Parameters of deformation of the excited states in sd-shell odd nuclei
The review of the major models used for the description of the deformed nuclei is given. The difficulties conditioned by the uniqueness of the choice of nuclear deformation parameters in rotational bands are traced. The calculations of the reduced probabilities of electromagnetic transitions in the...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1114762017-01-11T03:02:38Z Parameters of deformation of the excited states in sd-shell odd nuclei Sarana, V.D. Lutsai, N.S. Shlyakhov, N.A. Korda, L.P. Ядерная физика и элементарные частицы The review of the major models used for the description of the deformed nuclei is given. The difficulties conditioned by the uniqueness of the choice of nuclear deformation parameters in rotational bands are traced. The calculations of the reduced probabilities of electromagnetic transitions in the sd-shell odd nuclei performed within the modified Nilsson model show that the use of different values of deformation parameters for different excited states gives better fits to the experimental data. Приведено огляд основних моделей, використовуваних для опису деформованих ядер. Простежуються труднощі, пов'язані з однозначністю вибору параметра деформації ядра в обертальних смугах. Розрахунки наведених імовірностей у рамках модифікованої моделі Нільсона показують, що використання різної деформації для збуджених станів поліпшує згоду з експериментом. Розгляд ведеться для ядер sd-оболонки. Дан обзор основных моделей, используемых для описания деформированных ядер. Прослеживаются трудности, связанные с однозначностью выбора параметра деформации ядра во вращательных полосах. Расчеты приведенных вероятностей в рамках модифицированной модели Нильсона показывают, что использование разной деформации для возбужденных состояний улучшает согласие с экспериментом. Рассмотрение ведется для ядер sd-оболочки. 2011 Article Parameters of deformation of the excited states in sd-shell odd nuclei / V.D. Sarana, N.S. Lutsai, N.A. Shlyakhov, L.P. Korda // Вопросы атомной науки и техники. — 2011. — № 5. — С. 24-29. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 25.40. Lw, 23.20.-g http://dspace.nbuv.gov.ua/handle/123456789/111476 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Sarana, V.D. Lutsai, N.S. Shlyakhov, N.A. Korda, L.P. Parameters of deformation of the excited states in sd-shell odd nuclei Вопросы атомной науки и техники |
| description |
The review of the major models used for the description of the deformed nuclei is given. The difficulties conditioned by the uniqueness of the choice of nuclear deformation parameters in rotational bands are traced. The calculations of the reduced probabilities of electromagnetic transitions in the sd-shell odd nuclei performed within the modified Nilsson model show that the use of different values of deformation parameters for different excited states gives better fits to the experimental data. |
| format |
Article |
| author |
Sarana, V.D. Lutsai, N.S. Shlyakhov, N.A. Korda, L.P. |
| author_facet |
Sarana, V.D. Lutsai, N.S. Shlyakhov, N.A. Korda, L.P. |
| author_sort |
Sarana, V.D. |
| title |
Parameters of deformation of the excited states in sd-shell odd nuclei |
| title_short |
Parameters of deformation of the excited states in sd-shell odd nuclei |
| title_full |
Parameters of deformation of the excited states in sd-shell odd nuclei |
| title_fullStr |
Parameters of deformation of the excited states in sd-shell odd nuclei |
| title_full_unstemmed |
Parameters of deformation of the excited states in sd-shell odd nuclei |
| title_sort |
parameters of deformation of the excited states in sd-shell odd nuclei |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2011 |
| topic_facet |
Ядерная физика и элементарные частицы |
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http://dspace.nbuv.gov.ua/handle/123456789/111476 |
| citation_txt |
Parameters of deformation of the excited states in sd-shell odd nuclei / V.D. Sarana, N.S. Lutsai, N.A. Shlyakhov, L.P. Korda // Вопросы атомной науки и техники. — 2011. — № 5. — С. 24-29. — Бібліогр.: 12 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T02:13:16Z |
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| fulltext |
PARAMETERS OF DEFORMATION OF THE EXCITED
STATES IN sd-SHELL ODD NUCLEI
V.D. Sarana1, N.S. Lutsai1, N.A. Shlyakhov2∗, L.P. Korda2
1V.N.Karazin Kharkov National University, Kharkov, Ukraine
2National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received August 23, 2011)
The review of the major models used for the description of the deformed nuclei is given. The difficulties conditioned
by the uniqueness of the choice of nuclear deformation parameters in rotational bands are traced. The calculations
of the reduced probabilities of electromagnetic transitions in the sd-shell odd nuclei performed within the modified
Nilsson model show that the use of different values of deformation parameters for different excited states gives better
fits to the experimental data.
