Few-nucleon systems: status and results of investigations
The model-independent calculation of the nuclei ground state and the states of scattering can be carried out with due regard for realistic NN and 3N forces between nucleons and also, with the use of exact methods of solving the many- body problem. The tensor part of NN interaction and 3NF's gen...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1118572017-01-15T13:53:12Z Few-nucleon systems: status and results of investigations Lyakhno, Yu.P. Ядерная физика и элементарные частицы The model-independent calculation of the nuclei ground state and the states of scattering can be carried out with due regard for realistic NN and 3N forces between nucleons and also, with the use of exact methods of solving the many- body problem. The tensor part of NN interaction and 3NF's generate the lightest nuclei states with nonzero orbital momenta of nucleons. These states in the lightest nuclei are the manifestation of the properties of inter-nucleonic forces, and therefore, similar effects should be observed in all nuclei and also in all their excited states. In this paper primary attention is given to the investigation of the ⁴He nucleus. Модельно незалежнi розрахунки основних станiв ядер, а також станiв розсiяння можна провести на основi реалiстичних NN i 3N-сил мiж нуклонами та при застосуваннi точних методiв вирiшення богато-частинкової задачi. Тензорна частина NN-взаємодiїi 3N-сили приводять до появи в основних станах найлегших ядер станiв з ненульовими орбiтальними моментам и нуклонiв. Цi стани в найлегших ядрах є проявом властивостей мiжнуклонних сил, i тому подiбнi ефекти повиннi спостерiгатися у всiх ядрах а також у їх збуджених станах. У цiй роботi велика увага придiлена дослiдженню ядра ⁴He. Модельно независимые расчёты основных состояний ядер,а также состояний рассеяния можно провести на основе реалистических NN и 3N- сил между нуклонами и с применением точных методов решения задачи многих тел. Тензорная часть NN-взаимодействия и 3N- силы приводят к появлению в легчайших ядрах состояний с ненулевыми орбитальными моментами нуклонов. Эти состояния являются проявлением свойств межнуклонных сил и, поэтому, подобные эффекты должны наблюдаться во всех ядрах и во всех их возбуждённых состояниях. В этой работе большее внимание уделено исследованию ядра ⁴He. 2013 Article Few-nucleon systems: status and results of investigations / Yu.P. Lyakhno // Вопросы атомной науки и техники. — 2013. — № 3. — С. 136-146. — Бібліогр.: 81 назв. — англ. 1562-6016 PACS: 25.10.+s; 23.20.-g http://dspace.nbuv.gov.ua/handle/123456789/111857 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 Lyakhno, Yu.P. Few-nucleon systems: status and results of investigations Вопросы атомной науки и техники |
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The model-independent calculation of the nuclei ground state and the states of scattering can be carried out with due regard for realistic NN and 3N forces between nucleons and also, with the use of exact methods of solving the many- body problem. The tensor part of NN interaction and 3NF's generate the lightest nuclei states with nonzero orbital momenta of nucleons. These states in the lightest nuclei are the manifestation of the properties of inter-nucleonic forces, and therefore, similar effects should be observed in all nuclei and also in all their excited states. In this paper primary attention is given to the investigation of the ⁴He nucleus. |
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Article |
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Lyakhno, Yu.P. |
| author_facet |
Lyakhno, Yu.P. |
| author_sort |
Lyakhno, Yu.P. |
| title |
Few-nucleon systems: status and results of investigations |
| title_short |
Few-nucleon systems: status and results of investigations |
| title_full |
Few-nucleon systems: status and results of investigations |
| title_fullStr |
Few-nucleon systems: status and results of investigations |
| title_full_unstemmed |
Few-nucleon systems: status and results of investigations |
| title_sort |
few-nucleon systems: status and results of investigations |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы |
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| citation_txt |
Few-nucleon systems: status and results of investigations / Yu.P. Lyakhno // Вопросы атомной науки и техники. — 2013. — № 3. — С. 136-146. — Бібліогр.: 81 назв. — англ. |
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Вопросы атомной науки и техники |
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| fulltext |
FEW-NUCLEON SYSTEMS: STATUS AND RESULTS OF
INVESTIGATIONS
Yu.P.Lyakhno∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received January 17, 2012)
The model-independent calculation of the nuclei ground state and the states of scattering can be carried out with due
regard for realistic NN and 3N forces between nucleons and also, with the use of exact methods of solving the many-
body problem. The tensor part of NN interaction and 3NF’s generate the lightest nuclei states with nonzero orbital
momenta of nucleons. These states in the lightest nuclei are the manifestation of the properties of inter-nucleonic
forces, and therefore, similar effects should be observed in all nuclei and also in all their excited states. In this paper
primary attention is given to the investigation of the 4He nucleus.
PACS: 25.10.+s; 23.20.-g
1. INTRODUCTION
From the physical standpoint, to describe the nucleon
system, one must know the nucleon properties and
also inter-nucleonic forces. The world constants and
nucleon properties are known within sufficient accu-
racy, while inter-nucleonic forces are complicated in
character and are known to a less accuracy. Unlike
the atom, these forces cannot be described by the
1/r2 ratio (where r is the distance between nucleons)
or by more complicated expressions like the Woods-
Saxon potential [1]. The distinctive feature of inter-
nucleonic forces is that they depend not only on the
distance r, but also on the quantum configuration
of the nucleon system, which is determined by the
orbital momentum L, spin S and isospin T of this
system.
The realistic NN potential can be determined phe-
nomenologically from the experimental data on the
ground state of the two-nucleon system and on the
elastic (p,p), (n,p) and (n,n) scattering at nucleon
energies up to 500 MeV. At higher nucleon energies,
nonelastic processes come into play, and the potential
approach becomes inapplicable. However, the data
about the inter-nucleonic forces in this nucleon en-
ergy region are sufficient for the description of nu-
clear ground states, and also of nuclear reactions up
to the meson-producing threshold.
