Number of states with given spinIand isospinTfornfermions in ajorbit

Determining the number of states with a given spinIfor identical nucleons in a single-jshell [denoted by $ D_I(j, n) $ ] is a common practice in nuclear structure theory and atomic physics. In the nuclear shell model, $ D_I(j, n) $ is usually tabulated for relatively smalljandn. In cases with largerjorn, $ D_I(j, n) $ might be obtained by subtracting the number of states with total angular momentum projection $ M = I + 1 $ from that with $ M = I $ [1], whereMequals $ m_1 + m_2 + \cdots + m_n $ , and $ m_i $ [ $ i=1, 2, \cdots, n $ ] is the projection ofjon thez-axis for thei-th nucleon. Other recipes to extract $ D_I(j, n) $ include Racha's formulas in terms of the seniority scheme [2] and generating function method studied extensively by Katrielet al.[3] and Sunko and collaborators [4–6].

The first explicit and compact formula of $ D_I (j, n) $ was obtained specifically for $ I = 0 $ and $ n = 4 $ by Ginocchio and Haxton while studying the fractional quantum Hall effect [7]. In Ref. [8], $ D_I (j, n) $ was constructed empirically for $ n=3 $ and 4, and for a fewIvalues with $ n=5 $ . The formulas for $ n=3 $ was soon proved by Talmi, who introduced his recursive formulas [9]. These formulas were later derived based on the reduction rule fromSU(4) toSO(3) group [10]; the formula for $ n=4 $ was derived by the reduction rule fromSU(5) to theSO(3) group [11], with a demonstration that $ D_I (j, n) $ can be actually derived based on the reduction rule fromSU(n+1) toSO(3) group. The Ginocchio-Haxton formula of $ D_0 (j, 4) $ was also revisited by Zamick and Escuderos [12]. An explicit recursion formula from $ n - 1 $ particles tonparticles was obtained in Ref. [13] by introducing "pseudo" particles which allow fermions take integer spins. In Refs. [14,15], the number of states, denoted by $ D_I (j, n) $ , was applied to derive sum rules of six-jand nine-jsymbols, some of which were also revisited in Ref. [16]. In the last decade, Pain and collaborators extensively studied the odd-even staggering of $ D_I (j, n) $ [17,18], and compact formulas of $ D_I (j, n) $ for $ n=3, 4 $ , and 5 [19]. The enumeration of the number of states with given spin was also extended to boson systems in Ref. [20]. The study of $ D_I (j, n) $ motivated a number of studies related to a single-jshell; here, we mention Refs. [21–34] without providing further details for completeness.

Given that the nuclear shell model Hamiltonian respects the isospin symmetry, it is natural to generalize the enumeration of $ D_I(j, n) $ to the number of states with given spinIand isospinT, denoted by $ D_{IT} (j, n) $ , for nucleons in a single-jshell. In Ref. [35], Zamick and Escuderos found a few simple relations between $ D_{IT} (j, 4) $ with $ T=0 $ and $ D_{IT} (j, 4) $ with $ T=2 $ . In Ref. [36], compact and explicit formulas of $ D_{IT} (j, n) $ for $ n=3 $ and 4 were derived in terms of sum rules of six-jand nine-jsymbols involving the expression of $ D_I (j, n) $ given in Ref. [11].

Similar to the enumeration of $ D_I(j, n) $ , $ D_{IT} (j, n) $ can be obtained in terms of the number of states with given spin projectionMand isospin projection $ M_T $ . This number is denoted by $ N_{M M_T} (j, n) $ . Fornfermions in a single-jshell,

In this study, we investigate $ D_{IT} (j, n) $ in terms of $ N_{M M_T} (j, n) $ , which is obtained by generalizing Talmi's recursion formula of $ N_M(j, n) $ , in which isospin is not considered. Compact formulas of $ D_{IT} (j, n) $ are derived for $ n=3 $ as an exemplification. The rest of this paper is organized as follows. In Sec. II, we generalize Talmi's formulas for $ N_M (j, n) $ by further considering the isospin symmetry, and present recursion formulas of $ N_{M M_T} $ . In Sec. III, we apply our generalized recursion formulas of $ N_{M M_T} $ and extract $ N_{M M_T} (n, j) $ for $ n = 3 $ . In Sec. IV, $ D_{IT} (j, 3) $ is derived by using Eq. (1). In Sec. V, we summarize our study. In Appendix A, we provide an alternative mathematical proof of our generalized Talmi's recursion formula by using the generating function method. In Appendix B, we tabulate $ D_{IT}(j, n) $ with $ n=3 $ and 4, $ j =5/2, 7/2 $ , and 9/2.

