Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian
A new model-free approach for analyzing the intermediate-energy light nucleus-nucleus elastic-scattering differential cross-sections is proposed to extract the numerical dependencies of S-matrix modulus and nuclear phase on angular momentum directly from the experimental data via the evolutionary...
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| Published in: | Вопросы атомной науки и техники |
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| Date: | 2004 |
| Main Authors: | , , , , |
| Format: | Article |
| Language: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2004
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/80508 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian / А.N. Vodin, V.Yu. Korda, А.N. Dovbnya, A.S. Molev, L.P. Korda // Вопросы атомной науки и техники. — 2004. — № 5. — С. 48-54. — Бібліогр.: 32 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859610874195476480 |
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| author | Vodin, A.N. Korda, V.Yu. Dovbnya, A.N. Molev, A.S. Korda, L.P. |
| author_facet | Vodin, A.N. Korda, V.Yu. Dovbnya, A.N. Molev, A.S. Korda, L.P. |
| citation_txt | Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian / А.N. Vodin, V.Yu. Korda, А.N. Dovbnya, A.S. Molev, L.P. Korda // Вопросы атомной науки и техники. — 2004. — № 5. — С. 48-54. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | A new model-free approach for analyzing the intermediate-energy light nucleus-nucleus elastic-scattering differential cross-sections is proposed to extract the numerical dependencies of S-matrix modulus and nuclear phase on
angular momentum directly from the experimental data via the evolutionary algorithm. The refractive and absorptive properties of the ¹⁶О – ¹⁶О-interaction at Е = 22…44 МeV/nucleon are studied. The new approach based on a
genetic algorithm is presented to diagonalize the Hamiltonian matrix of the Nilsson-model for the deformed axially
symmetric nucleus. The tests witness that the larger the matrix size the more effective the genetic approach becomes
comparing to the traditional diagonalization methods.
Запропоновано новий безмодельний підхід для аналізу диференціальних перерізів пружного розсіяння
легких ядер проміжних енергій ядрами, що дає змогу визначати числові залежності модуля і ядерної фази
матриці розсіяння від орбітального моменту безпосередньо з експериментальних даних за допомогою
еволюційного алгоритму. Досліджені заломлюючі та поглинаючі властивості ¹⁶О – ¹⁶О-взаємодії при
Е = 22…44 МеВ/нуклон. Представлено новий метод діагоналізації матриці гамільтоніану в моделі Нільсона
деформованого ядра з аксіальною симетрією, що застосовує генетичний алгоритм. Тестування довело, що
чим більше розмір діагоналізуємої матриці, тим ефективнішим виявляється генетичний підхід порівняно з
традиційними методами діагоналізації.
Предложен новый безмодельный под
Предложен новый безмодельный подход для анализа дифференциальных сечений упругого рассеяния
легких ядер промежуточных энергий ядрами, позволяющий извлекать численные зависимости модуля и
ядерной фазы матрицы рассеяния от орбитального момента непосредственно из экспериментальных данных
с помощью эволюционного алгоритма. Изучены преломляющие и поглощающие свойства ¹⁶О – ¹⁶О-взаимодействия при Е = 22…44 МэВ/нуклон. Представлен новый метод диагонализации матрицы гамильтониана в
модели Нильссона деформированного ядра с аксиальной симметрией, использующий генетический алгоритм. Тестирование показало, что чем больше размер диагонализуемой матрицы, тем эффективнее оказывается генетический подход по сравнению с традиционными методами диагонализации.
|
| first_indexed | 2025-11-28T11:53:47Z |
| format | Article |
| fulltext |
APPLICATION OF EVOLUTIONARY ALGORITHMS
FOR THE MODEL-FREE DETERMINATION OF SCATTERING MA-
TRIX AND THE NUMERICAL DIAGONALIZATION
OF NILSSON-MODEL HAMILTONIAN
А.N. Vodin1, V.Yu. Korda2, А.N. Dovbnya1, A.S. Molev2, L.P. Korda1
1NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: vodin@kipt.kharkov.ua
2STC of Electrophysics Ukrainian National Academy of Sciences, Kharkov, Ukraine
e-mail: v_yu_korda@yahoo.com
A new model-free approach for analyzing the intermediate-energy light nucleus-nucleus elastic-scattering differ-
ential cross-sections is proposed to extract the numerical dependencies of S-matrix modulus and nuclear phase on
angular momentum directly from the experimental data via the evolutionary algorithm. The refractive and absorp-
tive properties of the 16О – 16О-interaction at Е = 22…44 МeV/nucleon are studied. The new approach based on a
genetic algorithm is presented to diagonalize the Hamiltonian matrix of the Nilsson-model for the deformed axially
symmetric nucleus. The tests witness that the larger the matrix size the more effective the genetic approach becomes
comparing to the traditional diagonalization methods.