The first phenomenological NN potentials were
calculated relying on the analysis of relatively small
arrays of the then existing experimental data on (p,p)
and (n,p) scattering. Gammel and Thaller [2], using
the data on differential cross-sections, nucleon polar-
ization and the deuteron data, have calculated the
central and tensor parts of NN potential, which were
represented with the help of 14 adjustable parame-
ters. The obtained results can also be represented as
singlet and triplet phases of elastic nucleon scatter-
ing. In the further analyses, as the experimental data
on (p,p) and (n,p) scattering were accumulated, ad-
justable expressions became more complicated, and
the accuracy of NN potential calculations got im-
proved. Among the most frequently used potentials,
the Hamada-Johnston potential [3], Reid potential [4]
and Paris potential [5] may be mentioned.
Nowadays, Argonne AV18 [6] and CD-Bonn [7]
appear to be the most accurate potentials. In the
construction of the charge-dependent CD-Bonn po-
tential in the range of laboratory-system nucleon
energies up to 350 MeV, 2932 (p,p)- and 3058 (n,p)-
scattering data were used. Adjustable expressions
were derived on the basis of the meson model of
strong nucleon interactions. Account was taken
of the π, ω and δ one-meson-exchange contribu-
tion, 2π -meson-exchange contribution, including ∆-
isobar configurations, and also of the πρ-exchange
contribution. The quantity χ2/datum was found
to be 1.02 at number of adjustable parameters
about 50. The results are presented in terms of
NN phase-shift and mixing parameters. Table 1
gives the classification of two-nucleon system states.
Table 1.Classification of two-nucleon system states.
Boldface type denotes the states with the data available
on them in the form of the CD-Bonn potential
S=0 S=1
J L-even L-odd L-even L-odd
T=1 T=0 T=0 T=1
0 1S0
3P0
1 1P1
3S1 + 3D1
3P1
2 1D2
2D3
2P3+2F3
3 3F1
3D3+3G3
3F3
4 1G4
3G4
3F4+3H4
5 1H5
3G5 + 3I5
3H5
∗Corresponding author E-mail address: lyakhno@kipt.kharkov.ua
136 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2013, N3(85).
Series: Nuclear Physics Investigations (60), p.136-146.
The following spectrometric notation is used for the
purpose: 2S+1LJ , J is the total momentum of the
system. The states, for which the dependence of NN
interaction in the form of the CD-Bonn potential
is obtained, are printed in bold type (up to J ≤4).
The total momentum J of the system may have even
higher values, however with an increasing J the con-
tribution of the states under decreases.
Arenhoevel et al. [8],[9] have calculated the struc-
ture functions of the reaction of polarized deuteron
disintegration by polarized electrons with measure-
ment of polarized nucleon recoil in different kine-
matic sectors. The measurement of these structure
functions permits to separate out the contributions
of π and ρ exchange, ∆-isobar configurations, and
relativistic effects. This is of great importance for a
more precise definition of the low-energy constants of
meson-nucleon interaction.
At present, it is found that 3N forces take action
in the nucleus. In the calculations, the 3N poten-
tials of types UrbanaIX [10] and Tucson-Melbourne
[11, 12] are most frequently used. Comparative study
of three-nucleon force models are held in Ref. [13].
It is hoped, that the accuracy of measurements of
realistic NN and NNN potentials would get further
better, in particular, at the expense of using the data
from double-polarization experiments [14]. A num-
ber of laboratories create polarized 3He nuclear tar-
gets [15]-[17]. The investigation of disintegration of
polarized 3He nuclei by polarized beams of particles
can provide some new information about 3N forces.
Along with the elaboration more precise defin-
ition of phenomenological potentials, important re-
sults were obtained through theoretical calculations
of inter-nucleonic forces within the framework of chi-
ral effective field theory (ChEFT). At present, the
calculation of chiral interactions is not as accurate
as that of phenomenological NN forces. Calculated
within the framework of the ChEFT, the NN poten-
tial parameters for partial waves with J ≤ 3 are in
satisfactory agreement with the experiment in the re-
gion of nucleon lab energy up to TN ∼290 MeV [18].
In the context of the ChEFT, Rozpedzik et al. [19]
estimated the effect of 4N forces and found the addi-
tional contribution of 4N forces to the binding energy
of the 4He nucleus to be about several hundreds of
keV.
The calculations in the context of ChEFT are of
particular importance for explaining the origin and
explicit representation of 3N and 4N forces. The rea-
son is that there are a good many experimental data
to determine the NN potential, whereas for determi-
nation of 3N and 4N forces these data are not nearly
enough. The origin and the explicit form of 3N and
4N forces it is a basic issue of few-nucleon systems.
Mathematically, to describe the nucleon system,
it is necessary to use the accurate methods of solv-
ing the many-nucleon problem. The solution of this
problem is an intricate theoretical task. The method
for solving the three-body problem in the limit of
the zero radius of action has been proposed in Ref.
[20]. To describe the three-body system in the case
of an arbitrary two-body potential, Faddeev [21] sug-
gested solving a set of connected integral equations.
At negative energy of the particle system these equa-
tions are homogeneous, while at positive energy the
equations are non-homogeneous. In a special case
of three different spinless particles, the system com-
prises three equations. Should the particles have
spins, then it is necessary to set up equations for
each possible quantum configuration of the system.
Later on, Yakubovsky [22] generalized this result for
the case of any number of particles in the system.
In this connection, that 3N forces participation in
the nucleus, for exact description of a many-nucleon
system it is necessary to solve the set of connected in-
tegral equations with due regard for the contribution
of NN and 3N forces. The solution of the problem
by the Faddeev-Yakubovsky (FY) method was first
reported by Gloeckle and Kamada (GK) [23].
The realistic NN and NNN forces were also used
in the calculations by the Lorentz integral transform
(LIT) method [24], [25] the hyperspherical harmonic
variational method (HHVM) [26], the refined resonat-
ing group model (RRGM) [27], [28] and others [29].
