Х А Б А Р Л А Р Ы

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ISSN 1991-346X (Print) ҚАЗАҚСТАН РЕСПУБЛИКАСЫ

ҰЛТТЫҚ ҒЫЛЫМ АКАДЕМИЯСЫНЫҢ

Х А Б А Р Л А Р Ы

ИЗВЕСТИЯ

НАЦИОНАЛЬНОЙ АКАДЕМИИ НАУК РЕСПУБЛИКИ КАЗАХСТАН

N E W S

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTAN

ФИЗИКА-МАТЕМАТИКА СЕРИЯСЫ

 СЕРИЯ

ФИЗИКО-МАТЕМАТИЧЕСКАЯ



PHYSICO-MATHEMATICAL SERIES

1 (311)

ҚАҢТАР – АҚПАН 2017 ж.

ЯНВАРЬ – ФЕВРАЛЬ 2017 г.

JANUARY – FEBRUARY 2017

1963 ЖЫЛДЫҢ ҚАҢТАР АЙЫНАН ШЫҒА БАСТАҒАН ИЗДАЕТСЯ С ЯНВАРЯ 1963 ГОДА

PUBLISHED SINCE JANUARY 1963 ЖЫЛЫНА 6 РЕТ ШЫҒАДЫ

ВЫХОДИТ 6 РАЗ В ГОД PUBLISHED 6 TIMES A YEAR

АЛМАТЫ, ҚР ҰҒА АЛМАТЫ, НАН РК ALMATY, NAS RK

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Б а с р е д а к т о р ы

ф.-м.ғ.д., проф., ҚР ҰҒА академигі Ғ.М. Мұтанов Р е д а к ц и я а л қ а с ы:

Жұмаділдаев А.С. проф., академик (Қазақстан) Кальменов Т.Ш. проф., академик (Қазақстан) Жантаев Ж.Ш. проф., корр.-мүшесі (Қазақстан) Өмірбаев У.У. проф. корр.-мүшесі (Қазақстан) Жүсіпов М.А. проф. (Қазақстан)

Жұмабаев Д.С. проф. (Қазақстан) Асанова А.Т. проф. (Қазақстан)

Бошкаев К.А. PhD докторы (Қазақстан) Сұраған Д. PhD докторы (Қазақстан) Quevedo Hernando проф. (Мексика), Джунушалиев В.Д. проф. (Қырғыстан) Вишневский И.Н. проф., академик (Украина) Ковалев А.М. проф., академик (Украина) Михалевич А.А. проф., академик (Белорус) Пашаев А. проф., академик (Əзірбайжан)

Такибаев Н.Ж. проф., академик (Қазақстан), бас ред. орынбасары Тигиняну И. проф., академик (Молдова)

«ҚР ҰҒА Хабарлары. Физика-математикалық сериясы».

ISSN 2518-1726 (Online), ISSN 1991-346X (Print)

Меншіктенуші: «Қазақстан Республикасының Ұлттық ғылым академиясы» РҚБ (Алматы қ.)

Қазақстан республикасының Мəдениет пен ақпарат министрлігінің Ақпарат жəне мұрағат комитетінде 01.06.2006 ж. берілген №5543-Ж мерзімдік басылым тіркеуіне қойылу туралы куəлік

Мерзімділігі: жылына 6 рет.

Тиражы: 300 дана.

Редакцияның мекенжайы: 050010, Алматы қ., Шевченко көш., 28, 219 бөл., 220, тел.: 272-13-19, 272-13-18, www:nauka-nanrk.kz / physics-mathematics.kz

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Г л а в н ы й р е д а к т о р

д.ф.-м.н., проф. академик НАН РК Г.М. Мутанов Р е д а к ц и о н н а я к о л л е г и я:

Джумадильдаев А.С. проф., академик (Казахстан) Кальменов Т.Ш. проф., академик (Казахстан) Жантаев Ж.Ш. проф., чл.-корр. (Казахстан) Умирбаев У.У. проф. чл.-корр. (Казахстан) Жусупов М.А. проф. (Казахстан)

Джумабаев Д.С. проф. (Казахстан) Асанова А.Т. проф. (Казахстан) Бошкаев К.А. доктор PhD (Казахстан) Сураган Д. доктор PhD (Казахстан) Quevedo Hernando проф. (Мексика), Джунушалиев В.Д. проф. (Кыргызстан) Вишневский И.Н. проф., академик (Украина) Ковалев А.М. проф., академик (Украина) Михалевич А.А. проф., академик (Беларусь) Пашаев А. проф., академик (Азербайджан)

Такибаев Н.Ж. проф., академик (Казахстан), зам. гл. ред.

Тигиняну И. проф., академик (Молдова)

«Известия НАН РК. Серия физико-математическая».

Собственник: РОО «Национальная академия наук Республики Казахстан» (г. Алматы)

Свидетельство о постановке на учет периодического печатного издания в Комитете информации и архивов Министерства культуры и информации Республики Казахстан №5543-Ж, выданное 01.06.2006 г.

Периодичность: 6 раз в год.

Тираж: 300 экземпляров.