PACS: 25.40. Lw, 23.20.-g
1. INTRODUCTION
The decay schemes at low excitation energies of sev-
eral odd nuclei in the sd-shell are usually inter-
preted with help of the generalized collective rota-
tional model [1]. The levels are grouped in the rota-
tional bands each of which is based on the internal
single-particle states calculated in the deformed nu-
clear potential. Assuming a cylinder symmetry, the
analysis starts with the determination of the form of
the nuclear shape from the measured quadrupole mo-
menta, magnetic moments or unbinding parameters.
When the value and the sign of deformation become
known the detailed comparison of the calculated and
measured values becomes possible by the use of the
eigenvalues and eigenfunctions corresponding to the
one-particle shell model. Those values are the level
excitation energies, the probabilities of gamma and
beta transitions, the reduced nucleon widths which
characterize such nuclear reactions as stripping and
nuclear pick-up, the magnetic and quadrupole mo-
menta of nuclear levels and their spins and parities.
Although the major striking achievements of the gen-
eralized collective model were associated with the
150 < A < 190 and A > 222 nuclei the remark-
able correlations with the data were found for the
sd-shell light odd nuclei as well [1]. The application
of the generalized model to the light deformed nuclei
is more complicated than to the heavier ones because
one must take into account the overlap of rotational
bands and the hole states in the low-energy part of
the spectrum. In general, the model with symmetric
spheroid core is applicable to this shell due to the mix
of two or more rotational bands with ∆K = ±1. In
this case the R.P.C. operator used and it has the form
ORPC = α(I1J1 + I2J2) ([7]). It is usually assumed
that the deformation does not depend on the excita-
tion energy. At the same time, the data point on the
existence of deformation of light nuclei in the excited
states as well as in the ground ones. of The shell
model built at the end of 1940-th, having explained
great amount of data associated with the ground and
weakly excited states of atomic nuclei, faced substan-
tial problems. Particularly, the measured values of
quadrupole momenta for several nuclei appeared to
be much higher the same values calculated due to the
shell model. On the basis of the method developed in
NSC KhIPT the nuclear deformation in the excited
states was accounted for, which allowed for better de-
scription of the reduced probabilities of electromag-
netic transitions [2]. We have considered the nuclei
with n = 11 nucleons of definite kind 21Ne, 23Na,
23Mg with pronounced rotational structure of lev-
els. We have confirmed the influence of deformation
on the values of the probabilities of electromagnetic
transitions to the ground state from both the one-
particle and collective initial states. For systematic
study the computer codes which calculate the matrix
elements of the electromagnetic transitions on the ba-
sis of the developed method which uses Nilsson‘s wave
functions are needed.
2. NILSSON’S MODEL
In Nilsson‘n model, the interaction of nucleons with
the nuclear field is described with help of the Hamil-
tonian of the form [3]
H = H0 + C · l · s + D · l2 , (1)
where H0 is the oscillatory potential to which the
spin-orbit potential C · l · s and D · l2 correction for
the states with higher angular momenta are added.
The weights of these two terms are chosen to repli-
cate the known sequence of one-particle states in the
shell model with spherical potential. Nilsson studied
∗Corresponding author E-mail address: shlyakhov@kipt.kharkov.ua
24 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N5.
Series: Nuclear Physics Investigations (56), p.24-29.
the role of deformation of the nuclear potential with
a cylinder symmetry. Neglecting for a while the l · s
and l2 terms, he found that the Hamiltonian with de-
formed potential could be divided into the spherically
symmetric term H0 and the Hδ term describing the
binding of the particle to the symmetry axis. The
representation is chosen to be diagonal with respect
to H0. All operators l2, lz, sz commute with H0. The
respective quantum numbers are denoted as l,
∧
, Σ.
