | 0.1721(16) | 7.31(23) | 0.448(40)
BeIA $\xrightarrow{\gamma}$Be |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 3/2${}^{-}(1/2)$ |  | 14392.2(18) | 14380.0(18) | 6.6(4) | 11.32(34) |  |  |
Li $\xrightarrow{\beta^{-}}$ Be1 | 9 |  3/2${}^{-}(3/2)$ $\rightarrow$ 5/2${}^{-}(1/2)$ | 5.135(44) |  |  |  |  | 0.214(11) | 8.26(42) | 0.760(98)
BeIA $\xrightarrow{\gamma}$Be1 |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 5/2${}^{-}(1/2)$ |  | 14392.2(18) | 11954.3(22) | 7.48(7) | 15.907(75) |  |  |
C $\xrightarrow{\beta^{+}}$ B | 9 |  3/2${}^{-}(3/2)$ $\rightarrow$ 3/2${}^{-}(1/2)$ | 5.320(12) |  |  |  |  | 0.1728(25) | 7.42(25) | 0.469(44)
BIA $\xrightarrow{\gamma}$B |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 3/2${}^{-}(1/2)$ |  | 14655.0(25) | 14462.2(25) | 6.97(42) | 11.54(35) |  |  |
C $\xrightarrow{\beta^{+}}$ B1 | 9 |  3/2${}^{-}(3/2)$ $\rightarrow$ 5/2${}^{-}(1/2)$ | 5.228(84) |  |  |  |  | 0.192(18) | 8.7(10) | 0.77(21)
BIA $\xrightarrow{\gamma}$B1 |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 5/2${}^{-}(1/2)$ |  | 14655.0(25) | 12301(11) | 7.3(11) | 15.1(11) |  |  |
C $\xrightarrow{\beta^{+}}$ B2 | 9 |  3/2${}^{-}(3/2)$ $\rightarrow$ 1/2${}^{-}(1/2)$ | 5.876(45) |  |  |  |  | 0.0912(47) | 7.8(24) | 0.28(21)
BIA $\xrightarrow{\gamma}$B2 |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 1/2${}^{-}(1/2)$ |  | 14655.0(25) | 11870(160) | 1.17(70) | 6.4(19) |  |  |
B $\xrightarrow{\beta^{-}}$ C | 13 |  3/2${}^{-}(3/2)$ $\rightarrow$ 1/2${}^{-}(1/2)$ | 4.0294(94) |  |  |  |  | 0.7632(83) | 2.804(94) | $-1.452(73)$
CIA $\xrightarrow{\gamma}$ C |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 1/2${}^{-}(1/2)$ |  | 15108.2(12) | 15098.8(12) | 22.1(14) | 27.82(88) |  |  |
B $\xrightarrow{\beta^{-}}$ C1 | 13 |  3/2${}^{-}(3/2)$ $\rightarrow$ 3/2${}^{-}(1/2)$ | 4.452(46) |  |  |  |  | 0.470(25) | 6.24(55) | 0.72(26)
CIA $\xrightarrow{\gamma}$C1 |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 3/2${}^{-}(1/2)$ |  | 15108.2(12) | 11418.2(12) | 17.9(25) | 38.1(27) |  |  |
O $\xrightarrow{\beta^{+}}$ N | 13 |  3/2${}^{-}(3/2)$ $\rightarrow$ 1/2${}^{-}(1/2)$ | 4.081(11) |  |  |  |  | 0.7202(92) | 3.11(10) | $-1.149(73)$
NIA $\xrightarrow{\gamma}$ N |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 1/2${}^{-}(1/2)$ |  | 15064.6(4) | 15055.2(4) | 24.0(14) | 29.12(85) |  |  |
O $\xrightarrow{\beta^{+}}$ N1 | 13 |  3/2${}^{-}(3/2)$ $\rightarrow$ 3/2${}^{-}(1/2)$ | 4.552(90) |  |  |  |  | 0.419(43) | 7.11(78) | 1.01(34)
NIA $\xrightarrow{\gamma}$N1 |  |  3/2${}^{-}(3/2)$ $\rightarrow$ 3/2${}^{-}(1/2)$ |  | 15064.6(4) | 11557(2) | 19.2(16) | 38.7(16) |  |  |
C $\xrightarrow{\beta^{-}}$ N1 | 15 |  1/2${}^{+}(3/2)$ $\rightarrow$ 1/2${}^{+}(1/2)$ | 4.1101(56) |  |  |  |  | 0.6949(45) | 3.06(38) | $-1.14(27)$
NIA $\xrightarrow{\gamma}$N1 |  |  1/2${}^{+}(3/2)$ $\rightarrow$ 1/2${}^{+}(1/2)$ |  | 11615(4) | 6316(4) | 1.6(4) | 31.9(40) |  |  |
O $\xrightarrow{\beta^{-}}$ F1 | 19 |  5/2${}^{+}(3/2)$ $\rightarrow$ 3/2${}^{+}(1/2)$ | 4.6303(97) |  |  |  |  | 0.3814(43) | 7.26(79) | 0.97(30)
FIA $\xrightarrow{\gamma}$F1 |  |  5/2${}^{+}(3/2)$ $\rightarrow$ 3/2${}^{+}(1/2)$ |  | 7539.6(9) | 5986(9) | 2.3(5) | 52.6(57) |  |  |
O $\xrightarrow{\beta^{-}}$ F2 | 19 |  5/2${}^{+}(3/2)$ $\rightarrow$ 7/2${}^{+}(1/2)$ | 3.865(16) |  |  |  |  | 0.918(17) | 6.31(64) | 1.47(59)
FIA $\xrightarrow{\gamma}$F2 |  |  5/2${}^{+}(3/2)$ $\rightarrow$ 7/2${}^{+}(1/2)$ |  | 7539.6(9) | 3161.9(9) | 1.5(3) | 110(11) |  |  |
Ne $\xrightarrow{\beta^{-}}$ Na | 23 |  5/2${}^{+}(3/2)$ $\rightarrow$ 3/2${}^{+}(1/2)$ | 5.2810(86) |  |  |  |  | 0.1804(18) | 9.38(37) | 0.842(68)
NaIA $\xrightarrow{\gamma}$ Na |  |  5/2${}^{+}(3/2)$ $\rightarrow$ 3/2${}^{+}(1/2)$ |  | 7891.19(25) | 7889.7(3) | 1.97(15) | 38.9(15) |  |  |
Ne $\xrightarrow{\beta^{-}}$ Na1 | 23 |  5/2${}^{+}(3/2)$ $\rightarrow$ 5/2${}^{+}(1/2)$ | 5.391(18) |  |  |  |  | 0.1588(32) | 8.16(42) | 0.548(67)
NaIA $\xrightarrow{\gamma}$Na1 |  |  5/2${}^{+}(3/2)$ $\rightarrow$ 5/2${}^{+}(1/2)$ |  | 7891.19(25) | 7449.9(3) | 0.97(9) | 29.8(14) |  |  |
Ni $\xrightarrow{\beta^{+}}$ Co1 | 53 |  7/2${}^{-}(3/2)$ $\rightarrow$ 9/2${}^{-}(1/2)$ | 4.52(22) |  |  |  |  | 0.44(10) | 2.26(57) | $-1.06(35)$
CoIA $\xrightarrow{\gamma}$Co1 |  |  7/2${}^{-}(3/2)$ $\rightarrow$ 9/2${}^{-}(1/2)$ |  | 4325(2) | 4325(2) | 0.11(2) | 52.2(47) |  |  |
$T=2$ |  |  |  |  |  |  |  |  |  |
$\xrightarrow{\beta,\gamma}$ |  |  |  |  level of $\gamma$ |  Eγ | $\Gamma_{M1}$ |  |  |  |
$T=1$ | $A$ | $J_{i}(T_{i})\rightarrow J_{f}(T_{f})$ | $\log ft$ | (keV) | (keV) | (eV) | $|b|_{\gamma}^{exp}$ | $|c|^{exp}$ | $|b/Ac|^{exp}$ | $M_{L}^{exp}$
Si $\xrightarrow{\beta^{+}}$ P | 32 |  0${}^{+}(2)$ $\rightarrow$ 1${}^{+}(1)$ | 8.230(55) |  |  |  |  | 0.00602(38) | 5.76(39) | 0.0063(24)
P1 IA $\xrightarrow{\gamma}$P |  |  0${}^{+}(2)$ $\rightarrow$ 1${}^{+}(1)$ |  | 5072.44(6) | 5072.00(6) | 0.00022(1) | 1.110(25) |  |  |