We use the convention that thez-axis projection of the isospin for a neutron and a proton equals $ 1/2 $ and $ -1/2 $ , respectively. Fornnucleons in a single-jshell, we denote the projections of total spinIand total isospinT, respectively, asMand $ M_T $ . With this convention, the neutron number equals $ n/2 + M_T $ , and the proton number equals $ n/2 - M_T $ . Let us denote thez-axis spin projections of neutrons and protons by using ( $ m_{\nu_1}, m_{\nu_2},..., m_{\nu_{2j + 1}} $ ) and ( $ m_{\pi_1}, m_{\pi_2},..., m_{\pi_{2j + 1}} $ ), respectively, with the convention that $ j = m_{\nu_1} > m_{\nu_2} >... > m_{\nu_{2j + 1}} = -j $ for neutrons and $ j = m_{\pi_1} > m_{\pi_2} >... > m_{\pi_{2j + 1}} = -j $ for protons. We define $ n_{\nu_i} = 1 $ (or $ n_{\pi_i} = 1 $ ) for $ i = 1,..., 2j + 1 $ if the orbit of $ m_{\nu_i} $ (or $ m_{\pi_i} $ ) is filled; otherwise, it equals zero.

According to the above conventions, we have

and the maximum ofIequals

We denote the number of states with givenMand $ M_T $ fornnucleons in a single-jshell by using $ N_{M M_T}(j, n) $ . Clearly, we can divide the values of ( $ n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}} $ ) into sixteen cases, as tabulated inTable 1. Here, we exemplify them by using ( $ n_{\nu_1} = 1, n_{\nu_{2j + 1}} = 1, n_{\pi_1} = 1, n_{\pi_{2j + 1}} = 1 $ ). In this case, there are ( $ n - 4 $ ) nucleons distributed in orbits for which the absolute values of $ m_i $ values equal or are below ( $ j - 1 $ ). Consequently, the number of states with givenMand $ M_T $ , i.e., $ N_{M M_T}(j, n) $ , equals the number of states of thesameMand $ M_T $ , but with ( $ n - 4 $ ) nucleons in a ( $ j - 1 $ ) shell, namely, $ N_{M M_T}(j - 1, n - 4) $ . According to this classification, we are able to obtain $ N_{M M_T}(j, n) $ by using the values of $ N_{M M_T}(j', n') $ , where either $ j' $ is smaller thanj, or $ n' $ is smaller thann, or both $ (j',n') $ are smaller than $ (j, n) $ . In other words, fromTable 1we have

