ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
SIMPLE NUMERICAL METHOD OF COMPUTING THE PROBABILITIES OF ANGULAR MOMENTUM Jv OCCURRING AMONG POSSIBLE VECTORS J RESULTING FROM n ANGULAR MOMENTUM
SUMMATION (^ = 1—n)
© 2004 M. Kaczmarczyk
Division of Nuclear Physics, University of Lodz, Poland Received February 7, 2003; in final form, November 19, 2003
A relatively simple numerical method of summing angular momentum vectors with maintaining space quantization rules of each summed angular momentum has been presented. The method enables the calculation of the values of probability p(Jv) of finding a definite angular momentum Jv among all vectors J being results of quantum summation of n angular momentum vectors = 1—n). It may be used, e.g., in the calculations of angular momentum of many particle states. Significance of the paper is connected with the possibility to take into account, in a simple way, the angular momentum conservation principle for a system, which consists of arbitrary number of excitons.
1. INTRODUCTION
The angular momentum j is the nucleon's quantum characteristic in the potential well according to the shell model1).
In the compound nucleus excitation process, e.g., as a result of slow neutron capture, various nucleon configurations are realized. According to formalism of semi-classical approach [1], we consider the target nucleus as a potential well with single-particle levels occupied by nucleons up to the Fermi level. After the capture of a slow neutron with kinetic energy T lower than neutron binding energy Sn, the composite nucleus is excited to the energy U ~ Sn. The initial configuration of the composite nucleus is the single-particle one (1p0h), as long as the captured neutron does not collide first with some of the target nucleons. The probability of collisions is higher than the probability for neutron escape from the nucleus and, as a rule, the 2p1h configuration is obtained as a result of the energy exchange in the two-body residual interaction. A sequence of two-body interactions between nucleons moving in an averaged potential well
!)In this paper it has been assumed that, e.g., symbol j = = 5/2 represents vector of the angular momentum, which
has the length equaling ^J§(§ + where 5/2 denotes
the maximum value of the quantum magnetic number, describing maximum angular momentum projection on the quantization axis. According to the quantization rule, vector j = 5/2 can assume 2j + 1, e.g., six orientations towards the given axis. Symbol j denotes the angular momentum quantum number.
leads to the redistributions of the excitation energy U among nucleons. In the semi-classical model [1], based on the description of the composite nucleus excitation, according to the exciton model [2] it has been accepted to characterize the configuration type adequately to the number of i particles raised over the target nucleus Fermi level and of i — 1 holes under this level. As an example, one of possible configurations is the configuration ip(i — 1)h, where i is contained in the range 1—k; k is called the complexity of the nuclear structure and denotes the maximum number of particles, which may be raised above the Fermi level. Suitable expression for k is placed in [1].
The complexity k values for particular nuclei excited as a result of slow neutron capture are determined by energy conservation principle, Pauli exclusion rule, and the presence of energy gaps separating nucleon shells in the potential well. The configuration ip(i — 1)h can be realized in many ways, as a result of possible creation of a large number of subconfigurations (each with i particles and i — 1 holes) in nucleus excitation process. The subconfigurations of the configuration ip(i — 1)h for defined excitation energy U arise, because particles and holes (excitons) may be located in the potential well on different single-particle levels with different quantum characteristics. However, these subconfigurations (for configuration ip(i — 1)h) are not equivalent to one another because excitons with different angular momentum values participate in them.
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One can expect that considering angular momentum conservation principle and using it in excitation analysis of i particles and i — 1 holes in potential well will limit, to a large extent, the number of possible subconfigurations for each configuration with i particles over the Fermi level.
There exists a need of working out a method for the calculation of reduction coefficients for the whole numbers of subconfigurations, which are specific for every configuration with i particles and i — — 1 holes. These reduction coefficients allow to bring these numbers to such values which result not only from energy conservation principle but from angular momentum conservation principle too. This requirement refers to every one from possible configurations ip(i — 1)h, where i = 1—k.
This paper presents a proposal for calculation of the probabilities of the angular momentum vector Jv occurring among possible J vectors, which are the result of many angular momentum summation. It is worth noticing that the presented numerical method provides a way of relatively simple calculating p(Jv) probability values. We can demonstrate this with a few simple examples. The calculation of p(Jv) is essential, e.g., in shell-model calculations; however, associated methods are not presented separately. Moreover, adoption of the basic formulae placed in a selected academic handbook [3] is not efficient.
It seems that the considered problem may have a significance for agreeing upon neutron resonance density values pcal calculated on the base of the semi-classical model with resonance level density pexp values determined in an experimental way.