Then the total Hamiltonian has the form
H = H0 + Hδ + C · l · s + D · l2 , (2)
in which the last three terms are treated as a per-
turbation. By the proper choice of the parameters
(shown here for convenience) Nilsson arrived to
H = H0 + kηω0R , (3)
where k = −C/2ηω0 and R were given as
R = ηR− 2ls− µl2 . (4)
Here µ = 2D/C and η = (δ/k)(ω0(δ)/ω0(0)). The
parameter δ is connected to the deformation of the
nuclear potential U and is defined from
Hδ = δηω0U = kη2ω0U , (5)
where
U = −4
3
√
π
5
r2Y20 . (6)
From the diagonalization of the operator R Nils-
son found the eigenvalues rNΩ(η) and the correspond-
ing eigenvalues of the total Hamiltonian
ENΩ = (N + 3/2)ηω0 + kηω0r
NΩ . (7)
The general quantum number N is the number of
of oscillator quanta and Ω is the quantum number
corresponding to the operator jz=lz+sz which com-
mutates with H. Thus, Ω is the component of the
total angular momentum along the nuclear axis. The
state vectors use are |NlΛΣ〉 represented as where Λ
and Σ are the eigenvalues of the operators lz and sz
and, thus, Ω=Λ+Σ.
For the d-shell which is the outer shell for the
considered nuclei Nilsson diagonalized R with µ=0
for N=0,1,2 when for the higher values of N the val-
ues of µ (0.35-0.55) were used. It seems valuable to
study the influence of µl2 term on the eigenvalues
and eigenfunctions for N=2. For N=2, Ω=5/2 (orbit
5) there exists only one eigenvector |222+〉 with the
eigenvalue r=2/3η-2-6µ. For N=2 and Ω=3/2 there
exist two eigenvectors |221+〉 and |221−〉. Because
the term µl2 is diagonal and each of these eigenvec-
tors has l=2 the influence is in the simple addition of
the constant diagonal matrix - the unity matrix - 6µ
times to matrix diagonalized by Nilsson. Therefore,
the new eigenvalues appear as those ones tabulated
by Nilsson and divided by -6µ and the eigenfunctions
are the same both for the orbit 8 and the orbit 7.
For N=2, Ω=1/2 however, the eigenvectors are
|220+〉, |200+〉 and |221-〉. The argument used for
Ω=3/2 is not valid any more and the matrix should
be diagonalized for each values of η and µ. The re-
sults of these calculations are presented in [4], giving
the eigenvalues and the eigenfunctions for the orbits
11, 9 and 6. In this work the influence of the term re-
lated to the centrifugal correction l2 for the sequence
of levels of the sd-shell nuclei is studied. The graphs
of the eigenvalues and the unbinding parameters for
the orbit Ω=1/2 are given. The other approach to the
modification of Nilsson‘s model is associated with the
use of the other representations of the eigenvectors of
the harmonic oscillator [3,5].
3. APPLICATION TO THE SCHEMES OF
EXCITED STATES
Now we are going to discuss several examples of the
decay schemes for which the calculations giving bet-
ter agreement with the data are presented.
3.1. 25Mg AND 27Al [6].
The configuration of the ground state for the nucleons
are taken in the form
π(O16; d5/2(1/22, 3/22))ν(O16; d5/2(1/22, 3/22, 5/21)) ,
(8)
and corresponds to the prolate deformation as found
for the neighboring nuclei 23Na and 27Al. The un-
binding parameter for the band built on the first ex-
cited state is found to be a=0.2. If one takes this
state with η¿+6 then the orbit 9 appears lower than
the orbit 5. Ignoring the strength of the spin-orbit
coupling k within the calculations of this kind can
reduce this discrepancy.
Nilsson‘s calculations are mostly suitable for this
case because they contain results for the eigenfunc-
tions for A=25. The energy states of the model for
A=25, are found by the extension of Nilsson‘s calcu-
lations of the single-particle energies in the spheroid
potential towards the higher values of η. With this
aim Nilsson‘s equation is rewritten as
Ej/0.75ηω0 = (Nj + 3/2)(1− 1/3ε2 − 2/27ε3)−1/3 +
krj(ε)− (1/3ηω0) 〈Uj〉 , (9)
where the following relations between ε, η and δ are
used
kη = ε(1− 1/3ε2 − 2/27ε3)−1/3 , (10)
and
ε = 3
√
1 + 2/3δ −
√
1− 4/3δ√
1− 4/3δ − 2
√
1 + 2/3δ
. (11)
The value of k which defines the weight of spin-orbit
coupling used in calculations is taken as 0.08. This
appears the better value for the light nuclei than 0.05.