PACS: 24.10.Ht, 02.60.Pn, 25.40. Lw, 23.20.-g
1. INTRODUCTION
Experimental and theoretical investigations of differ-
ential cross sections of the scattering of light nuclei by
nuclei at Е ≥ 15…20 МeV/nucleon are still in focus for
the analysis of such cross sections comprising clearly
pronounced refractive structures allows to obtain valu-
able information on inter-nucleus interaction on small
distances (see, e.g., [1–3]). Usually the differential cross
sections of intermediate energy nuclei-nuclei elastic
scattering are analyzed with help of complex optical po-
tentials or scattering matrices having particular forms
and a few number of free parameters the values of
which are determined from the fitting of available ex-
perimental data. Unfortunately there are a great variety
of models leading in several crucial cases to different
physical interpretations of the observed features of dif-
ferential cross sections. Thus, it feels actual to pose the
problem of extraction of the optical potential or the scat-
tering matrix directly from the experimental data with
making as less of model assumptions as possible. In the
course, the quality of fitting the data can be improved if
one chooses more flexible forms of optical potential or
scattering matrix than those of conventional ones. With
this in mind various initial forms were used to be simple
parameterizations or found from the microscopic calcu-
lations further modified with additional terms being the
full set function expansions (see, [4–6]). The real part of
optical potential can also be chosen in the form being
more general than the common Saxon-Woods or folding
ones, allowing the non-monotonic behavior. For in-
stance, in [7] the spline form potential assisted the fold-
ing one. These approaches turned out to be quit success-
ful in analyzing the experimental data but should be
treated as model dependent for they used a priory model
visions for the optical potential or the scattering matrix.
In this paper we expose the novel model-free ap-
proach in which the numerical dependencies of the S-
atrix modulus and nuclear phase shift on the angular
momentum are extracted directly from the experimental
data with help of the data fitting procedure using evolu-
tionary algorithm with smooth deformations of these de-
pendencies. On the basis of the proposed approach the
refractive and absorptive properties of the 16О–16О-inter-
action at Е = 22…44 МeV/nucleon are studied. Besides,
we present the novel approach based on the use of a ge-
netic algorithm which enables the numerical diagonal-
ization of an arbitrary Hermitian matrix, with asymptoti-
cally exponential convergence. With help of the ap-
proach developed we solve the full problem on eigen-
functions and eigenvalues of the Hamiltonian matrix of
Nilsson’s model describing the one-particle states of the
deformed nuclei, in which the nucleons move in the os-
cillator potential with axial symmetry.
2. EVOLUTIONARY ALGORITHM FOR
SCATTERING MATRIX DETERMINATION
The scattering matrix in the angular momentum rep-
resentation can be presented in the form
( )llll iiS σ+δη= 22exp , (1)
where ηl is the scattering matrix modulus, δl and σ l are
the nuclear and Coulomb phase shifts correspondingly.
We have developed the approach with help of which
the modulus ηl and the nuclear phase δl as functions of
angular momentum l are extracted directly from the ex-
48 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 48-54.
perimental data (the Coulomb phase σl is assumed to
poses the quasi-classic form of the point charge scatter-
ing by the uniformly charged sphere with radius RC [8],
which is legitimated by the high enough energy of scat-
tering) without assuming any additional model assump-
tions. In our approach the values of ηl and δl for each l
are treated as independent fitting parameters. Thus the
parameter space of the variation problem has high di-
mension and one has to choose an adequate optimiza-
tion method.
Evolutionary algorithm is the one of choice the ap-
plication of which turned out to be quite efficient in
solving such different and hard problems as the integra-
tion of differential equations [9], Thomson’s problem on
the minimum energy configuration of N point charges
on a unit sphere [10], the optimization of thermodynam-
ic analysis of phase transitions in ferroelectrics [11], the
construction of the light exotic nuclei bound state wave
functions [12, 13], the optimization of parameters of the
energy dependent optical potential for the elastic scat-
tering of heavy ions by nuclei [14], the analysis of spec-
tra of quantum systems [15], the optimization of kine-
matics of nuclear physics experiments [16], etc.
The only condition the modulus ηl and the phase δl
must obey is the requirement for them to have smooth
form. Alongside with that one should note that at
Е ≥ 15…20 МeV/nucleon the number of fitting parame-
ters usually exceeds the number of experimental points.
To avoid overfitting we have devised the special muta-
tion operator with dispersion. The developed algorithm
consists of the following steps:
1. The initial population of n individuals is created,
each of which including the pair of real-valued vectors
(ηl, δl) of dimension lmax with l = 0,…, lmax -1. The initial
dependencies ηl and δl are chosen in the analytical form
( ) ,exp2,),,(exp 2
2
∆
−µ=δ∆µ−=η
r
rlaaal
LLLg
,exp1),,(
1−
∆
−+=∆
a
a
aa
LLLLg (2)
where L = l + 1/2, the parameter µa determines the mag-
nitude of absorption, the parameters La and ∆a (µr and ∆
r) characterize the size of the region of absorption (nu-
clear refraction) in the angular momentum space. These
shapes of the modulus ηl and the nuclear phase δl meet
the condition of smooth alteration with the increase of l
(see [17]) and correctly account for the character of ab-
sorption and refraction in the process of scattering of
light nuclei by nuclei at intermediate energies. The val-
ues of the parameters µa, µr, La, ∆a and ∆r from Eq. (2)
are the random ones uniformly distributed in wide
range. Thus this choice of the initial representation of ηl
and δl does not influence the final result obtained with
the algorithm at hand.
