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ISSN 1063-7788, Physics of Atomic Nuclei, 2007, Vol. 70, No. 5, pp. 848–854. c Pleiades Publishing, Ltd., 2007.
Original Russian Text c L.I. Galanina, N.S. Zelenskaja, A.K. Morzabaev, 2007, published in Yadernaya Fizika, 2007, Vol. 70, No. 5, pp. 882–888.
NUCLEI
Theory
Role of Various Mechanisms in the Formation of a 12C Nucleus
in the Reaction 13C(3He, α)12C
L. I. Galanina1)* , N. S. Zelenskaja1), and A. K. Morzabaev2)
Received March 21, 2006; in final form, October 5, 2006
Abstract—The contributions of various mechanisms to the production of the final nucleus 12 C in
the reaction 13 C(3He, α)12 C are estimated. These are neutron stripping and the transfer of the
heavy cluster 9Be in the pole approximation or via a sequential transfer of virtual 8Be and a
neutron. It is shown that the sequential mechanism of heavy-cluster transfer must be taken into
account in order to describe correctly experimental data over the whole angular range.
PACS numbers: 24.10.-i
DOI: 10.1134/S1063778807050067
In this study, we perform a comprehensive analysis of
1. INTRODUCTION
Investigation into mechanisms of nuclear reactions on light nuclei remains one of the urgent
challenges in intermediate-energy nuclear physics. In
particular, it is of interest to study processes for which
it is assumed that they are governed by mechanisms
more complex than the one-step transfer of a nucleon
13 3
12
or a group of nucleons. The reaction C( He, α) C
+
+
(0 ; 2 , 4.443 MeV) is one of such processes. In [1,
2], where spin-tensor components of the density
12
matrix for the findal-state nucleus C
+
in the 2 state at 4.443 MeV were reconstructed at
projectile energies of about 7 to 10 MeV per
nucleon on the basis of correlation measurements,
it was shown that the neutron-pickup mechanism is
insufficient for describing the reaction in question. It
turned out that the components of rank k = 4 are
quite sizable, but they must vanish in the case of
the neutron-pickup mechanism, where the angularmomentum transfer is l = 1 [3]. In analyzing the
reaction in question, Lebedev et al. [2] used the
coupled-channel method, which takes into account
12
the reaction 13C(3He, α)12 C (0+ ; 2+, 4.443 MeV) for the
first time, taking into account not only the one-step
mechanisms of neutron pickup and the exchange of the
heavy cluster 9Be but also the mechanism of the
sequential transfer of virtual 8Be and a virtual neutron.
The two-step mechanism allowing for the delay of
interaction (box diagram) corresponds to the second
order of perturbation theory. The role of this mechanism
is determined by the structural features of the nuclei
involved in the reaction.
The ensuing exposition is organized as follows.
The general formulas used to calculate the amplitude
13 3
12
+
+
of the reaction C( He, α) C (0 ; 2 , 4.443 MeV)
for various mechanisms are presented in Section 2. In
Section 3, we discuss the results of our calculations
for the differential cross section with respect to angles
and for the spin-tensor components of the density
12
matrix for the final nucleus
C and compare these
results with experimental data. Intermediate formulas
used to calculate the matrix element for the two-step
mechanism corresponding to the box diagram are
presented in the Appendix.
the collective nature of the C nucleus and, hence,
the possibility of multistep (but not different-time)
nucleon exchange in the formation of this nucleus,
whereby they were able to relax the selection rule in
the angular-momentum transfer l. Their calculations
permitted qualitatively describing experimental data
for the forward hemisphere but did not lead to perfect
agreement with experimental data.