####  III.3.2 Nuclear matrix elements $M_{GT}$ and $M_{L}$

#####  III.3.2.1 Experimental values for $M_{GT}$ and $M_{L}$

Using the experimental values for the $b$ and $c$ form factors in Tables 22 to
25, and assuming the ratio $b/Ac$ to be positive (Sec. III.3.1), the matrix
element $M_{L}$ can be derived using Eq. (33). As we use for this the absolute
values of the $c$ form factor the sign obtained for $M_{L}$ is relative to
that of $M_{GT}$ (Eq. (33)). The resulting values for $M_{L}$ are listed in
the last columns of Tables 22 to 25.

#####  III.3.2.2 Shell model values for $M_{GT}$ and $M_{L}$

For a number of $\beta$ transitions in the $T=1$ multiplets shell-model
calculations of the matrix elements relevant for the different form factors
have again been performed. These transitions correspond to the ones considered
in an early version of this paper dating from before one of the authors
retired and before the data set was significantly extended by a thorough
inspection of the Nuclear Data Base ENSDF . For each case, several
calculations were performed with different effective shell-model interactions
(Table 26). In the p-shell ($p_{3/2}$ and $p_{1/2}$ orbits), three
interactions of Cohen-Kurath (i.e. CK6162BME, CK8162BME, and CK816POT, and
labelled as such in Table 26) Cohen and Kurath (1965), and a more recent one
from Brown-Warburton (PWBT) Warburton and Brown (1992) were used. In the sd-
shell ($d_{5/2}$, $s_{1/2}$, and $d_{3/2}$ orbits), we used three versions of
the universal sd-shell interaction, i.e. USD Wildenthal (1984), and two more
recent ones, USD-A and USD-B Brown and Richter (2006)). For A=16 and possibly
A=18 a cross-shell interaction is needed. Here, the Millener-Kurath (MK)
interaction Millener and Kurath (1975) that was designed to give 1p-1h matrix
elements (relative to a closed-shell 16O) and including the $p_{3/2}$,
$p_{1/2}$, $d_{5/2}$, $s_{1/2}$, and $d_{3/2}$ orbits was used, together with
USD for $sd$-shell interactions and one of Cohen-Kurath for $p$-shell
interactions. That left the 2p-2h interaction to be determined. This was
computed from the Millener-Kurath potential, but then adjusting its strength
so the excited $0^{+}$ state in $A=18$ (4$p$-2$h$ configuration) came at about
the right energy. The accuracy of this interaction for A=16 and A=18 might
therefore be inferior to the others.

