A few comments are merited concerning the impact of activation products on decay — heat calculations for fission-based reactors. The radioactive decay of the activation products generated from the structural materials of a thermal-reactor core represents an extremely minor contribution to the resulting decay heat. This source is more significant following shutdown of a fast reactor, with the formation of 22Na and 24Na by activation of the sodium coolant, and 58Co and 60Co from the activation of the nickel content of the steel structures in the reactor core:

58Ni(n, p)58Co(n, Y)59Co(n, y)60Co

Up to ~ 10% of the decay heat at 5 years cooling time arises primarily from 60Co. However, the production cross sections and decay data of all the main contributors are sufficiently well known that uncertainties in these parameters pose no problems in summation calculations. Hence, the activation products will be considered no further.

There are only 30 nuclear electricity-generating countries. Table I below shows the total electricity generating capacity in various countries in the world. Note that only 8 countries have total capacity of more than 100 GWe, and of these two of the largest population countries, China and India, have only a few percentage of nuclear to share. However, China and India currently have solid programs for nuclear power. The important part of the table is that there are many dozens of countries with a total capacity of 2 GWe and less, who need the power most. Because of their grid size, these countries cannot add a large plant of the size of 1GWe; plants for these countries would have to be smaller and more cost-effective (and hence more innovative) than existing large plants.

The worldwide operating experience of power reactors is tremendous. Overall 438 reactors were in operation in 2002. The breakdown of these reactors by types and generating capacity are shown in Table II.

As shown in Figure 1, there are currently 32 nuclear power plants under construction in 12 countries; 8 in China, 4 each in Ukraine and Republic of Korea, 3 in Japan, 2 each in India, Slovakia, Russia, Iran, and Taiwan, China, and 1 each in Romania, Czech Republic and Argentina. China is building six PWRs in the range of 640 to 1000 MWe from Framatome, Russia and their own design, and two 730 MWe PHWRs from Canada. Two 500 MWe PHWRs are under construction in India. India has also announced that four more 220 MWe PHWRs and 2 1000 MWe WWERs and a 500 MWe prototype fast breeder reactor will be under construction soon. In Ukraine Khmelnitski Units 2, 3 and 4 and Rovno Unit 4, all 1000 MWe WWERs, are under construction since 1985 through 1987. Large advanced PWRs and BWRs are being built in Republic of Korea, Japan and Taiwan. Mohovce Units 3 and 4 in Slovakia, WWER 440 plants, are under construction since 1985 and are currently on hold. Atucha Unit 2 in Argentia, 700 MWe Siemens PHWR, is under construction since 1981 but currently on hold. Cernavoda Unit 2, CANDU 700 MWe PHWR, is under construction since 1983. Bushehr Units 1 and 2 in Iran, WWER 1000, are currently replacing the original reactor designs. Temelin Unit 2 in Czech Republic, WWER

1000 further modernized by Westinghouse, is currently under startup testing. Figure 2 gives their size breakdown. It is important to note that primarily large size reactors are being built: 22 in the range of 900 — 1350 MWe. Then there are 6 in 600 — 700 MWe range, and 4 between 300- 500 MWe. Thus it is apparent that the utilities will build power plants as large as the grid size will tolerate because that is most economical. However, there is a need for both small and large reactors for flexibility in power management, to suit the grid size and investment capitals, and for remote or special situations such as small localities in Siberia.

TABLE II. REACTOR TYPES AND GENERATING CAPACITY IN THE WORLD AS OF JUNE 2002

Number of NPP under construction

(January 2002)

In the previous chapter various examples demonstrated the great flexibility of the resonating group method. Finally I will discuss some limitations of the method. Of a principle nature are breakup channels into three or more fragments. If the direct coupling to such channels is indeed strong, then the recipe of approximating these channels by effective two fragment channels will not yield good results. Attempts [52] to connect the resonating group method with Faddeev type approaches face the very complex interacting potentials between the fragments. A straightforward approach by expanding the wave function is not possible, because the boundary condition depends on the coordinates. Fortunately, however, most strong breakup channels proceed via sequential decay and thus can be well approximated.