$ n_{\nu_1} $	$ n_{\nu_{2j + 1}} $	$ n_{\pi_1} $	$ n_{\pi_{2j + 1}} $	Number of states
$ 1 $	$ 1 $	$ 1 $	$ 1 $	$ N_{M, M_T}(j - 1, n - 4) $
$ 1 $	$ 1 $	$ 1 $	$ 0 $	$ N_{M - j, M_T - 1/2}(j - 1, n - 3) $
$ 1 $	$ 1 $	$ 0 $	$ 1 $	$ N_{M + j, M_T - 1/2}(j - 1, n - 3) $
$ 1 $	$ 1 $	$ 0 $	$ 0 $	$ N_{M, M_T - 1}(j - 1, n - 2) $
$ 1 $	$ 0 $	$ 1 $	$ 1 $	$ N_{M - j, M_T + 1/2}(j - 1, n - 3) $
$ 1 $	$ 0 $	$ 1 $	$ 0 $	$ N_{M - 2j, M_T}(j - 1, n - 2) $
$ 1 $	$ 0 $	$ 0 $	$ 1 $	$ N_{M, M_T}(j - 1, n - 2) $
$ 1 $	$ 0 $	$ 0 $	$ 0 $	$ N_{M - j, M_T - 1/2}(j - 1, n - 1) $
$ 0 $	$ 1 $	$ 1 $	$ 1 $	$ N_{M + j, M_T + 1/2}(j - 1, n - 3) $
$ 0 $	$ 1 $	$ 1 $	$ 1 $	$ N_{M, M_T}(j - 1, n - 2) $
$ 0 $	$ 1 $	$ 0 $	$ 1 $	$ N_{M + 2j, M_T}(j - 1, n - 2) $
$ 0 $	$ 1 $	$ 0 $	$ 0 $	$ N_{M + j, M_T - 1/2}(j - 1, n - 1) $
$ 0 $	$ 0 $	$ 1 $	$ 1 $	$ N_{M, M_T + 1}(j - 1, n - 2) $
$ 0 $	$ 0 $	$ 1 $	$ 0 $	$ N_{M - j, M_T + 1/2}(j - 1, n - 1) $
$ 0 $	$ 0 $	$ 0 $	$ 1 $	$ N_{M + j, M_T + 1/2}(j - 1, n - 1) $
$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ N_{M, M_T}(j - 1, n) $

Table 1.Number of states corresponding to the available values of $ n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}} $ .

Here, the summation over $ \left\{n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}}\right\} $ refers to the sixteen cases listed inTable 1. Note that the above formula holds also for $ n = 1, 2, 3, 4 $ , with the convention that $ N_{M M_T}(j, n) = 0 $ if $ n < 0 $ and $ N_{M M_T}(j, 0) = \delta_{M, 0} \delta_{M_T, 0} $ . Note also that Eq. (3) can be proved alternatively based on the generating-function technique described in Ref. [3], as presented in Appendix A.

The value of $ N_{M M_T}(j, n) $ with $ n=1 $ or 2 is trivially determined as follows. For $M_T = \pm {1}/{2}$ and $ |M| \leq j $ , $ N_{M, M_T = \pm \frac{1}{2}}(j, 1) = 1 $ ; otherwise, it equals zero. For $ n=2 $ with $ M_T = \pm 1 $ , the highest value ofMequals $M_{\rm max} = 2j - 1$ and for $ M_T = 0 $ , $ M_{\rm max} = 2j $ . For $ M_T = \pm 1 $ and $ M \leq 2j - 1 $ ,

where $ " \left \lfloor "\; {\rm and} \; \; "\right \rfloor" $ means taking the largest integer without exceeding the value inside; for $ M_T = 0 $ and $ M \leq 2j $ ,

Otherwise, $ N_{M M_T}(j, 2) = 0 $ . With these results, we can construct explicit formulas of $ N_{M M_T}(j, 3) $ by using Eq. (3).

We first discuss $ N_{M M_T}(j, 3) $ with the requirement that $ M_T = \frac{1}{2} $ and $ j < M \leq M_{\rm max} $ . We recursively apply Eq. (3) to this term and obtain the value of $ N_{M M_T}(j, n) $ :

where $ \sideset{}{^{\prime}}\sum $ denotes summations that exclude the case $ \left\{n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}}\right\} = \left\{0, 0, 0, 0\right\} $ and decouple the term $ N_{M, M_T}(j - 1, n) $ [the last term in the summation of the sixteen terms ofTable 1];lis fixed with the requirement that $ N_{M M_T}(j - l, n) = 0 $ while $ N_{M M_T} (j - l + 1, n) \neq 0 $ , which means that

or equivalently,

For $ n=3 $ and $ M_T= 1/2 $ , the value oflbecomes

According to Eq. (6), with the conditions $ N_{M M_T}(j - l, n) = 0 $ and $ N_{M M_T} (j - l + 1, n) \neq 0 $ , we have