1)
Institute of Nuclear Physics, Moscow State University, Vorob’evy
gory, 119992 Russia
2)
Gumilev National Eurasian University, Astana, Kazakhstan
* E-mail: [email protected]
2. FORMULAS
FOR CALCULATING THE PROPERTIES
OF THE REACTION
13 3
C( He, α)12C (0+ ; 2+, 4.443 MeV)
WITH ALLOWANCE FOR VARIOUS
MECHANISMS
Figure 1 shows the diagrams illustrating various
13 3
12
+
+
mechanisms of the reaction C( He, α) C (0 ; 2 ,
4.443 MeV). The first of them (Fig. 1a) corresponds to
the direct neutron-pickup process, in which case
848
ROLE OF VARIOUS MECHANISMS IN THE FORMATION OF A 12C NUCLEUS
(a)
12
(b)
α
ë
α
12
9
n
13
3
ë
α
He
849
(c)
8
Be
ë
ë
12ë
α
12
n
Be
3
ë
13
3
He13ë
He
Fig. 1. Diagrams illustrating various mechanisms of the reaction 13 C(3 He, α)12 C: (a) neutron-pickup process, (b) exchange
of the heavy cluster 9Be in the pole approximation, and (c) two-step transfer of virtual 8 Be and a virtual neutron.
l = 1. The last two diagrams illustrate the exchange of
9
the heavy cluster Be either in the pole approx-imation
(Fig. 1b) or via the two-step mechanism according to
8
which virtual Be and a virtual neutron are transferred
sequentially in time (box diagram in Fig. 1c).
There is the problem of assessing the degree to
which it is of importance to include the mechanism taking
into account delay in the calculation of the matrix
13 3
12
element for the reaction C( He, α) C. The energy
difference ∆E between virtual transferred par-ticles is
known to be the main criterion of the realiza-tion of
mechanisms taking into account delay that are
represented by the box diagram [4]. In our case, ∆E is
great both because of the high binding energy ε in the n +
3
∗
He → α vertex and because of a wide band ∆E of levels
8
of the virtual nucleus Be that are allowed by selection
8
rules. Therefore, the wave functions for the Be cluster
and the neutron overlap only slightly, with result that the
probability of their transfer to the target nucleus at
different instants is great. These arguments give reasons
to assume a priori that the contributions of the one-step
pole mechanism and the mechanism of sequential
13 3
12
transfer in the reaction C( He, α) C are commensurate.
The matrix elements for the one-step transfer of
particle C (pole diagrams in Figs. 1a and 1b) can be
calculated in the distorted-wave Born approximation for a
finite interaction range (DWBA–FIR) [3]. We denote the
“elementary” and the composite particle in the entrance
(exit) channel by x (y) and A = y + C (B = x + C ), respectively.
The DWBA–FIR matrix element in terms of the angularmomentum transfers J1 and J2 and the orbital-angularmomentum trans-fer l is then given by
M
(1)
DWBA
2 +2
(2J1 + 1)(2J2 + 1)(−1) Jx Jy −M1−J1
=
J1 J2 l
involved in the reaction (i = x, A, y, B), Λ1 and Λ2 are the orbital angular
momenta of cluster relative motion in the decay vertices for nuclei B
and A, βΛ1 Λ2l (θy ) are terms that contain the entire body
of information on the reaction mechanism and which
depend on the emission angle θy of final particles,
and the structural factors Θ
Λ1 Λ2 l
β
(θ )Θ
Λ1Λ2 l y J1 J2
_J1
− M1J2M2|lm_,
Λ1 Λ2
PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007
Λ1Λ2 l
are related to the
J1J2
_
_
B→x+C
and
reduced decay widths of the nuclei (ΘΛ1
A→x+C
Θ
).
Λ2
The kinematical factors βΛ1 Λ2l (θy ) are determined by
the double integral of the invariant form factor F Lx Ly x y
(r , r ) and the partial distorted waves
Λ1 Λ2 l
χLx (kx rx) and χLy (ky ry ) of the initial and final reaction channels
with respect to radial variables.
The expression for the matrix element M♦ under
the assumption of the mechanism of two-step sequential transfer (Fig. 1c) is obtained as the convolution of the matrix elements of the two pole mechanisms that arise upon cutting the complete box diagram. In general, this expression contains
additional integrals featuring a variable upper limit
and involves additional coherent summation over
the angular mo-menta of the intermediate system
[4]. For the reac-tion being considered, it is
impossible to simplify the general formulas for
calculating the relevant matrix elements [5] since
the main condition under which simplifications are
∗
possible, ∆E _ ε, does not hold here.