#####  III.3.2.3 Comparison between theoretical and experimental matrix
elements

Results from the shell model calculations of the $M_{GT}$ and $M_{L}$ matrix
elements and the corresponding values for the $b$ form factor and the $b/Ac$
ratio (the latter being obtained from Eq. (33)) for the fourteen $\beta$
transitions considered, are listed in Table 26. Comparing the theoretically
calculated and experimentally obtained values for the Gamow-Teller matrix
element, the Cohen-Kurath interaction seems to perform best for $A=$ 6
(indicated by the boldface font in the fifth column), with there not being a
preferred choice for $A=$ 10, and with the PWBT interaction yielding results
closest to experiment for $A=$ 12\. As can be seen by comparing the last two
columns in Table 26, for all these cases the theoretically predicted value for
$|b/Ac|$ (listed in the last column) is significantly larger than the
experimental one (last but one column). This is due to the systematically too
low theoretical values for $|M_{L}|$ as is clear from a comparison of columns
6 and 7. The reason for this is not clear. For $A=$ 16 and 18, the MK
interaction seems to perform quite well, reproducing the value for
$|b/Ac|^{exp}$ always within about one standard deviation. For $A=$ 18 the USD
interaction is further found to provide similar results to the MK interaction.
For $A=$ 20 and 24 the USD-A interaction seems to reproduce the experimental
Gamow-Teller matrix elements slightly better than the USD and USD-B
interactions, whereas for $A=$ 30 the USD interaction provides best agreement.
Note that even for the case of 24mAl, the large value of
$|b/Ac|^{exp}=17.5(32)$, being due to the combination of a very small Gamow-
Teller matrix element and a relatively large M1 $\gamma$-decay width,
$\Gamma_{M1}$, is reproduced quite well.

Fig. 19 shows the theoretical and calculated Gamow-Teller matrix elements to
agree within about 20% (see also columns 4 and 5 in Table 26). The unweighted
average of their ratio yields $|M_{GT}|^{exp}/|M_{GT}|^{theo}=1.16(47)$ (or
1.04(11) when omitting the 16N $(1^{-})\rightarrow^{16}$O $(2^{-})$ transition
for which the calculated Gamow-Teller matrix element is about a factor three
lower than the experimental one). This average value being in agreement with
unity indicates that also for these $\beta$ decays from $T=$ 1 states, with in
general a more complex nuclear structure than the $T=$ 1/2 mirror nuclei, the
shell model reproduces the Gamow-Teller matrix elements on average quite well.
The unweighted average of the $|b/Ac|$ values for all 14 transitions (obtained
from columns 10 and 11 in Table 26 and using the $(b/Ac)^{theo}$ values
corresponding to the $M_{GT}^{theo}$ values listed in boldface in column 5) is
found to be $(b/Ac)^{exp}/(b/Ac)^{theo}=1.01(15)$, again showing good overall
agreement (within typically about 20%) between the shell model calculations
and experiment (Fig. 19). However, as was mentioned already, the $M_{L}$
values calculated for the transitions with $A=$ 6 to 12 are systematically too
low, while for the higher masses they do agree with experiment within about
one standard deviation (see Table 26). Evidently the weak magnetism form
factor, $b$, which depends on both the spin matrix element, $M_{GT}$, and the
orbital current matrix element, $M_{L}$, (see Eq. (33)) is nonetheless
dominated by the spin terms in these isovector transitions.