Section 2 presents the results of theoretical calcu-
lations of the ground states of few-nucleon nuclei, and
also, the examples of nuclear reaction calculations
based on the realistic NN and NNN forces. Section 3
describes the results of the study into the 4He nuclear
structure, and also, presents the multipole analysis
of the 4He(γ, p)T and 4He(γ, n)3He reactions, per-
formed on the basis of the experimental data about
the differential cross-sections and cross-section asym-
metry with linearly polarized photons. The possible
effects, determined by realistic inter-nucleonic forces
in nuclei with A > 4 are discussed in Section 4. The
conclusions are formulated in Section 5.
2. RESULTS OF THE THEORETICAL
CALCULATIONS
The characteristics of three- and four-nucleon sys-
tems by the FYGK method were calculated in works
[30]-[34]. In Ref. [31] calculation of the ground-state
of the α-particle is carried out. The calculations took
into account the contributions from the states of the
NN system having the total momentum up to J ≤6.
The consideration of large total-momentum values of
the two-nucleon system is necessary, for example, for
a correct calculation of short-range correlations. In
the calculation [31], account was taken of the states,
in which the algebraic sum of orbital momenta of
all nucleons of the 4He nucleus was no more than
lmax=14. The system comprised 6200 partial waves.
The authors of work [31] estimated their mistake in
the calculations of 4He nuclear binding energy to be
∼50 keV. Considering that the calculated value of
the binding energy is ∼200 keV higher than the ex-
perimental value, the authors have made a conclusion
about a possible contribution of 4N forces that are of
repulsive nature.
137
Table 2 lists the values of nuclear binding ener-
gies (in MeV) for 4He, 3H, 3He and 2H, calculated
with the use of NN potentials AV18 and 3NF’s Ur-
banaIX. It is evident from the table that without
taking into account the 3N forces, the nuclei appear
underbound, while with due regard for the forces
the agreement with experimental data is satisfactory.
Table 2. Binding energies (in MeV units) of 4He,
of 3H, of 3He and of 2H, calculated with Argonne
V18 and Argonne V18 + Urbana IX interaction
Inter-
action Method 4He 3H 3He 2H
AV18 FY -24.28 -7.628 -6.924
RRGM -24.117 -7.572 -6.857 -2.214
HHVM -24.25
AV18+ FYKG -28.50 -8.48 -7.76
+UIX RRGM -28.342 -8.46 -7.713 -2.214
HHVM -28.50 -8.485 -7.742
Exp -28.296 -8.481 -7.718 -2.224
Similar results were obtained with the use of the
NN potential CD-Bonn and the 3N potential Tucson-
Melbourne.
Table 3 gives the calculated root-mean-square
radii rrms of the 4He nucleus [26, 28]. The
agreement with experiment is also satisfactory.
Table 3. The 4He nucleus < r2 >1/2 radii (fm),
where r-distance between nucleons centers
Interaction Method 4He
AV18 RRGM 1.52
HHVM 1.512
AV18+UIX RRGM 1.44
HHVM 1.43
Exp 1.67
It should be also noted that the Coulomb in-
teraction between protons results in the produc-
tion of T=1 and T=2 isospin states of 4He. Ta-
ble 4 gives the probabilities of these states for
the 4He nucleus calculated in papers [26], [31].
Table 4.Contribution of different total isospin
states to the 4He nuclear wave function. The values
are given in %
Interaction Method T=0 T=1 T=2
AV18 FY 99.992 3·10−3 5·10−3
HHVM 2.8·10−3 5.2·10−3
The tensor part of NN interaction and 3NF’s gen-
erate the 4He nuclear states with nonzero orbital
momenta of nucleons. The measurement of prob-
ability for l 6= 0 states occurrences provides new
information about these properties of inter-nucleonic
forces. For illustration, Table 5 gives possible 4He
nuclear states allowed by the laws of conservation
of the total momentum and parity. The symbols
s and l denote the spin and orbital momentum of
the nucleon, respectively. According to the vector
addition rules, spins and orbital momenta of indi-
vidual nucleons are summed so that the total mo-
mentum J of the 4He nucleus and its parity π are
conserved and remain equal to 0+. The nuclear shell
model predicts only one state given in the first row
of Table 5. The probability of this state has been
calculated to be around 84%. The nuclear states of
4He, listed in Table 5, satisfy the Pauli principle.
Table 5. Scheme of possible quantum states
of the 4He nucleus
States of
the 4He p n p n L,S Jπ P,%
nucleus l,s l,s l,s l,s
0↑ 0↑ 0↓ 0↓ 0,0 0+ 84
1S0 1↑ 0↑ 0↓ 1↓ 0,0 0+
1↑ 1↑ 1↓ 1↓ 0,0 0+
2↑ 1↑ 1↓ 0↓ 0,0 0+
- - - -
1↑ 0↑ 0↑ 1↓ 1,1 0+ 0.7
2↑ 0↑ 0↑ 1↓ 1,1 0+
3P0 2↑ 2↑ 0↑ 0↓ 1,1 0+
3↑ 1↑ 1↑ 1↓ 1,1 0+
- - - -
0↑ 0↑ 1↑ 1↑ 2,2 0+ 16
2↑ 0↑ 0↑ 2↑ 2,2 0+
5D0 2↑ 1↑ 1↑ 2↑ 2,2 0+
3↑ 1↑ 1↑ 3↑ 2,2 0+
- - - -
Table 6 gives the probabilities of S, S′, P and D states
of the 4He and 3He nuclei calculated by Nogga et al.
[31], where S′ is a part of 1S0-states with nonzero or-
bital momenta of nucleons. The calculations gave the
probability of 5D0 states having the total spin S=2
and the total nucleon orbital momentum L=2 of the
4He nucleus to be ∼ 16%, and the probability of 3P0
states having S=1, L=1 to be between 0.6 and 0.8%.
It is obvious from Table 6 that the consideration of
the 3NF’s contribution increases the probability of
3P0 states by a factor of ∼2.