2. The elastic scattering differential cross section is
calculated and the value of χ2 is found for each of the
individual from the population. The procedure uses the
amplitude of elastic scattering in the form of the Legen-
dre polynomials expansion. The differential cross sec-
tion is determined by the squared modulus of this ampli-
tude.
3. According to the χ2 magnitude, two parent indi-
viduals are selected randomly from the population (the
lower the magnitude of χ2 the higher the probability to
be selected) to create two offsprings using the transfor-
mation
,
1
)1,0(25,01)(ln)(ln 2
0
'
−+
+η=η
d
ll
aN
ll
,
1
)1,0(25,01 2
0
'
−+
+δ=δ
d
ll
aN
ll (3)
where a is the mutation amplitude, N(0, 1) is the nor-
mally distributed one-dimensional random variable with
zero mean and one standard deviation, l0 is the random
point of mutation, d is the dispersion characterizing the
size of the mutation region.
The values of l0 and N(0, 1) are generated ones for
each string of η и δ. The control parameters a and d al-
ter in intervals [amin, amax] and [dmin, dmax] chosen by the
user. These parameters are automatically tuned by the
computational program in the specified intervals. The
mutation operator (3) is nonlocal and if the values of a
and d are set properly the transformed functions '
lη and
'
lδ will be smooth. In order to provide monotonicity of
the extracted functions ηl and δl we first find the mini-
mum value 2
monχ at which these functions still poses the
property. Then while χ2 decreases below 2
monχ we apply
the additional condition due to which the maximal rela-
tive alteration of the functions ηl and δl can not exceed
some small value (say 10-4, as in our case).
4. For each of two offsprings the elastic scattering
differential cross section is calculated and χ2 magnitude
is determined.
5. According to the χ2 magnitude, two individuals
are selected randomly from the population (the higher
the magnitude of χ2 the higher the probability to be se-
lected) to be substituted by the offsprings.
6. Go to step 3.
For the details of the selection procedure see exposi-
tion in [18].
3. RESULTS AND DISCUSSION
We have carried out the model-free study of the re-
fractive and absorptive properties of 16О – 16О-interac-
tion at Е = 22…44 МeV/nucleon. The choice of the
16О + 16О system is conditioned by the fact that there
exit valuable discrepancies between the S-matrix mod-
ule and deflection functions, characterizing absorptive
and refractive properties of interaction, respectively, ob-
tained with use of different models (phenomenological
49
optical model, folding model and S-matrix model with
Regge poles) (see [19, 20]). Therefore, it is important to
get new reliable information on the scattering matrix in
the cases under study.
0 20 40 60 80 100
10-3
10-2
10-1
100
3
2
1
Θ , deg
η
L
η
L
Θ , deg
0 20 40 60 80 100
-60
-40
-20
0
1
2
3
L
0 20 40 60 80 100
10-3
10-1
dc
3
2
1
0 20 40 60 80 100
-60
-40
-20
0
bа
2
1
3
L
Fig. 1. The modulus of scattering matrix (а, b) and
the deflection function (c, d) for elastic 16О – 16О-scat-
tering: b, d and curves 1 (а, c) are for Е = 350 МeV
while curves 2 (а, c) are for Е = 480 МeV and
curves 3 (а, c) are for Е = 704 МeV
The results of analysis of differential cross sections
of elastic 16О – 16О-scattering at Е = 350, 480 and 704
МeV carried out with help of the proposed approach are
exposed in Figs. 1 - 3. The values of control parameters
of the evolutionary algorithm are presented in Table
alongside the χ2 magnitudes and the total reaction cross
sections σr. Besides the boundary values for a and d, the
Table contains their initial (ai and di) and final (af and df)
values. The calculations utilized n = 100 and
RС = 0,95·2·161/3 [23]. The values of parameters of the
representations (2) altered in wide intervals: La = 30,0…
55,0; ∆a = 4,0…14,0; ∆r = 20,0…50,0; µa = 4,0…10,0,
µr = 5,0… 50,0 which provided for the various enough
initial conditions. The amount of χ2 calculations were
1,4⋅106, ⋅3,8⋅106 and 2,0⋅106 iterations for Е = 350, 480
and 704 МeV respectively.
The calculated differential cross sections were
symmetrized for the scattering of the identical nuclei.