The technical limitations fall into several classes. The most obvious one is the number of particles. Depending on the cluster structure the double coset classification of chapter 3 is feasible on modern workstations up to 12 or even 16 particles, as long as only the highest orbital symmetry is considered. In case of lower symmetries playing an essential role, which means that the number of clusters is increased, the maximum number is around 12. A typical example for such cases are the neutron-halo nuclei like 11 Li and 11 Be, which can be described by a—^H—n — n — (nn) and a — a — n — n — n configurations respectively. Larger nuclei can be much better described in the framework of the Generator coordinate method [53, 3], where shell-model techniques can be utilised to reduce the calculation of A-body matrix elements to 2-body matrix elements.

Another obvious limitation is the expansion of the bound state wave func­tion in terms of Gaussians. Even though the number of width parameters may be chosen quite high, the asymptotic behaviour is never that of an ex­ponential function. Hence, if the bound state wave function is needed far beyond the root-mean-square radius, like in the radiative capture at very low energies, approaches where the asymptotic form of the wave function can be utilised [54, 55] are preferable. These methods, however, have to solve the integro-differential equations and therefore have to calculate the kernels, so far by hand or computer algebra and are therefore restricted to mostly 2-, at most 3-cluster systems.

A further limitation lies in the expansion of the scattering wave function in the interaction region in terms of Gaussian functions. Since the Gaussians are all centered at the origin, they will become numerically dependent if too many of them are used. On the other side if the kinetic energy becomes higher and higher more and more zeros of the relative motion wave function move into the interaction region, thus requiring more and more Gaussians. In model studies we found numerically stable sets of Gaussians for up to three zeros. In principle one could use eq. (3.17) and multiply the Gaussians by even powers of r. In model studies this procedure works quite well in practical calculations, however, the following limitations do not allow to reach such an energy range.

With increasing energy usually the number of open channels increases rapidly. The diagonalisation of the resulting large matrices can be done easily. The threshold energies, however, are no more well reproduced, introducing some uncertainties. Furthermore, near thresholds the numerical procedures tend to be less stable, thus increasing the uncertainties. A typical example on this limit is the reaction 4He (2H, 3He) 3H studied in [56]. By carefully choosing the Gaussian parameters and the weight function, eq. (4.2), the situation can be improved appreciably.

The most serious limitation in energy is, however, related to the potential used. Since one wants to utilise the symmetry of the dominant structures, one needs simple wave functions for the lightest nuclei, i. e. deuteron, triton, 3He and 4He. In practice these light nuclei are just described by pure S — waves, using linear combinations of some Gaussian functions of the internal coordinates. For a nucleon-nucleon potential such wave functions only yield binding, if the short-ranged repulsive core is reduced appreciably. This pro­cedure on the other hand does not yield enough repulsion at higher energies, where the short range behaviour of the potential is tested more closely. The potential we usually employ [26] gives reasonable cross sections up to 35 till 50 MeV centre-of-mass energy above the lowest threshold, depending on the system. The simple cure of using a realistic nucleon-nucleon potential, like [39, 40], leads to such complicated bound state wave functions, that any calculation beyond A = 5 seems to be no more technically feasible [38].

In conclusion, the resonating group model provides the means to study a wide range of few body problems, if an interaction exists which allows to describe the systems with sufficient accuracy. So far nuclear systems have been studied predominantly, but also atomic [14] or molecular problems may be investigated.

Mann et al (1982) have advocated that accurate predictions can be made of P-decay properties using a statistical model with only one free parameter (implying that any intrinsic structure is not important). The beta-strength function for a transition to a daughter level at energy E via a multipole Я is given by the equation:

SЯ (E) dE = Z p(E, J, п) вЯ (E) dE / D

J, п

where p(E, J,n) is the density of levels with spin J and parity nat excitation energy E, рЯ is the average reduced transition probability per level for moment Я in the interval (E, E+dE), and D is the vector coupling constant.

