ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 5, с. 603-609
ЯДРА
INTERSHELL BETA- AND GAMMA-DECAY TRANSITION RATES IN THE MULTIPARTICLE SHELL MODEL
©2014 V. I. Isakov*
Petersburg Nuclear Physics Institute, Gatchina;
NRC Kurchatov Institute, Moscow, Russia Received May 29, 2013; in final form, October 31,2013
In the framework of the two-group configuration model we obtain formulas for the reduced transition rates for beta and gamma transitions in even—even, odd—odd, even—odd, and odd—even nuclei. We explored dependencies of the transition rates on the occupancies of the involved subshells, as well as on the spin values of the initial and final states. The obtained formulas are useful for the qualitative spectroscopic analysis of experimental data, particulary in the regions of magicity, including the regions of the "remote" nuclei.
DOI: 10.7868/S0044002714050092
Study of beta and gamma transitions is one of the powerful methods of nuclear spectroscopy which enables us to define quantum characteristics of nuclear states implicated in the decay. The magnitude of the corresponding reduced transition rate B(\) depends on the multipolarity of the transition, as well as on the structure of the wave functions of the initial and final states. This structure in turn depends on the number of active neutrons and protons, on the effective interaction between them, and on the values In of the involved states. We have the most definite knowledge on the structure of states for nuclei that are not very far from the closed shells, where at a first approximation it is described in terms of the multiparticle shell model with no configuration mixing. This approach offers a good reference point for precise calculations, see [1], where calculations for some nuclei just close to N = Z were performed in the representation of the total isospin, and is useful for qualitative evaluations. The corresponding formulas give us an opportunity to reveal the dependence of the reduced transition rates on the occupancies of single-particle levels and their single-particle characteristics, as well as on the values of the total angular momenta of the involved nuclear states.
In case of beta decay of spherical nuclei that are considered here, two nljtz orbitals are involved in the transition. For example, in [+ decay we have ji(p) ^ j2(n). In the general case, we have the transformation between the configurations
\i) = \jn(siaiJi),jn2(s2«2J2); Ii)a . (1)
\f ) = jini-1(s'1a'1J'1 )j+l(s'2aJ); I
f.
Here, we used the jj-coupling scheme and the neutron—proton representation, where j1(p) = n1,l1, ji,tz1 = -1/2, while j2 (n) = n2,l2,j2 ,tz2 = +1/2. In (1) "s" is seniority (number of unpaired particles), while "a" is an additional quantum number (if it is necessary for unambiguous classification of state). The proposed structure of the wave functions is a very good starting point in the case of short-range attraction between the identical nucleons. Wave functions \i) and \f) are antisymmetric with respect to all permutations, including those between particles of the {ji} and {j2} groups.
To obtain formulas for reduced transition rates we use the formalism of the fractional parentage decompositions. The corresponding algebra in the case of one-group configuration is represented in [2], while the necessary formalism for the two-group configurations one can find in [3—5] (in [3] for the case of LS coupling).
By making the necessary recouplings and permutations, we can represent the functions \i) and \ f) in the form, that is suitable for calculation of matrix elements of the single-particle transition operators:
\i) = jn1 (siaiJi).jn2 (S2a2J2); Ii )a = (2)
E
S3 a3J3J'
(-1)
П2 +JI+J2+J1+J'
4(2Ji + 1)(2J + 1) IJ3 ji Ji
E-mail: Visakov@thd.pnpi.spb.ru
(ni + П2)
Ii J2 J'
a
X
X
X
fni-1(ssasJs)jl Jj (slalJl) jHl-i(saa3 J3 (s2a2J2))a J J, ji; h)
+
+ ^ (-1)
Jl +J2-Ii
s4^4 J4J
(-1)
j4+j1-j
X (— 1)J'2 +J1+J2 + J x
n2(2J2 + 1)(2J + 1) I J4 h 'h
(ni + n2)
Ii Ji J
while
\f ) =
jn2 1(s4aJ)j2J2\}jn2(s2a2J2) x jit1 (siai Ji), jn2-l(sAaAJA)^a J, j2; Ii)
jn1 -i(sia'iJ[),j^2+i(s'2a'2J2); If ) =
/ a
= ( — 1)n2 + i+jl + J2 + J'l + J' x
s3a3J3J '
'(ni — 1)(2Ji + 1)(2J' + 1) I J3 ji Ji
(ni + n2)
if J2 J'
jni-2(ssasJs)jiJ[\}jni-i(s'i aJ )
jni-2(ssa3Js ),jn2+i (s^J ))o J ',j i ; If
+ ^ (—1)J1+J2-if —1)J4+J1 -J x s40.4 J4 J
X (— 1)j2 +J1+J2 +J x
+
x /(n2 + 1)(2J'2 + 1)(2J + 1) I J4 j2 4 > x V (n1+n2) j/ J
jn2(s^aJjA\}jn2+i(s^aJ) x
jn-i(si aJ ),jn (staJ)) J,j2 ; If) .