Figs. 2 and 3 show that the application of the developed
approach allows to obtain good agreement between the
calculated and measured cross sections. As in [19] the
fitting of differential cross sections was held with the
standard experimental error of 10%. The scattering
matrix (1) for the cases shown on Figs. 1, а, c is
characterized by smooth dependence on L since the
extracted smooth monotonic functions of L. The
boundary from
0 20 40 60 80
10-5
10-2
1
2
3
σ/σR
θo
σ/σR
0 20 40 60 80
10-4
10-2
3
1
2
bа
Fig. 2. The ratio of the differential cross section of
elastic 16О – 16О-scattering at Е = 350 МeV and its
components to the Rutherford one: curves 1 – 3 (а) are
the cross section and its refractive and absorptive com-
ponents calculated with the S-matrix to which the
curves 1 in Fig. 1, а, c correspond; curves 1 – 3 (b) are
the cross sections calculated with the S-matrix to which
the curves 1 – 3 in Fig. 1, b, d correspond. The points
are the experimental data taken from [21, 22]
0 20 40
10-3
2
1
3
σ/σR
θo
σ/σR
0 20 40 60
10-4
10-2
1
2
3
bа
Fig. 3. The same as in Fig 2, а but for
Е = 480 МeV (а) and Е = 704 МeV (b). а, b show the
results of calculations with the S-matrix to which the
curves 2 and 3 in Fig. 1, а, c correspond. The points are
the experimental data taken from [19]
the data module η(L) and nuclear phases δ(L) are
smooth monotonic functions of L. The boundary mo-
menta of strong absorption Lsa defined from the correla-
tion η2(Lsa) = 0,5 with help of the dependencies η(L)
shown on Fig. 1, а acquire the values: Lsa = 58,1; 66,3
and 76,9 at Е = 350, 480 и 704 МeV respectively. The
deflection functions Θ(L) = 2(δL – δL–1) + 2dσ(L)/dL in
the cases under study have the forms typical of the cases
of nuclear rainbow (see Fig. 1, c). The angles of nuclear
rainbow corresponding to the minima of Θ(L) are equal
to θR = 64; 41 and 26° at Е = 350, 480 and 704 МeV re-
spectively. Note that the values of θR found at Е = 350
50
and 480 МeV are in good agreement with the ones ob-
tained on the basis of the folding model with the density
dependent nucleon-nucleon interaction [19] and the
nine-parameter S-matrix model [24].
The control parameters of the evolutionary algo-
rithm used in the calculations of the differential cross
sections of elastic 16О – 16О-scattering at different ener-
gies
Parameter
Value
350 MeV 480 MeV 704 MeV
lmax 120 135 160
amin 10-2 10-3 10-5
amax 0,9 0,9 0,9
ai 0,9 0,9 0,9
af 1,3·10-2 1,2·10-2 1,6·10-4
dmin 5,5 7,0 9,5
dmax 100,0 100,0 100,0
di 80,0 90,0 100,0
df 6,5 7,0 11,5
χ2 2,5 1,9 1,1
σr, mb 1693 1581 1483
In order to clarify the degree to which the nuclear re-
fraction and the strong absorption are responsible for the
formation of different features of the elastic 16О –16О-
scattering in different regions of scattering angles θ
Figs. 2, а and 3 display the refractive and diffractive
components of the cross sections studied calculated due
to the method proposed in [25]. Figs. show that at
θ ≥ 20…25° the refractive components (curves 2) al-
most totally replicate the respective differential cross
sections, particularly they describe fine details of the
rainbow “tail” in the cases of the scattering at Е = 480
and 704 МeV. The diffractive components (curves 3)
dominate in the regions of Fraunhofer oscillations at
small angles θ.
When the values of dispersion dmin decrease to 2,0…
2,5 and the procedure of additional control of mono-
tonicity of the extracted dependencies η(L) and δ(L) is
switched off the behavior of the modulus η(L) in the re-
gion of small momenta becomes non-monotonic. In this
region there arise isolated structures (“humps”) like the
ones shown on Figs. 1, b (curves 1, 2) for the scattering
at Е = 350 МeV while the deflection function has the
similar form as in Fig. 1, c. Note the values of those pa-
rameters of calculations for the curves 1, 2 in Fig. 2, b
which differ from the ones mentioned in the Table:
dmin = df = 2,0, amin = 0,0, af = 6,4·10-11, σr = 1667 mb
(curve 1), dmin = 2,0, df = 2,6, amin = 10-3, af = 6,5·10-3, σ
r = 1678 mb (curve 2).
Fig. 2 shows that the use of the different forms of
the scattering matrix to which the curves 1 and 2 in
Fig. 1, b, c and the curves 1 in Fig. 1, а, c correspond
gives equivalent quality of fit of the data although in the
region of large enough scattering angles (θ > 70°) some
discrepancy between the differential cross sections ob-
tained is however observed. Therefore, the detailed
structure of η(L) in the region of small momenta con-
taining small “humps” the character size of which λ
h ≈ 7/k ≈ 0,9 fm (k is the wave number) is compared
with the wave length of the relative motion of colliding
nuclei λ = 2π/k ≈ 0,8 fm should not be assigned any
physical meaning and its existence is not necessary for
the correct explanation of the available experimental da-
ta. Thus the monotonic function (see Fig. 1, а) smoothly
altering from small values to unity with the increase of
L may be used as a modulus η(L) of scattering matrix
for the 16О + 16О system at the energy under study.
