From the ansatz of our wave function eq. (2.36) we see that the calculation of the necessary matrix elements can be performed in several steps, after decomposing the antisymmetrizer into a sum over all permutations acting on spatial and spin-isospin coordinates: Since the potential terms in the Hamiltonian can be written as a product of operators acting in coordinate space and spin-isospin space separately

^ = ^^(-1)q w° (k> — q)wiijT (k, q) (3Л)

i<j i<j kq

we can also calculate the respective matrix elements of each operator sepa­rately. The rank of the interaction is denoted by k, e. g. k = 1 for the spin orbit force. Since the multi-dimensional integration in coordinate space is usually by far the most elaborate part of the calculation, we first describe the essential parts of this calculation.

Actinide decay contributes a maximum of approximately 20-30% of the total decay heat for a wide range of irradiation conditions up to cooling times of ~ 108 sec. (Fig. 6). The most significant contributions to actinide decay heat arise from 239U and 239Np for cooling times < 106 sec, while the a decay of 238Pu, 241Am, 242Cm and 244Cm become more important at much longer times (Fig. 7). Assuming a decay-heat contribution of 30% by 239U and 239Np gives rise to a maximum uncertainty in the total decay-heat predictions of 2% for uranium-based thermal reactors at cooling times up to 106 sec. At longer cooling times, the actinide contribution to decay heat is dominated by 242Cm, and the contribution to the overall uncertainty in decay heat ranges from 0.5% to 6% over the cooling time 108 to 3 x 109 sec. Overall, there would appear to be no serious problems associated with the actinides in decay-heat calculations.

Fig. 4. Uncertainties in total decay heat for 235U(T) without taking correlation effects into account

j —— TOTAL

——- DECAY CONSTANT

——- INDEPENDENT YIELD

……… CUMULATIVE YIELD

——- ENERGY

Uncertainties in total decay heat for 238Pu(F) without taking correlation effects into account

T able 5: Main parameters contributing to the uncertainties in total

decay heat for 238Pu(F)

Fig. 6. Percentage contribution of actinides to total decay heat from typical fuel of different reactor systems

Fig. 7. Percentage contribution from individual actinides to total decay energy release rates [5 MW/Te(U) end of life rating 30 GWd/Te(U)] for gas-cooled reactor fuel (AGR)

Debu Majumdar*

Nuclear Power Technology Development Section,
Division of Nuclear Power, Department of Nuclear Energy,
IAEA, Vienna, Austria

Lectures given at the
Workshop on Nuclear Reaction Data and
Nuclear Reactors: Physics, Design and Safety
Trieste, 25 February — 28 March 2002

LNS0520004

D. Majumdar@iaea. org

Abstract

Global demand for energy is going to keep on increasing, especially in developing countries where per capita energy use is only a small fraction of that in industrialized countries. In this regard nuclear energy could play an important role, as it is an essentially unlimited source of energy. However, the nuclear option faces the challenges of increasingly demanding safety requirements, economic competitiveness and public acceptance. Worldwide, a significant amount of experience has been accumulated during development, licensing, construction, and operation of nuclear power reactors. This experience forms a sound basis for further improvements. Nuclear programs in many countries are addressing the development of advanced reactors, which are intended to have better economics, higher reliability, improved safety, and proliferation-resistant characteristics in order to overcome the current concerns about nuclear power. Advanced reactors, now being developed, could help to meet the demand for power in developed and developing countries, not only for electricity generation, but also for district heating, desalination and for process heat.

This paper reviews the status and trends in advanced nuclear power technology development around the world, discusses the challenges it faces, and summarizes the international approach and technical advances made with examples of new designs of reactors.