Table 6. S, S′, P, and D state probabilities for 4He and 3He
4He 3He
Interaction S% S′% P% D% S% S′% P% D%
AV18 85.45 0.44 0.36 13.74 89.95 1.52 0.06 8.46
CD-Bonn 88.54 0.50 0.23 10.73 91.45 1.53 0.05 6.98
AV18+UIX 82.93 0.28 0.75 16.04 89.39 1.23 0.13 9.25
CD-Bonn+TM 89.23 0.43 0.45 9.89 91.57 1.40 0.10 6.93
138
2.1. STATES OF SCATTERING
A full calculation of the nuclear reaction must take
into account the ground-state structure of the nu-
cleus, the contribution of the interaction of the probe
with nucleons and meson exchange currents (MEC),
quantum configuration of the off particles and the
final-state interaction of particles (FSI). These calcu-
lations were carried out for three-nucleon nuclei. In
the work of Gloeckle et al. [35], the analysis of elec-
tron scattering by 3He and 3H nuclei was performed.
Elastic charge Fch(q) and magnetic Fm(q) form fac-
tors, inclusive electron scattering, pd-breakup and
full-breakup of these nuclei were calculated with
the use of the AV18 NN force and the Urbana IX
3NF’s. The contribution of the π and ρ exchange
was taken into account according to Riska’s pre-
scription [36]. The calculations were performed by
the Faddeev scheme, that allowed one to analyze in
detail the 3NF’s, MEC and FSI contributions to dif-
ferent observable quantities. Fig. 1 shows the elastic
charge Fch(q) and magnetic Fm(q) form factors of
the 3He nucleus. The discrepancy between the calcu-
lation and the experiment at high-transfer momen-
tum values (q>3 fm−1) was attributed by the au-
thors of [35] to the contribution of relativistic effects.
Fig.1. Elastic charge Fch(q) and magnetic Fm(q)
form factors of the 3He nucleus. The experimental
data are taken from Ref.[37], the curve - from
Ref.[35]
Similar calculations were carried out for the radiative
proton-deuteron capture reaction (Golak et al. [38],
Kotlyar et al. [39]), three-nucleon photodisintegra-
tion of 3He (Skibinski et al. [40]).
The calculations of nuclear reactions in 4He nu-
cleus were performed by the RRGM method [28] with
using potentials AV18 and UrbanaIX, and with using
semi-realistic potential [41], by the LIT method with
realistic potentials [25] and with semi-realistic poten-
tial MT I-III [42], and also other methods. There is a
dispersion between different theoretical calculations.
Theoretical calculations [27], [43] and numerous
experiments were carried out to investigate hadronic
probe reactions with participation of three and four
nucleons [44], [45].
3. STRUCTURE IVESTIGATIONS OF 4He
BY MEANS OF PHOTOREACTIONS
For measuring the probability of 5D0 states of
the 4He nucleus, it is reasonable to investigate the
2H(~d, γ)4He reaction of radiative deuteron-deuteron
capture. The tensor analyzing power of the reac-
tion is sensitive to the contribution of 5D0 states
Weller et al. [46]-[48]. In these studies the differential
cross section, the vector and tensor analyzing power
of reaction measured in the deuteron energy range
0.7< Ed <15 MeV. In Ref. [48] was made a compari-
son of this data with theoretical calculation Wachter
et al. [49]. The agreement of the calculation with
the experiment was achieved in the assumption, that
the probability 5D0 states of the 4He nucleus com-
poses ∼ 2.2%. In this calculation the semi-realistic
NN potential was used. In work Mellema et al. [50]
has measured the differential cross-section, the vec-
tor and tensor analyzing powers of the reaction at
the deuteron energy Ed=10 MeV. In fitting, the best
agreement with the experimental data was obtained
in the assumption that 1D2(E2), 5S2(E2), 3P1(E1)
and 3P2(M2) were the basic transitions. The prob-
ability of the 5D0 state of the 4He nucleus was es-
timated around 15%. However, it is marked in this
work that at the calculation of this probability there
is a problem of account of the tensor-force effects
in the incident channel which also contribute to the
measured values of the multipole amplitudes. The
reaction was investigated at the deuteron energy Ed
=1.2 MeV [51], and also at Ed =20, 30 and 50 MeV
[52].
At low deuteron energies, the 5S2(E2) → 5D0
transition should dominate. This is due to the fact
that at a low deuteron energy the 1D2(E2), 5D2(E2),
5G2(E2) and 3F2(M2) transitions are suppressed
by the angular momentum barrier. The 3P1(E1)
and 3P2(M2) transitions to the final state 1S0 or
5D0 are suppressed because of the spin flip ∆S=1
[28]. Besides, in the reaction under discussion the
E1 and M1 transitions are suppressed according to
the isospin selection rules for self-conjugate nuclei
(∆T=1) [53]. The analysis of measured differen-
tial cross section, vector/tensor analyzing powers of
the reaction at the deuteron energy Ec.m.= 60 keV
Sabourov et al. [54] has shown the transition proba-
bilities to be 5S2(E2)=(55±8)%, 3P1(E1)=(29±6)%
and 3P2(M2)=(16±3)%. Significant cross sections
for 3P1(E1) and 3P2(M2) transitions may be due to
a greater contribution of 3P0 states of the 4He nu-
cleus than it follows from the calculations [26, 31]. In
turn, the last fact may be the result of a high sen-
sitivity to the peculiarities of NN and 3N potentials
139
[31]. The contribution of meson exchange currents
that may cause the spin flip ∆S=1 is also possible.
In view of this, the experimental data obtained from
the study of only one reaction appear insufficient for
calculating the probabilities of states with nonzero
orbital momenta of nucleons.
A new information about the l 6=0 states of 4He
can be obtained from studies of the 4He(γ, p)3H and
4He(γ, n)3He reactions, and also, the reactions of ra-
diative capture of protons or neutrons by tritium or
3He nuclei, respectively. In this case, the transi-
tions from 3P0 states of the nucleus to the final S=1
state occur without any spin flip. So, it may be ex-
pected that the comparison between the E1 and M1
transition cross-sections in (d, γ) and (γ,N) reactions
would provide new information on the contribution of
MEC.