Note that when the program of calculations automat-
ically selected the conditions for S(L) corresponding to
the case of week nuclear refraction we found the differ-
ential cross section (curve 3 in Fig. 2, b) for which χ
2 = 0,6 (df = 9,3, amin = 10-3, af = 9,2·10-3, σr = 1590 mb).
In the context the question remains on how deep physi-
cal meaning might have the oscillation structures in the
modulus of the scattering matrix and the deflection
function (see curves 3 in Fig. 1, b, c). Obviously this re-
sult should be considered as a physical nonsense and
only the smooth behavior of η(L) and δ(L) without any
additional structures characterizes correctly the effects
of absorption and refraction in the 16О – 16О-scattering
at the energies under study.
These conclusions contradict the results of the work
[20] in which the differential cross section of elastic
16О – 16О-scattering was analyzed on the basis of the
ten-parameter S-matrix model allowing for the pole fac-
tor and the strong influence of Regge poles and zeros of
S-matrix lying in the vicinity of the real axes in the
complex L plane on the refractive behavior of the stud-
ied cross sections at Е ≤ 480 МeV was found. The ac-
count for the poles and zeros of S-matrix strongly vio-
lates the smoothness of S(L). Due to that the character
of refraction and absorption at L ≤ Lsa in this approach
substantially differs from that of the discussed above
(see curves 1, 2 in Fig. 1, а, c).
4. GENETIC ALGORITHM BASED NUMER-
ICAL DIAGONALIZATION OF HERMITIAN
MATRIX. APPLICATION TO NILSSON
MODEL
The variability of shapes of atomic nuclei is an im-
portant property of nuclear matter. It is well established
that the majority of nuclei are deformed both in ground
and excited states. The study of the influence of the nu-
clear deformations in different states on various observ-
ables is permanently in the focus [26].
One of the most simple but physically rich ap-
proaches to the systematics of the energy level bands of
the deformed nuclei is Nilsson’s [27] model in which
51
the nucleons are placed in the self-consistent field de-
scribed by the oscillator potential with axial symmetry.
The key point in the calculations of the one-particle
states in this model is the diagonalization of the Hamil-
tonian matrix in the basis of the one-particle wave func-
tions of the spherical oscillator potential.
The full solution of the problem of searching eigen-
functions and eigenvalues of the Hermitian matrix can
be found using the most developed and effective Jaco-
bi’s method – the so-called plane rotations one. The
method consists in the successive transformation of the
initial matrix H = H0 into the more simple form by mul-
tiplying it with the matrix of plane rotation U. As a re-
sult there arises a series H0, H1, H2,… in which
Hk = UkHk-1Uk
-1 and Uk is selected to make the matrix Hk
not contain the non-diagonal element of the matrix Hk-1,
having the maximal absolute value. If Hk = ║hij
(k)║, │hpq
(k-
1)│ = max
ji ≠ │hij
(k-1)│ Uk = ║uij
(k)║, then for the real ma-
trix H we have
( ) ( ) ϕ== cosk
qq
k uu
pp , ( ) ( ) ϕ=−= sink
qp
k
pq uu ,
( )
( ) ( )11
12
2 −−
−
−
=ϕ k
qq
k
pp
k
pq
hh
h
tg , 4/π≤ϕ . (4)
The sequence of the matrices Hk asymptotically square-
ly converges to the diagonal matrix. Besides, all the di-
agonal elements of the final matrix Hk turn out to be the
approximate eigenvalues of H while the rows of the ma-
trix Uk = U1U2…Uk become the approximate eigenvec-
tors. Within this method the most time-consuming pro-
cedure is the search for the non-diagonal element of the
diagonalized matrix, having maximal absolute value.
Thus in practice various methods are employed to accel-
erate or optimize this process.
From the other hand, the full solution of the problem
of searching eigenfunctions and eigenvalues of the Her-
mitian matrix is provided with help of the unitary trans-
formation describing n - dimensional rotation (n is the
dimension of the diagonalized matrix) which, using the
generalized Euler angles [28], can be presented in the
form:
( ) ( )11 ...ggg n−= , ( ) ( ) ( )k
kk
kk ggg θθ= ...11 ,
( ) ( ) ( )
( ) ( )
θ⋅+θ⋅−=
θ⋅+θ⋅=
⇒θ
++
+
k
jj
k
jjj
k
jj
k
jjjk
jj xxx
xxx
g
cossin'
sincos'
11
1 ,
11 −≤≤ nk , kj ≤≤1 , (5)
π<θθ≤ 2...,,0 2
1
1
1 , π<θθθθ≤ ...,,...,,,0 2
3
1
3
2
2
1
2 . (6)
To this end the problem can be considered as the one of
finding the optimal set of n(n-1)/2 values of Euler an-
gles minimizing the sum of absolute values of all non-
diagonal elements of the matrix under diagonalization in
the rotated basis. To solve the problem we have applied
the genetic algorithm which varies the optimized param-
eters independently and in parallel and has the exponen-
tial convergence rate.