In this place we want to notice distinctly that the nuclear model [1] used for calculation of subconfiguration numbers as based on exciton model [2] assumes that particles and holes occupy equally-spaced single-particle levels separately in the nuclear potential well (Equidistant Spacing Model). So in such conception the Pauli principle is fulfilled automatically.
2. THE SUMMATION OF TWO ANGULAR MOMENTA ji AND j2 TO THE Jv VALUES WITHOUT NECESSITY OF USING CLEBSCH-GORDAN COEFFICIENTS
Below we have explained the rules of calculating probabilities p(Jv) of angular momentum Jv occurring in a set of vectors J resulting from summation of two angular momenta j1 and j2 when, e.g., j1 = 5/2 and j2 = 2.
According to the expectations of the quantum mechanics, vectors j1 and j2 may assume, respectively, 2j1 + 1 and 2j2 + 1 orientations, relatively to the quantization axis. For the considered case these are the numbers 6 and 5. Table 1 contains the presentation of magnetic quantum numbers for angular momenta j1 and j2 in the analyzed case.
As a result of summing magnetic numbers presented in Table 1, we obtain a collection of 30 magnetic numbers of angular momenta resulting from the summation. Being ordered, the collection of these 30 elements is written bellow:
7/2 5/2 3/2 1/2 — 1/2
7/2 5/2 3/2 1/2 — 1/2
5/2 3/2 1/2 — 1/2
3/2 1/2 — 1/2
1/2 — 1/2
3/2 — 5/2 — 7/2 — 9/2
3/2 — 5/2 — 7/2
3/2 — 5/2
3/2
As we can easily see on the basis of the above presentation, in the written set of 30 numbers there exist groups of angular momentum magnetic numbers, which are adequate for angular momenta: Ji = 9/2, J2 = 7/2, J3 = 5/2, J4 = 3/2, J5 = 1/2, resulting from summation.
Assuming that each set of quantum configurations is equally probable, we can calculate the probability p(Jv) of angular momentum Jv occurrence on the
base of formula
r
P( = (1)
In (1) rv denotes frequency of angular momentum occurrence (i.e., the number of magnetic numbers for angular momentum Jv), whereas N denotes the total number of all possible projections of all vectors J, which arise as summation results. In the analyzed case N = 30.
SIMPLE NUMERICAL METHOD OF COMPUTING THE PROBABILITIES
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Table 1. The presentation of magnetic quantum numbers mj for angular momenta ji = 5/2 and j2 = 2
ji = 5/2 h = 2
5/2 2
3/2 1
1/2 0
-1/2 -1
-3/2 -2
-5/2
Table 2. The presentation of two-vector summation results, when j1 = 5/2, j2 = 2, and of the appropriate p(Jv) values
J, VK 2J„ + 1 TV = + 1) p(iv)
1/2 1 2 2 2/30
3/2 1 4 4 4/30
5/2 1 6 6 6/30
7/2 1 8 8 8/30
9/2 1 10 10 10/30
We should notice that in the analyzed case (and always when only two angular momenta are being summed) the so-called multiplication factors j of angular momenta Jv occurrence are equal to one.
In Table 2 we have the presentation of vector summation results when j = 5/2 and j2 = 2.
On the basis of the data collected in Table 2 formula (1) can be written in the form
J (2 Jv + 1)
P(Jv
(2 J + 1)'
Jv
Table 3. The presentation of p(Jv) probability values in a few other cases of two angular momentum j1 +j2 summation
J,
J'l = 1/2 0 1/4
h = 1/2 1 3/4
ji = i 0 1/9
j2 = l 1 3/9
2 5/9
J'i = l 1/2 1/3
h = 1/2 3/2 2/3
the Clebsch—Gordan coefficients, makes the method encouraging and worth extending to n angular momentum summation.
2.1. Angular Momentum Statistical Coefficient g
It is worth noticing that formula (2) for the case of only two (n = 2) angular momentum summation ji + j2 can be written in the form
p(J) = (2J + 1)
■ J=jl+32
E (2J + 1)-
J = | jl 32 |
(3)
(2)
Taking into consideration the presented method, the p(Jv) values have been calculated for summing a few other angular momenta ji and j2; they have been presented in Table 3.
In formula (2) the summation comprises the values \j1 — j2\ <J < \j1 + j2\ with a unit step.
The results p(Jv) presented in Table 3, calculated according to the method proposed in this paper, are in full agreement with the received values if we use Clebsch—Gordan coefficients [4].
Great easiness in calculating p(Jv) probabilities in the case of two angular momentum summation with the method proposed in this paper, compared to the generally recognized and accepted method usi
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