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Z. Phys. A 351, 397-404 (1995) ZEITSCHRIFT FOR PHYSlKA 9 Springer-Verlag 1995 Particle emission from a hot, deformed, and rotating nucleus K. Dietrich ~, K. Pomorski 2, J. Richert 2'* Physik Department, TU Munich, James Franck Strasse, D-85748 Garching, Germany 2Department of Theoretical Physics, M. Curie Sktodowska University, ul. Radziszewskiego 10, 20-031 Lublin, Poland Received: 11 May 1994/Revised version: 14 December 1994 Abstract. The emission of nucleons from a hot, deformed and rotating nucleus is treated within the Thomas-Fermi approximation. We study in particular the dependence of the transmission coefficient on the deformation and the rotational frequency of the emitting nucleus. A tractable form of the transmission coefficient is given. N and Z is given by [2] c2 (E*, I, N, z ) = 2Sv+ 1 2~zhp(E*, li f dew~(e,l~;r)pR(E~,IR) " (1) IR--lI--lt~ ] a~-- Ae PACS: 25.85.-w; 24.75.+i 1. Introduction Whenever the excitation energy of a compound nucleus exceeds considerably the binding energy of a nucleon or a light composite particle (especially the c~-particle), the emission of this particle competes favorably with the Yemission. Therefore, a detailed understanding of the particle emission is of great importance in the early stage of the decay of highly excited nuclei. Such highly excited nuclei are generally deformed and in rapid collective rotation. Consequently, the dependence of the emission probability of neutrons, protons, and ~-particles on the shape and the angular momentum of the mother nucleus is of great physical interest. A specific example is the competition between particle emission and nuclear fission in the decay of highly excited medium heavy and heavy nuclei, where the neutrons, protons, and ~-particles emitted prior to fission carry interesting information on the fission process [1]. The famous formula by Weisskopf describing the decay widths FIB of a nucleus of excitation energy E*, total angular momentum I, and neutron and proton numbers This work is partly supported by the Polish State Committee for Scientific Research and by the European Economic Community (EEC) under contract No. ERBCIPACT93 1576. K.D. acknowledges support by the BMFT *Permanent address: Physique Th6orique, CRN, BP20, F-67037 Strasbourg C~dex 2, France The quantities e~,tp, S, denote the energy, the orbitM angular momentum, and the intrinsic spin of the emitted particle of type v 1. The quantities p and PR represent the level densities of the emitting and of the residual nucleus. The arguments E*, [R are the excitation energy and the angular momentum of the residual nucleus. Finally, the transmission coefficient ~'(e, l~; r) represents, for each type v of particles, the fraction of the flux penetrating the barrier with an angular momentum l~ and a final energy e. The argument r denotes all further parameters upon which the transmission coefficient depends. For the emission from a deformed, rapidly rotating nucleus, the transmission coefficient depends on the deformation and the rotational frequency, i.e. the argument r represents a deformation parameter and a rotational frequency. It is the purpose of this paper to investigate this dependence of the emission width within the Thomas-Fermi approximation. The transmission coefficients are either determined empirically from elastic scattering on spherical nuclei [3] or calculated on the basis of various assumptions, for instance the WKB expression for the penetrability of a barrier [4]. If the mother and daughter nucleus are deformed, not only the level densities p(E*, 1) and pe(E~, IR) depend on the deformation, but also the transmission coefficient #"(e,l~; r). The dependence of the level densities on the deformation of the nucleus is usually taken into account in an approximate way (see for instance ref. [5]). The dependence of the emission width on the deformation of the nuclear surface was considered in different 1The subscripts ~ and /3 are introduced, because in practice we calculate the emission probability for a finite set of discrete values for and I. ~ and fi denote a "cell" in a two-dimensional grid for e and I 398 approximations by various authors [-6-10]. In none of these publications the distribution of the neutrons and protons prior to emission was described by a Fermi-Dirac distribution as we shall do in the following chapter, but rather by classical approximations. A detailed discussion on the relation between our theory and the one by Aleshin will be given in Sect. 5. 