Адрес редакции: 050010, г. Алматы, ул. Шевченко, 28, ком. 219, 220, тел.: 272-13-19, 272-13-18, www:nauka-nanrk.kz / physics-mathematics.kz

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E d i t o r i n c h i e f

doctor of physics and mathematics, professor, academician of NAS RK G.М. Mutanov E d i t o r i a l b o a r d:

Dzhumadildayev А.S. prof., academician (Kazakhstan) Kalmenov Т.Sh. prof., academician (Kazakhstan) Zhantayev Zh.Sh. prof., corr. member. (Kazakhstan) Umirbayev U.U. prof. corr. member. (Kazakhstan) Zhusupov М.А. prof. (Kazakhstan)

Dzhumabayev D.S. prof. (Kazakhstan) Asanova А.Т. prof. (Kazakhstan) Boshkayev K.А. PhD (Kazakhstan) Suragan D. PhD (Kazakhstan) Quevedo Hernando prof. (Mexico), Dzhunushaliyev V.D. prof. (Kyrgyzstan) Vishnevskyi I.N. prof., academician (Ukraine) Kovalev А.М. prof., academician (Ukraine) Mikhalevich А.А. prof., academician (Belarus) Pashayev А. prof., academician (Azerbaijan)

Takibayev N.Zh. prof., academician (Kazakhstan), deputy editor in chief.

Tiginyanu I. prof., academician (Moldova)

News of the National Academy of Sciences of the Republic of Kazakhstan. Physical-mathematical series.

Owner: RPA "National Academy of Sciences of the Republic of Kazakhstan" (Almaty)

The certificate of registration of a periodic printed publication in the Committee of information and archives of the Ministry of culture and information of the Republic of Kazakhstan N 5543-Ж, issued 01.06.2006

Periodicity: 6 times a year Circulation: 300 copies

Editorial address: 28, Shevchenko str., of. 219, 220, Almaty, 050010, tel. 272-13-19, 272-13-18, www:nauka-nanrk.kz / physics-mathematics.kz

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N E W S

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTAN PHYSICO-MATHEMATICAL SERIES

ISSN 1991-346Х

Volume 6, Number 310 (2016), 5 – 12

N. Burtebayev¹, Zh.K. Kerimkulov^1,3, D.K. Alimov^1,2, A.M. Otarbayeva³, Y.S. Mukhamejanov^1,2, D.M. Janseitov^1,2

1INP, Almaty, Kazakhstan; ²al-Farabi KazNU, Almaty, Kazakhstan;

3L.N. Gumilev ENU, Astana, Kazakhstan, e-mail: [email protected]

STUDY OF ELASTIC SCATTERING

OF DEUTERONS FROM

⁶

Lі AT ENERGY 18 MeV

Abstract. Differential cross sections of elastic scattering of deuterons from ⁶Li nuclei at energy 18 MeV were measured at U-150M accelerator. The measurements were performed with an accuracy of no more than 10%. One minimum and one maximum of cross sections are clearly seen in the angular distributions at small angles. The obtained data were analyzed within optical model, distorted wave method with a finite interaction radius and coupled reactions channel method. The optimal values of the optical interaction potential and spectroscopic factor were determined. It is shown that the potential scattering forms cross section only at low and medium angles. In the range of large angles cross sections are formed by α-cluster transfer mechanisms.

Key words: elastic scattering, light charged particles, optical potential, FRESCO, cluster transfer, spectroscopic factor.

Introduction. The study of the interaction of charged particles with lithium nuclei is of considerable interest in the light of that role to be played by these nuclei in nuclear technology, fusion energy and astrophysics. So, 6Li nucleus is one of the most important elements of the fuel cycle in the most promising projects of fusion reactors using deuterium-tritium fusion. In tritium reproduction it is assumed that lithium is included in the nearest shell to the plasma combustion region. This technique requires highly accurate data on the sections of different particles interaction with lithium nuclei, which can be obtained as by experimentation, and by calculations within certain nuclear models. Astrophysical aspect of relevance is connected primarily with questions of nucleosynthesis of light nuclei at the initial stage of evolution of the universe and the problem of unexpectedly high prevalence of lithium (and beryllium and boron) in cosmic rays, has on order above, as opposed to their theoretical estimates.

Experimental procedure and measurement results. Experiments were carried out in the isochronous cyclotron U-150M [1] of the Institute of Nuclear Physics of the Republic of Kazakhstan. The differential cross sections of elastic scattering of deuterons on 6Li nuclei were measured at an energy of 18 MeV in the angular range from 10˚ to 170˚ in the center-of-mass system. The total error of the data did not exceed 10%.

Charged particles in a cyclotron are formed in the source, located in the central part of the camera in an arc discharge when applying the appropriate gas (hydrogen, deuterium, helium-3, helium-4). Their accelerating happens in the interpolar space of 1.5-meter magnet at the time of flight of the particles between the dees.

When installing the operating parameters of particle acceleration special attention is given to the operating mode of the ion source, its duty cycle, microstructure of the current pulse and beam quality of the wiring on the target. This optimization of spatial and temporal characteristics of the beam made it possible to significantly reduce the level of various noise, uneven loading of electronic equipment.