Table 26: Comparison of the experimental values for the Gamow-Teller matrix element, $M_{GT}$ (the $c$ form factor), the orbital current matrix element, $M_{L}$, the form factor $|b|$, and the ratio $|b/Ac|$, with theoretical values calculated in the shell model using the interactions mentioned in the text (Sec. III.3.2), for fourteen $\beta$ transitions from $T=1$ triplet states in Tables 22 to 25. The theoretically calculated values of $|M_{GT}|^{theo}$ in column 5 that agree best with the experimental results, $|M_{GT}|^{exp}$, listed in column 4 are indicated in boldface. For the transition from 10C to 10B no preference can be given. Values for $|b/Ac|^{theo}$ were obtained from the values of $|M_{GT}|^{theo}$ and $|M_{L}|^{theo}$ using Eq. (33). The value $g_{A}$ = 1 was used throughout. $\beta$ decay | $J_{i}\rightarrow J_{f}$ | shell model | $|M_{GT}|^{exp}$ | $|M_{GT}|^{theo}$ | $|M_{L}|^{exp}$ | $|M_{L}|^{theo}$ | $|b|^{exp}$ | $|b|^{theo}$ | $|b/Ac|^{exp}$ | $|b/Ac|^{theo}$
---|---|---|---|---|---|---|---|---|---|---
|  | interaction |  |  |  |  |  |  |  |
6He $\xrightarrow{\beta^{-}}$ 6Li |  0+ $\rightarrow$ 1+ | CK6162BME | 2.7802(20) | 2.348 | 1.72(12) | 0.17 | 68.2(7) | 67.3 | 4.088(42) | 4.78
|  | PWBT |  | 1.819 |  | 0.003 |  | 51.4 |  | 4.71
10C $\xrightarrow{\beta^{+}}$ 10B1 |  0+ $\rightarrow$ 1+ | CK6162BME | 2.3649(19) | 2.210 | 3.2(17) | 0.138 | 79(17) | 102.6 | 3.34(72) | 4.64
|  | CK8162BME |  | 2.141 |  | 0.288 |  | 97.9 |  | 4.57
|  | CK816POT |  | 2.213 |  | 0.241 |  | 101.7 |  | 4.60
|  | PWBT |  | 2.208 |  | 0.238 |  | 106.3 |  | 4.81
12B $\xrightarrow{\beta^{-}}$ 12C |  1+ $\rightarrow$ 0+ | CK6162BME | 0.73852(55) | 0.558 | 0.650(33) | 0.012 | 33.9(4) | 31.4 | 3.825(45) | 4.69
|  | CK8162BME |  | 0.573 |  | 0.050 |  | 31.8 |  | 4.62
|  | CK816POT |  | 0.554 |  | 0.065 |  | 30.5 |  | 4.59
|  | PWBT |  | 0.699 |  | 0.123 |  | 38.0 |  | 4.53
12N $\xrightarrow{\beta^{+}}$ 12C |  1+ $\rightarrow$ 0+ | CK6162BME | 0.69613(64) | 0.558 | 0.451(33) | 0.012 | 33.9(4) | 31.4 | 4.058(48) | 4.69
|  | CK8162BME |  | 0.573 |  | 0.050 |  | 31.8 |  | 4.62
|  | CK816POT |  | 0.554 |  | 0.065 |  | 30.5 |  | 4.59
|  | PWBT |  | 0.699 |  | 0.123 |  | 38.0 |  | 4.53
16N $\xrightarrow{\beta^{-}}$ 16O1 |  2- $\rightarrow$ 3- | MK | 0.4503(22) | 0.338 | 0.11(11) | 0.155 | 32.1(18) | 23.0 | 4.46(25) | 4.25
16N $\xrightarrow{\beta^{-}}$ 16O2 |  2- $\rightarrow$ 1- | MK | 0.2184(91) | 0.079 | 0.165(89) | 0.041 | 13.8(13) | 5.3 | 3.95(41) | 4.18
16N $\xrightarrow{\beta^{-}}$ 16O3 |  2- $\rightarrow$ 2- | MK | 0.521(17) | 0.595 | 1.49(30) | 1.181 | 63.1(42) | 63.7 | 7.57(56) | 6.69
18Ne $\xrightarrow{\beta^{+}}$ 18F |  0+ $\rightarrow$ 1+ | MK | 2.2328(32) | 2.252 | 2.3(12) | 1.060 | 230(22) | 209.8 | 5.72(55) | 5.18
|  | USD |  | 2.250 |  | 1.128 |  | 210.9 |  | 5.21
18F $\xrightarrow{\beta^{+}}$ 18O |  1+ $\rightarrow$ 0+ | MK | 1.2778(22) | 1.300 | 1.36(71) | 0.612 | 230(22) | 121.2 | 5.77(55) | 5.18
|  | USD |  | 1.299 |  | 0.651 |  | 121.8 |  | 5.21
20F $\xrightarrow{\beta^{-}}$ 20Ne1 |  2+ $\rightarrow$ 2+ | USD | 0.25493(45) | 0.246 | 0.91(11) | 0.851 | 42.1(21) | 40.2 | 8.26(41) | 8.16
|  | USD-A |  | 0.251 |  | 0.827 |  | 40.2 |  | 8.00
|  | USD-B |  | 0.244 |  | 0.868 |  | 40.3 |  | 8.27
20Na $\xrightarrow{\beta^{+}}$ 20Ne1 |  2+ $\rightarrow$ 2+ | USD | 0.2506(19) | 0.246 | 0.93(11) | 0.851 | 42.1(21) | 40.2 | 8.40(22) | 8.16
|  | USD-A |  | 0.251 |  | 0.827 |  | 40.2 |  | 8.00
|  | USD-B |  | 0.244 |  | 0.868 |  | 40.3 |  | 8.27
24mAl $\xrightarrow{\beta^{+}}$ 24Mg |  1+ $\rightarrow$ 0+ | USD | 0.101(15) | 0.139 | 1.29(38) | 1.020 | 42.3(44) | 40.2 | 17.5(32) | 12.05
|  | USD-A |  | 0.100 |  | 0.873 |  | 32.2 |  | 13.44
|  | USD-B |  | 0.199 |  | 1.009 |  | 46.7 |  | 9.78
30S $\xrightarrow{\beta^{+}}$ 30P |  0+ $\rightarrow$ 1+ | USD | 0.5429(98) | 0.532 | 0.75(18) | 0.593 | 99.1(52) | 92.9 | 6.08(34) | 5.82
|  | USD-A |  | 0.354 |  | 0.637 |  | 69.0 |  | 6.51
|  | USD-B |  | 0.473 |  | 0.606 |  | 84.9 |  | 5.99
30P $\xrightarrow{\beta^{+}}$ 30Si |  1+ $\rightarrow$ 0+ | USD | 0.2998(41) | 0.307 | 0.50(10) | 0.342 | 99.1(52) | 53.6 | 6.36(34) | 5.82
|  | USD-A |  | 0.204 |  | 0.368 |  | 39.9 |  | 6.51
|  | USD-B |  | 0.273 |  | 0.350 |  | 49.0 |  | 5.99

