c Pleiades Publishing, Inc., 2006. ISSN 1063-7788, Physics of Atomic Nuclei, 2006, Vol. 69, No. 7, pp. 1168–1172.
PLENARY TALKS
Quantum and Thermodynamical Aspects of Binary and Ternary Fission* V. E. Bunakov1)** and S. G. Kadmensky2) Received October 25, 2005
Abstract—A short review is given of the recent developments in the quantum theory of low-energy fission. It is emphasized that angular anisotropy and various angular correlations of fission products are possible only if the fissioning nucleus remains nonthermalized during all the stages of fission from the saddle point to scission. PACS numbers : 25.85.-w, 25.85.Ec DOI: 10.1134/S106377880607012X
1. INTRODUCTION Although the neutron-induced fission reaction was discovered at the dawn of nuclear physics, the essentially classical character of its theoretical description given by N. Bohr and J. Wheeler did not progress much in spite of the 60 years of intense investigations. One of the most typical quantum features, the interference of various resonance amplitudes, was observed in the total fission cross sections long ago [1, 2]. The most puzzling feature of this phenomenon was that the interference pattern “survived” after summing over billions of partial fission channels (with presumably random amplitude signs) performed automatically in experimental measurements. This fact was qualitatively explained by A. Bohr’s hypothesis of the transition states at the saddle point of the fissioning nuclei [3]. However, this hypothesis merely postulated the survival of interference between the resonances which fission via the same transition state rather than explaining the mechanism of this survival in the framework of quantum reaction theory [4]. Twenty years later, a similar problem was faced when P -violation effects in fission induced by polarized neutrons were observed [5]. This discovery stimulated the theoretical studies of possible P -violation effects in nuclear reactions induced by polarized neutrons. These studies [6–8] demonstrated that the P -violating effect observed in fission is just a particular case of P ∗
The text was submitted by the authors in English. Petersburg Nuclear Physics Institute, Russian Academy of Sciences, Gatchina, 188350 Russia. 2) Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006 Russia. ** E-mail:
[email protected] 1)
violation in various neutron-induced reactions. It was also demonstrated [7, 8] that, in the particular case of fission, P -violating effects are of most interest in understanding fission dynamics rather than in extracting information on the weak interaction in nuclei. The main peculiarity of the observed P -odd correlation in fission was again the above-mentioned “survival” of the P -odd interference effects (caused by the weak-interaction forces) after the experimental summation over billions of partial channels. The same “survival” of the interference effects was also observed for the P -even “right–left” and “forward– backward” asymmetries in neutron-induced fission (caused by the interference between resonances with opposite parities). Therefore, although the first theoretical explanations of the observed P -odd and P -even asymmetries already treated them as the results of quantum interference phenomena, some assumptions and educated guesses had to be made because of the lack of a quantum fission theory. A new approach to this problem of the “survival” of interference effects in fission was suggested in [9] which is based on the approximate validity of the total helicity quantum number in fission reactions and on its identity with the K quantum number of A. Bohr’s transition states. However, the main point which hindered the creation of the quantum theory of neutron-induced fission was our poor understanding of the physics of transition states. The basis of a purely quantum theory of lowenergy and spontaneous fission was developed in [10– 16]. Since it operates with all the objects of the quantum reaction theory (wave functions, partial waves, phase shifts), it allows one to properly describe the interference phenomena. It also analyzes the approximations which allow one to understand the unique properties of A. Bohr’s transition states and