The energy and the energy dispersion in the beam is determined by measuring the energy spectrum of the particles elastically scattered by thin gold targets set out in the laboratory cell scattering of low-energy

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nuclear reactions of the INP RK [2]. In this case, measurements at a small angle (about 10 °) avoids errors due to inaccurate knowledge of the target thickness and angular dispersion of particles in the beam. For absolute calibration of the energy scale the "triple" alpha source is used (^241,243Am+²⁴⁴Cm).

Transporting scheme of the beam of accelerated ions from the cyclotron chamber to the scattering chamber, located at 24 meters from the exit of the beam is shown in Figure 1. It includes a quadrupole lens system, two rotary, diluting, two targeting magnets and the collimator system. All these installetions together with the elements of targeting and correction, provide at the target a charged particle beam of angle dulution of not more than 0.4 ° and a diameter of 3 mm. Adjusting the collimator and the scattering of the camera relative to the center axis of the ion guide was carried out by an optical method and was monitored using quartz twelve screens and television cameras, image transmission to remote control of the cyclotron.

м-3 м-2

м-1

км кр

л

л л

л

Л – quadrupole lenses; М-1, М-2 – deflecting magnets;

М-3 – diluting magnet; КМ – correcting magnet;

КР – scattering chamber

Figure 1 - the transportation scheme of the ion beam of the cyclotron to the scattering chamber

The measurements were made using a Е-Е method of registration and identification of particles, based on the simultaneous measurement of specific losses of charged particle energy in dE/dx substance and its total kinetic energy E. The method is based on the Bethe-Bloch equation, binding energy of the emitted charged particle with its specific ionization in the matter:

dE kMz

2

dx  E

where k – constant, weakly dependent on the types of particles, М and z – mass and charge of the emitted particles, Е – energy of the incident particle.

Figure 2 shows the Е-Е distribution of products of the interaction of deuterons with nuclei ⁶Li In the experiment, as targets there were used ⁶Li thin film (thickness of 700-1100 ug/cm²) deposited on a substrates of aluminum oxide (Al2O3) of 30-40 ug/cm² thick. In the measurements there were used silicon surface barrier detectors with a thickness of 10-100 microns (Е - detector) and 1,000-2,000 microns (E - detector). Beam current was varied in the range of 1-100 nA, depending on the scattering angle and the load of the electronic equipment. All measurements were carried out on the measuring and computing complex lab, which serves as the basis for a system of multivariate analysis processes based on electronic blocks ORTES and PC/AT [3]. Figure 3 shows the elastic scattering spectra of deuterons on ⁶Li nuclei at two corners.

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Fig

Отсчеты

Analy the potent experimen and detaile In the of imagina problem o

gure 2 - Е-Е d

0 5

0 200 400 600 800 1000 1200

Figure 3

ysis and disc tials of the ntal data on e

ed mathemat e optical mod

ary absorbin of scattering

Lower loci - si distribution of in

50 100

Кан E_d = 25 MэВ,

6Li(d,d')⁶Li 2.186 МэВ

12С(d,d')¹²С 4.439 МэВ

- the spectrum

cussion of th particles int elastic scatte tical formula del the effec ng part in the on a many-p

ingly charged p nteraction produ

150 200

налы

_л.с. = 30 град

12С(d,d

6Li(d,d) i

В

of the scattered

he results. T teraction wit ering on the b ation of whic ct of inelastic e interaction

particle syste

particles, upper ucts of deuteron

250

(а)

д

d)¹²С )⁶Li

0 500 1000 1500 2000 2500 3000 3500 4000

d deuteron mea

The most dev th atomic nu basis of the o ch are expose c channels is potential bet em - core, is

loci - doubly ch ns with 6Li nuc

0 50

0 0 0 0 0 0 0 0 0

E_d

6Li 2.1

asured at angles

veloped meth uclei is the optical mode e in a numbe

s considered tween the co s reduced to

harged particles clei (scattering a

Каналы

100 150

d = 25 MэВ, 

6

(d,d')⁶Li 86 МэВ

of 30 (a), 41 (b

hod of extrac phenomeno el of the nucl

r of studies [ d by phenom

olliding nucle a simple pro

s

angle - 24 degr

ы

200 2

(б

л.с. = 41 град

12С(d,d)¹²С

6Li(d,d)⁶Li

b) degrees

cting informa logical analy leus, the argu [4].

enological in ei. In this app ocess - scatte

rees)

50

)

ation about ysis of the umentation ntroduction proach, the ering in the

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field of integrated optical potential, the shape and size of which are determined by optimizing the design values of the model parameters with the corresponding experimental data. Technically, this procedure is associated with the solution of the Schrödinger equation

2 , ,

2



_E U r( )_ , 0

 

    



with a complex potential U(r). Here µ=mAрAt/(Aр+At) – reduced mass of the colliding nuclei, Ар and Аt – mass numbers of the incident nucleus and the target nucleus, m – nucleon mass, Е – kinetic energy of the relative motion in the center-of-mass system (c.m.s.).

