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ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
THE FORMATION OF DEUTERONS IN 16Op COLLISIONS AT 3.25 A GeV/c
Bazarov Erkin
Candidate of Science, of Physical - Technical Institute of Academy of Sciences of Republic Uzbekistan
Olimov Kasim
doctor of sciences in physics and mathematics of Physical - Technical Institute of Academy of Sciences of Republic Uzbekistan
Yuldashev Bekhzod Institute of Nuclear Physics of Academy of Sciences of the Republic of Uzbekistan doctor of sciences in physics and mathematics, professor, academician
ABSTRACT.
The results of a brief review of experimental studies on the formation of deuterons in 16Op collisions at 3.25 A GeV/c are presented. It is shown that the average multiplicity of deuterons depends on the degree of excitation of the fragmenting nucleus. It is the largest at the maximum excitation of the oxygen nucleus and decreases with its decrease. The mean value of the total, longitudinal and transverse moments of deuterons do not depend on the degree of excitation of the fragmenting nucleus. Formation of deuterons and other light fragments andpions occurs independently.
Key words: deuteron, nucleus, collision, excitation, fragment, multiplicity.
Study of the production of light fragments - p, d, t and 3He in the fragmentation of relativistic nuclei is of particular interest, as these fragments, with the inclusive cross section, commensurate with the inelastic cross section of the reaction can be witnessed almost all stages of hadron-nucleus collisions - quasielastic knockout of their primary particle from the nucleus, the Fermi-collapse of the excited residual nucleus and evaporation intermediate weakly excited residual nucleus.
This paper deals with some analysis of studies on the formation of deuterons in the interactions of oxygen nuclei with protons at 3.25 A GeV/c. As is well known [1,р.1388], deuterons can be formed by the Fermi-collapse of the excited residual nucleus oxygen from the decay of the intermediate excited nuclei such as 6Li*(a + d), 10B*(2a + d), 16O(14N + d) (in brackets are the decay channels of excited nuclei), the merger of cascade nucleons - protons and neutrons, as well as the destruction of a-cluster (2d) under the influence of the proton target associated with the a-cluster structure of the oxygen nucleus. The
experimental data were obtained by the 1 m hydrogen bubble chamber LHE JINR, irradiated by oxygen nuclei with momentum 3.25 A GeV/c in the Dubna synchrophasotron, and are based on a fully measured 8712 inelastic 16Op events. Methodical features of the experiment associated with the separation of fragments by mass, and the determination of their kinematic characteristics are given in [2,р.497,3,р.336]. The experimental results are compared with the predictions of the cascade-fragmentation evaporation model (CFEM) [4,15р.,5,р.649]. As part CFEM for interactions of light nuclei and nucleons main mechanism of fragments (except nucleons) is the collapse of the excited residual nucleus thermalized after the intranuclear cascade. For light nuclei, such as 16O evaporative mechanism of formation of fragments in the model is ignored, even including the nucleons. Table 1 shows the experimental and calculated on CFEM average multiplicities and inclusive cross sections for light fragments with mass numbers A < 3.
Table 1
The average multiplicity <nf> and inclusive cross sections the yield of light fragments 'H, 2H, 3H and 3He in the experiment and
CFEM
Type of fragment 2H 3H 3Не
<nf> (exp) 1.78 ± 0.331 ± 0.141 ± 0.142 ±
<nf> (CFEM) 1.75 ± 0.249 ± 0.108 ± 0.152 ±
s 594.5 ± 110.6 ± 45.0 ± 45.4 ±
s 584.5 ± 83.2 ± 34.5 ± 48.6 ±
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As can be seen from Table 1 average multiplicity (inclusive cross sections) deuterons in the experiment is 1.33 times higher than the estimated average multiplicity (inclusive cross sections) for CFEM. In good agreement with experiment are predicting CFEM nuclei 'H and 3He. Inclusive cross section for deuteron in the experiment is about 19% cross section of protons and -2.4 times the cross sections of the mirror nuclei 3H and 3He. Experimental data for the mirror nuclei 3H and 3He within statistical errors very well coincide with each other. The big difference with the predictions KFIM and experiment is observed also in the formation of nuclei 3H. It also shows that in the model of the 3He isotope has a ~1.4 times more born than neutron-rich nucleus 3H. This apparently indicates that the model charge proton participates in the formation of the target fragments nucleon, and is not involved in the experiment. Thus, the mechanisms of formation of few-nucleon fragments embedded in axiomatic CFEM are clearly insufficient for a quantitative description of experimental data on the formation of nuclei with mass numbers A<3.
It is interesting to consider the dependence of the average multiplicity of deuterons on the degree of excitation of the oxygen nucleus. This dependence has been studied in [6,р.51]. As a measure of the degree of excitation of the fragmenting nucleus of oxygen in [6,р.51] was adopted by the total charge (Q) fragments with z > 2 (see. Fig. 1).