The first experimental data on spin-triplet tran-
sitions have been obtained from studies of the reac-
tion of radiative capture of protons by tritium nu-
clei. When investigating the 3H(~p, γ)4He reaction
on a polarized protons beam of energies between 0.8
and 9MeV, Wagenaar et al. [55] came to the con-
clusion that 3S1(M1) was the basic transition. At
the same time, from the studies of the same reac-
tion but at polarized protons energy Ep=2 MeV Pitts
[56] has stated 3P1(E1) to be the basic transition.
This reaction was investigated also at protons energy
Ep=80 keV [57]. In this work the conclusion that
a basic transition with the spin of S=1 is 3S1(M1).
These contradictory statements are due to the fact
that the experimental data obtained had significant
errors. At higher energies, the measurements are
complicated by the necessity of considering the am-
plitude 3D1(M1), which is suppressed at low photon
energies by the angular momentum barrier.
Nowadays, were made more than ten experimen-
tal works on measurement total and differential cross-
section reactions 4He(γ, p)3H and 4He(γ, n)3He in en-
ergies range at the giant dipole resonance peak. The
results of these measurements can be found in the
work Tornow et al. [58]). The data obtained by dif-
ferent laboratories on the total cross section these
reactions attains a factor of ∼2.
To measurement the S=1 transition cross-sections
for the reactions 4He(γ, p)3H and 4He(γ, n)3He, one
needs the experimental data about the cross sections
of these reactions in the collinear geometry, and also
the polarization observable quantities.
In Refs. [59, 60], chambers placed in the mag-
netic field were used to detect the reaction products.
This has permitted measurements of the differential
cross section of the reactions in the range of polar
nucleon-exit angles 00 ≤ θN ≤ 1800. However, the
number of events registered in those experiments, was
insufficient for measuring the reaction cross-sections
in collinear geometry.
Jones et al. [61] have measured the differ-
ential cross section of the 4He(γ, p)3H reaction
at the energy of tagged bremsstrahlung photons
63≤ Eγ ≤ 71MeV. The reaction products were regis-
tered by means of a large-acceptance detector LASA
with electronic information retrieval. The measure-
ments were performed in the interval of polar proton-
exit angles 22.50 ≤ θN ≤ 145.50. The absence of the
data at large and small angles has led to significant
errors in the cross section measurements of this reac-
tion in the collinear geometry.
Nilsson et al. [62] investigated the reac-
tion 4He(γ, n) in the energy range of tagged
bremsstrahlung photons 23≤ Eγ ≤70MeV. The
time-of-flight technique was employed to determine
the neutron energy. However, the measurements of
this reaction at large and small angles were not per-
formed either.
In the work Shima et al. [63] used a mono-
energetic photon beam in the energy range from
21.8 to 29.8 MeV and nearly 4π time projection
chamber, were measured the total and differen-
tial cross-sections for photodisintegration reactions
of the 4He nucleus. Authors received that the
M1 strength is about (2±1)% of the E1 strength.
Fig.2. The total cross-section in collinear geometry
reactions 4He(~γ, p)T and 4He(~γ, n)3He. Triangle;
Jones et al. [61], filled circle; Shima et al. [63],
open circles; Nagorny et al. [66]. The errors are
statistical only.
Differential cross sections for two-body (γ, p) and
(γ, n) reactions have been measured in works Arka-
tov et al. [64], [65] in the photon energy range from
the reaction threshold up to Eγ=150MeV. The re-
action products were registered with the help of a
diffusion chamber. Later, these data were processed
for the second time in the work of Nagorny et al.
[66]. The data are based on the statistics of ∼ 3 · 104
events in each of the (γ,p) and (γ,n) reaction chan-
nels. The cross sections were measured with a 1 MeV
step up to Eγ=45MeV, and with a greater step
at higher energies, as well as with a 100 step in
the polar nucleon-exit angle in the c.m.s. Unfor-
tunately, authors published data about differential
cross-sections only at photon’s energies 22.5, 27.5,
33.5, 40.5, 45, and 49MeV. As a result of multipole
analysis Voloshchuk [67], the total cross sections for
electric dipole and electric quadrupole transitions
with the S=0 spin in the final state of the particle
140
system, and also, the phase shift between E1 and
E2 amplitudes were calculated from the differential
cross sections. The total cross-section in collinear
geometry reactions 4He(~γ, p)T and 4He(~γ, n)3He are
shown in Fig. 2.
The angular dependence of cross-section asym-
metry in the 4He(γ, p)3H and 4He(~γ, n)3He reactions
with linearly polarized photons of energies 40, 60 and
80MeV was measured by Lyakhno et al. [68, 69]. The
beam of linearly polarized photons was produced as
a result of coherent bremsstrahlung of 500, 600 and
800MeV electrons, respectively, in a thin diamond
single crystal. The reaction products were registered
with the use of a streamer chamber located in the
magnetic field [70]. The observed data on the angular
dependence of the cross-section asymmetry are pre-
sented in Fig. 3. Here the square represents the data
obtained with semi-conductor detectors by the ∆E-E
method [71]. The dashed curve is the calculation by
Mel’nik and Shebeko [72] made in the plane-wave
impulse approximation with consideration of the di-
rect reaction mechanism and the mechanism of recoil.