Usually calculations are limited to central potentials depending only on the distance between the centers of mass of the colliding nuclei. This is justified by the fact that, as shown by the detailed theoretical study, the spin-orbit interaction has virtually no influence on the differential cross section of elastic scattering in front corners. Thus, the optical potential can be written as

( ) _C( ) ( ) ( _V( ) _S( )) U r V r V r i W r W r

The first member is the Coulomb potential. Since scattering is not sensitive to the particular form of the charge distribution, and therefore, there is no need to consider its diffuse edge, then for practical purposes it is sufficient to take the Coulomb potential of a uniformly charged sphere as

2

2 2

2

( ) (3 / ) для

2

( ) для

C

C p t

C C

C p t C

Z Z e

V r r R r R

R Z Z e V r

vvvvv v vvvvvvvvvvvvv v r R

r v v vv









  

 

where

R

_c

 r A

_o( ^1/3_p 

A

_t^1/3) - Coulomb radius, ZP and Zt – charge of the incident particle and the target nucleus. Other members of formula

2 , ,

2



_E U r( )_ , 0

 

    

 describe the nuclear force.

Usually as a nuclear it is taken Woods-Saxon potential with such set of phenomenological parameters, at which there is the best agreement with experiment, or the potential, calculated theoretically on the basis of fundamental nucleon-nucleon interaction.

In the first case the real part is given as

1

( ) 0 1 exp ^V

V

V r V r R

a

  

  

  

 

   imaginary volumetric

1

( ) 0 1 exp

V W

W

W r W r R

a

  

  

  

 

   and imaginary surface

1

( ) 4 _D _D 1 exp ^D

S

D

d r R

W r a W

dr a

  

  

  

 

   

As can be seen from formulas the radial dependence of the nuclear potential is determined by the Woods-Saxon form factor

1

1 exp ⁱ

i

r R a

  

  

  

 

  , where R_iand a_i – corresponding radius and diffuseness, which characterizes the decay rate of the potential. Woods-Saxon parameterization corresponds to the assumption that the internuclear interaction corresponds to the distribution density of the nucleons in the nucleus of the target.

Imaginary potential can be volumetric (WV ≠ 0, WD = 0), surface (WV = 0, WD ≠ 0) or mixed (WV ≠ 0, WD ≠ 0).

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Theoretical calculations were performed on SPIVAL program. OP parameters were chosen so as to achieve the best agreement between the theoretical and the experimental angular distributions.

Automatical search of the OP optimal parameters was made by minimizing the value ²/N of the least squares method. The initials were the potential parameters proposed in [5]. To reduce the ambiguity we tried not to go far from the recommended values of geometrical parameters (r_V, a_V) of the real potential.

For a better agreement with the experimental data depth of the imaginary part (W_D) decreased only slightly. The final potential parameters are shown in Table 1.

To describe the direct mechanisms in the mid 50-ies it was developed the method of distorted waves (MDW) and the Born approximation with distorted waves (DWBA). This is the most common, but not the only model for the description of direct nuclear reactions [6].

Table 1 - Optimal parameters of optical potentials ща еру process ⁶Li(d,d)⁶Li at incident deuteron energy 18 MeV V, MeV rV,

fm aV,

fm WD, MeV rD,

fm aD,

fm VSO, MeV rSO,

fm aSO,

fm

70.56 1.17 0.85 9.19 1.325 0.69 6.76 1.07 0.66

MDW can be considered as a generalization of the optical model to inelastic channels. Studying nuclear reactions, it cannot be, as in the case of elastic scattering, ignored the internal structure of the interacting particles. The wave function in each reaction channel is represented as (for example, for the input channels)

i a Ai

    

where _a и  _A- the wave function describing the incident particle and a target nucleus, χ_i - the wave function describing the relative motion of the particles in the channel.

The MIV uses the fact that the incident particle transfers its energy and impulse to a small number of degrees of nucleus freedom. This makes it possible to obtain an approximate solution of the many-particle Schrödinger equation using perturbation theory. Full Hamiltonian of the system can be written as

H = H⁰ + H^res

where H⁰ - Hamiltonian of the system consisting of two particles which interactions are described by optical potential V^opt, H^res – Hamiltonian of residual interaction, which is regarded as a small perturbation, transforming the system into the final state.

The process of interaction thus splits into three stages:

1. The motion of the incident particle in the "distorting" optical potential of the target nucleus;

2. The transfer of nucleons under the influence of the residual interaction;

3. The movement of the emitted particles in the field of the final nucleus.

The amplitude of the scattered wave has the form

( , )

2

( ) ( )

2

b res

a b f b i a

f k k  k H k

   

   



where



_b– reduced mass,

k 

_a

and

k 

_b

– the wave vectors of the input and output channels, _i( )k_a и

f( )kb

  – wave functions in the input and output channel having the structure (H = H⁰ + H^res), with

f( )kb

  – optical wave function. In the Born approximation, the exact wave function



_i

( ) k 

_a

is replaced by an optical wave function. The expression for the cross section has the form:

( , )2

DWBA a b

a b

b a

d k f k k

d k

 







 

All the above mentioned formulas of the method of distorted waves were innate in the DWUCK5 program, which is calculated using the theoretical section. Figure 4 schematically shows the transmission mechanism of α-cluster.