The scheme of application of a genetic algorithm to
the problem under study is as follows. Each of the opti-
mized Euler angles is presented via the binary encoding.
The population consists of the fixed number of sets of
these angles. Initially all the angles are set randomly in
the specified intervals (6). Then for each set of angles
the unitary transformation matrix (5) is calculated and
applied to the diagonalized matrix and the sum of abso-
lute values of all non-diagonal elements of the trans-
formed matrix is determined. Further two parent sets of
angles are selected from the population according to the
prescription – the less the sum of absolute values of
non-diagonal elements the greater the probability to be
selected. The selected sets of angles are replicated to
produce offsprings. During the replication, some bits of
the binary codes representing the angles are flipped with
some probability (mutation). After that, the exchange of
the complementary portions of bits between the parent
sets of angles takes place with some probability
(crossover). For each offspring the sum of the absolute
values of all non-diagonal elements of the transformed
primary matrix is calculated. If the value of offspring’s
sum becomes less the largest value of individual’s sum
in the population, the latter is substituted by the former.
The same way the next pair of parents is selected and
the process proceeds until the required quality of diago-
nalization is achieved. Then the diagonal elements of
the matrix H’ = gHg-1 are the approximate eigenvalues
of H and the rows of the matrix g are the approximate
eigenvectors.
Using the diagonalization method presented above,
we have independently replicated the one-particle ener-
gy states scheme found by Nilsson via Jacobi’s method.
The comparative analysis of the developed approach
and the conventional ones (e.g., Jacobi’s) shows that the
higher the dimension of the matrix under diagonaliza-
tion the higher the efficacy of the genetic approach.
5. CONCLUSIONS
The proposed model-free approach allows to extract
the numerical dependencies of the modulus of S-matrix
and the nuclear phase on the angular momentum direct-
ly from the measured differential cross sections of elas-
tic scattering. This approach is free from the a priori vi-
sions of the shape of the nuclear part of scattering ma-
trix. The results of analyses of the differential cross sec-
tions of elastic 16О – 16О-scattering at 22…44 МeV/nu-
cleon featured by the pronounces refractive picture of
rainbow scattering witness that the quantitative descrip-
tion of the available experimental data is achieved with
the use of the scattering matrix determined by the mod-
ulus and the nuclear phase being smooth monotonic
functions of angular momentum while the deflection
function has the form typical of the case of nuclear rain-
bow.
It should be noted that we have studied the case of
the elastic scattering of identical nuclei limited in angu-
lar distributions by θ < 90° so that one can miss some
important details of rainbow structures. From this point
52
of view it seems reasonable to extract numerical depen-
dencies of the S-matrix modulus and nuclear phase shift
on angular momentum from the differential cross sec-
tions for elastic 16О – 12C-scattering at E(16О) = 230…
281 MeV with help of the fitting procedure based on the
evolutionary algorithm and find out whether these char-
acteristics are smooth monotonic functions in the angu-
lar momentum space.
Nilsson’s model allows successful description of the
ground state properties of light nuclei with 4<А<32, in-
cluding spins, parities, magnetic and quadrupole mo-
ments, etc [29]. With help of the modified Nilsson’s
model [30-32] which account for the different deforma-
tions of a nucleus in different excited one-particle states
the experimentally measured probabilities of electro-
magnetic transitions between the excited one-particle
states of light nuclei have been more adequately ex-
plained. Within the specified model it is actual to recal-
culate the energies and the quantum characteristics of
the excited one-particle states of light nuclei, being the
ones equilibrium in deformation, and find out whether it
is possible to agree the calculated and the measured data
if we take into account the dependencies on nuclear de-
formation of the weights with which the operators of the
spin-orbit interaction and the interaction proportional to
the square of angular momentum enter the Hamiltonian.
This study was partly supported by The State Fund
of Fundamental Research of the Ukraine (grant
02.07/372).
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ПРИМЕНЕНИЕ ЭВОЛЮЦИОННЫХ АЛГОРИТМОВ ДЛЯ БЕЗМОДЕЛЬНОГО ОПРЕДЕЛЕНИЯ
МАТРИЦЫ РАССЕЯНИЯ И ЧИСЛЕННОЙ ДИАГОНАЛИЗАЦИИ ГАМИЛЬТОНИАНА
В МОДЕЛИ НИЛЬСCОНА
А.Н. Водин, В.Ю. Корда, A.Н. Довбня, А.С. Молев, Л.П. Корда
Предложен новый безмодельный подход для анализа дифференциальных сечений упругого рассеяния
легких ядер промежуточных энергий ядрами, позволяющий извлекать численные зависимости модуля и
ядерной фазы матрицы рассеяния от орбитального момента непосредственно из экспериментальных данных
с помощью эволюционного алгоритма. Изучены преломляющие и поглощающие свойства 16О – 16О-взаимо-
действия при Е = 22…44 МэВ/нуклон. Представлен новый метод диагонализации матрицы гамильтониана в
модели Нильссона деформированного ядра с аксиальной симметрией, использующий генетический алго-
ритм. Тестирование показало, что чем больше размер диагонализуемой матрицы, тем эффективнее оказыва-
ется генетический подход по сравнению с традиционными методами диагонализации.