2. Emission process in the Thomas-Fermi approximation (TFA) In the TFA the nucleus is described as a gas of free neutrons and protons which are confined in the finite nuclear volume and satisfy the Fermi-Dirac statistics. The volume f2 with the surface Z may have any shape. In addition we assume that the deformed volume is in rigid rotation. For simplification we restrict ourselves to an axially symmetric nucleus which rotates perpendicular to its symmetry axis with a rotational frequency co. We denote coordinates referring to the body-fixed reference frame K ' by a prime' and we orient the 3'-axis in the direction of the symmetry axis. The nucleus is assumed to rotate around the Y-axis. The excited nucleus is described by a temperature T. In the TFA the distribution of the neutrons and protons in the phase space is described by single-particle Wigner functions of the following form 2 00(~') f,(2',~'; T ) = ~5. 1 + exp(-~(~-~ - Vo - coll- m)) 2 (2) In fact, strictly speaking, the "Thomas-Fermi approximation" doesn't imply this self-consistent procedure. Replacing the self-consistent potential ((2)) by a square well of given form means that we replace the Thomas-Fermi model by the simple Fermi-gas model. Since we want to describe the emission from deformed nuclei, a self-consistent treatment would have to include shell corrections, i.e. we would have to go beyond the TFM. Therefore, we choose to stay with the simpler Fermi gas picture to start with. The chemical potentials #v for neutrons and protons are determined from the conservation of the average neutron number N and proton number Z jf~(2', fi') d3x ' d3p ' = N, (6) ~fnO~:, ~') dax ' d3p ' = Z. (7) The quantity . . .3. -- x 3 p 2 l'1 := Xzp represents the orbital angular momentum of the nucleon along the axis of rotation. The rotational frequency co is related to the total angular momentum Ih by the condition [h = ~ d 3 x ' 5 (3) d3p'll. (f,(2', ~') + fp(2', ~')) E*= f d3x'~ d3p '[(fi'2-L\2mVo)(L(2',P',T)-f,,(Y',F';0)) + \ ~ m - Vo + Vcb (fp(21,f; T ) - f p ( 2 ' , f i ' ; 0 ) ) where the step function 0o(2') is equal to 1 inside the nuclear volume 0 and 0 outside of it 0o(2') {10 f~ 2 ' ~ O = otherwise" (4) Henceforth, we leave away the argument T except in Eq. (10) where it is important. The parameter V0 ( ~ 50 MeV) represents the depth of the nuclear well. The term - V o could be absorbed in the chemical potentials g~,/~p. Since we shall discuss emission processes, it is more convenient to incorporate - V o explicitly. The term Vcb(2') represents the average Coulomb potential felt by a proton at point Y' (co = elementary charge) Vcb(-~') = y dBp ' y d3 . . . . . eg Y f p ( Y , P ) i.~, ~ Y'I" (5) We shall see that this term influences the emission probability for protons. Of course, instead of replacing the nuclear potential by a finite square well, we could also determine the average potential V, acting on the neutron selfconsistently on the basis of some effective nucleonnucleon interaction V,(2') = ~ d~p'~ d3x'v(2', y ') [ f , ( y ' , ~') +fp(y',,F')] (9) implying that the average contribution from the intrinsic spin is negligible. Finally, the temperature T is determined by the average excitation energy E* of the total nucleus by the relation 0o(2') 1 + e x p ( ~ ( ~ -- Vo + Vcb(x ) --col~ -- #~,)) (8) (10) We notice that the integration over the space coordinates becomes trivial whenever we go to the limit co = 0 and neglect Vcb(2'). In this case, there is no influence of the deformation left given the fact that the volume f2 of the nucleus should not depend on its shape. In this limit the relation (10) takes approximately the form E* = (a,, + ap) T 2, (11) where a, and ap are the "level density parameters" for the neutrons and protons (see f.i. ref. I-5]). Let us emphasize at this point that, apart from fulfilling the requirements of the Pauli principle, the TFA treats the dynamics classically. The true quantum-mechanical Wigner function is not a probability and is not positive-definite as a consequence of the fact that momentum and position cannot simultaneously have sharp values. If the Wigner functions occurs in integrals over space and (or) momentum coordinates, its deviation from the classical distribution function often turns out to be less relevant than if we were to consider it locally in phase space. It is with this optimistic