1. INTRODUCTION

An examination of the global energy use shows that fossil fuels account for nearly 80%, and nuclear power provides only 7%, of our current energy supply. Additionally, around 83% of nuclear power is produced only in a dozen industrialized countries out of 30 nuclear power producing countries. The demand for an increase of standard of living and population growth in developing countries are asking for a considerable increase of this energy supply. However, many factors come into play in specific countries in providing energy to the people — economics, infrastructure, and government policy being the most important factors. The effect on the environment is another crucial factor whose importance, however, has not yet received adequate attention in the energy mix.

The population of the earth, the prime reason for energy use, is increasing although the birth rate has decelerated since the early 1990s. Present trends suggest that total population may not exceed 8 billion people around 2050 and may start to decline shortly thereafter1. This is still a large increase from today’s population of 6 billion, and energy for these people must be provided. It is important to note that virtually all of this growth will occur in developing countries. Industrialized country populations have peaked or will do so shortly. Moreover, the greater part of the population increase will be urban. The proportion of people living in rural areas has already peaked and will decline in future. An indication of urbanization is that today there are five mega cities of more than 15 million habitants (Tokyo, Mexico City, Mumbai, Sao Paulo and New York), but in 20 years there will be 15, mostly located in developing countries1. In energy terms, already we have nearly 2 billion people without access to a regular electricity supply. Even with lower population projections, the challenge to achieve access to energy for all is clearly substantial. An issue here is that concentration of people requires large sources of energy nearby; this needs to be solved in a way that does not create an environmental problem for the city dwellers.

The environmental issues have received prominence since the 1990s, particularly with respect to greenhouse gas emissions, climate change possibilities and their effect on our living conditions. The Third Assessment Report of the Inter­Governmental Panel on Climate Change (February 2001)2 presented the strongest evidence yet that climate change is occurring (for example, temperatures have risen in the lowest 8 km of the atmosphere, snow and ice cover have decreased, and the sea level has risen between 0.1 and 0.2 meters in the last century). The report also finds that concentrations of atmospheric greenhouse gases have continued to increase as a result of human activities. However, the nations of the world have not unified in their response to this phenomenon.

Nuclear energy is one way to provide bulk electricity supply without greenhouse gas emissions; it is supported by ample uranium resources worldwide and can be made to last almost forever by using the breeder option. The nuclear industry accumulated 10,000 reactor years of operating experience. But nuclear is not without its problems. The challenges facing nuclear power include (1) continuing to assure the highest level of safe operation of current plants, (2) implementing disposal of high level waste, (3) establishing and convincing the public of a sound basis for nuclear power for sustainable development, (4) achieving further technological advances to assure that future nuclear plants will be economically competitive with fossil alternatives, especially in deregulated and privatized electricity markets, and (5) developing economical and non-proliferating small and medium sized reactors to provide nuclear power to countries with small electricity grids and also for non­electric applications such as seawater desalination.

This paper will discuss the status and trends of advanced nuclear reactors, which could help in the solution of the energy problem of the world and, at the same time, address the issues raised by the nuclear critics.

The next heavier standard neutron cross sections are 10B(n, a)7Li and 10B(n, a, y)7Li from thermal energies to 250 keV. As can be seen from the energies of light nuclei [51] there are many resonances in the compound system 11B close to the 10B-n threshold. There are already many channels open, like 10Be-p, 8Be — 3H, and 7Li -4He together with the first excited state, and furthermore the 11B nucleus has many particle stable states below the

lowest break-up threshold. Therefore describing the reactions 10B (n, а0;1) microscopically poses a major problem. For the potential described in the previous section this task seems to be feasible, but the outcome is unknown.

Klapdor (1983) has argued that the neglect of any structure within beta-strength functions is an oversimplification that is inconsistent with experimental observations. Structure is found at high-level densities, and can significantly affect the half-lives and branching ratios of P—delayed processes. Klapdor and Metzinger (1982a and 1982b) included this structure when determining the electron and antineutrino spectra generated after the fission process. Their method involved microscopic calculations of Sp(E) for all fission products:

Sp (E) dE=£ Bt (Et) dE / D

i

where B(E) is the reduced beta transition probability to a state at excitation energy Ei in the daughter nucleus, and D is the vector coupling constant. However, a major source of uncertainty in the calculation of the electron and antineutrino spectra is the effect the fission products with unknown or poorly-defined decay schemes will have on the shape of the beta-strength function.