Fig.3. Angular dependence of cross-section
asymmetry of linearly polarized photon reactions
4He(~γ, p)T and 4He(~γ, n)3He. The points represent
the results of Refs. [68, 69]: a) Epeak
γ =40MeV,
b) Epeak
γ =60MeV, c) Epeak
γ =80MeV. The square
shows the data of Ref. [71]. The errors are statis-
tical. The solid curve - calculation [66], the dashed
curve - calculation [72]
It was indicated in Ref. [72] that without consider-
ation of the mechanism of recoil the cross-section
asymmetry in the (~γ, n) channel would be equal to
zero. This is due to the fact that in the mentioned ap-
proximation the 4He(γ, n)3He reaction is contributed
only by the magnetic component of the interaction
Hamiltonian. So, the asymmetry Σ(θn) in the (~γ, n)
channel measured in the experiment to be close to
unity confirms an essential role of the mechanism of
recoil. The solid curves represent the calculation [66]
that meets the requirements of covariance and gauge
invariance. The calculation took into account the
contribution of a number of diagrams corresponding
to the pole mechanisms in s-, t- and u-channels, the
contact diagram c, and also a number of triangular
diagrams. A satisfactory fit of the calculation to the
experiment confirms an essential role of the direct
reaction mechanism, the mechanism of recoil and the
final-state rescattering effects.
3.1. MULTIPOLE ANALYSIS OF 4He(γ, p)T
AND 4He(γ, n)3He REACTIONS
In the E1, E2 and M1 approximations, the laws
of conservation of the total momentum and parity
for two-body (γ, p) and (γ, n) reactions of 4He nu-
clear disintegration permit two multipole transitions
E11P1 and E21D2 with the spin S=0 and four tran-
sitions E13P1, E23D2, M13S1 and M13D1 with the
spin S=1 of final-state particles. The differential
cross section in the c.m.s. can be expressed in terms
of multipole amplitudes as follows [73, 74]:
dσ
dΩ
=
λ−
2
32
{sin2 θ[18|E11P1|2 − 9|E13P1|2
+9|M13D1|2 − 25|E23D2|2
−18
√
2Re(M13S∗1 M13D1) + 30
√
3Re(M13D∗1 E23D2)
+30
√
6Re(M13S∗1 E23D2)
+ cos θ(60
√
3Re(E11P∗1 E21D2)
−60Re(E13P∗1 E23D2))
+ cos2 θ(150|E21D2|2 − 100|E23D2|2)]
+ cos θ[−12
√
6Re(E13P∗1 M13S1)
−12
√
3Re(E13P∗1 M13D1) + 60Re(E13P∗1 E23D2]
+18|E13P1|2 + 12|M13S1|2 + 6|M13D1|2
+50|E23D2|2 + 12
√
2Re(M13S∗1 M13D1)
−20
√
6Re(M13S∗1E23D2)− 20
√
3Re(M13D∗1 E23D2)} ,
(1)
where λ− is the reduced wavelength of the photon.
It is known that the cross-section asymmetry of
the linearly polarized photon reaction is described by
the following expression [69]:
Σ(θ) = sin2 θ{18 | E11P1|2 − 9|E13P1|2
−9|M13D1|2 + 25|E23D2|2
+18
√
2Re(M13S∗1 M13D1) + 10
√
3Re(M13D∗1 E23D2)
+10
√
6Re(M13S∗1 E23D2)
+ cos θ[60
√
3Re(E11P∗1 E21D2)
−60Re(E13P∗1 E23D2)]
+ cos2 θ[150|E21D2|2 − 100|E23D2|2]}/ 32
λ−2
dσ
dΩ
. (2)
The differential cross section can be presented as:
dσ
dΩ
= A[sin2 θ(1 + β cos θ + γ cos2 θ) + ε cos θ + ν] .
(3)
In the same terms, the cross-section asymmetry of
the linearly polarized photon reaction can be repre-
sented as follows:
Σ(θ) =
sin2 θ(1 + α + β cos θ + γ cos2 θ)
sin2 θ(1 + β cos θ + γ cos2 θ) + ε cos θ + ν
. (4)
141
The coefficients A, α, β, γ, ε, and ν are unambigu-
ously connected with multipole amplitudes. As it is
obvious from relation (3), only 5 independent coef-
ficients can be calculated in the long-wave approx-
imation using the data on the differential reaction
cross-section. So, an improvement in the accuracy of
measuring only the differential reaction cross-section
gives no way of obtaining information about subse-
quent multipole amplitudes. In this case, the num-
ber of unknown parameters in the right side of eq.
(1) would increase much quicker than the number of
found coefficients in the left side of the equation. In
this connection, in order to obtain information on the
succeeding multipole amplitudes, polarization exper-
iments or other data sources are required. As it can
be seen from relation (4), the experimental data on
the asymmetry of the linearly polarized photon re-
action cross-section enable one to calculate the sixth
independent coefficient.
It can be demonstrated that on the assumption
that σ(E23D2) À σ(M1), from expressions (1) and
(2) we obtain:
α =
50|E23D2|2
18|E11P1|2 − 9|E13P1|2 − 25|E23D2|2 > 0 . (5)
If we assume that σ(M1)À σ(E23D2), then we have
α =
�−18|M13D1|2
+36
√
2|M13S1||M13D1| cos[δ(3S1)− δ(3D1)]}/
{18|E11P1|2 − 9|E13P1|2 + 9|M13D1|2
−18
√
2|M13S1||M13D1| cos[δ(3S1)− δ(3D1)]} . (6)
From the phase analysis of elastic (p,3He) scattering
Murdoch et al. [75] have determined the phase dif-
ference to be δ(3S1) − δ(3D1) > 900. Therefore, the
both components in the numerator of expression (6)
enter with the minus sign, and the coefficient α must
be negative.
As a result of the least-squares fit of expressions
(3) and (4) to the experimental data on the differ-
ential cross section [66] and cross-section asymmetry
of linearly polarized photon reactions, the coefficients
A, α, β, γ, ε, and ν were calculated [69]. Since the
coefficients enter into relations (3) and (4) in linear
fashion, the solution was unambiguous.
Since only phase differences enter into formu-
las (1) and (2), these relations comprise 11 un-
known parameters. The currently available ex-
perimental data on the (γ, p) and (γ, n) reactions
are insufficient for determining all the parameters.