expectation in mind that we are now going to describe the emission of 399 particles within the TFA. As we shall see, it has the great advantage that the dependence of the transmission coefficient on the deformation and the rotation of the nucleus can be described in a simple and transparent way. For simplicity, we only consider neutron emission in what follows. The treatment of the proton emission is completely analogous and slightly complicated by the presence of the Coulomb field. In the spirit of the simple Fermi gas model the nuclear potential V,~d(Y') is assumed to have a constant depth Vo in the nuclear interior and to be zero outside The body fixed frame K' rotates with frequency co around the (common) 1-axis of the laboratory frame K (unit vectors 0'i). The intuitively simple classical relation (14) between the normal velocities of the incident and emitted particle can be derived from the energy conservation during emission htl _ ~t2 __ - - 2m __ V 0 -- (o(x;2p' 3 - X'o3Pt2) - Vnucl(~') = -- VoOo(~',) as we anticipated already in the form (2), (3) of the distribution functions. A neutron which hits the surface Z of the nucleus at a point 2; with a normal velocity gi(Y;) is classically emitted if the kinetic energy of its motion perpendicular to the surface exceeds the well depth V0 m > Vo 03) and otherwise elastically reflected. We denote velocity components of the emitted particle by a tilde (~). After the emission the velocity perpendicular to the surface v-+i(2;) is reduced due to the loss of kinetic energy v• tXo) = 5 v• txo) - Vo, (14) whereas the velocity parallel to the surface is unchanged. If ~'(Y;) is a unit surface vector at the surface point 2; pointing outward, the velocity components of the neutron before emission are given by < (~;):= ~'-~'(~;), 1)j_(.XT0) .= := .. v• n (Xo), . . . . . . - (15) (16) (17) and after emission by ff'(Y~a) = v, (Xo) g'(N;) + vjl(Xo), (18) g,i(Y;) = glT(~;). (19) The velocity ~' = ~' and the canonical momentum fi! are related by the equation --~/ --+! ~' = m~' + moo x x (20) and correspondingly for the emitted neutron ~' = mg--'+ mc~' x Y'. ~ (12) (21) ~ ' --- cog = toe'1. (22) Note that g' is the velocity relative to the body-fixed reference frame K ' spanned by the unit vectors e-~[ 3 -g'= ~ 2jell i=1 and equally for v'. (23) t ~t = 2m - (o(x0~p3 - ! ~i (24) x o 3 P 2 ). Assuming that the tangential momentum P~I!is unchanged in analogy to (19): = (25) The normal and tangential momentum vectors P~,Pll, • are defined in complete analogy to the Eqs. (i5)-(18). We define the local classical transmission factor -(2 = 00 ) (Xo) - Vo , (26) where 0o is the Heaviside function {~ 00(~) = for ~ > 0 for ~ < 0' (27) The total number n of neutrons emitted per time unit is then given by l 3 l ~i ,l ~t .cl l -+t n = ~z da 5 d p fn(X0, fi') • (xo) . wo [o l (xo) 3, (28) where do-' is the infinitesimal surface element, and where the canonical momentum fi' and the velocity g' are related by (20). Since the integration variable is fi', one has to express the normal velocity vl in (28) in terms of the normal momentum. The reason why we introduced the velocity g' in addition to the canonical momentum ~" is that the classical emission probability (26) can be formulated more simply in terms of the normal velocity. We now evaluate the probability per unit time that a neutron is emitted with given final energy g = i=1 Here, the angular velocity vector o3' is given by ,2 2m' where Pl, P2, PB are the components of the final neutron momentum in the laboratory frame K. Once the neutron is emitted and thus beyond the range of the nuclear potential, its momentum components in the laboratory frame are constants of motion. Thus the neutron assumes the final values of its momentum immediately after emission contrary to the proton which is still subject to the long range Coulomb field. Since the transformation between the space-fixed frame K and the rotating frame K' does not change the 400 absolute value of a vector, the total energy of the emitted neutron is given by &= -- = i=12m ~=~2m' (29) where the momentum components/~[ in K ' can be taken at any time after the emission. Choosing the time t = 0 just after emission, the momentum components/~[(0) of the neutron just after emission are related to its components Pl just before by the solutions (14)-(22). dn The number of neutrons G Ae emitted per time unit with an energy in the interval 8= - ~ < e < e~ + ~ is defined by the expression dY/ _.