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d

6Li

d



Figure 4 - transmission of alpha-cluster in the ⁶Li(d,⁶Li)dprocess

Accounting for exchange cluster transmission mechanism was held in the framework of the connected reaction channels using the FRESCO program [7]. In this method, the system of A nucleons, represented in the input channel with A = Ap + At, configuration, is replaced by N-related systems dividing them into two clusters (A = Ap’,k + At’,k). Here the indices p and t refer, respectively, to the incident particle and the target nucleus, and the index k varies from 1 to N. The total wave function in this case is the sum of products of pairs of internal basic wave functions of clusters φpk, φtk and wave function Φk,, describing relative motion of clusters in the channel k:

1

( )

N

pk tk k k

k

 



    R

where R_k – the radius vector between the fragments in the channel k. The corresponding radial functions f_α(R_k) to the relative wave function Φ_k(R_k) are found by solutions of the system of coupled equations:

( ' )

' '

', 0 ( ' )

' ' ' ' '

, 0 0

( ) ( ) ( ) ( ) ( )

( , ) ( )

m

L L

k kL k k k k k k

R

L L k k k k

E T R U R f R i V R f R

i V R R f R dR

  

 

 

 











 

 



    

 

where

2 2

( 1)

( ) 2

kL

k

d L L

T R  dR R

 

 

 

    

– the kinetic energy operator. The α value is a composite index comprising the channel number k and the quantum numbers - the spins of the incident particle and the target nucleus (J_p, J_t), partial wave (L) and the total spin (J_T), ie α = (k, (LJ_p)J, J_t; J_T); U_k(R_k) - interaction potential in the k channel, including nuclear and Coulomb part; E_k - asymptotic kinetic energy of the channel k: E_k = E + Q_k – ε_pk – ε_tk, где Q_k, ε_pk, ε_tk – Q - reactions and the excitation energy in the channel k;

V

_^ _'

( ) R

_k - local interaction for transitions in discrete states of nuclei with multipolarity λ (transferred orbital angular momentum);

V

__'

( R R

_k, _k_'

)

- nonlocal interaction, connecting channels with the transfer of one or more nucleons.

In case of d⁶Li-scattering, we took into account only two channels (N = 2): d+⁶Li and ⁶Li+d.

Switching between channels, carried out through the transfer of alpha-cluster was calculated by distorted waves with a finite interaction radius. Thus, the elastic scattering and the reaction with alpha-cluster transmission were included in circuit communication channels. In the calculations of the transmission mechanism we used prior-representation. Cluster (d + α) wave functions for the ground state of the nucleus ⁶Li were calculated using the standard method of adjusting the depth of the real part of the Woods-Saxon potential, which gives the desired energy cluster communication. Geometric potential parameters (radius and diffuseness) have fixed values: r = 1,25 fm, a = 0,65 fm. Cluster spectroscopic amplitudes (S_A = 0,85) were found from the calculated cross sections fit to the experimental data, agree well with cluster theoretical amplitude S_A = 1,02, calculated in the framework translationally invariant model [8].

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0 20 40 60 80 100 120 140 160 180 10^-2

10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵

6Li(d,d)⁶Li E_d = 18 МэВ

d/d [мбн/ср]

_ц.м. [град]

Squares - experimental data, solid curve - cross sections, calculated by FRESCO program taking into account the interference of the elastic scattering cross-sections and sections of the transmission mechanism of α-cluster (coupled-channel method); dotted

curve - cross sections, calculated by Spival program (optical model), dashed curve - cross sections, calculated by the DWUCK5 program (method of distorted waves)

Figure 5 - the angular distributions of elastic deuteron on 6Li nuclei at the energy of 18 MeV

It can be seen that the optical model reproduces the experimental sections up to 130˚ (dotted curve), the method of distorted waves describes the area of large angles (dashed line) and coupled-channel method, which takes into account the connection of both the above processes, in accordance with their interference, achieves a description of the experimental data in the full angular range.

Conclusion. The experiments on the elastic scattering of deuterons on ⁶Li nuclei at energy Е_d= 18 MeV in the range of angles from 10˚ to 170˚ in the center of mass with the use of Е-Е - technique were carried out. The differential cross sections at angles of 40˚ and 60˚ have the minimum and maximum.

Next comes the gradual decline to 135˚. In large angles area it is observed the rise of cross-sections associated with a pronounced cluster structure of ⁶Li nucleus.

From the analysis of experimental data in terms of the optical model of the nucleus there are found the best, physically reasonable parameters of the optical potential interaction, which are in good agreement with literature data. In the method of distorted waves method and coupled-channel method it was an analysis of the elastic scattering, taking into account the contribution of the transmission mechanism of α-cluster, which showed that for the studied process in large angles area, the effect of this mechanism on the formation of the scattering cross sections is considerable.