Figure 19: Ratio of theoretically calculated and experimental Gamow-Teller
matrix elements (top) and $b/Ac$ values (bottom) for the triplet $\beta$
transitions in Table 26. If not explicitly shown, error bars (based on the
experimental $M_{GT}$ and $b/Ac$ values only) are smaller than the size of the
symbols.

###  III.4 General discussion of the weak magnetism results

####  III.4.1 Relation to nuclear structure

#####  III.4.1.1 $\beta$ transitions from the $T=1/2$ mirror nuclei

For the $T=1/2$ mirror $\beta$ transitions for which CVC relates the weak
magnetism form factor $b$ to the difference between the mirror pair magnetic
moments, a clear sub-shell behaviour related to the single particle Schmidt
values of the magnetic moments is observed, despite significant variation in
the absolute values of $b$. This behaviour is further accentuated in the
values of $b/Ac$, in part due to the slight decrease of the size of the Gamow-
Teller form factor, $c$, with increasing mass (Fig. 6 and Sec. III.2.2).

For the mirror nuclei up to mass $A=45$ the $b$ and $c$ form factors, which
are related to the $M_{GT}$ and $M_{L}$ matrix elements, were also calculated
in the nuclear shell model (Sections III.2.3.2 and III.2.4). For this, well-
established interactions that give excellent fits to experimental spectra were
used. Calculations were limited to mass $A=45$ as for higher masses the
truncation of the shell model space required, due to calculation power
limitations, was too important to yield reliable results. Good agreement
between the experimental and theoretically calculated values for the form
factor ratio $b/Ac$ was obtained (Fig. 8). Further, the shell model calculated
values for the $M_{GT}$ and $M_{L}$ matrix elements also show good
correspondence with the experimental values extracted from the experimental
$b/Ac$ ratio when using the impulse approximation (Figs. 9 and 10). This is
also the case for the special cases of 33Cl and 35Ar with large oblate
deformation, although the shell model could not reproduce the sign of $b/Ac$
for 35Ar. However, an extreme single particle calculation in a deformed basis
could reproduce the sign as well as the magnitude of $b/Ac$ for both cases
within a factor of about 2.

#####  III.4.1.2 $\beta$ transitions from $T=1$ and $T=3/2$ states

The $b/Ac$ values for the $\beta$ transitions from $T=1$ and $T=3/2$ states
(Fig. 20) do not exhibit the clear single-particle related systematic that is
obeyed by the $T=1/2$ mirror transitions. Of course, many of the $T=1$ and
$T=3/2$ transitions involve excited states with often strongly mixed
configurations, implying the involvement of many shell-model orbitals, each
with a fractional occupation. One may nevertheless hope that the transitions
to ground states of even or even-even nuclei, and transitions to high-lying
first-excited $2^{+}$ states, which in general all exhibit a rather ’simple’
nuclear structure, do show some simple shell structure related systematic as
well. Table 27 therefore lists all $T=$ 1/2, 1 and 3/2 transitions to ground
states or high-lying first-excited $2^{+}$ states. As can be seen (but of
course statistics is rather limited) the $T=1$ and $T=3/2$ transitions in this
table indeed turn out to follow the trend of $|b/Ac|$ being typically large
for the subshells with $j=l+1/2$ and smaller for $j=l-1/2$. The (unweighted)
averages of $(b/Ac)^{exp}$ including all transitions in a given sub-shell turn
out not to differ significantly from the values calculated for only the mirror
$\beta$ transitions. Note, however, that many transitions from the higher
isospin multiplets discussed here cannot be included in this comparison as
they involve odd-odd nuclei with the odd proton in a $j=l+1/2$ orbital and the
odd neutron in a $j=l-1/2$ orbital, or vice versa.