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to include them (with certain corrections) in the framework of the reaction theory. In the present paper, we shall start with pointing out the drawbacks of the classical approach to fission reactions. Then we shall demonstrate how these drawbacks can be eliminated in the quantum approach and give a brief review of the main results which were already obtained in the quantum theory. We shall also give a general analysis of the sources of various anisotropies in the angular distributions of fission products and demonstrate the new implications of this analysis for the physical mechanisms of binary and ternary fission. 2. DRAWBACKS OF THE CLASSICAL APPROACH TO FISSION The standard theory of neutron-induced fission is essentially based on classical mechanics. This is especially evident from A. Bohr’s expression [17] for the angular distribution of fission fragments in (n, f ) reactions: dσ J 2 J 2 ∼ aM bK {|DM K (ω)| + |DM −K (ω)| }. dΩ M,K
(1) Here, J, M , and K stand for the spin of the fissioning nucleus and its projections onto the z axis of the laboratory system and onto the symmetry axis of the nucleus. Expression (1) implies that fission fragments fly along the symmetry axis z of the fissioning nucleus, that is, along a straight line in the internal coordinate system. D functions are used simply to make a transformation into the laboratory system. As we shall show below, straight line motion is a classical idealization. Moreover, since the centers of fragments move apart along this line, the angular momentum of their relative motion should be L = pb → 0, because the impact parameter b → 0 (p is the linear momentum of the fragments’ relative motion). In contrast to this, in quantum mechanics, the angular distribution of the reaction products is a result of the partial wave interference. For example, in the simplest case of elastic scattering it looks like lmax √ √ dσ ∼ 2l + 1 2l + 1 Yl0 (θ) dΩ
(2)
l,l
× Y (θ) sin δl sin δl cos(δl − δl ). l 0
Thus, we see that, for the quantum-mechanical description of the angular distribution, one needs to know the number of partial waves lmax and all the partial phase shifts δl , which are absent in A. Bohr’s PHYSICS OF ATOMIC NUCLEI Vol. 69
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theory. It is also well known that the particular case of l = 0 gives the isotropic angular distribution. As was mentioned above, the necessity of quantum theory became especially evident after the discovery [5] of P violation in fission induced by polarized thermal neutrons. The first attempts to introduce partial waves with l = 0 and corresponding partial phase shifts were done in [7, 8]. The theoretical attempt [9] to construct quantum theory in the helicity representation still used A. Bohr’s classical assumption about the fragments’ motion along a straight line. In the next section, we shall briefly mention the main results obtained in the framework of a purely quantum theory of low-energy fission [10–16] and analyze in more detail the quantum corrections to A. Bohr’s formula and the general sources of anisotropies and of various asymmetries in the fragments’ angular distributions. 3. QUANTUM THEORY 1. One of the main results of the quantum approach was the correction of A. Bohr’s expression (1) for the fragments’ angular distribution. Straight line motion in quantum mechanics might be expressed as a superposition of an infinite number of partial waves: δ(cos θ ± 1) =
∞ √
2l + 1 Yl0 (θ )Pl (±1).
(3)
l=0
θ
is the angle between the radius vector R Here, of the fragments’ relative motion and the symmetry axis z in the internal coordinate system; Pl (±1) is a Legendre polynomial. Of course, an infinite number of partial waves is a classical idealization. In reality, one should substitute instead a finite value lmax . In this case [12], the motion is limited by a cone around the z axis with the opening angle δθ which obeys the uncertainty relation δθ ∼ 1/lmax . Thus, a finite value of lmax might spoil A. Bohr’s formula (for instance, lmax = 0 would give an isotropic distribution). Therefore, the question is how large is the angular momentum of the fragments’ relative motion lmax in fission? It turned out that there are several independent ways to answer this question: 1.1. It is known experimentally that fission fragments have large spins J1 ≈ J2 ≈ 12, which are perpendicular to R. By neglecting the compoundnucleus spin (J ≈ 0), one can use the conservation law J1 + J2 + L ≈ 0.
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This distribution is isotropic if
This leads to the conclusion that J1 is parallel to J2 , while the angular momentum L ≈ lmax ≈ 25 and is antiparallel to the fragments’ spins. 1.2. In [13], the quantum description of the “pumping” mechanism for J1 , J2 , and L was given, caused by the interaction of nonspherical fragments at the scission point. Assuming realistic parameters, this mechanism gives the value L ≈ lmax ≈ 30.