= 5 d a ' 5 d s p f., ( .x o. , p. . ). v .l ( X. o. ) w o, D• , ~, d&" z "618~'--~//~/(0)21". 2m ] (30) Using the energy conservation at the surface point Y; (Eq. (24)) we may rewrite the argument of the a-function in the form e= - ~ 1)[(0),2 ~ p[2 - - &+ 17o + (ll - ~1) co, i=* 2m i = 1 2ram (31) barrier region as compared to the neutron, and thus also the value of the Wigner function. The form of the angular momentum constraint in (33) arises from the fact that the absolute value le of the angular momentum of the emitted particle does not depend on the choice of the coordinate system. As the chemical potentials for neutrons and protons are roughly equal in not too exotic nuclei, the Wigner functionfv of a proton is smaller than the Wigner function f, of a neutron at the same given point of phase space: .... p ) L(~o, (34) - f , ( x.o. ,. . p ) < O. The difference (34) of the Wigner functions at the nuclear surface represents the main difference between the emission probabilities for neutrons and protons. In general, this reduces the emission probability of protons compared to the one for neutrons. The reduction is largest around the waist of a spheroidal nucleus and smallest in the vicinity of the poles. We denote the total number of emitted protons by ~t, the number per time unit in the energy interval 4~ drt (e~) . & - T < 8 < & + ~ by ~ a8 and the number per time unit emitted into this energy interval with an angular momentum 1~ by d2Tg(&'la) AsAI. de~dl~ with (11 - 8) o = ~&(p; - ~;) - 2&(pl - Pl). (32) For small values of co, the last term in (31) can be neglected which simplifies the evaluation of the integral (30) considerably. The number Then the quantities ~z,a7s d~t and ~d2~ are given by 3 ! --+l rc = [. d a ' 5 d p L ( x o , --+t el r ~') v; (Xo)Wo [v• ~t (35) t- --=! dzc de~ da' , ~, p- , )vl(Xo)~ofV~(xo)] , ~, ct , -+, ~d 3p'f,,(Xo, d2n - - de, dl;; AsAI of neutrons emitted per time unit with an energy and 9 " * A~ angular momentum comprised between the hmlts (~= - T, zi,s zJl Al 9 . & + T) and (le - g, l~ + T) is obtained from the expression d 2rc de~dl~ (36) [ t _ ~ d a , I -u3 p:~,tXo,~')v• . . . . . . . . d2/,/ d&dl := ~da' ardS"'~c:2 ...... wCot Fv,(Xo)] ' ~' u:,,t o,p)vl(Xo) wSt [v•, ~, Z 9a 8~ - v ;3;2(-~176 a[;~ - IT( ootl]. ~ 2m J (37) ,5 [ 9a 8 ~ - ~ - m j. E;e-12gx~'(0)l], (33) where one ought to choose l: = 0, 1,2, ... ,h and Al = lb. The form of the angular momentum constraint in (33) arises from the fact that the absolute value of the angular momentum l~ of the neutron remains constant after emission9 The classical treatment of proton emission proceeds analogously. It is however slightly more complicated, because the protons continue to feel the long-range Coulomb field of the deformed nucleus after emission. Thus the momentum components p~ in frame K continue to be functions of time after emission. The dominant effect of the Coulomb field is to change the potential in the The formulae are analogous to the formulae (28), (30) and (33) for neutrons and differ only by the explicit form of the constraints. The final energy & and the final orbital angular momentum of the emitted proton are only attained at time ~ = oo when the Coulomb potential has become zero. The initial values for the trajectory calculation are 2; and ~'(0), where/~;(0) are related to the momentum components Pl of the proton inside the nucleus in the same way as for the neutron 9 The kinetic energy of the emitted proton at infinity and just after emission differ by the Coulomb potential at the surface point 2o: /3,2(o0) 2m i ;3?(o) - ~-2m + Vcb(2'o). (381 401 One can thus reformulate (36) in the form 67s dc.~ - z ~ f da ~d3p'fp(xo, p )v•177 96 s~ ~ 2m VCb(2;) , (39) which shows that it is not necessary to perform a trajectory calculation at all. The change of the orbital angular momentum of the proton on the way from the nuclear surface to infinity is due to the fact that the Coulomb potential is produced by a deformed rather than a spherical nucleus. If the deformation is not too large, weexpect this change to be small. In this case we may replace I(oQ) in Eq. (37) by the angular momentum just after emission. In this approximation, (37) may be given by the expression dZrc ! ~! (d~r ! ~d 3 pfp(xo, p- + l )v2(2'o) w0cl D•, -~, z ds~d/~ (40, Here, too, we need no evaluation of the trajectories of the emitted protons. Comparing the formulae (39) and (40) for the