In this connection it seems valuable to note that the
d5/2−d3/2 splitting at 5.08 MeV in 17O shall require
k=0.13. Due to the indefiniteness in the values of k
25
for the case A=25 all the respective curves are given
as functions of η because this conditions the neces-
sity to consider the alteration of the weight of the
spin-orbit interaction.
The values of η used in the respective comparisons
alter from +3 to +5. It is senseless a priori to expect
that η shall be constant for each rotational band and
the very fact that different values of η give better fits
to the data probably means that the neglect of the
RPC effect is important.
4. THE BAND MIXING MODEL (RPC)
For the odd masses the Hamiltonian of the axial
symmetric nuclei which describes the single particle
motion, the nucleus rotation and the particle-rotation
coupling is given in the form [7]
H = Hintr +
η2
2J ′′
(I2 − 2I
′
0J
′
0) +
η2
J ′′
(I
′
1J
′
−1 + I
′
−1J
′
1) , (12)
and if one neglects the last term in (12) the major set
of wave functions are
IMKα =
{
2I + 1
16π2
}1/2
(DI
MKXα
K +
(−1)I−JDI
M−KXα
−K) , (13)
where J ′′ is the momentum of inertia, I is the total
angular momentum of nucleus, J is the internal an-
gular momentum related to the motion of the odd
nucleon, I′0,∓1 and J′0,∓1 are their spherical compo-
nents in the rest frame, K is the projection of I or J on
the symmetry axis of the nucleus, M is the projection
of I on the fixed axis z, XK are Nilsson‘s wave func-
tions DI
MK and α denote different eigenstates with
the same value of K.
In the representation (14) only the last term in
(12) is not diagonal with respect to K. It gives the
diagonal contributions only in the case when K=1/2.
This term couples the states with one and the same
parities and total angular momenta for which ∆K =
±1 or ∆K=0 if K=K′=1/2. The diagonal matrix el-
ements of the Hamiltonian (12) are equal to
E(NIKα) = E0
K +
η2
2J ′′
((I(I + 1) + α(−1)I+1/2 ×
(i + 1/2)δK1/2)− (K(K + 1)− αδK1/2)) ,
(14)
where
E0
K = E0 +
η2
2J ′′
(K −K2 − αδK1/2) . (15)
The non-diagonal matrix elements can be written in
J −K representation as
AI
ββ′ =
〈
IMK ′α′| η2
2J ′′
(I ′1J
′
−1 + I
′
−1J
′
1)|IMKα
〉
=
−η2/2J
′′√
(I + K1)(I −K1+I + 1)(1− δKK′ +
(−1)I−1/2δK′1/2δK1/2)Aββ′ , (16)
where KI > K, K
′
; β and β
′
are the numbers of Nils-
son‘s states and
Aββ′ =
∑
J
C
′
JCJ
√
(J + K1)(I −K1 + 1)×
(1− δKK′ + (−1)I−1/2δK′1/2δK1/2) ,
(17)
or in the l-Λ representation for K=K
′
=1/2
Aββ′ = (−1)l
∑
l
(a
′
l0al0 +
√
l(l + 1)(a
′
l0al1 +a
′
l1al0)) ,
(18)
(if β = β
′
, Aββ = a equals to the unbinding parame-
ter) and for
Aββ′ =
∑
l
(a
′
lK1−1/2(alK1−1/2 + (a
′
K1−1/2(alK1−1/2×
√
(l + K1 − 1/2)(l −K1 + 3/2) + a
′
lK1−1/2(alK1+1/2×√
(l −K1 + 1/2)(l + K1 + 1/2)) .
(19)
The coefficients alΛ are given by Nilsson [3]. The cou-
pling parameters Aββ′ (16) are calculated using (18)
and (19). By diagonalization of the energy matrix
(12) the wave functions
Φ(IM) =
∑
Kα
CKαΨ(IMKα) . (20)
And their eigenvalues λI can be found for each mixed
state with I spin.