On the right hand side of the above formula, there are many terms that break the requirement of Eq. (2) for $ n \leq 2 $ . Let us exemplify this by using two cases: $ \left\{n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}}\right\} = \left\{0, 0, 0, 1\right\} $ and $ \left\{1, 1, 1, 0\right\} $ . For the first case, $ \left\{n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}}\right\} = \left\{0, 0, 0, 1\right\} $ , the corresponding contribution to $ N_{M, M_T = \frac{1}{2}} (j, 3) $ on the right hand of Eq. (10) is $ N_{M + (j - i), M_T = 1}(j - i - 1, 2) $ . Given that $ M > j $ here, the value of $ M + (j - i) $ is larger than the maximum $ M_{\rm max} $ of two nucleons in a single $ j - i - 1 $ shell, i.e., $ M_{\rm max} = 2j -2i -2 $ . Similarly, for $ \left\{n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}}\right\} = \left\{1, 1, 1, 0\right\} $ , the corresponding contribution on the right hand side is $ N_{M - (j - i), M_T = 0} (j - i - 1, 0) $ . Given that $ M - (j - i) > 0 $ , $ N_{M - (j - i), M_T = 0} (j - i - 1, 0) = \delta_{M - (j - i), 0} = 0 $ . As a result, we obtain

The summation of $ \left \lfloor \dfrac{3j - 3i - 1 - M}{2} \right \rfloor $ is tedious but straightforward. One has

When $ 3j - M = 6k + 1 $ , the summation becomes

This summation for the case $ 3j - M = 6k + 2, 3, 4, 5, 6 $ can be obtained similarly, as follows:

Substituting these results into Eq. (10), we obtain

where $\eta(3j - M) = \left(0, \dfrac{1}{4}\right)$ if $ \mathrm{mod} (3j - M, 2) = (0, 1) $ , respectively.

Let us now address $ N_{M, M_T = \frac{1}{2}}(j, 3) $ with $ T_z = 1/2 $ and $ 0 < M \leq j $ . In this case, Eq. (3) is applied recursively for $ j - M + 1 $ times. As a result, we have

where, again, $ \sideset{}{^{\prime}}\sum $ denotes summation among the sixteen sets of $ \left\{n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}}\right\} $ excluding $ \left\{n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}}\right\} = \left\{0, 0, 0, 0\right\} $ . For $ M_T = \frac{1}{2} $ and $ n = 3 $ , the above formula is reduced to

where the first term " $ 1 $ " is given by $ N_{M = 0, M_T = 0}(j - i - 1, 0)= 1 $ with $ \left\{n_{\nu_1}, n_{\nu_{2j + 1}}, n_{\pi_1}, n_{\pi_{2j + 1}}, i\right\} = \left\{1, 1, 1, 0, j - M\right\} $ ; the second term $ N_{M, M_T = \frac{1}{2}}(M - 1, 3) $ is given in Eq. (12) with $ j = M - 1 $ , i.e.,

The summations in Eq. (14) equal zero when the upper limit of the given summation index is larger than its lower limit. On the right hand side, the summations of $ \left \lfloor \frac{j - i + M - 1}{2} \right \rfloor $ and $ \left \lfloor \frac{j - i - M - 1}{2} \right \rfloor $ are classified into two cases, $ j - M = 2k $ and $ 2k + 1 $ . For $ j - M = 2k $ , we obtain

The summation with $ j-M = 2k+1 $ is obtained similarly. The results of these two cases are unified as follows:

Following a similar procedure, we obtain

Substituting Eqs. (15)–(17) into Eq. (14), we obtain $ N_{M, M_T = \frac{1}{2}}(j, 3) $ with $ 0 < M \leq j $ in a compact form:

The results for $ N_{M M_T}(j, n) $ are reduced to $ N_{M} (j, n) $ for identical particles, and Eq. (3) is reduced to Talmi's recursion formula. For this case, $ N_{M, M_T = \frac{3}{2}}(j, 3) $ was obtained in Ref. [19], which we cite for completeness. For $ j < M \leq M_{\mathrm{max}} $ , according to Eqs. (2.11a)–(2.11b) in Ref. [19],

where

for $ \mathrm{mod}(3j - M, 6) = 0, 1, 2, 3, 4, 5 $ , respectively. For $ 0< M \leq j $ , according to Eqs. (2.19a)-(2.19b) in Ref. [19],

where

for $ \mathrm{mod}(M - \dfrac{1}{2}, 3) = 0, 1, 2 $ , respectively.