Figure 20: Experimental $b/Ac$ values for the beta transitions from $T=$ 1 and $T=$ 3/2 states (Tables 22 to 25). Table 27: Overview of the values for $(b/Ac)^{exp}$ for all ground state-to-ground state and ground state-to-first-excited 2+ state transitions with $A<39$ discussed here (’mir’ = mirror $\beta$ transition). Values of $(b/Ac)^{exp}$ for $\beta$ transitions from $T=$ 1 and $3/2$ states are in italics. The last column lists the average values for all transitions in a subshell. Because of widely differing error bars unweighted averages were calculated. When for a given mass value the $(b/Ac)^{exp}$ value is available for both the $\beta^{-}$ and $\beta^{+}$ transitions in a $T=$ 1 multiplet, sharing the same analog $\gamma$ transition (i.e. for $A=$ 18, 20, 24, 24, and 30), the average of both $(b/Ac)^{exp}$ values is used to calculate the unweighted average for the respective sub-shell. Averages in square brackets indicate the averages for the mirror transitions separately (from Table 14). Note that the $f_{7/2}$ sub-shell is not shown here, but that the value $(b/Ac)^{exp}=6.2(14)$ for the 42Sc $\rightarrow^{42}$Sc1 transition from a $T=$ 1 state (see Table 24) agrees with the $f_{7/2}$ sub-shell average of 8.1(10) for the mirror transitions (Table 14). $\beta$ transition | $A$ | $J_{i}^{\pi}\rightarrow J_{f}^{\pi}$ | sub-shell | $j$ | type |  log $ft$ | $(b/Ac)^{exp}$ | $(b/Ac)^{exp}$
---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  | sub-shell avg.
|  |  |  |  |  |  |  | [only mir]
H$\xrightarrow{\beta^{-}}$He | 3 |  1/2+ $\rightarrow$ 1/2+ | $s_{1/2}$ | $l+1/2$ | mir | 3.0532(38) | 4.2012(23) | 4.145(80)
He$\xrightarrow{\beta^{-}}$Li | 6 |  0+ $\rightarrow$ 1+ |  |  | $T=1$ | 2.905753(61) | 4.088(42) | [4.2012(23)]
Li$\xrightarrow{\beta^{-}}$Be111The $A=$8 multiplet was not included in the analysis performed here since error bars on the log $ft$ values are difficult to estimate because broad levels of 8Be participate in the $\beta$ decay ENSDF ; Barker (1989). The value for $(b/Ac)^{exp}$ listed here was obtained from $\beta$-ray angular distribution measurements Sumikama  _et al._ (2011b). | 8 |  2+ $\rightarrow$ 2+ | $p_{3/2}$ | $l+1/2$ | $T=1$ | 5.72 | 7.5(2) | 7.04(69)
B$\xrightarrow{\beta^{+}}$Be11footnotemark: 1 | 8 |  2+ $\rightarrow$ 2+ |  |  | $T=1$ | 5.77 | 7.5(2) | [6.2506(29)]
Li$\xrightarrow{\beta^{-}}$Be | 9 |  3/2- $\rightarrow$ 3/2- |  |  | $T=3/2$ | 5.3246(80) | 7.31(23) |
C$\xrightarrow{\beta^{+}}$B | 9 |  3/2- $\rightarrow$ 3/2- |  |  | $T=3/2$ | 5.320(12) | 7.42(25) |
C$\xrightarrow{\beta^{+}}$B | 11 |  3/2+ $\rightarrow$ 3/2+ |  |  | mir | 3.59294(21) | 6.2506(29) |
N$\xrightarrow{\beta^{+}}$C | 13 |  1/2+ $\rightarrow$ 1/2+ | $p_{1/2}$ | $l-1/2$ | mir | 3.67037(45) | 3.1712(23) | [2.93(21)]
O$\xrightarrow{\beta^{+}}$N | 15 |  1/2+ $\rightarrow$ 1/2+ |  |  | mir | 3.64368(58) | 2.7557(20) |
F$\xrightarrow{\beta^{+}}$O | 17 |  5/2+ $\rightarrow$ 5/2+ | $d_{5/2}$ | $l+1/2$ | mir | 3.36006(36) | 6.0416(30) |  6.5(15)222The values for the transitions from the 24mNa and 24mAl isomeric states were not included.
Ne$\xrightarrow{\beta^{+}}$F | 18 |  0+ $\rightarrow$ 1+ |  |  | $T=1$ | 3.0955(11) | 5.72(55) | [5.82(42)]
F$\xrightarrow{\beta^{+}}$O | 18 |  1+ $\rightarrow$ 0+ |  |  | $T=1$ | 3.5783(23) | 5.77(55) |
Ne$\xrightarrow{\beta^{+}}$F | 19 |  1/2+ $\rightarrow$ 1/2+ |  |  | mir | 3.23591(25) | 4.8807(20) |
F$\xrightarrow{\beta^{-}}$Ne1 | 20 |  2+ $\rightarrow$ 2+ |  |  | $T=1$ | 4.98364(31) | 8.28(41) |
Na$\xrightarrow{\beta^{+}}$Ne1 | 20 |  2+ $\rightarrow$ 2+ |  |  | $T=1$ | 4.9972(64)) | 8.40(42) |
Na$\xrightarrow{\beta^{+}}$Ne | 21 |  3/2+ $\rightarrow$ 3/2+ |  |  | mir | 3.60991(40) | 5.5233(30) |
Mg$\xrightarrow{\beta^{+}}$Na | 23 |  3/2+ $\rightarrow$ 3/2+ |  |  | mir | 3.67225(70) | 6.4164(55) |
Ne$\xrightarrow{\beta^{-}}$Na | 23 |  5/2+ $\rightarrow$ 3/2+ |  |  | $T=3/2$ | 5.2810(86) | 9.38(37) |
Na$\xrightarrow{\beta^{-}}$Mg2 | 24 |  4+ $\rightarrow$ 4+ |  |  | $T=1$ | 6.12609(46) | 4.8(12) |
Al$\xrightarrow{\beta^{+}}$Mg2 | 24 |  4+ $\rightarrow$ 4+ |  |  | $T=1$ | 6.129(57) | 4.8(13) |
Na(m)$\xrightarrow{\beta^{-}}$Mg | 24 |  1+ $\rightarrow$ 0+ |  |  | $T=1$ | 5.936(88) | 20.7(30) |
Al(m)$\xrightarrow{\beta^{+}}$Mg | 24 |  1+ $\rightarrow$ 0+ |  |  | $T=1$ | 5.79(13) | 17.5(32) |
Al(m)$\xrightarrow{\beta^{+}}$Mg1 | 24 |  1+ $\rightarrow$ 0+ |  |  | $T=1$ | 5.921(50) | 15.8(21) |
Al$\xrightarrow{\beta^{+}}$Mg | 25 |  5/2+ $\rightarrow$ 5/2+ |  |  | mir | 3.56972(37) | 6.5875(38) |
Si$\xrightarrow{\beta^{+}}$Al | 27 |  5/2+ $\rightarrow$ 5/2+ |  |  | mir | 3.61665(28) | 7.6551(33) |
P$\xrightarrow{\beta^{+}}$Si | 29 |  1/2+ $\rightarrow$ 1/2+ | $s_{1/2}$ | $l+1/2$ | mir | 3.67802(72) | 5.7631(51) | 5.76(46)
S$\xrightarrow{\beta^{+}}$P | 30 |  0+ $\rightarrow$ 1+ |  |  | $T=1$ | 4.324(16) | 6.08(34) | [5.43(21)]
P$\xrightarrow{\beta^{+}}$Si | 30 |  1+ $\rightarrow$ 0+ |  |  | $T=1$ | 4.839(12) | 6.36(34) |
S$\xrightarrow{\beta^{+}}$P | 31 |  1/2+ $\rightarrow$ 1/2+ |  |  | mir | 3.68127(48) | 5.2988(33) |
Cl$\xrightarrow{\beta^{+}}$S | 33 |  3/2+ $\rightarrow$ 3/2+ | $d_{3/2}$ | $l-1/2$ | mir | 3.74802(74) | -0.4561(21) | 0.6(12)
Ar$\xrightarrow{\beta^{+}}$Cl | 35 |  3/2+ $\rightarrow$ 3/2+ |  |  | mir | 3.75548(46) | -0.8684(11) | [0.5(14)]
K$\xrightarrow{\beta^{+}}$Ar1 | 36 |  1+ $\rightarrow$ 2+ |  |  | $T=1$ | 4.778(40) | 0.189(42) |
K$\xrightarrow{\beta^{+}}$Ar2 | 36 |  1+ $\rightarrow$ 0+ |  |  | $T=1$ | 4.902(52) | 1.37(31) |
K$\xrightarrow{\beta^{+}}$Ar | 37 |  3/2+ $\rightarrow$ 3/2+ |  |  | mir | 3.66383(52) | 2.104(10) |
Ca$\xrightarrow{\beta^{+}}$K | 39 |  3/2+ $\rightarrow$ 3/2+ |  |  | mir | 3.63180(61) | 1.2316(10) |