Thus, the matrix element M♦ for the two-step
9
mechanism of the transfer of the Be cluster is obtained as the convolution of the amplitudes for the two
mechanisms (neutron pickup and the exchange of the
8
× _Jx MxJ1M1|JB MB __Jy My J2M2|JA MA_
×
where Ji (Mi ) are the total spins (their projections) of the nuclei
heavy cluster Be) with the respective Green’s function. Two independent coordinate spaces arise upon
cutting the complete box diagram in Fig. 1c. Figure 2
shows the corresponding pole diagrams. Expanding
850
GALANINA et al.
(a)
(b)
y=α
α
ëρ
B=
σ
12
8
n
3
x = He
13
A= ë
12ë
12
ë
Be
σ α
ρ
Fig. 2. Pole mechanisms arising upon cutting the complete box diagram.
l1 l2 l
the Green’s function, we obtain
M♦ = −2µρσ ikρσ
Λ
× _J1 − M1J2M2|lml _θ
J1 J2
∞2
r
r
drr
0
jΛ
2
_
_
dr M1
M2
_2 _
0
h
2
×
L
x
j
l2
r drM1 Λ M2 Λ ,
r dr
_lml Ly − ml |Lx0_
0
r
+
Lx +Ly
Lx Ly Λ
(2)
∞
i
×
hΛ
l1
Λ
Ly
l
P
(θ )I
,
Ly |ml | y Lx Ly Λ
ll l
is expressed in
where the spectroscopic factor θ 1 2
0
J1J2
where µρσ and kρσ are, respectively, the reduced mass of the
virtual system ρ + σ and the momentum of its relative motion,
zΛ
terms of the DWBA–FIR structural factors as
√
l1l2 l
θ
zΛ
and M1 (M2 ) corresponds to the matrix elements of the
bottom (top) half of the com-plete box diagram featuring the
interaction Vi (Vf ) for the virtual state characterized by an
orbital angular moment Λ and a cylindrical function
J1J2
= (2l + 1)(2l + 1)
1
(−1)2Jρ +2Jσ +J12 +J22−J2+l+l2+l1
×
J11J12 J21 J22
(2J11 + 1)(2J12 + 1)(2J21 + 1)(2J22 + 1)
×
J
describing the relative motion of the particles involved, zΛ (kρσ
Λ
r) (zΛ = jΛ , h ). Specifically, we have
M
zΛ
x
1
M
Λ11 Λ12 l1
_
3
=d r dΩ
zΛ
=d
r
z (k r )Y ∗ (Ω
Λρσ
Λµ
r
i
Θ
11 J12
J
J21J22
The matrix elements for the top and bottom
halves of the complete box diagram are calculated
within the DWBA–FIR scheme. The corresponding
expres-sions are presented in the Appendix.
In order to complete the calculation of the matrix
element for the process featuring the delay of the
interaction, it is necessary to multiply the
z
z
expressions for the matrix elements M1 Λ and M2 Λ
and to perform summation over the projections of
the intermediate angular momenta in the products
of the Clebsch– Gordan coefficients involved. As a
result, the matrix element M♦ for the mechanism
corresponding to the box diagram with
unidirectional arrows is calculated by the formula
(−1)2Jx +2Jy −J1−M1
(3)
× _Jx MxJ1M1|JB MB __Jy My J2M2|JA MA_
J2
l
22
2
The multidimensional integral ILxLy Λ has the form
√
I
= ( 1)
r dr χ
×
x
Λ
kxky
x
x
r
_ _
rdr
r dr
0
(5)
0
_
(k r ) F Lx Λ
Lx
x
∞
4 2π
−
Lx Ly Λ
_
)J (k r )
(r , r
x
Λ11 Λ12 l1
Λρσ
ΛLy
×r dr χ (k r )F
y
y Ly
y y
(r, r )h (k r)
y
Λ21 Λ22 l2
∞
Λ
ρσ
r
_ _
r dr
+
0
r dr χ
×
x
x
rdr
0
_
(r , r
(k r ) F Lx Λ
Lx
x
x
Λ11 Λ12 l1
_
)h (k r )
x
Λρσ
ΛLy
(2J1 + 1)(2J2 + 1)
J1J2lml
J
l1
l
12
× U (Jx J11JB J21 : Jσ J1)U (Jy J22JAJ12 : Jρ J2).
f
♦
21
J1
ry dΩrzΛ (kρσ r)YΛµ(Ωr)Vf Φ∗ (ry ).