3.1. All the values of aM are equal to 1/(2J + 1).
1.3. The analysis of the fragments’ angular distribution [16] in photofission showed a noticeable deviation of the experimental results from those predicted by A. Bohr’s formula. This deviation allowed one to extract the value lmax ≈ 30. 1.4. The theoretical analysis of the fragments’ angular distribution [15] for polarized nucleus spontaneous fission demonstrated that it is also sensitive to the values of lmax . This analysis was performed for the planned experiments with spontaneous fission of 255 Es and 257 Fm nuclei polarized at superlow temperature. Thus, we see that A. Bohr’s expression “survives” in the quantum approach but needs quantum corrections, which might be noticeable in some experiments and allow one to estimate the value lmax of the fragments’ relative angular momentum. 2. The correct definition of the phase shifts allowed one to perform consistent studies of the P -odd [12], P -even [14], and T -odd [18, 19] asymmetries observed in binary and ternary fission. The following essentially new results were obtained: 2.1. It was shown [14] that nonsequential (simultaneous) neck breaking might take place in ternary fission, contrary to the previous conclusion of [20], based on the existing experimental data on the absence of right–left P -even asymmetry in ternary alpha-particle emission. The theoretical estimate of this asymmetry for the nonsequential process demands [14] an increase in the present experimental accuracy by an order of magnitude. 2.2. It was also shown [18, 19] that the observed effect of T -odd asymmetry in ternary fission can be explained by the Coriolis interaction of the ternaryparticle orbital momentum with the spin of the rotating fissioning nucleus. 3. Consider now the general sources of anisotropy in the fragments’ angular distribution and of various asymmetries (P -odd, P -even, and T -odd). In order to do this, we turn back to A. Bohr’s formula (1): dσ J 2 J 2 ∼ aM bK {|DM K (ω)| + |DM −K (ω)| } β=θ . dΩ M,K
Here, β is the Euler angle and θ is the angle with respect to the incident beam direction.
Therefore, for asymmetry, we need to fix the orientation of J in the laboratory system. Such an orientation might be performed in several ways. One might polarize the fissioning nucleus by the external magnetic field (see the case of spontaneous fission above). One might also use the polarized neutron beam (as is done in experiments on P -odd, P -even, and T -odd correlation). In the case of photofission, one selects M = ±1 since the photon helicity is ±1. 3.2. All the values of bK are equal to 1/(2J + 1). Therefore, for anisotropy, we need to fix the orientation of J with respect to the symmetry axis of the fissioning nucleus (in the internal coordinate system). This means that one needs to select one (or a few) values of the quantum number K. In A. Bohr’s theory, this is done for low-energy fission by the transition states at the saddle points of the fission barrier. Since the system at the saddle point is cold, the barrierpenetration factors TK for different values of K are different. This serves as a filter of only a few K values providing the orientation of J with respect to the symmetry axis and therefore causing the anisotropy effects. It is important to note that A. Bohr’s theory implies further conservation of the thus “filtered” K as integrals of motion during all stages of fission up to the moment (shortly after scission) when the direction of relative motion for fragments is already established. 3.3. However, there are sources of K violation. The main one is Coriolis interaction HCor of the rotating nucleus spin J with the internal degrees of freedom. ˆ Cor on the eigenfunction The action of the operator H J J ΨK transforms it into ΨK±1 , thus mixing the different K values and causing K violation. The statistical analysis of neutron resonance distributions at excitation energies E ∗ ≈ 6 MeV shows that K is not an integral of motion. The explanation of this fact was given in [21, 22]. It was shown that, for the K-admixture coefficients c of compound resonances, there is the same effect of dynamical enhancement as in the case of P violation (see, e.g., [23]): c=
ˆ Cor |ΨJ √ ΨJK±1 |H K ≈ c0 N . EK±1 − EK
(4)
Here, N ≈ 106 is the number of simple shell-model components contributing to the wave functions of compound resonance, while c0 ≈ 0.03 is the admixture coefficient for the ground states. Thus, for completely thermalized (i.e., with very large N and small level spacing of d ≈ 1−10 eV), compound resonances