Fig. 1 shows that in the model and quantity <n> monotonical-ly decreases with increasing Q, that with a decrease of the degree of excitation of the oxygen nucleus. Thus, for values of Q
Fig. 1. The dependence of the average multiplicity of deuterons on the total charge of fragments with z > 2 in the experiment (•) and CFEM (o).
> 5 (in peripheral collisions) the experimental values and the predictions CFEM within statistical errors are the same. Such a coincidence, apparently due primarily to the law of conservation of electric and baryon charges, leading to a decrease of the number of particles that accompany the formation of multiply charged fragments. Along with a decrease in the absolute value of the contribution of the Fermi-collapse occurs also reduce the contribution of other possible mechanisms for the formation of deuterons in the experiment, unrecorded in CFEM. In the Q < 4 experimental values of <nd> systematically higher than in CFEM. The biggest difference of mean multiplicities of deuter-ons in the experiment and CFEM observed at the highest levels of excitation fragmenting nucleus oxygen (Q < 2). In such excitations fragmenting nucleus is natural to expect, along with an increase in the contribution of the Fermi-collapse and increasing contributions unrecorded in CFEM fracture mechanisms a-clus-ter decay of the excited compound nucleus 6Li, as well as the growth contribution of formation of deuterons by fusion of protons and neutrons.
In [6,р.51] also investigated the dependence of the mean value of the total momentum of deuterons <p> on the value of Q in the rest frame of the oxygen nucleus (see Fig. 2). Fig. 2 shows that the experimental values of <p>, making within the statistical errors on average 345 MeV/c, almost independent of the value of Q.
400 r
>
ш 280
360
320
v 240
+ I
0
200 -
+ I
О
0
о
160 -
0 2 3 4 5 6 7
Q
Fig. 2. Dependence of the mean value of the total momentum of deuterons on the total charge of fragments with z > 2 in the rest frame of the nucleus of oxygen in the experiment (•) and CFEM (o).
To determine the cause of the independence of the average of the total momentum of deuterons on the degree of excitation of the fragmenting nucleus was analyzed momentum spectra of deuterons Group with different values of Q. The momentum spectrum of deuterons was conditionally divided [7,р.1451] into three parts, corresponding to the appearance, mainly the collapse of the Fermi mechanism (0 < p < 275 MeV/c) coalescence (p > 535 MeV/c) and 275 < p < 535 MeV/c - a superposition of the mechanisms of formation of deuterons from the destruction of a-cluster and the collapse of the excited nucleon systems with the structure (a + 2H, 2a + 2H and 3a + 2H). Calculated under these assumptions the share of deuterons in three areas momen-
tum is shown in Fig. 3. There dashed lines show the values of these shares, averaged over Q for each of these mechanisms. Fig. 3 shows that within the statistical errors experimental values of these shares do not depend on Q. Thus, the independence of the average of the total momentum of deuterons on the degree of excitation of the fragmenting nucleus in our approach is related to the constancy of the relative contributions of the above mechanisms of their formation. For more information about the mechanisms of formation of deuterons interesting to investigate of the multiplicity of other types particles accompanying the birth of deuterons. Consider the associative multiplicity of secondary fragments
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0,60-
0,45-
I
а: о,зо-
0,15-
+
I
■+
i
+
Fig. 3. The experimental dependence of the relative contributions of different mechanisms of formation of deuterons on the total charge of fragments with z > 2 in the collapse of a Fermi (■), in the intermediate region with 0.275 < p < 0.535 MeV / c (o) and in the coalescence of the proton and neutron (•).
0,00-1---1----,---1----,----1----,---1----,---1----,----1----,---1
0 2 3 4 5 6 7
Q
depending on the presence or absence of the deuteron in the event [7,р.1451]. Table 2 shows the average multiplicities of light fragments ('H, 3H, 3He and 4He) in events with and without deuteron. Also given and calculated data on CFEM.