M
J
11
x
3
2
=
Λ21Λ22 l2
× ΘJ
)V Φ (r ),
i
(4)
2l + 1
2
r dr χ (k r )F
y y Ly y y Λ21 Λ22 l2
(r, ry )jΛ (kρσ r).
The spin-tensor components of the density matrix
for the final nucleus B produced in the binary reaction
×
PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007
2
ROLE OF VARIOUS MECHANISMS IN THE FORMATION OF A 1 C NUCLEUS
Structural factors for various mechanisms of the reaction
13
3
12
+
851
+
C( He, α) C(0 , 2 )
Reaction mechanism
J1
J2
Λ1
l
Λ2
ΘΛ1Λ2 l
J 1J 2
13
Neutron pickup
13
Neutron pickup
12
C→n+
12
C→n+
+
C(2 )
+
C(0 )
8
12
+
12
+
Exchange of the heavy cluster Be: α + C(2 ) = α + C(2 )
8
12
+
12
9
12
+ 13
+
Exchange of the heavy cluster Be: α + C(0 ) = α + C(0 )
9
12
+
13
9
Exchange of the heavy cluster Be for C(0 ): C → Be + α
with unpolarized particles are given by
√
µ µ k
xA yB x
ρkκ(JB ) =
4π 2
J
(1) B
y
−M
−
×
2JB + 1
k
(2Jx + 1)(2JA + 1)
_
JM
B
_B
J
M
B
B
−
B
|
3/2
0
1
1
−0.3424714
1/2
1/2
0
1
1
−0.2635027
2
2
0
2
0
2
0
0
0.20966
2
2
1
0.11483
2
2
4
0.07739
0
2
0
2
0
0
0.47602
0.33028
4
4
0
0.20129
0
Exchange of the heavy cluster Be for C(2 ): C → Be + α
9
1/2
0
0.06425
3
3
1
1
3
3
0
2
1
1
0.05603
5
1
5
2
3
−0.07866
1
1
1
1
0
2
1
1
−0.10461
0.06735
0.18604
mechanism of heavy-cluster stripping and the twostep sequential-transfer mechanism corresponding to
(6) the box diagram. It should be noted that no additional
normalization factors were introduced in calculating
kκ
the matrix element for neutron pickup and exchange
_
processes.
MB MB
M (M )M ∗ (M _ ).
×
if
B
if
3. CALCULATION OF THE DIFFERENTIAL
CROSS SECTION FOR THE REACTION
B
MxMA My
In this case, the normalization factors is chosen
_
in such a way that ρ00(JB ) = SpρJB (MB MB ) =
dσ/dΩ.
Expressions (1) and (3)–(6) make it possible
to obtain spin-tensor components of the density
13C(3He, α)12 C AND OF SPIN-TENSOR
COMPONENTS OF THE DENSITY MATRIX
FOR THE PRODUCT 12C(2+) NUCLEUS
AND COMPARISON OF THE RESULTS
WITH EXPERIMENTAL DATA
matrix for the 12C nucleus produced in the reaction
The table summarizes the maximum values that
we calculated by formulas (4), (A.4), and (A.6) for
pickup mechanism and the mechanism of the ex- the structural factors, assuming various mechanisms
9
13 3
12
+ +
change of the heavy cluster Be are realized in the
of the reaction C( He, α) C (0 , 2 ), such as the
neutron-pickup process 13C → n + 12C (0+, 2+), the
pole approximation and via the sequential transfer of
13C(3He, α)12 C(0+, 2+), for which the neutron-
8
virtual Be and a virtual neutron, so that
n
Mif = M
+ M 9Be ;
M 9 Be = M
DWBA
9 Be
DWBA
+M.