According to the experimental data obtained (see
Fig.4), αp and αn are the minus coefficients, and
hence, the least amplitude that enters into expres-
sions (1) and (2) is the E23D2 amplitude. After
the E23D2-comprising components are excluded, ex-
pressions (1) and (2) still comprise 9 unknown pa-
rameters: |E11P1|, |E21D2|, cos[δ(1P1) − δ(1D2)],
|E13P1|, |M13S1|, |M13D1|, cos[δ(3S1) − δ(3D1)],
cos[δ(3S1) − δ(3P1)] and cos[δ(3P1) − δ(3D1)].
Fig.4. Coefficients αp and αn. The errors are
statistical only
It is known [53] that according to the isospin selec-
tion rules for self-conjugate nuclei the isoscalar parts
of E1 and M1 amplitudes are essentially suppressed.
In view of this, using the Watson theorem, the last
three phase differences were calculated from the data
of phase analysis of elastic (p,3He) scattering [75].
The coefficients A, α, β, γ, ε, and ν are expressed
in terms of the multipole amplitudes as:
A = λ−
2
/32{18|E11P1|2 − 9|E13P1|2 + 9|M13D1|2 −
18
√
2|M13S1||M13D1| cos[δ(3S1)− δ(3D1)]}; (7)
α = {−18|M13D1|2 +
36
√
2|M13S1||M13D1| cos[δ(3S1)− δ(3D1)]} /
32
λ−2 A ; (8)
β = 60
√
3|E11P1||E21D2| cos[δ(1P1)− δ(1D2)] /
32
λ−2 A ;
(9)
γ = 150|E21D2|2 /
32
λ−2 A ; (10)
ε = {−12
√
3|E13P1||M13D1| cos[δ(3P1)− δ(3D1)]−
12
√
6|E13P1||M13S1| cos[δ(3P1)− δ(3S1)]} /
32
λ−2 A ;
(11)
ν = {18|E13P1|2 + 12|M13S1|2 + 6|M13D1|2 +
12
√
2|M13S1||M13D1| cos[δ(3S1)− δ(3D1)]} /
32
λ−2 A.
(12)
The experimentally observable quantities are ex-
pressed in terms of multipole amplitudes in a bilin-
ear fashion. Therefore, there must exist two different
sets of multipole amplitudes, which satisfy these ex-
perimental data. With the help of programs of the
least square method (LSM) one positive solution of
the problem can be calculated. The second solution
can be found, for example, by the lattice method.
Since both positive solutions have the same χ2 val-
ues, an additional information is necessary to choose
142
the proper solution. It should be also noted that
if the difference between the solutions is comparable
with the amplitude errors, then the LSM errors of the
amplitudes may appear overestimated.
The amplitude values were calculated from the
derived set of six bilinear equations (7-12) with six
unknown parameters using the random-test method
[69]. To calculate the errors in the amplitudes, 5000
statistical samplings of A, α, β, γ, ε, and ν values
with their errors were performed. The errors in the
coefficients were assumed to be distributed by the
normal law. After each statistical sampling the set of
equations was solved, the calculated amplitude val-
ues were stored and then their average values and
dispersions were calculated.
According to Ref. [56], one can assume that with
the photon energy increase the M13S1 transition
cross-section decreases as 1/V, where V is the nucleon
velocity. Therefore, at MeV nucleon energies the con-
tribution of the M13S1 transition can be neglected.
In this connection, out of the two found solutions of
the system of equations (7-12) the choice has been
made on the solution, where σ(E13P1) > σ(M13S1).
Fig.5. Total cross sections of spin S=1 transitions
of 4He(γ, p)T and 4He(γ, n)3He reactions: 3–data
from Ref. [56]; 4–data of Ref. [55]; •–data of Ref.
[69]. The errors are statistical only
The findings of the experiment aimed to determine
the total cross sections of S=1 transitions are pre-
sented in Fig.5. The triangles represent the data of
Wagenaar et al. [55], the diamond shows the data
of Pitts [56] obtained from studies of the reaction of
radiative capture of protons by tritium nuclei. The
points represent the data of Lyakhno et al. [69] from
the studies of two-body (γ, p) and (γ, n) reactions of
4He disintegration. The existing experimental data
on the total cross-sections of electromagnetic tran-
sitions with the spin S =1 in the 4He(γ, p)T and
4He(γ, n)3He reactions have considerable statistic
and systematic errors.
If for a certain photon energy range it can be as-
sumed that σ(M13S1) ∼ σ(E23D2) ∼0, then the
amount of the available experimental data on the
differential cross-section and cross-section asymme-
try with linear polarized photons is sufficient for cal-
culating the cross-sections of other transitions with
the spin S=1 of the final state of the particle system
without invoking the data on elastic (p,3He) scatter-
ing.
4. ROLE OF THE SPIN-ORBIT
INTERACTION IN NUCLEI
The investigation into the structure of few-nucleon
nuclei is of considerable importance for an under-
standing of the structure of other nuclei. The oc-
currence of states with nonzero orbital momenta of
nucleons in the lightest nuclei is a manifestation of
the properties of inter-nucleonic forces and, hence,
such effects should be observed without exception in
all nuclei as well as in all their excited states.
One can suppose that the contribution of the ef-
fects connected with the tensor part of NN potential
and 3NF’s increases with a growth of the atomic num-
ber. Firstly, it can be seen from the fact that the D-
state contribution in the deuteron is about 5%, while
in the 4He nucleus it makes ∼16%.
Secondly, at the calculation of the probabilities
of the outside shell states, for example, in the 12C
nucleus, similarly to the case with the 4He nucleus,
we must bear in mind that the nucleus spin of 12C
can take the values 0≤ S ≤6, the total orbital mo-
mentum of the nucleons of 12C can be 0≤ L ≤6, and
the orbital momenta of separate nucleons can take
on any values, which are not forbidden by the Pauli
principle. In other words, the ground state of the
nucleus 12C can be of the 1S0, 3P0, 5D0, 7F0, 9G0,
11H0, or 13I0 states. Nowadays, probability of these
states is not held due to their extreme complication.