####  III.4.2 Averages and their role for $\beta$ decay experiments and the
reactor neutrino anomaly

#####  III.4.2.1 $\beta$ decay experiments

In Fig. 21 the experimental values for $b/Ac$ for all $\beta$ transitions up
to mass $A=61$ considered here are shown on a single plot. For the $\beta$
transitions from $T=$ 1, 3/2 and 2 states the sign of $b/Ac$ is assumed
positive (see Sec. III.3). As can be seen, all values are in the same range,
with a maximum of about +10.0 and a minimum value of about -1.0. The
transitions from 14O and 14C are again excluded, while for the nine cases
where both the $\beta^{-}$ and $\beta^{+}$ transition in a $T=1$ triplet yield
a value for $|b/Ac|$ only the average of both values is used. The unweighted
average of the 27 $T=$ 1/2 transitions with mass $A=$ 3 to 61, showing a clear
subshell dependence related to their single-particle structure (Table 6 and
Fig. 6), is $(b/Ac)^{exp}=5.6\pm 2.7$. This is in good agreement with the
overall unweighted average of the 57 transitions from $T=$1, 3/2 and 2 states
(Fig. 20), i.e. $|b/Ac|^{exp}=5.3\pm 3.2$. The average for the entire set of
84 $\beta$ transitions discussed here is $b/Ac=5.4\pm 3.1$. In all cases the
one standard deviation error is rather large due to the spread in the
individual values.