proton emission with the formulae (30) and (33) for neutron emission one sees that for given momentum ~'(0), the proton ends up with a final energy e which is larger than the one of the neutron by the Coulomb potential Vcb(2;), an energy of the order of a couple of MeV. This implies that the spectral distribution of the emitted protons is shifted to higher energies by about this amount compared to the spectral distribution of the neutrons. Quantum effects tend to smoothen this effect. W~ 9 d2~t ~ ~ ~ d2~ ne expressmn ~ / 3 s ~ t t~AsA1) represents the probability of emission per unit time of a neutron (proton) with given final energy s~. and given final angular momentum I~. It is related to the Weisskopf formula in the following way Fig. I. Emission of a particle through the surface 2;. ~ symbolizes the tangent plane at the emission point 2'o. The body-fixed coordinate frame is represented by its unit vectors (~"~,e"2,d'3). See further explanations in the text a plane wave by a potential step which coincides with the tangential plane at the surface point 2;. We introduce Cartesian coordinates (d,~7,~) with the origin Ox at the surface point 2;, the d- an ~l-axis in the tangential plane (g and (-axis coinciding with the surface vector if'(2;) (see Fig. 1). If the rotational frequency co is not t o o large, one may neglect the term -co[',. In this case the hamiltonian for the neutron has the form h 2 fi~ = - 2 5 Z ( 0 ~ + e , , , + o : : ) - VoOo(-r (43) We consider a plane wave hitting the tangential barrier from the side of the nuclear interior. Only the component propagating perpendicular to the potential step is modified by the barrier: @'~(~) = [e O'~/h + ~ e -~162 "0o(-~,) + J-e O;~/h 0o(0. (44) Neglecting the influence of the rotation, the energy conservation implies d2n - - ds~d/~ A~AI = F'~, dZ~z AsAI de~dl~ - (41) P22 -. - - - V o = -/~2z 2m • r~. (42) Let us note that in our derivation the spin degeneracy factor (2S~ + 1) = 2 is contained in the definition of the Wigner function (2) ((3)). The factor ~ in Eq. (1) enters the Weisskopf formula as quantum unit of phase space similarly as in our Eq. (2) ((3)). 3. Quantum-mechanical correction of the classical transmission factor The classical transmission factor w~~cannot be expected to be realistic whenever the energy of the nucleon is close to the threshold of emission. We calculate a quantum-mechanical correction for the local transmission coefficient wCol(26) by considering the transmission and reflection of (45) 2m The amplitudes .~ and -Y- of the reflected and transmitted wave are obtained in the standard way. The ratio ,A,q,,, ,vo of the transmitted current to the total incident current is given by q,, = ~l.y_12 W 0 p• 4p• 02 + ~,)2 0~o(/~i). (46) It is only non-vanishing, if the normal momentum/~2 of the transmitted wave as obtained from (45) turns out to be a real (positive) number. We indicate this explicitly by the Heaviside function 0~o(/52). If we are to take the rotational motion into account, the quantum mechanical calculation becomes much more complicated, because the eigenfunctions of the hamiltonian /~" = h~ - co]'1 (47) 402 are not plane waves. Given the fact that we only aim at a quantum correction of an otherwise classical theory, we replace the operator l~ in (47) by its classical value. In this approximation, the eigenstates of/~" continue to be plane waves with the momentum fi' for the incident wave/~' for the transmitted one. From the conservation of energy (24) and the constancy of the momentum component Pll parallel to the tangent plane (see eqn. (25)) we find the following relation between the normal momentum components Pi and p• -- 2m - Vo - c o p l b(~'o) = - - 2m - cof'• b ( X ; ) , (48) where the quantity b is defined by (49) b(~;) = ~6~(~'. ~;) - x ; 3 ( ~ ' - ~ i ) . g' is the unit vector perpendicular to the surface S at the surface point Y6 and g[ are the unit vectors in the direction of the axes of the rotating frame. From (48) we obtain the normal momentum /~j_ of the emitted neutron as a function of the normal momentum fi'~ of the incident one P'l = mco(2'o) + ,/(,Pi - mcob) 2 - 2mVo. (50) The quantum-mechanically corrected form of the transmission coefficient continues to be given by formula (46). The evaluation of a corresponding quantum correction for the proton is more complicated because of the appearance of the Coulomb potential VCb in the hamiltonian h ~ of the proton h2 h~ = ~m (~?