The half-life can be expressed by the following equation:

Qe

ty = 1/ Zf SpE fp (Z, Qp-E) dE

where f contains the kinematic factors and Fermi function. Similar integrals can be derived for the mean beta and gamma energies. Energy-dependent рЯ values were found for all of the allowed transitions in the — decay of the fission products (log ft values of 4.3 to 5.6), while рЯ corresponded to log ft of 7.1 for first forbidden transitions. Beta rates were calculated by multiplying the level density parameter by N/(N+Z), where N is the number of neutrons and Z is the number of protons in the daughter nucleus; this factor represents the only free global parameter in this model.

a Exposure time of 1.5 d. b Exposure time of 3 y. c Infinite exposure time.

Fig. 22 shows the experimental data compared with the predicted half-lives of a series of Rb nuclides:

(i) calculated by gross theory (Takahashi et al, 1973);

(ii) determined on the basis of the microscopic structure of the beta — strength function (Klapdor, 1983);

(iii) from statistical analysis (Mann et al, 1982).

The simple statistical method can be seen to generate data in closer agreement with the measurements than the more rigorous theoretical approaches.

Water is essential for living but over a billion people, approximately 20% of the world’s population, lack safe drinking water, and three billion lack access to adequate sanitation5 for lack of water. Unfortunately, 94% of the world’s water is salt water and only 6% is fresh6, and less than 1% of the fresh water is easily accessible (27% being in the glaciers and 72% underground). As the standard of living increases all over the globe, the demand for both energy and water is also increasing. In this regard, the development and use of water desalination technologies7 are helping tremendously. Desalination of water requires energy but, as shown in Table I, it can be done at relatively low temperatures. Waste heat from power plants is sufficient for this purpose. Nuclear power can play a significant role, particularly in a dual capacity, by providing water in addition to greenhouse-gas-free energy. Many years of successful operation have proved the technical feasibility and reliability of nuclear plants for producing fresh watert.

Desalination technologies have evolved over the past 50 years to large-scale commercial processes. The major commercially available processes are of two kinds: (a) thermal processes, where heat is used to vaporize and distill fresh water from saline water; these are multi-stage flash distillation (MSF), multiple-effect distillation (MED), and vapor compression (VC), and (b) membrane processes where suitable membranes are used for the separation of salts such as the mechanism of reverse osmosis (RO). There are also other minor processes such as freezing and solar evaporation. Globally about 26 million m3/d of fresh water is produced by desalination (including both brackish and seawater plants). The maximum is produced in Saudi Arabia, about 21%. The U. S. produces approximately 17%, 80% of which is achieved by membrane processes.

The possibility of using nuclear energy for desalination of seawater was realized as early as the 1960s. Experience with nuclear desalination now exceeds 150 reactor-years. Table IV gives a list of the nuclear plants, which have been used for desalination of water; it also provides information about the reactor types, desalination technologies employed and the fresh water capacity of the plant8. The Kazakhstan nuclear plant was shut down in 1999 and was the only power reactor in the world supplying heat for industrial-scale desalination9. It produced 80,000 m3/d of potable water for municipal use. The Diablo Canyon Nuclear Power Plant in the U. S. also produces 4500 m3/d of fresh water from the sea for in-plant use; they use RO membrane technology10. Table V gives details of operating experience of LWRs in Japan, a PHWR in Pakistan, and the LMR in Kazakhstan. It should particularly be noted that there was no incidence of radioactive contamination of the water produced.

^ Nuclear desalination can be described as production of potable water from seawater or brackish water in a facility in which a nuclear reactor is used as the source of energy for the desalination process.

An example of how nuclear heat is used for desalination is shown in figure 1. In this example steam is produced in a secondary loop for generation of electricity and then another tertiary loop is used to heat the seawater for desalination. The salt water is in the 4th loop. This makes the production of fresh water far removed from the radioactive isotopes of the first, primary loop.