Figure 21: Overview of the experimental $b/Ac$ values for all 85 $\beta$
transitions from isotopes with $A=$ 3 to 61 considered here (Tables 6 and 22
to 25). The unweighted average of all values shown is $b/Ac=5.4\pm 3.1$.

An overview of all averages and spreads of the weak magnetism term, $b/Ac$,
and the matching matrix elements, $M_{GT}$ and $M_{L}$, mentioned in this text
is given in Table 28.

Table 28: Overview of $b/Ac$ averages and ratios of experimental and shell model calculated values for $M_{GT}$, $M_{L}$ and $b/Ac$ mentioned in the text (the relevant sections and figures are mentioned in the last column). | quantity | mass region or isospin value | average | Section / Figure
---|---|---|---|---
$T=1/2$ | $(b/Ac)^{exp}$ |  $A=$ 3, 11-61; 27 transitions |  5.6 $\pm$ 2.7 |  Sec. III.4 / Fig. 6
mirror | $(b/Ac)^{exp}/(b/Ac)^{theo}$ |  $A=$ 3, 11-31, 37-45 |  0.96(11)303030The values for 33Cl and 35Ar (i.e. resp. 2.22 and -5.63), which deviate significantly from all other values (see Table 16), were not included. |  Sec. III.2.3.2 / Fig. 8
transitions | $M_{GT}^{exp}/M_{GT}^{theo}$ |  $A=$ 3, 11-45 | 0.97(8) |  Sec. III.2.4 / Fig. 9
| $M_{L}^{exp}/M_{L}^{theo}$ |  $A=$ 11-29, 33-45 | 1.04(34) |  Sec. III.2.4 / Fig. 10
$T=1,3/2,2$ |  $(b/Ac)^{exp}$, $T=1$ |  $T=1$; $A=$ 6-42; 42 transitions |  4.9 $\pm$ 3.5 |  Sec. III.3.1 / Fig. 20
transitions |  $(b/Ac)^{exp}$, $T=3/2$ |  $T=3/2$; $A=$ 9-53; 15 transitions |  6.3 $\pm$ 2.4 |  Sec. III.3.1 / Fig. 20
|  $(b/Ac)^{exp}$, $T=1,3/2,2$ |  $T=$ 1, 3/2, and 2; $A=$ 6-53; 58 transitions |  5.3 $\pm$ 3.2 |  Sec. III.4 / Fig. 20
| $(b/Ac)^{exp}/(b/Ac)^{theo}$ |  $T=1$; $A=$ 6-30; 14 transitions | 1.01(15) |  Sec. III.3.2 / Fig. 19
| $M_{GT}^{exp}/M_{GT}^{theo}$ |  $T=1$; $A=$ 6-30; 13 transitions |  1.04(11)313131Including the very large value of 2.76 for the 2- $\rightarrow$ 1- transition 16N $\xrightarrow{\beta^{-}}$ 16O2 (see Table 26), which was omitted here, shifts the average to 1.16(47). |  Sec. III.3.2 / Fig. 19
all | $(b/Ac)^{exp}$ |  $T=$ 1/2, 1, 3/2, and 2; $A=$ 3-61; 85 transitions |  5.4 $\pm$ 3.1 |  Sec. III.4 / Fig. 21

The weak-magnetism form factor provides in most cases the major recoil
correction to the $\beta$-spectrum shape and $\beta$-correlation coefficients
for allowed transitions with ’normal’ strength (i.e non-retarded transitions
with $\log ft$ values less than about 6.7). This is true for both analog
transitions, i.e. between members of an isospin multiplet such as the $\beta$
transitions within the $T=1/2$ isospin doublets discussed here, and non-analog
transitions, as e.g. the ones from states with isospin $T=$1, 3/2 and 2. The
above analysis of weak magnetism form factors for the $T=1/2$ mirror $\beta$
transitions allows taking this form factor explicitly into account in the
analysis of experimental data, as was done in the determination of the
$\beta\nu$-correlation with 21Na Vetter  _et al._ (2008) and of the
$\beta$-asymmetry parameter for 37K Fenker  _et al._ (2018). In this way
higher precision and sensitivity can be reached when searching for new physics
or when determining the $V_{ud}$ quark mixing matrix element in experiments in
$\beta$-decay correlation measurements with the mirror nuclei.

The observation that for non-analog allowed $\beta$ transitions the $b/Ac$
value is approximately nucleus independent, within certain limits, enables
estimating its effect on $\beta$ decay correlation results for allowed
transitions for which no experimental nor theoretical data leading to $b/Ac$
are readily available. This was done already in the analysis of e.g. the
$\beta$-asymmetry parameters in the decay of 114In Wauters  _et al._ (2009)
and the recoil-asymmetry parameter in the decay of 80Rb Pitcairn  _et al._
(2009). The present analysis now offers for non-analog transitions, for which
one cannot rely on CVC, the value of $b/Ac=5.4\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 3.1$ obtained as an average over a large number of
$\beta$ transitions for nuclei with masses ranging from $A=$ 3 to 61 (Table
28).