ЗАСТОСУВАННЯ ЕВОЛЮЦІЙНИХ АЛГОРИТМІВ ДЛЯ БЕЗМОДЕЛЬНОГО ВИЗНАЧЕННЯ
МАТРИЦІ РОЗСІЯННЯ І ЧИСЕЛЬНОЇ ДІАГОНАЛІЗАЦІЇ МАТРИЦІ ГАМІЛЬТОНІАНУ
В МОДЕЛІ НІЛЬСОНА
О.М. Водін, В.Ю. Корда, A.М. Довбня, О.С. Молєв, Л.П. Корда
Запропоновано новий безмодельний підхід для аналізу диференціальних перерізів пружного розсіяння
легких ядер проміжних енергій ядрами, що дає змогу визначати числові залежності модуля і ядерної фази
матриці розсіяння від орбітального моменту безпосередньо з експериментальних даних за допомогою
еволюційного алгоритму. Досліджені заломлюючі та поглинаючі властивості 16О – 16О-взаємодії при
Е = 22…44 МеВ/нуклон. Представлено новий метод діагоналізації матриці гамільтоніану в моделі Нільсона
деформованого ядра з аксіальною симетрією, що застосовує генетичний алгоритм. Тестування довело, що
чим більше розмір діагоналізуємої матриці, тим ефективнішим виявляється генетичний підхід порівняно з
традиційними методами діагоналізації.
54
1NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
3. RESULTS AND DISCUSSION
REFERENCES
А.Н. Водин, В.Ю. Корда, A.Н. Довбня, А.С. Молев, Л.П. Корда
О.М. Водін, В.Ю. Корда, A.М. Довбня, О.С. Молєв, Л.П. Корда
|
| id | nasplib_isofts_kiev_ua-123456789-80508 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-28T11:53:47Z |
| publishDate | 2004 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Vodin, A.N. Korda, V.Yu. Dovbnya, A.N. Molev, A.S. Korda, L.P. 2015-04-18T16:23:51Z 2015-04-18T16:23:51Z 2004 Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian / А.N. Vodin, V.Yu. Korda, А.N. Dovbnya, A.S. Molev, L.P. Korda // Вопросы атомной науки и техники. — 2004. — № 5. — С. 48-54. — Бібліогр.: 32 назв. — англ. 1562-6016 PACS: 24.10.Ht, 02.60.Pn, 25.40. Lw, 23.20.-g https://nasplib.isofts.kiev.ua/handle/123456789/80508 A new model-free approach for analyzing the intermediate-energy light nucleus-nucleus elastic-scattering differential cross-sections is proposed to extract the numerical dependencies of S-matrix modulus and nuclear phase on angular momentum directly from the experimental data via the evolutionary algorithm. The refractive and absorptive properties of the ¹⁶О – ¹⁶О-interaction at Е = 22…44 МeV/nucleon are studied. The new approach based on a genetic algorithm is presented to diagonalize the Hamiltonian matrix of the Nilsson-model for the deformed axially symmetric nucleus. The tests witness that the larger the matrix size the more effective the genetic approach becomes comparing to the traditional diagonalization methods. Запропоновано новий безмодельний підхід для аналізу диференціальних перерізів пружного розсіяння легких ядер проміжних енергій ядрами, що дає змогу визначати числові залежності модуля і ядерної фази матриці розсіяння від орбітального моменту безпосередньо з експериментальних даних за допомогою еволюційного алгоритму. Досліджені заломлюючі та поглинаючі властивості ¹⁶О – ¹⁶О-взаємодії при Е = 22…44 МеВ/нуклон. Представлено новий метод діагоналізації матриці гамільтоніану в моделі Нільсона деформованого ядра з аксіальною симетрією, що застосовує генетичний алгоритм. Тестування довело, що чим більше розмір діагоналізуємої матриці, тим ефективнішим виявляється генетичний підхід порівняно з традиційними методами діагоналізації. Предложен новый безмодельный под Предложен новый безмодельный подход для анализа дифференциальных сечений упругого рассеяния легких ядер промежуточных энергий ядрами, позволяющий извлекать численные зависимости модуля и ядерной фазы матрицы рассеяния от орбитального момента непосредственно из экспериментальных данных с помощью эволюционного алгоритма. Изучены преломляющие и поглощающие свойства ¹⁶О – ¹⁶О-взаимодействия при Е = 22…44 МэВ/нуклон. Представлен новый метод диагонализации матрицы гамильтониана в модели Нильссона деформированного ядра с аксиальной симметрией, использующий генетический алгоритм. Тестирование показало, что чем больше размер диагонализуемой матрицы, тем эффективнее оказывается генетический подход по сравнению с традиционными методами диагонализации. This study was partly supported by The State Fund of Fundamental Research of the Ukraine (grant 02.07/372). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Ядерная физика и элементарные частицы Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian Застосування еволюційних алгоритмів для безмодельного визначення матриці розсіяння і чисельної діагоналізації матриці гамільтоніану в моделі Нільсона Применение эволюционных алгоритмов для безмодельного определения матрицы рассеяния и численной диагонализации гамильтониана в модели Нильсcона Article published earlier |