♦
exchange of the heavy cluster
13
9
9
Be via the process
C → Be + α, and the exchange of the heavy cluster 8Be via the process 12C(0+ , 2+) → 8Be + α. From
The matrix elements for neutron pickup and the the table, one can see that these terms are on the same
exchange of the heavy cluster 9 Be are summed in- 12 order of magnitude for each state of the final nucleus
coherently, since they are associated with different
C; therefore, we can expect that the contributions
of all three mechanisms to the reaction cross section
fore, there is no interference between them. In turn, will be sizable.
the matrix element for 9Be exchange is determined
Figure 3 shows the calculated differential cross
13 3
12
by the coherent sum of the amplitudes for the pole sections for the reaction C( He, α) C induced by
cluster configurations of the initial nucleus; there-
PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007
852
GALANINA et al.
dσ/dΩ, mb/sr
10
0
10
–1
10
–2
10
0
10
–1
10
–2
12
C (2 )
+
12
C (0 )
+
–3
10
–4
10
60
20
100
140
160
θα(c.m.), deg
Fig. 3. Differential cross sections for the reaction 13 C(3 He, α)12 C occurring at EHe = 18 MeV and leading to the
formation of a 12 C nucleus in the ground (0+ ) and an excited (2+ ) state according to calculations assuming various
reaction mechanisms: (dotted curve) results for the neutron-stripping mechanism, (dashed curve) results for the
mechanism involving the transfer of the heavy cluster 9Be within the DWBA–FIR scheme, (dash-dotted curve) results
for the transfer of the heavy cluster 9Be with allowance for both the pole and the box diagram, and (solid curve) sum
of all of these contributions. The displayed experimental data were borrowed from [1].
18-MeV projectile helium ions that is considered for
the cases where the final nucleus is produced in the
+
+
ground (0 ) and an excited (2 ) state and under the
assumption of various reaction mechanisms. It is
obvious from this figure that the pole mechanism of
neutron stripping is dominant in the forward hemiArb. units
0.2
0.1
0
– 0.1
⟨T10⟩
⟨T20⟩
0.1
0
action C( He, α) C (0 , 2 ).
The spin-tensor components of the density matrix make it possible to estimate the orientation of
12
+
various multipole moments of the product C (2 )
– 0.1
⟨T40⟩
0.1
0
– 0.1
– 0.2
sphere, but that the exchange mechanism of heavycluster transfer determines the cross section in the
backward hemisphere. The contribution of the twostep mechanism allowing for delay is significant at all
emission angles of alpha particles. Moreover, one can
12
see that, in the case of ground-state C, this mechanism increases the cross section for the exchange
process in the forward hemisphere. This makes it
possible to improve considerably agreement between
the theoretical cross section and experimental data [1]
over the whole angular range. In other words, it is
necessary to take into account the two-step mechanism of heavy-cluster transfer in order to describe
correctly the experimental cross sections for the re13 3
12
+
+
nucleus with respect to its symmetry axis. In the xyz
coordinate frame chosen in such a way that x||kB and z ⊥
20
60
100
140
160
θα(c.m.), deg
Fig. 4. Polarization tensors of the 2+ excited state of
the product nucleus 12 C under the assumption of
vari-ous reaction mechanisms. The notation for the
curves is identical to that in Fig. 3.
[kx × ky ], the mean values of the polarization tensors
1
_Tkκ_ =
ρ
kκ
2JB + 1 ρ00
characterize the spin orientation along the quantization axis and in the plane orthogonal to the reaction
plane [6]. The polarization tensors of lowest rank have
a clear physical meaning: the tensor _T10_ is
PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007
ROLE OF VARIOUS MECHANISMS IN THE FORMATION OF A 12C NUCLEUS
proportional to the polarization vector and characterizes the induced polarization of the system in the
plane orthogonal to the reaction plane. Apart from
normalization, the tensors _T20_ and _T22_ coincide
with, respectively, the longitudinal and the
transverse component of the alignment tensor.