Provided the results of $ N_{M M_T}(j, 3) $ , i.e., Eqs. (12), (18), (19), and (20), we can readily substitute them into Eq. (1) to obtain the number of states with givenIandT, i.e., $ D_{I T}(j, 3) $ .

We first consider the case $T = {1}/{2}$ . When $ I > j $ and $T = {1}/{2}$ , we obtain

According to Eqs. (12) and (19), the above formula is reduced to

For $ I = j $ and $ T = {1}/{2} $ ,

By using Eqs. (12), (18), (19), and (20), we have

Similarly, for $ 0< I < j $ and $T = {1}/{2}$ ,

By using Eqs. (18) and (20), the above formula is reduced to

Eqs. (24)–(26) can be rewritten as follows. For $ I \geq j $ and $T = {1}/{2}$ ,

and for $ 0 < I \leq j $ and $T = {1}/{2}$ ,

These results [Eqs. (27)–(28) in this paper] are consistent with Eqs. (25)–(26) in Ref. [36], which were obtained in terms of sum rules of six-jsymbols, except that the formulas derived in this paper are expressed in a more transparent and understandable form.

The $T = {3}/{2}$ case is much simpler than the $ T=1/2 $ case. One uses Talmi's recursion formulas for $ N_{M=I, \frac{3}{2}} (j, n) $ and obtains $ D_{I, T = \frac{3}{2}}(j, 3) $ straightforwardly. The resulting formulas are consistent with those given in Ref. [8] and Ref. [19]. We show them next for completeness. According to Eq. (1) in Ref. [8], for $ 0< I \leq j $ and $T = {3}/{2}$ ,

and according to Eq. (2) in Ref. [8], for $ I \geq j $ and $T = {3}/{2}$ ,

where

In principle, one can follow the above procedure as explained in Eqs. (1)–(3) for largernvalues. However, this procedure becomes formidably tedious with lengthy formulas and tables for largernvalues. Instead of those formulas, we calculate $ D_{IT}(j, n) $ for $ n=3 $ and 4 and $ j=5/2 $ , 7/2, and 9/2 and tabulate them in Table B1 and Table B2 in Appendix B.

To summarize, in this paper we generalize Talmi's recursion formula of the number of states with given spinIby further consideration of the isospin symmetry. This generalization is also proved alternatively based on the generating function method. We also exemplify the generalized Talmi's recursion formulas obtained in this paper by applying our generalized Talmi's recursion formulas in constructing the number of states with given spin and isospin of three nucleons in a single-jshell.

In this Appendix, we present an alternative proof of the recursion formula presented in Eq. (3) based on the generating-function method.

A generating function is a polynomial function for which the expansion coefficients are related to the number of states with a given requirement. We define the generating function corresponding to $ N_{M M_T}(j, n) $ , similar to that in Ref. [17],

which yields

with

The generating function is expanded in powers ofzas follows:

with

Using the Leibniz formula, one obtains

By using Eqs. (A1), (A5), the above relation yields

Given that $ g_{j}(x, y, z) $ is a fourth-order polynomial, the only derivatives that are nonzero correspond to

Eq. (A7) holds for any independent variablesxandy; therefore, the coefficients of homogeneous terms on both sides are equal, which immediately leads to Eq. (3).

In this Appendix, we tabulate $ D_{IT} (j, n) $ with $ j=3/2 $ – 9/2 andn=3–4, for convenience.