| spellingShingle | Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian Vodin, A.N. Korda, V.Yu. Dovbnya, A.N. Molev, A.S. Korda, L.P. Ядерная физика и элементарные частицы |
| title | Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian |
| title_alt | Застосування еволюційних алгоритмів для безмодельного визначення матриці розсіяння і чисельної діагоналізації матриці гамільтоніану в моделі Нільсона Применение эволюционных алгоритмов для безмодельного определения матрицы рассеяния и численной диагонализации гамильтониана в модели Нильсcона |
| title_full | Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian |
| title_fullStr | Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian |
| title_full_unstemmed | Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian |
| title_short | Application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of Nilsson-model Hamiltonian |
| title_sort | application of evolutionary algorithms for the model-free determination of scattering matrix and the numerical diagonalization of nilsson-model hamiltonian |
| topic | Ядерная физика и элементарные частицы |
| topic_facet | Ядерная физика и элементарные частицы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80508 |
| work_keys_str_mv | AT vodinan applicationofevolutionaryalgorithmsforthemodelfreedeterminationofscatteringmatrixandthenumericaldiagonalizationofnilssonmodelhamiltonian AT kordavyu applicationofevolutionaryalgorithmsforthemodelfreedeterminationofscatteringmatrixandthenumericaldiagonalizationofnilssonmodelhamiltonian AT dovbnyaan applicationofevolutionaryalgorithmsforthemodelfreedeterminationofscatteringmatrixandthenumericaldiagonalizationofnilssonmodelhamiltonian AT molevas applicationofevolutionaryalgorithmsforthemodelfreedeterminationofscatteringmatrixandthenumericaldiagonalizationofnilssonmodelhamiltonian AT kordalp applicationofevolutionaryalgorithmsforthemodelfreedeterminationofscatteringmatrixandthenumericaldiagonalizationofnilssonmodelhamiltonian AT vodinan zastosuvannâevolûcíinihalgoritmívdlâbezmodelʹnogoviznačennâmatricírozsíânnâíčiselʹnoídíagonalízacíímatricígamílʹtoníanuvmodelínílʹsona AT kordavyu zastosuvannâevolûcíinihalgoritmívdlâbezmodelʹnogoviznačennâmatricírozsíânnâíčiselʹnoídíagonalízacíímatricígamílʹtoníanuvmodelínílʹsona AT dovbnyaan zastosuvannâevolûcíinihalgoritmívdlâbezmodelʹnogoviznačennâmatricírozsíânnâíčiselʹnoídíagonalízacíímatricígamílʹtoníanuvmodelínílʹsona AT molevas zastosuvannâevolûcíinihalgoritmívdlâbezmodelʹnogoviznačennâmatricírozsíânnâíčiselʹnoídíagonalízacíímatricígamílʹtoníanuvmodelínílʹsona AT kordalp zastosuvannâevolûcíinihalgoritmívdlâbezmodelʹnogoviznačennâmatricírozsíânnâíčiselʹnoídíagonalízacíímatricígamílʹtoníanuvmodelínílʹsona AT vodinan primenenieévolûcionnyhalgoritmovdlâbezmodelʹnogoopredeleniâmatricyrasseâniâičislennoidiagonalizaciigamilʹtonianavmodelinilʹscona AT kordavyu primenenieévolûcionnyhalgoritmovdlâbezmodelʹnogoopredeleniâmatricyrasseâniâičislennoidiagonalizaciigamilʹtonianavmodelinilʹscona AT dovbnyaan primenenieévolûcionnyhalgoritmovdlâbezmodelʹnogoopredeleniâmatricyrasseâniâičislennoidiagonalizaciigamilʹtonianavmodelinilʹscona AT molevas primenenieévolûcionnyhalgoritmovdlâbezmodelʹnogoopredeleniâmatricyrasseâniâičislennoidiagonalizaciigamilʹtonianavmodelinilʹscona AT kordalp primenenieévolûcionnyhalgoritmovdlâbezmodelʹnogoopredeleniâmatricyrasseâniâičislennoidiagonalizaciigamilʹtonianavmodelinilʹscona |