Table 2
Average multiplicity of light fragments in the events with the formation of the deuteron (nd i
The fragment type
'H 3H 3He 4He
О II s'3 Exp 1.28± 0.087± 0.088± 0.476±
CFEM 1.52± 0.096± 0.137± 0.359±
ndi Exp 2.24± 0.232± 0.253± 0.806±
CFEM 2.64± 0.156± 0.211± 0.356±
As can be seen from Table 2, the average multiplicities of light fragments correlated with the presence in the event of the deuteron. In events with the formation of the deuteron multiplicity all considered light fragments larger than in its absence. This fact indicates that the formation of deuterons occurs mainly in the processes with a strong destruction of the original nucleus and of his fragmentation to light few-nucleon fragments. The process looks as if there is an interaction with the a-proton nucleus cluster of oxygen, which then decays into two deuterons, or one of the mirror nuclei (3He or 3H) and the corresponding nucleon. In the second case the deuteron is formed due to the pick-up missing the nucleon. This is supported by the table 3 and fact that in the events with formation of the deuteron, the 4He nucleus appears almost 2 times more likely. Comparison with CFEM shows that in the experiment the average multiplicity of fragments 'H and 4He in the events of the birth of the deuteron in -1.75 times more
than in the events without the deuteron. To mirror nuclei 3H, 3He is the difference exceeds 2.7 times. It should be noted that the average of the plurality of nuclei within the statistical error, regardless of the birth event deuteron coincide with each other. In CFEM positive correlation between the average multiplicity of fragments and the presence of the deuteron in the event there are only fragments with mass number A < 3, whereas for 4He within statistical error correlations are absent. We also note that in the model, in contrast to the experiment, matching the average multiplicities for mirror nuclei is observed; the average multiplicity of 3He in -1.4 times more than the average multiplicity isotope 3H. CFEM regardless of the deuteron in the event of the birth of overestimates and underestimates the formation of proton nuclei 4He that, apparently, due to the neglect of a model a
Table 3
The average multiplicity of fragments in events with and without deuteron
nd The charge of fragment
1 2 3 4 5 6 7
0 1.37± 0.56± 0.063± 0.035± 0.076± 0.216± 0.247±
>i 2.47± 1.06± 0.12± 0.054± 0.067± 0.126± 0.022±
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Here, just as in the Table 2, the correlation between the average multiplicity of fragments and the presence of the deuteron in the event: for fragments with zf < 4 are positively correlated, and for fragments with 5 < zf < 7 -
Table 4
Average multiplicity of light fragments in events with emission of deuteron back and forward in the rest frame of the nucleus of oxygen
J Type of fragment
1H 3H 3He 4He
<90° 2.31± 0.215± 0.235± 0.746±
>900 2.35± 0.242± 0.251± 0.753±
The data in Table 4 show that, within the statistic error, the mean multiplicity of the considered fragments do not depend on the angle of emission of the deuteron. That means (given the significant difference in the mechanisms of formation "deuteron-back" and "forward-deuteron") that the formation of associated lung fragments is not related to the manner in which they are born deuterons. In other words, it can be argued that the mechanisms of light fragments independent. A similar result was obtained recently in the analysis of momentum features of light fragments - protons, deuterons, tritium and helium-3 in the events of production and no production a-particles in 16Op collisions at 3.25 A GeV/c. It has been shown that the average values and widths of the spectra of light fragments on the full and transverse
momenta were independent of the presence or absence in the event of a-particles. The dependence of the average multiplicity and kinematic characteristics of the deuteron in the event on the presence of the charged pion [6,р.51]. To avoid the influence of the charge of the proton target on the studied correlations, consider peons from the projectile, ie will deal with the fast (P > 0.5 GeV/c) n+ - and п-mesons produced mainly as a result of inelastic charge of one or more nucleons in the nucleus of oxygen during the intranuclear cascade (n ^p+ n ,p^n+ n + Table 5 shows the average multiplicity and average values of the total and the transverse momentum of the deuteron and subject to availability in the event of fast n+ or n-
Table 5
The average multiplicity and average values of the total and the transverse momentum of the deuteron in the rest frame of the nucleus of oxygen depending on availability in the event of a charged pion
Value The presence of charged pions in the event
n = 0 n > 1 n-= 0 n->1
П+
<n> 0.287 ±0.008 0.412 ±0.018 0.292 ±0.007 0.441 ±0.023
<Р>, MeV/с 343 ±6 345 ±9 340 ±5 358 ±11
<РЛ> MeV/с 252 ±5 253 ±8 250 ±5 263 ±9
From Table 5 shows that the average multiplicity of deuterons in the event of production of charged pions in -1.45 times more than in the events without their education. It still does not indicate a possible link mechanisms of formation of deuterons and charged pions. This is apparently due to the fact that the events with pion implemented on average a higher level of excitation of the nucleus fragmenting oxygen than their formation without event. Proof of the lack of communication between the mechanisms of formation of deuterons and charged pions are independent of mean values of full and transverse momentum of deuterons from the presence or absence of the charged pion in the event.