The obtained experimental and theoretical data will be used in studies of the processes taking place in the stars, in the development of new theoretical models of nuclear physics, and will also be useful for the characterization of the processes occurring in the high-temperature plasma fusion reactors.

REFERENCES

[1] Arzumanov A.A., Nemenov L.M., Anisimov O.K., Batalin S.S., Volkov B.A., Gromov D.D., Kravchenko E.T., Nigmanov M.H. Popov Y.S., Prokofiev S.I., Ribin S.N. Izochronnii cyclotron s reguliruemoi energiei ionov // Izvestiya AN KazSSR, Seriya fisiko-matematicheskaya, – 1973, – № 4, – S. 6-15.

[2] Artemov S.V., Bazhazhin A.G., Baktibayev M.K., Burtebayev N., Duisebayev A., Duisebayev B.A., Zarifov R.A., Kadirzhanov K.K., Karahodzhayev A.A., Sahiev S.K., Satpayev N.K., Sargaskaev A.M., Seitimbetov A.M. Kamera rasseyaniya dlya izmereniya sechenii yadernih reakcii v predelno malih uglah na vivedennom puchke izohronnogo ciklotrona U-150M //

Izvestiya NAN RK, Seriya fisiko-matematicheskaya, – 2006, – № 6, – S. 61-64.

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[3] Burtebayev N.Т., Vinogradov А.А., Vongay А.D., Duisebayev А.D., Kurashov А.А., Mazurov I.B., Paramanov V.V., Prokovev S.I., Sakuta S.B., Sanichev V.I., Sytin N.P., Chesalov А.А., Chuev V.I. Sistema mnogomernogo analiza dlya issledovaniya yadernih reaksiy na siklotrone INP АNН КаzSSR // Izvestiya АN КаzSSR, Seriya fizikо-matematicheskaya. – 1975. – №2. – S.65-68.

[4] Hodgson P.E. The nuclear optical model. // Report of Progress in Physics. – 1971. – V.34. – P.765-819.

[5] Daehnick W.W., Childs J.D., Vrcelj Z. Global optical model potential for elastic deuteron scattering from 12 to 90 MeV // Physical Review C. – 1980. – Vol.21. – P. 2253-2274.

[6] Zelenskaya N.S., Teplov I.B. Metod iskazhennih voln v reaksiyah so slozhnimi chastisami // ЭЧАЯ. – 1979. – Т.11, № 2. – S.342-410.

[7] Thompson Ian J. Coupled reaction channels calculations in nuclear physics // Computer Physical Reports. – 1988. – Vol.

7. – P. 167-212.

[8] Nemec O.A., Neudachin V.G., Rudchik A.T., Smirnov Yu.F., Chuvilski Yu.M. Nuklonnie associacii v atomnih yadrah b yadernie reakcii mnogonuklonnih peredach. – Kiyev: Naukova dumka, – 1988, – 488 s.

ƏОЖ: 539.172.13

Н. Буртебаев¹, Ж.К. Керимкулов^1,3, Д.К. Алимов^1,2, А.М. Отарбаева³,Е.С. Мухамеджанов^1,2, Д.М. Джансейтов^1,2

1ЯФИ, Алматы қ., Қазақстан; ²əл-Фараби атындағы ҚазҰУ, Алматы қ., Қазақстан;

3Л.Н. Гумилев атындағы ЕҰУ, Астана қ., Қазақстан

18 МэВ ЭНЕРГИЯЛЫ ДЕЙТРОНДАРДЫҢ ⁶Li ЯДРОЛАРЫНАН СЕРПІМДІ ШАШЫРАУЫН ЗЕРТТЕУ

Аннотация. 18 МэВ энергияға ие дейтрондар ⁶Li ядроларынан серпімді шашырауының дифференциалдық У-150М үдеткішінде қимасы өлшенді. Өлшеулер 10 %-дан жоғары емес қателіктен жүргізілді. Бұрыштық таралулардың кіші бұрыштық аймағында қиманың бір минимумы жəне бір максимумы көрінеді. Алынған мəліметтер ядроның оптикалық үлгісі, бұрмаланған толқындар əдісі жəне реакцияның байланысқан арналар əдісі төңірегінде талданды. Əсерлесу опти- калық потенциалының жəне спектроскопиялық фактордың оптималды мəндері табылды. Потенциалдық шашырау тек қиманың кіші жəне орта бұрыштарында болатындығы көрсетілді. Қиманың үлкен бұрыштар аймағында α-кластер ауысу механизмі болатыны көрінеді.

Түйін сөздер: серпімді шашырау, зарядталған жеңіл бөлшектер, оптикалық потенциал, FRESCO, кластер ауысу, спектроскопиялық фактор.