0. 1 1 10

cooling time (s)

Fig. 18. Calculated P-and y-ray components of 235U decay heat compared with experiments at very short cooling times (Yoshida and Tachibana, 2000)

cooling time (s)

Fig. 19. Calculated P — and y-ray components of 238U decay heat compared with
experiments at very short cooling times (Yoshida and Tachibana, 2000)

Despite a lack of reliable decay-scheme data for a significant number of fission — product nuclides, P—spectra have been calculated by Davis et al (1979), Avignone and Greenwood (1980), Kopeykin (1980) and Vogel et al (1981) on the assumption that the resulting beta-strength function is smooth. These particular calculations do not reproduce the precise P-spectrum measurements of the fission products from 235U fission (Schreckenbach et al, 1981). Klapdor and Metzinger (1982b) undertook a microscopic analysis of Sp(E) for all fission products with unknown and uncertain decay schemes: considerable improvement was obtained against the measurements,
with a deviation from experiments of less than 4%. Figs. 20 and 21 compare the electron and antineutrino spectra obtained from the various methods of calculating Sp(E), normalised against the equivalent measurements of Schreckenbach et al (1981); calculation/experiment ratio (C/E = R) for method 4 (i. e., Klapdor and Metzinger, 1982b) is much closer to unity over the full energy range. Table 9 lists the different sets of data as a function of electron and antineutrino energy. Studies have also been made of the thermal fission of 239Pu, with similar results (Klapdor and Metzinger, 1982a).

Further microscopic modelling studies by Hirsch et al (1992) have resulted in the successful use of the proton-neutron quasiparticle random phase approximation (pn — QRPA) to calculate the beta-strength functions. Single particle energies are calculated, taking into account nuclear deformation and pairing interaction; subsequent RPA calculations include proton-neutron residual interactions. The resulting theoretical half-lives and mean beta and gamma energies are in good agreement with experimental measurements across the full range of Z, and these data have been adopted to extend the range of radionuclidic coverage of a number of decay-data libraries.

Debu Majumdar*

Nuclear Power Technology Development Section,

Division of Nuclear Power, Department of Nuclear Energy,

IAEA, Vienna, Austria

Lectures given at the
Workshop on Nuclear Reaction Data and
Nuclear Reactors: Physics, Design and Safety
Trieste, 25 February — 28 March 2002

LNS0520005

D. Majumdar@iaea. org

Abstract

As the standard of living increases globally, the need for fresh water and industrial products is also increasing; they require energy for production and hence, the demand for energy — both electric and non-electric, is also increasing. Nuclear energy provides now only about 7% of global energy use; fossil fuels which degrade the environment provide the rest. Nuclear energy has the potential to provide an abundance of greenhouse-gas-free energy for mankind. Currently, nuclear energy is mainly used for electricity production. This paper discusses non-electric applications of nuclear energy, summarizing the global status and enumerating the areas where it could be used.

1. INTRODUCTION

Some of the first civilian reactors in the world were used to supply heat, e. g., Calder Hall in UK (1956) and Agesta in Sweden (1963). Calder Hall provided electricity to the grid and heat to a fuel reprocessing plant, and Agesta provided hot water for district heating of a suburb of Stockholm. The first nuclear power station in Russia (1954) was also a multi-purpose facility providing electricity and heat to the closed city of Obninsk in Kaluja region, near Moscow. Currently less than 1% of the heat generated in nuclear reactors is used for non-electric applications1. Direct use of heat energy is more desirable from an energy efficiency point of view and nuclear energy is an enormous source of greenhouse-gas-free energy. However, nuclear power has remained primarily a source for electricity generation. Presently about 30% of the world’s primary energy is used for electricity production, and approximately 2/3 of this energy is thrown away as waste heat. Yet despite past and current use models, it is possible to optimise the use of nuclear heat for both electric and non­electric applications, thereby making more efficient use of nuclear energy. Experience in co-generation of nuclear electricity and heat has been gained in Bulgaria, Canada, China, Hungary, Kazakhstan, Russia, Slovakia and Ukraine2. This paper examines the scope of non-electric applications of nuclear energy3 .