Figure 4 presents the contributions of various reaction mechanisms to some of the polarization ten+
12
sors for the 2 excited state of the C nucleus. From this
figure, one can see that the pole mechanism of neutron
stripping is important for the emergence of the induced
polarization of the product nucleus. Po-larization tensors
of higher rank are determined by the mechanism of
heavy-cluster exchange, the influence of the sequentialtransfer mechanism being the most significant in the
region of intermediate angles.
APPENDIX
CALCULATION OF THE MATRIX
ELEMENTS FOR THE TOP AND BOTTOM
PARTS OF THE BOX DIAGRAM
In order to systematize the notation, all
interme-diate momenta over which summation is
performed are labeled with indices “1n” or simply
“1” for the diagram in Fig. 2a and “2n” or simply “2”
for the diagram in Fig. 2b.
z
Let us represent the matrix element M1 Λ ≡ M1
corresponding to the diagram in Fig. 2a in a form
analogous to that in (1); that is,
j
3
12
+
=
MΛ
Thus, we have used here different features of the
13
853
(A.1)
(2J11 + 1)(2J12 + 1)
1
J11J12 l1
+
reaction C( He, α) C (0 ; 2 , 4.443 MeV) (dif-ferential
cross sections for elastic and inelastic scat-tering,
spin-tensor components of the density matrix, induced
vector and tensor polarizations) to demon-strate the
role of various mechanisms in the produc-tion of the
final nucleus. The results indicating the importance of
taking into account the mechanism of sequential
virtual-cluster transfer (box diagram) are the most
interesting here. The contribution of this mechanism
proved to be sizable even in calculating the cross
section for elastic and inelastic scattering; moreover,
the inclusion of the two-step mechanism even leads to
qualitative changes in polarization prop-erties,
especially in the region of intermediate emis-sion
angles of alpha particles. This circumstance emphasizes once again that the mechanism of nuclear
reactions involving alpha particles is nontrivial, radically differing from the pole mechanisms of stripping or
pickup. Moreover, the inclusion of exchange processes requires considering second-order two-step
effects in addition to the mechanism of direct heavycluster transfer. The contribution of these effects is
determined by the structural features of the nuclei
involved in the reaction. In the reaction considered
here, these structural features are such (see Introduction) that the role of all mechanisms examined in this
study proved to be significant, so that only by combining these mechanisms were we able to describe
13
3
12
+
The work was supported by a grant from the
Presi-dent of the Russian Federation for Support
of Leading Scientific Schools (no. 5365.2006.2)
and a grant from the Scientific and Technological
Program Universi-ties of Russia (no. 02.02.506).
PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007
J11J12
Λ11 Λ12
× _JxMxJ11 M11|Jσ Mσ __Jρ Mρ J12M12|JA MA_
× _J11 − M11J12M12|l1m1_.
The expression for the kinematical factor βΛ11 Λ12 l1 can be
obtained by introducing the invariant form factor F
_
Lx Λ
(r , rx ) and performing integration with
Λ11 Λ12 l1
respect to the angular variables. We have
1
4π
β
=
µ
kr
Λ11 Λ12 l1
xx
2l1 + 2 (−1)
√
(A.2)
i _LxµxΛ − µ|l1 − m1_YL∗x µx (kx)
Lx
×
Lx µx
r dr χ
×
x
(k r ) F Lx Λ
Lx
x
x
x
_
(r , r
_
)j (k r ).
x
Λ11 Λ12 l1
Λρσ
Substituting (A.2) into (A.1), we find for the matrix
element M
jΛ
that
1
j
4π
MΛ=
(A.3)
( 1)2Jx +2Jρ −M11 −J11
−
kx J11 J12l1
1
Λ11 Λ12 l1
(2J + 1)(2J + 1)Θ
×
11
12
J11J12
× _JxMxJ11 M11|Jσ Mσ __Jρ Mρ J12M12|JA MA_
× _J11 − M11J12M12|l1m1_
µ Lx
(−1) i _LxµxΛ − µ|l1 − m1_YL∗x µx (kx)
×
ACKNOWLEDGMENTS
ΘΛ11Λ12 l1
Λ12 l1
Λ11
+
the properties of the reaction C( He, α) C (0 ; 2 ,
4.443 MeV).
β
−M11
( 1)2Jx +2Jρ −J11
×−
Lx µx
r dr χ
×
x
x
(k r ) F Lx Λ
Lx
x
x
Λ11 Λ12 l1
_
(r , r
_
)j (k r ).