Here,
jn- i (s' a'J ')jJ\}jn saJ
are the single-
(If Mf \rn(Xf)\IlMl)
= (—1)
If-Mf
If X Ii
(If\\r(X)\\Ii ),
—Mf f Mi (If\\m(X)\\Ii ) = (—1)Ii-If (Ii\\m(X)\\If ),
while the reduced transition rate is
=<!»!. (5,
Then, by using formulas (2)—(5) and the Racah algebra, we obtain the formula for the f3 + decay reduced transition rate of the multipolarity A:
B(X; Ii ^ If) = n 1 (n2 + 1)(2Ji + 1)x (6)
x J + 1) (2If + 1) (2j 1 + 1) x
jn1- 1 (si al J )jlJl j (s aJ)
j! (s2a2J2)j2J2\}j?+i (s^J )
/ ^ 2
x <
(3)
J1 J2 Ii J1' J2' If j1 j2 X
> Bsp(X; ji ^ j2).
Here, Bsp(X; j1 ^ j2) is the reduced transition rate for the single-particle transition. Formula (6) is applicable for any values of the entering parameters, but it is not visual, as in a common case the coefficients of fractional parentage are defined from the numerical calculations (or borrowed from the tables, see [6]). However, for some simple cases it is possible to represent the entering fractional parentage coefficients in an obvious algebraic form. In this way, we find from [2] that
jn-i(si = 1, Ji = j)j J = 0\}jn(s = 0,J = 0)
n is even,
and
jn-i(si, Ji even)j J = j\}jn(s = 1 J = j)
= 1, (7)
(8)
particle fractional parentage coefficients (fpc) of the configuration \jn). We define the reduced matrix element by the relation
n is odd.
On the other hand, it follows from [6] that j 2j+1-n (si aiJi)jJ\}j 2+2-n (saJ)
= (—i)Ji+3-J
(9)
(4)
n
2j + 2 — nV 2J + 1
2Ji + 1
jn i(saJ)jJi|}jn(siaiJi) From (8) and (9) we find that jn-i(s = 1 J = j)jJi\}jn(s = 2, Ji =0 even)
x
X
2
X
X
X
X
X
X
X
12(2j + 1 -n) n(2j - 1)
_ ni(n2 + 1) ~ (2J2 + 1)
n is even.
From formulas (6)—(10), we obtain the expressions for the transition rates in cases when the entering values of seniorities are not more than two (note that additional quantum numbers "a" are absent here). These cases are of the most interest as the corresponding nuclear states have the lower energies.
1. At first, we consider transitions, when both n1 and n2 are even. Here arise options:
For Ji = J2 = si = S2 = Ii = 0, and J[ = ji, J2 = j2, si = s'2 = 1, If = A, we have
B(A; Ii = 0 - If = A) = ni(2j2 + 1 - n2)
(11
(2j2 + 1)
Bsp(A; ji - j2).
(2j2 - 1)
A j2 J2
Bsp(A; ji - j'2).
2(2ji + l-rn)(2j2 + l-w2) (2j! - l)(2j2 + 1)
(2ji + 1) x
If Ji = si = 0, J2 = j2, s2 = 1, Ii = j2, and Ji = = ji. s'i = 1, J2 = 0 (even), s'2 = 2, then we obtain
B(A; Ii = j2 - If)= (16)
2n1(2j2 - n2)
(2j2 - 1)(2j2 + 1)
(2J2 + 1)(2If + 1) x
For Ji = si =0, J2 = 0 (even), s2 = 2, Ii = J2, and J = ji J2 = j2, si = s'2 = 1. we have
B(A; Ii = J2 - If)= (12)
^ 2
2nin2(2If + 1) I j2 If ji
xJ A Bsp(A; ji - j2>-
For Ji = 0 (even), si = 2, J2 = j2, s2 = 1, and J = ji, si = 1, J2 = s2 = 0, If = ji, we have
B (A; Ii - If = ji)= (17)
2(2ji + 1- ni)(n2 + 1),
(2ji - 1)(2j2 + 1)
-(2Ji + l)(2ji + 1) x
2
If Ji =0 (even), si = 2, J2 = s2 = 0, Ii = Ji, and J[ = ji, J2 = j2, si = s'2 = 1, we obtain
B(A; Ii = Ji - If)= (13)
I ^ Bsp(A; ji - j2).