There are four areas where nuclear heat can be utilized: for desalination of salty and waste water, district heating of residence and commercial buildings in cold countries, industrial process heat supply, and fuel synthesis. Primary experience of non-electric applications of nuclear energy is in the first two categories. There are more than 150 reactor-years of operating experience with nuclear desalination, particularly in Japan and Kazakhstan. District heating systems from nuclear power plants have operated reliably in many countries, particularly in Eastern Europe. Fuel synthesis has evolved in recent years because nuclear energy can generate high temperature heat; this heat can be used for hydrogen production, coal gasification and production of other fuels. The heating requirements of different industrial processes vary. The temperature requirements for the principal applications are shown in Table I. They vary from low temperature applications for hot water to high temperature industrial processes.

TABLE I. TEMPERATURE NEEDS OF VARIOUS TYPES
OF INDUSTRIAL PROCESSES

Various types of reactors are designed with different ranges of inlet and outlet coolant temperatures, and hence will be useful for different applications. Table II shows the range of coolant temperatures for different reactor types. A nuclear plant can provide steam or process heat from about 100 C for district heating or desalination to about 1000 C for very high temperature industrial applications. Table III shows the characteristic parameters of steam that could be produced by various reactor types4. Water reactors can provide steam in the range of 250 to 300 C at about 5 to 7 Mpa pressure, while liquid metal and gas cooled reactors can generate steam at higher temperature and pressure. LMFBRs can provide steam at approximately 500 C and gas cooled reactors at somewhat higher temperatures.

Tests have been performed on the optical, resonance and levels segments. A number of misprints and erroneous coding have been detected and corrected.

Several RIPL participants tested the preliminary version of the levels database by using the data in calculations. A new simple test was worked out for checking nuclear temperature (T) derived from the analysis of cumu­lative plots of discrete levels, yielding temperature values which are remark­ably similar to the T(A) function obtained in the global fitting procedure. Ignatyuk tested the performance of the T(A) function by comparing the results with the temperature obtained by Gilbert and Cameron; he found reasonable agreement and recommended the use of T(A) in cases for which no direct estimation is possible. Herman has extensively tested Nmax values for nearly 500 nuclei using the Gilbert-Cameron procedure and level densi­ties specific to the EMPIRE code. Perfect fits were obtained for about 50% of all analyzed cases, fair agreement was found for about 25%, and poor for the remaining 25%. The quality of the fit depends on the model used for level densities. No formatting errors were detected while reading files with discrete levels.