x
Λρσ
In calculating the spectroscopic factor, we consider
that a neutron plays the role of the transferred particle C . The corresponding expression then takes the
854
GALANINA et al.
form
ement of the top half of the box diagram (Fig. 2b),
which corresponds to the exchange of the heavy clusΛ11Λ12 l1
(2J11 + 1)(2J12 + 1)(2l1 + 1)
=
Θ
J11J12
2Λ
11
8
ter
+1
Be, is written as
(A.4)
M
h
Λ
k
( 1)−Jx −2Jρ +JC +J12 +J11+Sσ +l1+Λ11
2
−
×
4π
=
ρ
J
A
C
11
Λ12 SC J12
L
ΛJl
S
A
(2J21 + 1)(2J22 + 1)
J21 J22
J
12121
×_
_
|
i− y _ΛµLy µy |l2 − m2_YLy µy (ky )
L
×
J J J
Ly µ y
σ 11 x
S
S
C
Λ
,
σ
r dr
×
11
y
Λ21 Λ22 l2
Θ
L
=
(−1)
yy
) h ( k r) .
(r , r
Λ21 Λ22 l2
y
Λρσ
, we consider that x, y ≤ α (Lx = Ly = 0,
Jx = Sx, Jy = Sy ). We then have
+L
B
y
_
(k r )F ΛLy
Ly
J21 J22
In just the same way as in (A.3), the matrix el-
Λ21 Λ22 l2
χ
In the expression for the spectroscopic factor
where Li and Si (i = A, B, C, σ, ρ) are the orbital angular momenta
and spins of the nuclei involved.
J21 J22
__
|
A
Θ
JB MB Jy My J22 M22 Jρ Mρ
× _J21 − M21J22M22|l1m1_
J Λ S
ρ
11
×
Θ
Jσ Mσ J21M21
× ΘΛ Θ Λ
12 11
ρ
J21 J22l2 −
y
×
_ A→C+ρ _ σ→C+x
S
(A.5)
1)2Jσ +2Jy −M21 −J21
Λ21 Λ22 l2
× (2LA + 1)(2SA + 1)(2Lσ + 1)(2Sσ + 1)(2Jρ + 1)
L
(
+L
A
J
C
−
S +Λ
21 − C
_ B→x+C _ A→y+C
+Λ
22
21
(A.6)
Θ Θ
Λ21 Λ22
LL SA LB SB LC SC
×
(2LB + 1)(2SB + 1)(2LA + 1)(2SA + 1)(2l2 + 1)(J21 + 1)(2J22 + 1)
S L J
×
C
B 21
J B Sx SB
SC
L J
A 22
J A Sy SA
Expressions (A.3)–(A.6) solve completely the
problem of calculating the matrix elements for the
top and bottom halves of the box diagram.
REFERENCES
1. O. I. Vasil’eva et al., Izv. Akad. Nauk SSSR, Ser.
Fiz. 48, 1959 (1984).
2. V. M. Lebedev, N. V. Orlova, and A. V. Spassky, Yad.
Fiz. 62, 1546 (1999) [Phys. At. Nucl. 62, 1455 (1999)].
3. N. S. Zelenskaya and I. B. Teplov, Features of Excited Nuclear States and Angular Correlations in
LB l2 LA
Λ L Λ
22
C
21
LB l2 LA
J S J
22
.
C 21
´
Nuclear Reactions (Energoatomizdat, Moscow,
1995) [in Russian].
4. N. S. Zelenskaya, Yad. Fiz. 13, 734 (1971) [Sov. J.
Nucl. Phys. 13, 417 (1971)].
5. L. I. Galanina and N. S. Zelenskaya, Izv. Akad.
Nauk, Ser. Fiz. 69, 1741 (2005).
6. L. I. Galanina and N. S. Zelenskaya, Izv. Akad.
Nauk, Ser. Fiz. 63, 47 (1999).
Translated by E. Kozlovsky
PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007
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