Economic studies performed at the IAEA indicate that nuclear energy can be competitive for desalination compared with fossil-fuelled energy sources11. The desalination costs range from $0.40 to $1.90 per m3 of fresh water produced. It was generally found that (a) MSF processes cost higher than RO and MED processes, (b) RO and MED processes costs are in general comparable, (c) RO is economically more favourable for less stringent drinking standards, and (d) desalination costs are higher for smaller reactors.

Various research and construction project studies are being performed in several countries for nuclear desalination. Currently major activities are taking place in India, Pakistan, Russia, and China; Canada, Republic of Korea and France are also involved in nuclear desalination research. India12 is constructing a 6300 m3/d combined MSF-RO nuclear desalination demonstration plant connected to two 170 MWe PHWR units at Kalpakkam. Installation of the RO section has been completed and the MSF section is expected to be done in 2003. Pakistan has initiated a feasibility study for coupling a 4500m3/d desalination plant with the 137 MWe PHWR KANUPP. Russia has investigated various coupling schemes for 35 MWe KLT-40, 55 MWt RUTA, and 70 MWe NIKA reactors. Russia and Canada are working on a floating nuclear desalination plant with the KLT-40 reactor and an optimised Canadian desalination system involving reverse osmosis. The construction of the KLT-40 on a barge is expected to be completed in 2006. China is investigating a nuclear desalination project in Shandong peninsula with a 200 MWt nuclear heating reactor (NHR-200). Their plan is couple it to an MED process to produce 160,000 m3/d of potable water. The study was finalized in 2001, and Tsinghua University is setting up a test system to verify the design performance of the MED process. Candesal Technologies in Canada is developing a unique approach to the design and operation of RO system to improve energy efficiency and reduce the life cycle cost of potable water. The Republic of Korea (ROK) is developing an integrated desalination plant with the SMART reactor for dual purpose application. They aim to provide 90 MWe and 40,000 m3/d of fresh water. A 65 MWt pilot version of the SMART reactor is expected to be built in the ROK by 2008. ROK and Indonesia are also investigating the feasibility of nuclear desalination in Madura Island, Indonesia. France is working with Tunisia for a site-specific desalination study for La Skhira in Tunisia.

Nuclear desalination is thus a matured technology and can be installed in many nuclear plants to provide fresh water to solve regional water shortage problems. The desalination capacities of the world have been doubling each decade and hence there is a tremendous potential for nuclear desalination. Efforts are now primarily directed towards reducing production cost of desalinated water through innovations and technological enhancements.

The work on interfaces between selected nuclear model codes and RIPL — 2 segments has been facilitated by the standard RIPL-2 format. The two optical model codes (ECIS and SCAT2) and two statistical model codes (EMPIRE-II and UNF) use RIPL-2 library to a significant extent. Interface codes preparing inputs for ECIS and SCAT2 have been written by Young and is available in the optical segment.

The statistical model code UNF (PR China) makes use of RIPL optical potentials, masses, levels, level densities and GDR parameters. EMPIRE-II accesses RIPL-2 database directly and retrieves optical model parameters, discrete levels and microscopic level densities (HF-BCS). Built-in system — atics for GDR parameters and prescriptions for y-strength functions follow RIPL-2 recommendations. EMPIRE-II library of masses and ground state deformations is numerically identical to the mass-frdm. dat in the mass seg­ment of RIPL-2. TALYS uses a dedicated format for the input parameter library but numerical data are based on RIPL-2.

An interface code (OM-RETRIEVE) is provided to generate input files for SCAT2000 and ECIS96 from the OMP library. Utility codes for editing and summarizing the OMP library content are also available.

The Computer Index of Neutron DAta (CINDA) contains bibliographic references to measurements, calculations, reviews and evaluations of neutron reaction and spontaneous fission data. Proposed extensions include charged-particle and photon — induced reaction data. CINDA is also the index for EXFOR entries and the evaluated data libraries available from the Data Centres. Information contained in the CINDA file is available as annually-updated books, and by direct retrievals through the Internet.

• Work force policy — In general at the time of final plant shutdown of one unit the work force is between 200 and 500 people. If the unit is isolated on the site and if the utility has no other nuclear operating plant, the social situation may become acute, also in areas industrially developed. Therefore the worker decrease curve shall be carefully studied and their useful employment in the decommissioning process shall be planned, even considering their requalification.