Global testing of the RIPL-2 database has been performed in three sep­arate exercises. Large numbers of nuclear reaction cross sections were calcu­lated by means of the nuclear model codes EMPIRE-II, UNF and TALYS. Herman performed calculations for the most important neutron-induced re­actions on 22 targets from 40Ca up to 208Pb in the energy range from 1 keV up to 20 MeV. The 2-17-beta version of the statistical model code EMPIRE — II has been used with all default parameters except those differentiating the 3 series of runs. In all cases TUL MSD and Heidelberg MSC models were used for pre-equilibrium emission of neutrons, and exciton model (DEGAS) for pre-equilibrium emission of protons and ys. These studies were comple­mented with Hauser-Feshbach calculations including widths fluctuations at incident energies below 5 MeV (HRTW model). The results were converted into ENDF-6 format and compared with experimental data available from the EXFOR library. Three sets of calculations were performed in order to test new levels segment, Koning’s global optical potential and HF-BCS level densities. No problems were encountered while processing the new RIPL-2 files, which indicates that the files are formally correct. Comparison with experimental data shows reasonable overall agreement for most of the calcu­lations. There is a clear indication that calculations using the new RIPL-2 files fit experimental data better than those with default EMPIRE-II param­eters, which demonstrates the improvements brought about by RIPL-2. The HF-BCS microscopic level densities were found to perform comparably to the phenomenological level densities and in some cases even better. However, significant discrepancies among the results of the three sets of calculations were observed in a number of cases. These findings illustrate the importance of the model parameters and prove the practical usefulness of the RIPL-2 library for basic research and applications. The second exercise was carried out by the Beijing group, using the recently developed UNF code to study 103 nuclei from the mass region 69-160 in the incident energy range from

0. 1 to 20 MeV. All input parameters were taken from the RIPL database. Agreement with the experimental data was found to be very good for total and elastic cross sections (within 3%). For other main reaction channels, calculations reproduced the shape, but some parameter adjustments were necessary in order to fit the absolute cross sections. TALYS calculations were performed for various neutron-induced reactions on 5 isotopes from 52Cr to 208Pb. Default input parameters originated from RIPL-2. This exercise concentrated on the comparison of Ignatyuk-type and microscopic level densities and provided very reasonable agreement with experimental data for both formulations.

EXFOR is the EXchange FORmat used by the Data Centres to exchange compiled experimental data. Designed initially for neutron reaction data, the format is flexible enough to be extended to the compilation of a great variety of data types: neutron, charged-particle and photon-induced reactions, as well as spontaneous fission data. More details can be obtained in brochures from the Data Centres or directly from the appropriate Web pages.

The flexibility of the format as well as the possibility to store all experimental details for evaluator access make EXFOR ideal for the worldwide compilation of experimental data. Furthermore, all the fission yield data from the Meek and Rider files that were missing in EXFOR have been converted and added to the EXFOR data base, which makes the latter almost complete with regard to fission yields.

• Radioactive inventory — Depending on the amount and characteristics of the radioactive inventory the strategy can change. For example, plants which experienced accidents with radioactive release or incidental spilling on the floors, or contamination of insulation, may require special attention. On the other side plants which have been prematurely shutdown may have a lower total inventory and a long Safe Storage period may not be justified.

• Presence of other operating units on the same site or in the country — This situation may imply the maintenance of a high level of nuclear technology that will be available for a longer time. To have them on the same site means also that the site cannot be released anyway and that work force can be easily re-employed. Therefore, generally, it is convenient to wait some time before dismantling.

• Availability of a national waste repository — Decommissioning and dismantling means essentially to cut in pieces a NPP and to concentrate and package the radioactivity. This is of little advantage if all the radioactive material shall remain on-site. In addition, if a repository does not exist, uncertainties are present on the specification of packaging that will be finally required by the repository, once it will made available.

• Clearance levels and waste disposal costs- Clearance levels are of fundamental importance to decide which strategy and which technologies shall be used. Since clearance levels can vary even by one order of magnitude and more, correspondingly the amount of radioactive wastes to be generated, classified as such, can vary by even more than one order of magnitude. This, in connection with the cost of waste final disposal, can force the decision in favor or against the preliminary decontamination of systems and structures.

• Plant layout — Difficulties in the dismantling process due to very compact layouts may lead to decisions about preliminary decontamination and different cutting strategies

• Safety conditions of structures and systems — Costs of maintaining the safe storage condition for decades shall be low to make the SAFSTOR strategy convenient. This means that structural conditions and corrosion conditions of all components shall be good and capable to withstand the expected conditions for several decades, without any need for major refurbishment.