4.1. PROBABILITIES OF
ELECTROMAGNETIC TRANSITIONS IN
THE BAND MIXING MODEL
The formulae for the reduced probabilities M1 and E2
transitions are given in [7] for the case of two mixed
rotational bands. Using the wave function (20) and
the magnetic and electric multipole operators given
by Nilsson one can obtain [8] the expressions for the
reduced probabilities of M1 and E2 transitions when
the large number of bands are mixed. When for M1
transitions
B(M1, I → I ′) =
3
16π
(
η
2Mω0
)2
×
∑
K′,α′,K,α
C ′K′,α′CK,α [(I1KK ′ −K|I ′K ′)+
δK/2δK′/2(−1)I′−K′
bKK′
αα′ (M1)×
(I1K −K ′ −K|I ′ −K ′)] GKK′
αα′ (M1) . (21)
and for E2 transitions
B(E2, I → I ′) =
5e2
4π
(
η
Mω0
)2
×
(C ′K′α′CKα ×
∑
K′α′Kε
((I2KK ′ −K|I ′K; ) +
(−1)I′+K′
bKK′
αα′ (E2)× (I2K −K ′ −
|I ′ −K ′)GKK′
αα
′ (E2)))2. (22)
26
Here the diagonal matrix elements GKK
′
αα′ (E2) are
the own quadrupole momenta Q0 for each K in the
units of 2(η/Mω0). The one-particle matrix ele-
ments GKK
′
αα′ (E2) and bKK
′
αα′ (E2) are given by Nils-
son [3]. The value η/Mω0 was determined assum-
ing η/Mω0=41A−1/3 MeV. The own quadrupole mo-
mentum is assumed one and the same for all four
one-particle states which totally correspond to the
known quadrupole momenta of the ground states.
The mean self life of the excited states and the rela-
tion of branching are calculated using (12) and (13)
and are compared with the data.
5. MODIFIED NILSSON’S MODEL
One of the most important problems is the investi-
gation of a shape of a nucleus in the excited states.
However, this task is not well tested, especially in the
range of light nuclei. Up to now, the existence of de-
formation of light nuclei in both ground and excited
states is experimentally proven. At the same time,
it is usually assumed that the deformation does not
change with the excitation energy. In our opinion,
this assumption conditioned the failure of attempts
to explain the probabilities of electromagnetic transi-
tions in the framework of generalized model. Bearing
this in mind, we use the new approach developed in
[2]. Analyzing the probabilities of electromagnetic
transitions, we treat the nuclear deformation as a
variation parameter, meaning that the initial and fi-
nal states are assumed to have different deformations.
Thus, during the transition, the state of core nucle-
ons changes alongside the state of the odd nucleon.
Using the 1d2s-shell nuclei, we have studied the in-
fluence of the changes in nucleus deformation on the
probabilities of electromagnetic transitions. The ini-
tial and final states of a nucleus have been consid-
ered for different deformations and the contribution
of one-particle part of wave function to the probabili-
ties of electromagnetic transitions in light nuclei have
been analyzed.
5.1. PROBABILITIES OF
ELECTROMAGNETIC TRANSITIONS IN
THE MODIFIED NILSSON’S MODEL
In order to determine the matrix element of the one-
particle multipole transition operator
M =
∑
i=1
t̂(i) , (23)
we consider the systems of one-particle wave func-
tions for the initial and the final states: ϕ1, ϕ2,. . . ..
ϕA; ψ1, ψ2,. . . .. ψA. Here A denotes the number
of nucleons in the nucleus; indices 1, 2, A denote the
numbers of occupied orbits in the initial and the final
states. The formula takes place:
(ψf , Mψi) =
A∑
s=1
detBs
j , (24)
where the determinant elements Bs
j are as follows:
Bs
j = (ψi, t̂ϕj), i = s ,
Bs
j = (ψi, ϕj), i 6= s . (25)
In the case under study, the operator t̂ is:
t̂ = e
[
1 + (−1)λ Z
Aλ
]
rλYλµ(ϕ,ψ) . (26)
If λ < K + K
′
, then the reduced probability of elec-
tric multipole transitions between the initial and the
final state with IK and I
′
K
′
taken at different defor-
mations η and η
′
and is equal to:
|M(Eλ)|2 = (27)
= e2
[
1 + (−1)λ Z
Aλ
]2 (
h̄
Mω0
)λ 2λ + 1
4π
×
×
∣∣∣∣∣〈IλKK ′ −K|I ′K ′〉
Z∑
s=1
det(ψs
i , ϕj)
∣∣∣∣∣
2
.