At last, for J1 = 0 (even), s1 = 2, J2 = j2, s2 = = 1, and J[ = ji si = 1, J2 =0 (even), s'2 = 2, there follows the result
I3(A; /j —»If) = 1 ~ ^ x
x (2If + 1)i ji If Bsp(A; ji - j2).
1 A ji Ji
Finally, if both proton and neutron groups in the initial state have seniorities two, we have (J1 =0 (even), s1 = 2, J2 = 0 (even), s2 = 2, and J = j1, J2 = j2, si = s2 = 1)
1 ' f) (2ji - l)(2j2 - 1) ^ j x (2Ji + 1)(2J2 + 1)(2ji + 1)(2If + 1) x ( \2 J1 J2 Ii x < j1 j2 I^ Bsp(A; ji - j2).
j j2 A ^
2. Now, consider the case when n1 is even, while n2 is odd. Here, we have the following options:
For Ji = si = 0, J2 = j2, s2 = 1, Ii = j2, and Ji = ji, si = 1, J2 = s2 = 0, If = ji, we have
B(A; Ii = j2 - If = ji)= (15)
(2ji - 1)(2j2 - 1) x (2Ji + 1)(2J2 + 1)(2ji + 1)(2If + 1) x Ji j2 Ii
(18)
X <
ji J2 If
ji j2 A
> Bsp(A; ji - j2).
3. Consider now cases when n1 is odd, while n2 is even.
If Ji = ji, si = 1, J2 = s2 = 0, Ii = ji, and Ji = = si = 0, J2 = j2, s2 = 1, If = j2, then
B(A; Ii = ji - If = j2)= (19)
(2ji +2- ni)(2j2 + 1- n2)
(2 ji + 1)(2j2 + 1)
Bsp(A; ji - j2).
For Ji = ji, si = 1, J2 = s2 = 0, Ii = ji, and Ji =0 (even), si =2, J2 = j2, s'2 = 1, we obtain
B(A; Ii = ji - If )= (20)
= 2(m - 1)(2j2 + 1 - n2)(2J( + 1)(2// + 1) (2j1-1)(2J2 + 1)
xJ IT B!p(A" - j2>-
2
For J1 = j1, s1 = 1, J2 = 0 (even), s2 = 2, and Ji = 0, s1 = 0, J2 = j2, s'2 = 1, If = j2, we have
B(X; Ii — If = j2)= (21)
_ 2(2ji + 2 -m)n2(2J2 + 1)
(2J2 " 1)
* J » jS Bsp(A; j1 - j2).
If J1 = j1, s1 = 1, J2 = 0 (even), s2 = 2, and J[ =0 (even), s1 =2, J2 = j2, s'2 = 1, we obtain
B(X; Ii
r 4(m - 1)n2(2J[ + 1)(2J2 + 1)(2If + 1)(2j1 + 1)
If) = -—-ttt^--- X
(2j1 - 1)(2j2 - 1)
(22)
^ 2
x <
j1 J2 Ii J1 j2 If j1 j2 X
> Bsp(X; j1 — j2).
—►
/
4. After all, we consider cases when both n1 and n2 are odd.
For J1 = j1, s1 = 1, J2 = j2, s2 = 1, Ii = X, and J[ = J!2 = s'1 = s2 = If = 0, we have
B(X; Ii = X — If = 0)= (23)
= 1 (n2 + l)(2ji +2 -m)
(2A + 1) (2j2 + 1)
x Bsp (X; j1 — j2).
B(X; Ii — If = J[)= (24)
2(m-l)(ri2 + l)(2Ji + l)(2ji + l)
(2j1-l)(2j2 + l)
x < j1 j Ii \ Bsp(X; j1 — h). X J1 j1 I
,„ . -t j ■ , J, ,n/ x For J1 = j1, s1 = 1, J2 = j2, s2 = 1, and J1 = 0,
If J1 = 3U s1 = 1, J2 = J2, s2 = 1 and J1 =0 (even), _/ = 0 j, =0 (even) _/ = 2 t = j, we have1 s1 = 2, J> = 0, s2 = 0, If = J1, then s1 = 0 J =0 ^^ s2 = 2, lf = J, we have
B(X; Ii — If = J2)
2(2j1 + 2 - n{)(2j2 - n2)J + 1) ( j2 j1 Ii
(2j2 - 1)
X J2 j2
Bsp (X; j1 —
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.