• Expectations about development of licensing rules and of technologies for decommissioning — Licensing rules and available technologies change usually more quickly than expected. This is the experience of the last years. Extrapolations of the situation in 50 or 100 years are extremely difficult to be made and, more important, extremely uncertain. This means that in the case of the safe storage strategy it is wise to use contingencies in cost calculations and to leave open the introduction to new technologies.

• Connections between conventional and nuclear safety — In a NPP there are also some conventional safety issues and a production of conventional waste (such as asbestos). Complications may arise when the same waste is at the same time radioactive and toxic, such as in the case of contaminated asbestos. The extent of such situation may lead to the use of special techniques of dismantling and waste packaging.

• Status of plant configuration documentation — In the lack of a good documentation about plant design bases, plant modifications and plant conditions, the knowledge of the plant staff might lead to the decision to perform as many as possible activities, while the people are still available.

Worker doses are not included in the previous list, since it is assumed that with proper preparation and the use of proper tools, including remote operations and decontamination activities, the worker dose can always be reduced to acceptable levels. Also environment and public health impact levels have not been included in this list, since they are generally so low that do not present any serious concern. Of course also these elements have to be taken into account in a complete strategy evaluation.

In the second group, the following elements have to be considered:

• Availability of funding — Generally the fund accumulation during plant operation is based on a cost evaluation, which, in turn, is based on a specific strategy. Any change of the strategy would imply the identification of different fund sources.

• Expectations about cost evolutions — Calculated costs are affected by a certain level of uncertainties. Uncertainties grow with the extension of calculation extrapolation, because of the uncertainties in the evolution of technologies and the evolution in the safety rules and in the costs of waste disposal. The need to reduce uncertainties implies a reduction of decommissioning duration

• Expectations in terms of inflation rate — Funds tend to grow, because, with proper investments, they will produce a net interest. Assumptions about this net interest (that usually range from 2% to 5 %) are very sensitive for the definition of the convenience of different strategies

• Assumptions in terms of contingencies — Excessive contingencies intended as provisional funds to cover unexpected situations may tend to increase the needed funds. In general, the amount of contingencies may be reduced, obviously, when the decommissioning time is shortened.

The spatial part of our wave function eq. (2.36) consists of Gaussian func­tions and products of solid spherical harmonics. To keep the notation as simple as possible, we disregard in this section the coupling of the various angular momenta. Therefore a single term on the right-hand side of the matrix elements (marked by the index r) is of the structure

N-1

L, a>= П П У, m (SN-n, r +j)

i =

where we have converted the single particle coordinates ri in eq. (2.34) into Jacobian coordination si via an orthogonal matrix. The numbering of the Jacobians starts with the internal ones, see fig. 2. The number of clusters on the right-hand side is denoted by nCr. Note that sN, proportional to the coordinate of the center-of-mass, is absent in eq. (3.2) due to the translational invariance of our wave function. The index Lra > is just a reminder of the fragmentation and the angular momentum structure of the

ket wave function. The function on the left-hand side of the matrix element Lia’ > can be expressed in an analogous way by the Jacobian coordinates si on the left-hand side, which differ in general from those on the right-hand side. So the general spatial matrix element is of the form

JLla’Lra(P) = < Lla PwijLra > (3.3)

The orbital operators w®® contain the coordinates in the form of eq. (3.2), but in addition to that also differential operators may occur, like in the spin — orbit force. We have put the permutation P to the left of the symmetric interaction for convenience, see the discussion below. Since the evaluation of all interactions can be reduced to the calculation of certain overlap matrix elements [16], [17], [4], see also section 3.4, we restrict our considerations to the norm, because there all the essential steps become apparent. We can express the norm matrix element of eq. (3.3) by choosing the Jacobian coordinates on the left-hand side as independent variables in the form

z

Уі„т„ (Qn) = Г1іті…lzmz

Since we used an orthogonal transformation from the single particle coor­dinates r to the Jacobian coordinates s, whose index l we suppressed for simplicity, no determinant appears in the integral. The matrix p in the ex­ponent results from applying the permutation P to the coordinates on the right-hand side and then expressing these coordinates by those on the left — hand side. The vectors Qn are the intercluster coordinates pk, see eq. (2.35) and fig. 1, on the left — and right-hand sides, after applying the permutation P onto the latter. Again these can be expressed as linear combinations of Jacobi coordinates s, the former are just some of these coordinates.