For i = s:
(ψi, t̂φj) =
∑
l′,l
〈
N ′l′
∣∣rλ
∣∣ Nl
〉 √
2l + 1
2l′ + 1
〈lλ00|l′0〉 ×
×
∑
Λ′,Λ,Σ′,Σ
δΣ′Σa′l′Λ′alΛ 〈lλΛK ′ −K|l′Λ′〉 ,
(28)
while when l 6= s we have:
(ψi, ϕj) = δN ′N
∑
l,Λ
a
′
l′Λ′alΛ . (29)
For the case of magnetic multipole transitions, the
sum of N determinants over all orbits occupied by
nucleons can be divided on two sums - over N orbits
occupied by neutrons and Z ones taken by protons.
Thus for L < K + K
′
we find:
|M(Mλ)|2 =
(
eh̄
2Mc
)2 (
h̄
Mω0
)λ 2λ + 1
16π
×
∣∣∣∣∣〈IλKK ′ −K|I ′K ′〉
{
det(ψipϕjp)
Z∑
sn=1
det(ψsn
in
, ϕjn)
−det(ψinϕjn)
Z∑
sp=1
det(ψsp
ip
, ϕjp)
∣∣∣∣∣∣
2
. (30)
The determinant elements for in = sn and ip = sp
correspond to the values GML for the neutron and
the proton from ref. [3].
6. APPLICATION OF THE MODIFIED
NILSSON’S MODEL TO THE ODD
NUCLEI OF sd SHELL
In refs. [9-12], using M1 and E2 transitions, the
influence of the changes in nucleus deformation on
the probabilities of electromagnetic transitions in
21,23Na, 25,27Al nuclei have investigated (table). The
calculated value of the matrix element depends on
27
two deformation parameters corresponding to the ini-
tial and the final states of the nucleus. Therefore, per-
forming the comparison between the theory and the
experiment, in the two-dimensional space of the de-
formation parameters we obtain the regions in which
the theoretical and experimental matrix elements co-
incide. Analyzing the bands of transitions from one
and the same level or onto one and the same state, we
are able to reduce the regions of possible values of de-
formation parameters for some levels and sometimes
even give exact values.
The nucleus deformation parameters in the region 21 < A < 27,
extracted via comparing the experimental and theoretical values of B(σl)
for the case L < K + K ′
M1 transition
Nucleus Ei → Ef , Jπ
i → Jπ
f B(σl)exp, B(σl)cm, ηi ηf
MeV W.u W.u
21Na 2.432 → 0 1/2+ → 3/2+ 0.043(5) 0.038 2 4
23Na 2.982 → 0 3/2+ → 3/2+ 0.09(2) 0.014 2 4
→ 0.44 3/2+ → 5/2+ 0.014(4) 0.022 2 4
E2 transition
Nucleus Ei → Ef , Jπ
i → Jπ
f B(σl)exp, B(σl)cm, ηi ηf
MeV W.u W.u
23Na 2.982 → 0 3/2+ → 3/2+ 1.3(2) 0.082 2 4
25Al 0.451 → 0 1/2+ → 5/2+ 3.0(5) 2.82 0 4
25Al 2.486 → 0 1/2+ → 5/2+ 0.8(3) 0.07 0 4
27Al 0.843 → 0 1/2+ → 5/2+ 7.5(5) 5.75 2 4
27Al 3.673 → 0 1/2+ → 5/2+ (∼ 0) 0 −2 4
The joint analysis of the transition matrix ele-
ments allowing for deformations in the initial and fi-
nal states and the other data on the low-lying lev-
els depending on the deformation (the position, the
quadrupole momenta etc.) also help to determine
the values of deformation for these levels more ex-
actly. The deformation parameters of the nuclei in
the ground and excited states are usually extracted
either from the data or from the theoretical calcu-
lations. In both cases the nucleus is treated as a
deformed object so that the extracted information is
model dependent. The statements about the defor-
mation parameters made due to the calculated prob-
abilities of the transitions between the levels lying
in the rotational bands witness that the deformation
parameters are different not only for the transitions
between one-particle states but also between the ro-
tational levels in the band.