• Pressures by central or local authorities — Pressures may exist from local and central authorities to clean up the site as soon as possible to solve the occupational issue mentioned above, to improve the "image" of a certain area with touristic or agricultural objectives, and so on.

Detailed planning of the decommissioning activity is also a difficult task. Very few activities are routine activities and some last for years.

Dismantling can be approached in several ways, depending on the specific circumstances. For example the process can be followed “room by room”, dismantling everything present in a specific room of the plant, possibly starting from the most contaminated components to reduce the overall doses, or from the easiest components, in order to facilitate the remaining operations. Or it may proceed, system by system, keeping operating some support systems that may be still useful in the operations. In the general case the solution is a combination of the two strategies, to be decided on a case by case basis.

It is also necessary to mention that planning is also difficult because the plant configuration is different day by day. To keep under control the status of the plant and its configuration is an heavy task, addressed generally with complex and specific planning tools.

6. Waste management

Wastes related to plant decommissioning come only from the structures of the plant which have been either irradiated or contaminated with radioactive isotopes.

Criteria for waste classification are not standardized worldwide and therefore a consensus classification is not possible. Criteria may be related to the type of isotopes, to their concentration and/or to the total amount of radioactivity in a package. However, in general, 3 categories of wastes are identified:

1. Low level radwaste — these are waste that would not be radioactive (i. e. their radioactivity will be below the clearance level[14]) in a period that can last from few days to some decades. This type of waste is not the waste that concern most in the decommissioning. In the cases in which also concentration is important, then it becomes a very important issue for decommissioning. In fact, one of the isotopes that may influence the entire decommissioning process is Cobalt-60, which has an half-life of 5,3 years. Since it is present in significant quantities in all plants and since it is dangerous to workers, because it decays with a strong gamma, it may lead to very large quantities of waste. Therefore the need for concentration of radioactivity, the appropriate treatment and the waste form is strictly related to the country regulations, definitions and disposal costs for such a type of waste.

2. Intermediate level radwaste — this is the waste which needs up to some centuries to decay below the clearance threshold level

3. High level radwaste — All other radioactive waste, that do not fit in the 2 previous categories are classified as high level waste. This category may include activated materials, components contaminated with transuranic isotopes, Carbon-14 isotopes, such as graphite blocks of Magnox reactors, etc. Vitrified residues of reprocessing certainly fall into this category.

The need for proper processing indeed exists. However, in many cases the required technologies and the goals are the same as those for operational wastes. In particular, the goals to be achieved and optimized should be minimization of the quantities and volumes at the origin, stabilization, concentration, conditioning, sorting and packaging. Among the technologies used for waste conditioning we may recall: nitrification, bitumization, polymerization, cementation, super-compaction, incineration, vitrification, etc.

In fig.2 below a flow diagram is depicted to show the process of producing, treating, characterizing and packaging of decommissioning wastes.

Figure 2 — Decommissioning waste processing

7. Decontamination technologies

The decontamination process is defined as the removal of contamination from the surfaces of installation structures and from internal and external surfaces of piping and equipment.

The major categories of techniques are washing, heating, chemical action, electrochemical action and mechanical action

The objectives of a decontamination process are:

• Reduce worker doses

• Reuse of materials and equipment

• Reduce the amount and volume of radioactive wastes to be disposed

• Remove radiological restraints in all or part of the plant

• Eliminate removable contamination and fix the other one

• Reduce the time after which a material can be freely released

Decontamination objectives can vary according to the specific strategy chosen and according to the specific phase of the decommissioning process. For example in the SAFSTOR strategy the decontamination can be reduced to eliminate the easy removable contamination and to minimize the doses to workers at the end of the safe store period.

In the definition of the most appropriate decontamination strategy considerations of cost-benefit must be applied. Decontamination itself causes doses to the operators and produces secondary wastes to be evaluated in terms of quantity and typology. This “cost” shall be compared with the corresponding savings expected as a consequence of the activity.