In case of a genuine interaction its radial dependence in Gaussian form is also included into p and its angular dependence is then an additional spherical harmonic in eq. (3.4). For treating explicitly other radial dependencies, e. g. the Coulomb interaction, see ref. [4] and the discussion below. The number of angular momenta z is thus the sum of relative coordinates on the left-hand side and those on the right-hand side plus possibly one from the interaction.

Except for the solid spherical harmonics the matrix element eq. (3.4) is just a multi-dimensional Gaussian integral, which can be evaluated by bringing the matrix p into diagonal form. For treating the solid spherical harmonics we use their generating function [18]

L

(b • r)L = bL ^2 CLm b-m YLm (r) (3.5)

m=-L

with the vector b = (1 — b2, г(1 + b2), —2b) being a null vector with respect to a real scalar product, i. e. b • b = 0 and the scalar product of two vectors given by

bn • bn = 4bnbn’ — 2(ЬП + b2n) (3.6)

The coefficients CLm are given by [18]

CLm = (“2)LL! y^ + 1)(L — m)(L + m) ‘ (3J)

To evaluate the matrix element eq. (3.4) we now consider the generating integral

(3.8)

Expanding the expression expQ^ anbn ■ Qn) into a power series in an and bn we find

taking into account eq. (3.5).

Since the generating integral eq. (3.8) is just a Gaussian integral it can be done explicitly and the result can again be expanded in a power series in an and bn to find the desired integrals via eq. (3.9). The transformation

N-i

sn = ^ T^tA » = 1,…,^ — 1 (3.10)

A=^

with TAA = 1 and Ty = 0 for X> X’ brings the matrix p into diagonal form. This transformation results in

I (aibi… az bz) = J dti… dtN-i exp ^- ^ ^ PnaJon ■ tAj j

where we expressed the vector Qn as

N-i

Qn — 52 Pnntn

n=i

and used the properties of the transformation eq. (3.10). Employing the method of completing squares the integral yields

and taking into account eq. (3.6). Expanding the exponentials into a power series and ordering the terms in the form of eq. (3.9) yields the final re­sult [16], [4]

The sums over gnn>, hnn’ ,knn> run over all possible combinations of nonneg­ative integers, which fulfill the following relations for n1 = 1,…,z

/ t (gnn’ + hnn’ + knn’ + gn’n + hn’n + kn’n) ln’

Eq. (3.16 a) results from comparing the exponent of an in eq. (3.13) and eq. (3.9), whereas eq. (3.16 b) results from that of bn>. In these relations gnn’ = hnn’ = knn’ = 0 if n < П. If there are more than two angular mo­menta different from zero, e. g. more than 2 clusters, eqs. (3.16) allow many solutions, which have to be found by trial and error. Starting from group theoretical considerations, Stowe [19] developed a general scheme to find all solutions, which is realized in a very efficient Fortran program. Since the solution of eqs. (3.16) is independent of the permutation and the width pa­rameters used to describe cluster internal and cluster relative wave functions, it need only be done once.