j	T	$ I^D $
$ \frac{5}{2} $	$ \frac{1}{2} $	$ (\frac{1}{2})^{1}, (\frac{3}{2})^{1}, (\frac{5}{2})^{2}, (\frac{7}{2})^{2}, (\frac{9}{2})^{1}, (\frac{11}{2})^{1}, (\frac{13}{2})^{1} $
$ \frac{5}{2} $	$ \frac{3}{2} $	$ (\frac{3}{2})^{1}, (\frac{5}{2})^{2}, (\frac{9}{2})^{1} $
$ \frac{7}{2} $	$ \frac{1}{2} $	$ (\frac{1}{2})^{1}, (\frac{3}{2})^{1}, (\frac{5}{2})^{2}, (\frac{7}{2})^{3}, (\frac{9}{2})^{2}, $
		$ (\frac{11}{2})^{2}, (\frac{13}{2})^{2}, (\frac{15}{2})^{1},(\frac{17}{2})^{1}, (\frac{19}{2})^{1} $
$ \frac{7}{2} $	$ \frac{3}{2} $	$ (\frac{3}{2})^{1}, (\frac{5}{2})^{2}, (\frac{7}{2})^{3}, (\frac{9}{2})^{1}, (\frac{11}{2})^{1}, (\frac{15}{2})^{1} $
$ \frac{9}{2} $	$ \frac{1}{2} $	$ (\frac{1}{2})^{1}, (\frac{3}{2})^{1}, (\frac{5}{2})^{2}, (\frac{7}{2})^{3}, (\frac{9}{2})^{3}, (\frac{11}{2})^{3}, $
		$ (\frac{13}{2})^{3}, (\frac{15}{2})^{2}, (\frac{17}{2})^{2}, (\frac{19}{2})^{2}, (\frac{21}{2})^{1}, (\frac{23}{2})^{1}, (\frac{25}{2})^{1} $
$ \frac{9}{2} $	$ \frac{3}{2} $	$ (\frac{3}{2})^{1}, (\frac{5}{2})^{1}, (\frac{7}{2})^{1}, (\frac{9}{2})^{2}, (\frac{11}{2})^{1}, (\frac{13}{2})^{1}, (\frac{15}{2})^{1}, (\frac{17}{2})^{1}, (\frac{21}{2})^{1} $

Table B1. $D_{IT}(j,\,n) $ forj=5/2-9/2, withn=3.I^Din this Table represents that there areDstates with spinIfor thejⁿconfiguration.

j	T	$ I^D $
$ \frac{5}{2} $	$ 0 $	$ 0^{2}, 2^{3}, 3^{1}, 4^{3}, 5^{1}, 6^{2}, 8^{1} $
$ \frac{5}{2} $	$ 1 $	$ 1^{2}, 2^{2}, 3^{3}, 4^{2}, 5^{2}, 6^{1}, 7^{1} $
$ \frac{5}{2} $	$ 2 $	$ 0^{1}, 2^{1}, 4^{1} $
$ \frac{7}{2} $	$ 0 $	$ 0^{3}, 2^{4}, 3^{2}, 4^{5}, 5^{2}, 6^{5}, 7^{2}, 8^{3}, 9^{1}, 10^{2}, 12^{1} $
$ \frac{7}{2} $	$ 1 $	$ 1^{3}, 2^{3}, 3^{5}, 4^{4}, 5^{5}, 6^{4}, 7^{4}, 8^{2}, 9^{2}, 10^{1}, 11^{1} $
$ \frac{7}{2} $	$ 2 $	$ 0^{1}, 2^{2}, 4^{2}, 5^{1}, 6^{1}, 8^{1} $
$ \frac{9}{2} $	$ 0 $	$ 0^{3}, 2^{6}, 3^{2}, 4^{7}, 5^{4}, 6^{7}, 7^{4}, $
		$ 8^{7}, 9^{3}, 10^{5}, 11^{2}, 12^{3}, 13^{1}, 14^{2}, 16^{1} $
$ \frac{9}{2} $	$ 1 $	$ 1^{4}, 2^{4}, 3^{7}, 4^{6}, 5^{8}, 6^{7}, 7^{8}, $
		$ 8^{6}, 9^{6}, 10^{4}, 11^{4}, 12^{2}, 13^{2}, 14^{1}, 15^{1} $
$ \frac{9}{2} $	$ 2 $	$ 0^{2}, 2^{2}, 3^{1}, 4^{3}, 5^{1}, 6^{3}, 7^{1}, 8^{2}, 9^{1}, 10^{1}, 12^{1} $

Table B2.The same asTable B1except for $ n=4 $ .