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By making use of the method for treating the pairing force, which conserves the number of particles, we have analysed the influences of pairing force on the various properties of actinide nuclei, including the intrinsic spectra of odd-A nuclei and even-even nuclei, β-decays and α-decays, etc. The general formula expressing the influences of pairing force on the various types of α-decay, are given.1. First, Nilsson's parameters, which, after the pairing force being taken into account, can account for the observed intrinsic spectra of odd-A nuclei, are determined. It is found that they may be chosen as follows: K～0.050, η～4.4-5.4 (δ～kη)～0.22-0.27) μ～ 0.70(N = 5), μ～ 0.62-0.66(N = 6), for protons, μ～0.43-0.45(N = 6), μ～0.41-0.43(N = 7), for neutrons. 2. Subsequently, with such realistic single-particle spectra, the intrinsic spectra and β-decays of neighbouring even-even nuclei (Th 228 ,Th 230 ,U 232 ,U 234 , Cm 244 ) are analysed in detail. Some observed levels (E≥1 MeV.) may be explained as the pair-broken states or the pair-excitation states. (See Table 14, and Figs. 15, 16.) According to calculation, there exist still many levels in the energy region, E～l-2 MeV, but they have not been observed yet. In the β-decay experiments, the majority of these levels can not be detected easily, but some of them, easily. The systematically occurring vibrational bands in the spectra of even-even nuclei could not be considered as the pair-broken states or the pair-excitation states. They areγ-vibrational states, K π =2 + , E～0.8-1.0 MeV;β-vibrational states, K π =0 + , E～0.7-0.9 MeV;octupole b-vibrational states, K π =0 - , E～0.5-0.7 MeV. 3. The influences of pairing force on the α-decay probabilities are treated both with the BCS method and with the present method. The results, especially for the favoured a-decay, are quite different in the two methods. For example, when about 10 levels are involved, the retardation factors for the favoured α-decay of even-even nuclei are:R = R Z R N ≈200 (BCS wave function), R = R Z R N ≈ 70 (exact solution). Compared with the exact solution, the occupation probability of the v-th levels, v v 2 , in the BCS method converges too slowly. In the BCS wave function, the portion, in which the number of particles is equal to the average number of particles, is about one third. After the influence of pairing force is taken into account, the nuclear radius constant derived from the a-decay is about r 0 ～1.35fm. The favoured a-decay of odd-A nuclei and the unfavoured a-decay are also discussed. The so-called F-factors are calculated and compared with experiments.

In this paper the various properties of the rare-earth even-even nuclei (intrinsic excitation spectra, β-decay, and even-odd mass differences, etc.) are analysed with the new method presented in reference [10]. For this purpose, the single-particle levels have to be determined. It is found that on the basis of Nilsson's level scheme, with suitably chosen parameters μ and k, the spins and parities of the low-lying intrinsic spectra and the level spacing of odd-A nuclei may be explained, after taking account of the influence of pairing force. The parameters chosen are as follow: k～0.067-0.072, η～4-4.6 μ～0.59(N = 4), μ～0.45(N = 5) for proton μ～0.40-0.45(N = 5), μ～0.33(N = 6) for neutron. The variation of the energy eigenvalue induced by the change of parameter μ is estimated with the perturbation method, and the results are close to the exact eigenvalues. From the analyses of Er 166 , Er 168 , Yb 170 , Yb 172 , Yb 174 , Hf 178 , W 182 , Dy 160 and Dy 162 the following several conclusions may be drawn: 1. The majority of the low-lying intrinsic excited states of even-even nuclei may be considered as the pair-broken excited states. The calculated first pair-broken states, in spite of the large variation in their spins and parities, are observed in experiments systematically (Fig. 30). Their excitation energies range from 1.2 MeV to 1.6 MeV. The calculated variation of these energies with different nuclei agrees with experiments ap- proximately. These states correspond to the two quasi-particle excited states pointed out by Gallagher and Soloviev.2. Near the first pair-broken excited state, appears the K π =0+ pair excitation state. In Er 166 , Hf 178 and W 182 these pair excitation states have been observed (Fig. 32). In other nuclei, these pair excitation states have not been observed yet, because the β-decays from the neighbouring odd-odd nuclei to these states are highly forbidden or strictly forbidden. 3. The systematically occured K π = 2+ states (Fig. 31) in even-even nuclei (E～0.8-1.2 MeV) can not be considered as pair-broken excited states. Theoretically there does not exist in these nuclei (except Yb 172 ) the K π = 2+ intrinsic state below 2 MeV. 4. The splittings in energy △E between the pair-broken doublets, K = |Ω 1 ± Ω 2 |, vary violently with different nuclei and with different pair-broken states. △E～(40-700) KeV. 5. The variation of even-odd mass differences with different nuclei is analysed also and it is found that the even-odd mass difference depends sensitively on the single-particle level scheme. The fact that the calculated and the observed variation of even-odd mass differences agree with each other (Fig. 29) indicates that the single-particle level scheme determined in this paper is approximately correct. The average strength constant G deduced from the even-odd mass differences is close to that deduced from the energy spectra. In the calculation the cut-off energy C is chosen to be ≈ 4 MeV. (C》G). With increasing C the influences of the pairing force on the low-lying spectra may be compensated by decreasing the average strength constant G.

A new approximate method for treating nn and pp pairing force in nuclei is presented. In this method the difficulties connected with the application of BCS-Bogolubov method to nuclei such as the non-conservation of the number of particles, the apperance of spurious states, and non-orthogonality between wave functions of different excited states are avoided. But the method can be conviently adopted only when the number of particles n and the number of levels Ω considered are small (n ≤10, Ω≤10 ). Also the approximation of this method is good onlv under such conditions. In this method the many-body nature of the excitation is displayed explicitly without introducing the conception of quasi-particles. The excitation of the system can be divided into two classes. The first class of excitation is a kind of single particle excitation embedded in a many-body system under the action of the pairing force. This would correspond to the excitation of a single quasi-particle a v + a (v 0 ) or to two quasi-particle excitation of the type α v + β v′ + ,α v + α v′ + (v′≠v) etc. in the BCS-Bogolubov method. The second class of excitation is a pair excitation in which the distribution of the conjugated pairs of particles among the levels is altered. This type of excitation would correspond to superposition of quasi-particle excitations of the type α v + β v + ,α v + β v + α μ + β μ + , Also examined is the effect of pairing force on the β- and γ-transition probability in the new formalism. In general, β-transition is retarded by the pairing force while in γ-transition the effect of pairing force varies according to the nature of the relative excitation between the initial and final states. When the relative excitation is of the first class, the transition is retarded to various extent, depending on the nature of the γ-transition. If the relative excitation is of the second class, the transition is forbidden when the pairing force is not in action, and is allowed only for electric transition arising from the charge-current distribution when the pairing force is present. Finally this method is applied to investigate the effect of pairing force on the intrinsic spectra and β-transition in 72 Hf 178 as an example.