With this formalism we can now calculate each individual overlap matrix element from eq. (3.15) using eqs. (3.7, 3.10 — 3.12, 3.14 and 3.16). Straight­forward extensions cover potentials, whose spatial form is a linear combina­tion of products of Gaussian functions, solid spherical harmonics and powers of r2. In case of a Gaussian radial dependence, this dependence has to be put into the construction of the matrix p^> in eq. (3.4), which then modifies the transformation T^ in eq. (3.10) and hence also the diagonal elements fix (eq. (3.11)) and the final coefficients ann> (eq. (3.14)). Forces containing ex­plicitly a solid spherical harmonic, like the tensor force, are taken care of by
increasing the number of angular momenta by one by adding an additional vector Qn. With the relation [20]

2 i

C = 2ІТТ £ (-irJWr№-„.(r) (3.17)

m=-l

We can treat even powers of r2 by just increasing the number of angular momenta by two. Obviously all three radial dependencies can be treated simultaneously by modifying the matrix p and the number of angular mo­menta accordingly. The results are always of the form of eq. (3.14), i. e. of the type norm integral.

The Coulomb potential ZZ2/r is apparently not of the above form, but it can be written as

і Гв f°°

-=20— dkexp(—k2{3 r2) (3.18)

r VO0

yielding again a Gaussian radial dependency and an additional integration over k. The explicit reduction of the matrix elements of the Coulomb inte­gration in terms of overlap integrals is given in ref. [4]. Other negative powers of r are easily obtained by modifying the integral in eq. (3.18) by the appro­priate monomial in k2. For the tensor potential in the non-relativistic quark model, which contains an ^ term, an explicit expression is given in [21].

For potentials containing differential operators, like the spin orbit interac­tion, the differential operators are applied to the wave function on the right — hand side, resulting in additional polynomials in vectors r — rj, which can be treated by the method described above. For an explicit calculation of the kinetic energy, see [16]. All the potential terms are finally reduced to linear combinations of certain norm-type matrix elements. Each of these integrals can be evaluated explicitly by hand, the sheer number of matrix elements, however, necessitates the use of a computer program. As will be shown below, we need between 15 and 20 different Gaussian width parameters in order to facilitate the expansion of the Coulomb functions in the interaction region. So for one permutation we have typically 200 — 400 different ma­trix elements per structure taken into account (neglecting the symmetry of entrance and exit channel). On the other side the number of terms in the antisymmetriser grows rapidly with the increasing number of particles and furthermore in general the number of important configurations grows too.

Therefore one is forced to avoid multiple calculations of the same matrix elements. For this purpose the symmetry of the ansatz for the internal wave function, eq. (2.36), can be utilized. Permutations inside a cluster will not modify the wave function eq. (2.36). Furthermore, interactions between different pairs of particles might yield identical spatial matrix elements.

Let us consider 12C as an illustrative example. Without any symmetry the 12-nucleon antisymmetrizer contains 12! = 479 001 600 terms. Since protons with spin up are distinguishable from protons with spin down, we can reduce the huge number to (3!)4 = 1296 terms. If we group the nucleons in 12C together into three alpha-particles, then only 120 terms survive. Allowing for a two-body interaction, each of the above numbers has to be multiplied by 12 • 13/2 = 78, the number of possible interactions. Realizing that each of the two interacting particles can be in one of the 3 clusters initially and after interaction (and permutation) in some other one yields 9 • 10/2 = 45 possibilities, where we assumed a spatially symmetric interaction and did not check if this combination is possible for the permutation considered, thus arriving at 120 • 45 = 5400 possible interaction terms. The actual calculation yields 2601 different terms under the assumption that the 3 clusters have different width parameters.

What was shown here by an example is a quite general decomposition of a finite group into double cosets of appropriate subgroups [22]. Here we comply with the cluster symmetry of the wave function eq. (2.36) by considering subgroups Sni x Sn2 x… of the symmetric group SN with ^щ = N. Each double coset can be characterized uniquely by one permutation. It is easy to convince one-self [22] that the overlap matrix elements eq. (3.15) for all permutations belonging to one double coset are identical because the matrix p^i is unchanged. In the above example we consider the subgroup S4 x S4 x S4, yielding 120 double cosets. Also the interacting particles can be easily marked within the double coset expansion, which is described in some detail in the following section.