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c 1 and the wave functions are statistical mixtures of all possible K values: J J CK ΨnK . ΨJn = K J coefficients are random numbers which are Here, CK distributed around zero according to the Gaussian law with dispersion: 1 J 2 |CK . | = 2J + 1
4. IMPLICATIONS FOR FISSION MECHANISMS 1. Thus, we have established two very important points: A. All kinds of angular anisotropies of fragments result from the fact that K values “filtered” by the transition states remain integrals of motion up to the moment that the direction of the fragments’ motion is fixed. B. Coriolis interaction will not destroy K only if the system is cold and its average level spacing d 10 keV. Therefore, at the stages of evolution from the saddle point to scission, the fissioning nucleus should be cold. Otherwise, the strong Coriolis K mixing would cause a complete loss of memory about the transition states and the disappearance of any anisotropy in the fragments’ angular distributions, including the experimentally observed P -even, P -odd, and T -odd correlations. This seems to be a challenge to the standard statistical approach developed in [24], which postulates the nonadiabatic descent from barrier to scission, friction, and thermodynamic equilibrium at the scission point with temperatures ≈1 MeV for binary fission and ≈2.5 MeV for the neck in ternary fission. 2. Our conclusions and suggestions: (i) One can use a statistical approach with the concept of temperature only after scission. Of course, after scission, the thermalization of the internal excitation and of deformation energies happens during the time t ∼ 10−21 s and the heated fragments (with different temperatures) evaporate neutrons and gammas. (ii) Excitations caused by the nonadiabatic descent to the scission point have no time for temperature equilibration and exist in the form of fewquasiparticle states with low level density (“doorway states”). (iii) For ternary fission, we suggest [19] the nonevaporational mechanism similar to the “shakeoff” mechanism of atomic electrons in α and β decays. The nonadiabatic changes of nuclear potential, which PHYSICS OF ATOMIC NUCLEI Vol. 69
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are strongest near the scission point, excite the ternary particles up to the states above their binding energies and Coulomb barriers. The small number of nucleons (A ≈ 30) in the neck allows one to treat it as a light nucleus, where the cluster formation mechanism is enhanced [25, 26]. An analogous mechanism was suggested in [27–29]. Our latest studies [19] of the T -odd effect in ternary fission show that this mechanism allows one to obtain the experimental value of the effect. (iv) The coldness of the fissile nucleus provides for the existence of superfluid pairing correlations in it. This leads to enhancements in the formation of fragments and ternary particles similar to the superfluid enhancements in α decay. These enhancements might explain the even–odd effects in the yields observed experimentally for fission fragments [30] and for ternary particles [31]. ACKNOWLEDGMENTS We appreciate the support of INTAS (grant no. 0351-6417) and the Russian Foundation for Basic Research (project no. 03-02-17469). REFERENCES 1. V. L. Saylor, in Proceedings of the International Conference on Peaceful Uses of Atomic Energy, Geneva, 1955 (United Nations, New York, 1955), Vol. 4, p. 199. 2. J. E. Lynn and N. J. Pattenden, in Proceedings of the International Conference on Peaceful Uses of Atomic Energy, Geneva, 1955 (United Nations, New York, 1955), Vol. 4, p. 210. 3. A. Bohr, in Proceedings of the International Conference on Peaceful Uses of Atomic Energy, Geneva, 1955 (United Nations, New York, 1955), Vol. 2, p. 220. 4. J. E. Lynn, The Theory of Neutron Resonance Reactions (Clarendon, Oxford, 1968). ´ 5. G. V. Danilyan et al., Pis’ma Zh. Eksp. Teor. Fiz. 26, 298 (1977). 6. O. P. Sushkov and V. V. Flambaum, Usp. Fiz. Nauk 136, 3 (1982) [Sov. Phys. Usp. 25, 1 (1982)]. 7. V. E. Bunakov and V. P. Gudkov, Nucl. Phys. A 401, 93 (1983). 8. V. E. Bunakov and V. P. Gudkov, Z. Phys. A 321, 277 (1985). 9. A. L. Barabanov and V. I. Furman, Z. Phys. A 357, 411 (1997). 10. S. G. Kadmensky, Yad. Fiz. 65, 1424 (2002) [Phys. At. Nucl. 65, 1390 (2002)]. 11. S. G. Kadmensky, Yad. Fiz. 65, 1833 (2002) [Phys. At. Nucl. 65, 1785 (2002)]. 12. S. G. Kadmensky, Yad. Fiz. 66, 1739 (2003) [Phys. At. Nucl. 66, 1691 (2003)]. 13. S. G. Kadmensky, Yad. Fiz. 67, 167 (2004) [Phys. At. Nucl. 67, 170 (2004)].
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