However, the rough estimation of these probabilities
can be achieved in the following way. Let us sup-
pose the 12C nucleus consist of three weakly bound
α-clusters. The total momentum and parity conser-
vation laws do not forbid, and the tensor part of NN
interaction and 3NF’s initiate states with the orbital
momenta larger than predicted by the nuclear shell
model (NSM) [75, 76] in every cluster, in two clusters
at one time or in all three clusters. In the result, the
probability of these states in the 12C can be higher
than in the 4He nucleus.
Using this supposition one can explain the row of
the well-known nuclei properties. Particularly, it is
possible to explain the significantly higher contribu-
tion of the spin-orbit interaction in the nuclei, than
calculated in the NSM frames. Contribution of the
spin-orbit interaction A nucleons in the potential en-
ergy nucleus can be assessed by the relation:
USO = −λ
(
h̄
Mc
)2 A∑
i=1
1
ri
∂Vi
∂ri
(~li · ~si) , (13)
where M is the nucleon mass, Vi is the spherically
symmetrical potential, l is the orbital momentum, s
is the spin of the nucleon. However, for the agreement
143
of the experimental data into the expression (13) was
put a constant, which is λ ∼10.
The appearance of the fitting constant λ can be
partially explained as follows. Let k-number of the
nucleons, with the orbital momenta in accordance
with the NSM, and other A-k nucleons have orbital
momenta bigger than it is predicted by the NSM.
Then the expression (13) can be refined as [78]:
USO = −
(
h̄
Mc
)2
[
k∑
i=1
1
ri
∂Vi
∂ri
P (lsh
i )(~lsh
i · ~si)
+
A∑
i=k+1
1
ri
∂Vi
∂ri
P (li > lsh
i )(~li · ~si)
]
, (14)
where P (li) is the probability for the i-th nucleon to
have the orbital momentum li. The sum of probabili-
ties is
∑A
i=1 P (li)=1. Thus, in the case of the lightest
nuclei second summing of the expression (14) leads to
the small but not equal zero impact of spin-orbit in-
teraction to the potential energy of the nucleus. In
medium and heavy nuclei nucleons are situated, gen-
erally, in l > lsh states. In other words, the tensor
part of NN potential and 3N forces push the nucle-
ons outside of nuclear shells, with the rise of atomic
number the role of this effects is rising at that. The
second summing, which is not predicted by NSM, can
give a significant additional contribution to potential
energy of a nucleus.
In particular, calculations [79, 80], made on the
basis of the Woods-Saxon potential, can give the over-
estimation of the protons number Z for the position
of the island of stability of the superheavy nuclei.
In the frames of semiempirical shell models [81] at
the extrapolation of fitting expressions for the area
of the heavy nuclei to the area of the superheavy nu-
clei, apparently, the according corrections should be
also counted.
5. CONCLUSIONS
Modern methods of the decision of a many-nucleon
problem make it possible to calculate the character-
istics of few-nucleon nucleus to an accuracy, which
is determined by the accuracy the measurement of
NN potential, and also 3NF’s and 4N forces. In this
connection the 4He nucleus is an ideal laboratory for
investigating the properties of these forces.
The measurement of total cross-sections for the
electromagnetic transitions with spin S=1 in the fi-
nal state of the particle system in 4He(γ, p)3H and
4He(γ, n)3He reactions, and, in addition to data
about radiative deuteron-deuteron capture, can pos-
sibly allow to separate the effects specified by the nu-
clear ground state structure from the effects specified
by the reaction mechanisms. Theoretical calculations
wanted.
The states with non-zero orbital momenta of the
nucleons of the lightest nuclei are the manifestation
of the properties of inter-nucleonic forces and, conse-
quently, such effects should be observed in all nuclei
and in all their excited states. One can suppose, that
the tensor part of NN potential and 3N forces push
the nucleons outside of nuclear shells, with the rise
of atomic number the role of these effects is rising at
that. This can lead to the additional contribution of
the spin-orbital interaction to the potential energy of
the nucleus in comparison with an estimation on the
nuclear shell model.
Author gives the gratitude to Yu.P. Stepanovsky
for important advice and discussion over the article
material, and to A.V. Shebeko for the sequence of
critical remarks.
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МАЛОНУКЛОННЫЕ СИСТЕМЫ: СОСТОЯНИЕ И РЕЗУЛЬТАТЫ
ИССЛЕДОВАНИЯ
Ю.П.Ляхно
Модельно независимые расчёты основных состояний ядер, а также состояний рассеяния можно про-
вести на основе реалистических NN и 3N - сил между нуклонами и с применением точных методов
решения задачи многих тел. Тензорная часть NN- взаимодействия и 3N- силы приводят к появлению
в легчайших ядрах состояний с ненулевыми орбитальными моментами нуклонов. Эти состояния яв-
ляются проявлением свойств межнуклонных сил и, поэтому, подобные эффекты должны наблюдаться
во всех ядрах и во всех их возбуждённых состояниях. В этой работе большее внимание уделено иссле-
дованию ядра 4He.
МАЛОНУКЛОННI СИСТЕМИ: СТАН I РЕЗУЛЬТАТИ ДОСЛIДЖЕННЯ
Ю.П.Ляхно
Модельно незалежнi розрахунки основних станiв ядер, а також станiв розсiяння можна провести на
основi реалiстичних NN i 3N- сил мiж нуклонами та при застосуваннi точних методiв вирiшення богато-
частинкової задачi. Тензорна частина NN- взаємодiї i 3N- сили приводять до появи в основних станах
найлегших ядер станiв з ненульовими орбiтальними моментами нуклонiв. Цi стани в найлегших ядрах
є проявом властивостей мiжнуклонних сил, i тому подiбнi ефекти повиннi спостерiгатися у всiх ядрах
а також у їх збуджених станах. У цiй роботi велика увага придiлена дослiдженню ядра 4He.
146
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