References
1. A.Bohr and B.R. Mottelson. Collective and
individual-particle aspects of nuclear struc-
ture //Kgl. Danske Videnskab. Selkab. Mat.-fys.
Medd. 1953, v.27, p.1-174.
2. A.N.Vodin, E.G. Kopanets, L.P.Korda, and
V.Yu.Korda. The influence of nuclear deforma-
tions on the probabilies of electromagnetic tran-
sitions in 1d2s-shell nuclei // PAST, Ser.: ”Nu-
clear Physics Investigations”, 2003, N.2, p.66-71
(in Russian).
3. S.G.Nilsson. Binding states of individual nucle-
ons in strongly deformed nuclei //Kgl. Danske
Videnskab. Selskab, Mat. Fys. Medd. 1955, v.29,
N.16, p.1-68.
28
4. G.R.Bishop. Application of the collective model
to some nuclei in the d-shell // Nucl. Phys.
1959/60, v.14, p.376-388.
5. B.E.Chi. Singl-particle energy levels in a Nilsson
well // Nucl. Phys. 1966, v.83, p.97-144.
6. A.E. Litherland, H. McManus, E.B. Paul, et al.
An interpretation of the low-lying excited states
of 25Mg and 25Al // Can. J. Phys. 1958, v.36,
p.378-404.
7. A.K.Kerman. Low-lying excited states of nucleus
Mat. Fys. Vedd. Dan. Vid. Selsk. 1956, v.30, p.1-
44.
8. H. Lancman, A. Jasinski, J.Kownacki, et al. The
decay scheme of 23Ne // Nucl. Phys. 1965, v.69,
p.384-400.
9. E.V. Inopin, E.G.Kopanets, L.P.Korda, et al.
Electromagnetic transition in nuclei between
states with different deformations // PAST . Se-
ries: ”Physics of High Energies and Atomic Nu-
cleus”. 1975, N.3(15), p.31-33, (in Russian).
10. E.G.Kopanets, E.V. Inopin, L.P.Korda. Electro-
magnetic transitions in nuclei between states
with differrent deformations for the case
L>Ki+Kf Izvestya AN USSR, Physics Series.
1980, v.44, p.1947-1949 (in Russian).
11. L.P.Korda, E.G. Kopanets, E.V. Inopin. Consid-
eration of Coriolis interaction in calculations of
B(E2) between states with different deforma-
tions // PAST, Series: ”General and Nuclear
Physics”. 1984, N.2(27), p.63-65 (in Russian).
12. L.P.Korda, E.G. Kopanets. To the deformation
of light nuclei // PAST, Series: ”General
and Nuclear Physics”. 1979, N.2(16), p.3-6 (in
Russian).
ПАРАМЕТРЫ ДЕФОРМАЦИИ ВОЗБУЖДЕННЫХ СОСТОЯНИЙ sd-ОБОЛОЧКИ В
НЕЧЕТНЫХ ЯДРАХ
В.Д. Сарана, Н.С. Луцай, Н.А. Шляхов, Л.П. Корда
Дан обзор основных моделей, используемых для описания деформированных ядер. Прослеживаются
трудности, связанные с однозначностью выбора параметра деформации ядра во вращательных поло-
сах. Расчеты приведенных вероятностей в рамках модифицированной модели Нильсона показывают,
что использование разной деформации для возбужденных состояний улучшает согласие с эксперимен-
том. Рассмотрение ведется для ядер sd-оболочки.
ПАРАМЕТРИ ДЕФОРМАЦIЇ ЗБУДЖЕНИХ СТАНIВ sd-ОБОЛОНКИ В НЕПАРНИХ
ЯДРАХ
В.Д. Сарана, Н.С. Луцай, М.А. Шляхов, Л.П. Корда
Приведено огляд основних моделей, використовуваних для опису деформованих ядер. Простежуються
труднощi, пов’язанi з однозначнiстю вибору параметра деформацiї ядра в обертальних смугах. Розра-
хунки наведених iмовiрностей в рамках модифiкованої моделi Нiльсона показують, що використання
рiзної деформацiї для збуджених станiв полiпшує згоду з експериментом. Розгляд ведеться для ядер
sd-оболонки.
29
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