In the choice of a specific technology attention must be given to the specific geometry, surface characteristics and materials of the parts to be decontaminated.

Chemical techniques use diluted or concentrated solvents which come in contact with the radioactive substances to be dissolved. The dissolution may imply also the dissolution of part of the base material or simply of the radioactive deposit film on the surface. This last way is adopted when there is an interest in maintaining the integrity of the base metal such as in the case of operating plants, where the decontamination is applied only to reduce worker doses during maintenance activities. Chemical decontamination is applied by a continuous flushing in intact piping, creating a closed loop, and it is preferred for areas where access is difficult and for decontaminating the internal surfaces of piping. Chemical decontamination can be also successfully used for large areas such as floors and walls.

Mechanical decontamination (automatic or manual, locally or remotely controlled) is based on purely physical processes. It includes washing, flushing in closed loops, pipe swabbing, foaming agents and latex-peelable coatings. Most aggressive mechanical processes include wet or dry abrasive blasting, surface grinding, concrete spalling.

As we have seen in the previous section, the calculation of the spatial matrix elements might be quite involved, therefore we try to reduce the number of spatial matrix elements to be calculated as far as possible but still keeping the procedure so general that different systems can be calculated by the same program without any necessary modifications, i. e. just a new input. With
respect to example 12C we consider only a cluster decomposition into three clusters of four nucleons each, but do not take into account their identity.

The first step is to go from the general matrix element

^J’M’A ^ wj фш ) (3.19)

i<j

to the reduced one. Here фJM denotes a single term in the total wave function eq. (2.12) allowing for the various fragmentations, the possible different components of the internal wave function eq. (2.36) and the various square integrable functions ф in eq. (2.12). The wave function фJM consists of spatial and spin (-isospin) part according to

ФJM = ^(LmSoJM^Lm(space^Se(spin — isospin) (3.20)

me

where the coupling of the total orbital angular momentum L and the total spin S of the nucleons is explicitly given by the Clebsch-Gordan coefficient (LmSffjJM). Using Racah algebra the matrix elements eq. (3.19) can be expressed in terms of reduced matrix elements for fixed interacting particles i and j and a given rank of the interaction k [20]

jj*$мм'(-1)l+2s’+s Ji l SL’ ^(-1)P

L S k P

< L’ || PwO(k) || L >< S’ || Pwj(k) || S > (3.21)

where j L* J is a 6j-coefficient. Equation (3.21) has to be evaluated for all permutations by employing the symmetry of the orbital wave function. We will sum the spin-isospin matrix elements over all permutations, which yield the same orbital matrix.

In nuclear physics the spin-isospin operators wj are products of the isospin operators 1 = identity resp. ri-Tj with the spin operators 1 (norm and central potential), ai ■ Oj (central potential), (ai + Oj)q the spherical component q (spin-orbit potential), and oiq Ojq* (tensor potential). According to eq. (3.21), we have to calculate the reduced spin-isospin matrix element. Using

Wigner-Eckhart’s theorem we find [20]

< 5’S’Pwfjr(k, S’ — S)SS >

(3.22)

By using maximal projections of the spin functions the matrix element is guaranteed to be different from zero, if the triangular conditions are ful­filled. The easiest way to calculate the r. h.s. of eq. (3.22) is to decompose the coupled spin functions SS > into linear combinations of products of elementary single particle spin functions by using again Clebsch-Gordan co­efficients. Then the operators acting on the product wave functions yield (linear combinations of) product wave functions. The permutation P can be easily applied to a product and the matrix element is straightforward to evaluate. Since usually many of these matrix elements vanish, it is more eco­nomic to start from the product functions in bra and ket and determine all permutations P with non-vanishing matrix element, for details see [16], [17]. The reduced matrix element itself requires only the sum over the known Clebsch-Gordan coefficients. Restricting our considerations for the moment to the overlap matrix element, where Wj = 1, we can sum all spin-isospin matrix elements belonging to one double coset including the sign of the per­mutation. Thus eq. (3.21) reduces to