It is also seen that the average multiplicity and average impulse responses deuteron within the statistical error does not depend on the sign of the charge of fast pions. Thus, the comparison of the experimental and calculated data on the formation of deuterons in 16Op collisions at 3.25 GeV/c can be concluded that:
- as in the experiment and in CFEM highest average multiplicity deuterons observed at the maximum excitation of the oxygen nucleus and decreases with its decrease. The average multiplic-
ity of deuterons do not depend on the type of mirror nuclei 3H and 3He accompanying the birth of the deuteron, which is a consequence of the deuteron isoscalar nucleus;
- there are positive correlations between the multiplicities of charged pions, and deuterons associated with a high level of nuclear excitation of oxygen in the formation of pions;- Experimental values of the mean total and transverse momenta of deuterons, and their emission angles are independent of the degree of excitation of the fragmenting nucleus of oxygen, which is associated with persistence share of contributions of different mechanisms of their formation; - CFEM qualitatively describing the dependence of the degree of excitation, predicts significantly less than the average total momentum, transverse momentum and emission angle of deuterons. This fact indicates that CFEM, apparently, there is a higher dissipation of excitation energy between nucleons fragments. This leads to the fact that the in model the other fragments, including deuterons, have less kinetic energy, which is observed in the experiment;
- mechanisms of formation of deuterons and other light fragments and charged pions are independent.
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3. V V. Glagolev, K.G. Gulamov, M. Yu. Kratenko et al.//JETP Letters. v. 59. number 6. 1994. Р 336 -338.
4. A.S. Botvina, A.S. Lanin et al.// Statistical modeling of disorder of easy nucleuses in адрон-nuclear reactions. Preprint. Insti-
tute of Nuclear Physics. Moscow. 1990-П0657.15р.
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О ЕДИНСТВЕННОСТИ РЕШЕНИЯ КРАЕВОЙ ЗАДАЧИ ДЛЯ УРАВНЕНИЯ ТРЕТЬЕГО ПОРЯДКА С КРАТНЫМИ ХАРАКТЕРИСТИКАМИ
Балкизов Жираслан Анатольевич
Научный сотрудник отдела Уравнений смешанного типа Федерального государственного бюджетного научного учреждения Институт прикладной математики и автоматизации, г. Нальчик
АННОТАЦИЯ
Методом интегралов энергии в работе получено достаточное условие однозначной разрешимости краевой задачи для уравнения третьего порядка с кратными характеристиками в конечной прямоугольной области.
ABSTRACT
The method of energy integrals in a sufficient condition for the unique solvability of boundary value problem for third order equation with multiple characteristics in the final of the rectangular area.
Ключевые слова: уравнение с кратными характеристиками, краевая задача, интеграл энергии, непрерывность функции и ее производных.
Keywords: equation with multiple characteristics, boundary value problem, the integral energy and continuity of its derivatives.
В прямоугольной области D = {(x, y): 0 < x < l, 0 < y < h) евклидовой плоскости точек z = (x, y) рассматривается
h = uxxx + b2 (z)uy + b1(z)uy + a2 (zК + a1(z)ux + a0 (z)u = 0
где ai (z) (i = 0,1,2) и bj (z) (j = 1,2) заданные функции из класса ai - искомое решение.
(z Ь C (d ), bj (z )<= Cy (d )
а u
(z ) = u(x, y)
Уравнение (1) относится к уравнению параболического типа [1, с. 72]. В работе [2] методами теории потенциалов и интегрального преобразования Лапласа была изучена краевая задача для уравнения (1) в случае, когда коэффициент b2 (x, y)= 0 . Исследованная в работе [2] в настоящее время называется задачей Каттабрига.
С помощью фундаментальных решений уравнения (1), полученных в [2], в монографии [3, с. 132] была построена функция Грина задачи Каттабрига для уравнения (1) и получены оценки фундаментальных решений и их производных различных порядков. Также с помощью функции Грина в [3, с. 135] построено решение задачи Каттабрига для уравнения (1) в замкнутом виде. Локальная, нелокальная и общие краевые задачи для уравнения (1) в случае, когда коэффициент b2 (x, y) = 0 были исследованы в работах [4]-[5]. Различные краевые задачи для уравнения вида (1) при b2 (x, y ~)ф 0 как в ограниченной так и в неограниченной областях изучались в работах [6-13]. Регулярным в области D решением уравнения (1) назовем любую функцию C (d )n Cl,2y (D ) из класса C (D )n> C^y (D), подстановка которой обращает уравнение (1) в тождество.
В данной работе исследуется следующая
В данной работе исследуется следующая Задача А. Требуется найти регулярное в области D решение уравнения (1) из класса
u(x, y)е Cl(D u {x = 0} u {x = l}),
удовлетворяющее краевым условиям
ux (0,y) = a(y)u(0, y) + A(y) ux (l,у) = e(y)u(l,у) + A (y),’
(3(0, у ) = A3(у ),
u(x,0) = 0, u (x, h) = 0,
, (4)
0 < y < h
, (2)
0 < y < h,
0 < x < l
где a(y), P(y). A(y)€ C[0,h] (k = 1,2,3*, причем
Ak (°) = Ak (h) = 0