УДК 539.172.13

Н. Буртебаев¹, Ж.К. Керимкулов^1,3, Д.К. Алимов^1,2, А.М. Отарбаева³,Е.С. Мухамеджанов^1,2, Д.М. Джансейтов^1,2

1ИЯФ, Алматы, Казахстан; ²КазНУ, Алматы, Казахстан; ³ЕНУ им. Л.Н. Гумилева, Астана, Казахстан

ИЗУЧЕНИЕ УПРУГОГО РАССЕЯНИЯ ДЕЙТРОНОВ НА ЯДРАХ ⁶Li ПРИ ЭНЕРГИИ 18 МэВ

Аннотация. На ускорителе У-150М измерены дифференциальные сечения упругого рассеяния дейтронов на ядрах

6Li при энергии 18 МэВ. Измерения выполнены с погрешностью не более 10 %. В угловых распределениях, в области малых углов, четко проявляется один минимум и один максимум сечений. Полученные данные проанализированы в рамках оптической модели ядра, метода искаженных волн с конечным радиусом взаимодействия и метода связанных каналов реакций. Найдены оптимальные значения оптического потенциала взаимодействия и спектроскопического фак- тора. Показано, что потенциальное рассеяние формирует сечения лишь в области малых и средних углов. В области больших углов сечения формируют механизмы передачи α-кластера.

Ключевые слова: упругое рассеяние, легкие заряженные частицы, оптический потенциал, FRESCO, передача кластера, спектроскопический фактор.

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N E W S

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTAN PHYSICO-MATHEMATICAL SERIES

ISSN 1991-346Х

Volume 6, Number 310 (2016), 13 – 19 UDC 517.956.223, 519.62

1,2D.S. Dzhumabaev, ^1,3S.М. Temesheva

1Institute of Mathematics and Mathematical Modeling MES RK, Almaty, Kazakhstan;

2International Information Technology University, Almaty, Kazakhstan;

3Al-Farabi Kazakh National University, Almaty, Kazakhstan [email protected], [email protected]

APPROXIMATION OF PROBLEM FOR FINDING THE BOUNDED SOLUTION TO SYSTEM OF NONLINEAR

LOADED DIFFERENTIAL EQUATIONS

Abstract. On the whole axis the system of nonlinear loaded differential equations is considered. The questions of existence and approximation bounded solution to the system are studied. The definition of «limit as t» solution to the system of nonlinear loaded differential equations is introduced. Sufficient conditions for the existence of bounded solution to the system of nonlinear loaded differential equations and convergence of the function sequence composed by the bounded solutions to the linearized system of loaded differential equations are obtained.

Regular nonlinear two-point boundary value problem for the system of nonlinear loaded differential equations on the finite interval is constructed, which approximate the problem of finding bounded solutions to the original system of loaded differential equations. It is given an estimate of the difference between the solution to initial singular problem and the solution to the approximating regular two-point boundary value problem.

Keywords: singular problem, nonlinear loaded differential equation, bounded solution, approximation.

Questions of existence and construction of approximate methods for finding of nonlinear ordinary differential equations, restricted on the whole axis, are considered by many authors [1-11]. Various problems for loaded differential equations and methods for their solutions are studied in [12-17].

In this article, nonlinear loaded differential equations is considered on R(,)

|,

| max

||=

||

, )),

( , ), ( ), ( , ( ) , (

= f t x f₀ t x _m x _m ₁ x _m x Rⁿ x x_i

dt

dx  _ _ _ _   (1)

where f :Rⁿ^1Rⁿ, f₀ :R²ⁿ^2 Rⁿ are continuous,



__m <



__m_₁ <...<



₀ =0<



₁ <...<



_m. The aim of this research is to find conditions for the existence of system solutions of nonlinear loaded differential equations restricted on the whole axis (1) and the construction of the regular two-point boundary value problems on a finite interval, which allows determining the narrowing of the decision on the final interval with given accuracy.

In work [11] there was introduced the definition of "limit at t" solution of nonlinear ordinary differential equations, and proved that, if the system is linearized along such solution is exponentially dichotomous on semi-axis, the "limit at t" decision has the attractive property. This result allowed building approximate two-point boundary value problems on a finite interval for the singular boundary value problems for nonlinear ordinary differential equations on the whole axis. Methods and results [11]

are used to find conditions for the existence of the equation solutions restricted on the whole axis (1) and for the construction of approximating regular boundary value problems on a finite interval.

The following symbols are used:

) ,

~( _n R J

C – space of continuous and restricted on J R functions x:J Rⁿ with norm )

(

1 sup x t

x

tR

 ;

(14)

) , (J Rⁿ

C – set of continuous on J functions;

} ),

,

~( )) ( ) ( ( : ) , ( ) ( { ) , ), (

(x₀ t J r x t C J R x t x₀ t C J R x x₀ ₁ r

S   ⁿ   ⁿ   , где x₀(t)C(J,Rⁿ);

}

||

) (

||

, : ) , {(

) , ), (

( x

₀

t J r t x t J x x

₀

t r

G    

;

} , ,

||

) (

||

, : ) ,....

, {(

) , ), (

(

₀ ₀

0

x t J r t v v t J v x r k m m

G 

__m _m



_k

 

_k

  

.