UJJ omm’ ^ l’ s’ k j

•^2 <L II PdcwOj(k) II L > CSS(dc) (3.23) where Pdc is any permutation representing the double coset dc and CSjS (dc) contains the sum over spin matrix elements. In case of an interaction Wij = 1 one has to extend the double coset decomposition and mark the interacting particles i and j. In the following we will not review the decomposition into double cosets in general, this can be found in [22], [23], but rather discuss an illustrative example, which shows all the complexity but is still transparent. Let us consider the 7Li nucleus, which can be described very well by a fragmentation into 4He and 3H. A small admixture of 6Li and neutron, however, improves the description of excited states appreciably, see [24]. To be specific, let us decompose a matrix element eq. (3.19) into double cosets where the r. h.s. is the 4He—3H fragmentation and the l. h.s. the 6Li — n configuration. This corresponds to a matrix element of the standard reaction

Li(n, t).

In general the decomposition into double cosets can be illustrated by sym­bols in matrix form, which are called dc-symbols [22]. In our example we decompose into S4 x S3 for the r. h.s. and into S4 x S2 x Si, for the l. h.s., because the 6 Li containing 6 nucleons has to be described by 2 clusters at least, the main component being 4He — d. Therefore we have a dc-symbol containing 3 rows and 2 columns:

where the decompostion into clusters reflects the symmetry of the spatial wave functions in bra and ket. In the example of a dc-symbol given above, the fact that particle numbers are conserved so that they can just be exchanged from a cluster in the bra (ket) into all clusters in the ket (bra) is taken care of by the sum of the entries in a row (column) being equal to the number of nucleons in that cluster. There is a one-to-one correspondence of dc-symbols and double cosets [22], therefore we can use the dc-symbol to construct a permutation characteristic of the double coset. For this we write the digits 1 to N row-wise into the dc-symbol, as many digits as indicated per site, and then read this scheme column-wise. Writing the digits found in this procedure below the digits 1 to N in natural order we find a permutation representing the double coset. In our example this will read

We can now interpret this result: There is at least one permutation in the double coset, maybe the one given, which maps the spin functions in the ket onto those of the bra. The entries in (3.24) indicate how many of the particles are exchanged from the cluster given on the left side into the cluster started above by the permutation. Note that exchanging particles and permuting wave functions are inverse operations to each other. To construct the dc — symbol from the permutation, we write the digits 1 to N in natural order and group them according to S4 x S2 x S1 , i. e. 1234 56 7 and then group the second line of the Pdc in (3.25) according to S4 x S3, i. e. 1257 346 and ask how many digits in the various combinations of clusters in bra and ket

agree. For example the 7 in cluster 3 of the bra agrees with the 7 in cluster 1 of the ket.

If we consider an interaction, we have to mark the interacting particles either before or after applying the permutation. We have put the permutation after the interaction, because the potential could contain an exchange operator, therefore it is more convenient to mark the permuted digit of the interacting particles with a point. For a two-body interaction we find for interacting particles 4 and 5 in our example:

7(= Pdc(4)) and 3(= Pdc(5))

(3.26)

where we have omitted the group configurations for convenience. The above example shows that also interacting particles 6(= P-(4)) and 4 would yield the same orbital matrix element. We sum the spin matrix elements belonging to these two terms. Thus we arrived at a classification scheme for matrix elements of any two body interaction in terms of 2-point dc-symbols, over which the sum in eq. (3.23) runs. The above classification scheme can be extended easily to one-body or three-body interactions.

Various equations and model parameters have been derived from studies of the systematic trends in measured yield distributions. Although these theoretical data are not sufficiently reliable for applied purposes, they have been used to obtain numerical values where no yields have been measured, or to check and adjust experimental data to the expected distribution. Wahl (1987 and 1988) has carried out a thorough evaluation of independent fission yields in order to obtain best values for a set of empirical model parameters. These model parameters have been subsequently used by fission-yield evaluators to calculate charge distributions and estimate unmeasured yields.