We take a continuously differentiable on R function x₀(t) so that ) ,

~( )) ( , ), ( ), ( , ( ) ) ( , ( )

( ₀ ₀ ₀ ₀ ₁ ₀

0 n

m m

m x x C R R

x t f t x t f t dtx

d 



 



   _ _ _ _  (2)

Restricted on Rsolution of the system of loaded differential equations (1) is defined as the limit of a sequence of functions compiled using linearized solutions of systems of loaded differential equations restricted on the whole axis. Therefore, we consider the linear loaded ordinary differential equation

, ,

), ( ) ( ) ( )

(

=

R t R x t f x

t A x

t dt A

dx _n

j j

m

m j











(3)

where matrix ^A

 

^t ^, A_j

 

t (j =m,m) and vector-function f(t) are continuous and restricted on R. Restricted solution of equation (3) is called solution of problem 1.

Definition 1. Task 1 is called correctly solvable if for any continuous and restricted on Rfunction )

, ( )

(t C R Rⁿ

f  equation (3) has only one restricted on R solutionx^*(t)and ||x^*||₁|| f ||₁,inequality is carried out, where  does not depend on f(t).

Definition 2. Continuously differentiable on R function x₀(t) is called limit at t by equation solution (1), if

. 0

= )) ( , ), (

), ( , ( )) ( , ( ) (

lim ||

0 0 0 0 _m 0 _m 1 0 _m

||

t

x  t f t x t f t x  x   x 

   





 

The following conditions should be carried out:

(А). Function f(t,x) is continuous and has uniformly continuous derivatives f(t,x)

x

 in

) , ), ( (x₀ t R r

G , where x₀(t) – limit at t equation solution (1), and the following limit relations are correct

 

t x f

 

x f

 

t x f

 

x

f t

t 



 



 , = , lim , =

lim ⁽⁴⁾

 

= , lim

 

⁼ ^,

lim 0 0 



 



 x t x x t x

t

t (5)

where x_, x_ are the solutions of systems of nonlinear equations f_

 

x =0, f_

 

x =0, respectively.

(B). Function f₀(t,v__m,....v_m) is continuous and has uniformly continuous derivatives )

,....

,

0( m m

k

v v t

v f ^



 (k m,m) in G₀(x₀(t),R,r) and for all



t,v__m,,v_m



G₀



x₀

 

t ,R,r



^has

relation point



sup



 , , ,    , sup

₀

 , , , 

₀

  ,

) , 0 [

, 0

T v

v t f T

v v

t

f

_m _m

T m t



  



_



 









 

  = 0, lim  , , ,  = 0, = , .

lim

0

f

0

t v v k m m

T v

_m _m

t k

T





 













(С). Task 1 for the linearized loaded differential equation

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 



,

 

, , ,



, ,

=

) (

= ) ( 0 =

= 0

0 0

n k

x v

x m v m k

m

m k

R y y J v

v t v f y

t x t x f dt dy

m m

m

m 













 

 





 



 ^





 ⁽⁶⁾

can be correctly solved, where ^J ^y

   

^t ^y _k ^k ^m ^m

k =



, = ,

 .

(В). Functions f_(x), f_(x) in S(x_,r), S(x_,r) respectively, have derivatives f_(x), f_(x) and uniformly relative x is correct limit relations

, ) , ( ,

) (

= ) , (

lim f t x f x x S x r

x

t   



 

, =

 

,



,



,

lim f t x f x x S x r

x

t   



and f_(x_)= A_{ }_ , Re



^_j 0, where



^_j - the eigenvalues of A_{ }_ , j=1,n.

Theorem 1. Functions f(t,x) and f₀(t,v__m,....v_m) are continuous and have uniformly continuous derivatives f(t,x)

x

 and ₀(, _m,.... _m)

k

v v t

v f ^



 respectively in G(x₀(t),R,r) and G₀(x₀(t),R,r). At

any xˆ(t)S(x₀(t),R,r) task 1 for linearized loaded differential equation



, ˆ( )

 

, , ,



ˆ( ), , ,

=

) ˆ(

= ) ( ˆ

= 0

=

R t R y t f y J v

v t v f y

t x t x f dt

dy _n

k x

v x v m m k

m

m k

m m

m

m   













 

 





 



 ^





 ⁽⁷⁾

can be correctly solved with constant



. Then, at inequality solutions

r x

x x

t f t x t f t

x₀( ) ( , ₀( )) ₀( , ₀( __m), ₀( __m_₁), , ₀( _m)) ₁

||

  

||



_ _

there is a such number



1, the sequence of continuously differentiable on R functions

,..., 1 , 0 ),

( )

1

(    



t x t x t n

x

_n _n _n (8)

where x_n(t) – restricted on R solution of linear equation

   















 

 





 



 _v ^f ^t ^v ^v ^J ^y

y t x t x f dt dy

k x

v x m v m k

m

m k n

m n m

m n

m 



 ) (

= ) ( 0 =

=

, , , )

( ,

=





^, ⁽ ⁾

 

^, ⁽ ^), ^, ⁽ ⁾



^, ^, ^,

) 1 (

0 t x x y R t R

f t x t f t dt x

d _n

m n m

n n

n   



 



  

 _ 

 ^ ⁽⁹⁾

by norm C~(R,Rn) is consistent to x^*(t) of the equation solution (1) in S(x₀(t),R,r). Corroboration. In equation (1) we substitute u xx₀(t), then have