~ + ann +acr - VoOo( - 0 + Vcb -- co'J1, (51) where Vcb is to be calculated at the space point defined by the coordinates (~, ~/, r which determine its position with respect to the surface point Y6. As the Coulomb potential is smooth and changes most strongly in the direction normal to the surface, one may approximate Vcb in (51) by its value for the coordinates (~ = 0, q = 0; ~): (52) v~ ~ v~(xo + Cn (xo)), the function Vcb being given by Eq. (5). Introducing again the classical approximation for the term -col~ in (51), we arrive at the problem of a 1dimensional barrier penetration described by the effective hamiltonian he~ff h2 hC~ff - 2m ar162+ Vcb(Y'o + ~g') + [ - Vo - co(x62p'3 - X'o3P'2)] @o( co(xo~p~ - x o ~ p ~ ) 0o(~). ~) (53) The influence of the co-dependent terms is again expected to be negligible for small enough co. One can use the WKB for evaluating the transmission coefficient w~. 4. Averaging transmission coefficients over the surface and results The evaluation of the integrals (30) and (33) is technically complicated, because it involves 4- and 3-dimensional integrations. Therefore, as a first crude approximation, we replaced the distribution functionf"(2~, fi') in eqn. (33) by a constant. Since we used a deformed square well for the nuclear potential, f" is by definition independent of )7' apart from the small Coriolis term (see Eq. (2)). Neglecting the momentum dependence off"(U, fi') is certainly a poor approximation, which is expected to falsify the magnitude of the average transmission factor #"(a~,lp). We hope, however, that it does not influence appreciably the dependence of #" on the deformation. The simple approximation for f, makes us lose the temperature-dependence of the average transmission factor. Certainly, this implies that the present results are only meaningful for large temperatures where the dependence o f f . on the momentum ~' is smooth. Expression (46) gives the local value of the emission probability at a given point of the surface X and a given value of the momentum fi' of the particle. As expressed in (1), we need the transmission probability for a particle with fixed energy a and angular momentum I in the laboratory frame whereas the calculations have been performed in the body fixed frame. Notice however the fact that e = e' and ll = l] (e, I and the projection of-/on the 1-axis correspond to the laboratory frame, the primed quantities to the body-fixed frame). Hence we can calculate the probabilities by proceeding in the following way. We fix e, I and we take all values of l[ in the interval [ - l , +l]. For each set {e, l,l~} and fixed point 26 on the surface we determine ps (k = 1 to 3) compatible with the fixed energy and angular momentum. Knowing (.~', 2;) we calculate w through (46). The value of the transmission coefficient is averaged over all allowed orientations of the angular momentum of emitted particles. This calculation is repeated for fixed e and l over a dense mesh of points covering the whole surface. The transmission coefficients u?~(a,l; r) appearing in (1) are obtained by averaging the values obtained for w over the whole surface over which this quantity is different from zero (here r stands for the angular velocity co, the surface deformation parameters, the total number of particles). In a similar way one may also determine an average value <11> of the angular momentum projection and its square <l~> which must be known for the calculation of the rotation energy of the emitting nucleus on its fission path. Expression (46) for fixed e, l and co has been used in order to get the transmission coefficients for the emission of neutrons from 126Ba. The shape of this nucleus has been defined by using the Trentalange-Koonin parametrization of the surface [11]. Calculations performed in the deformed case correspond to a ratio of the lengths of the half-axes equal to 1.7. We present calculations corresponding to a nucleus at rest (co = 0) and a nucleus which rotates with hco = 0.8 MeV which corresponds to an angular momentum of approximately 60 h. The depth of the mean potential Vo is fixed to 50 MeV. Fig. 2 shows the behaviour of @ for an angular momentum l = 0 of the emitted particle. The transmission 403 1.0 1.0 , , , l l , , . ' , l , , , , i , , i , (Q) 0.8 j~ ,~ 0.6 0.6 j/// Ii/11 / 0.4 0.4 / 0.2 0.2 / 0.0 , I=0 , , , . | 1 1 1 , t , , I , 1 . , , , 0.0 5 10 15 20 ~//'/ / / , i I | " , 9 5 , kll ,,I I (MeV) 10 ] 15 I " I 20 1.0 (b) 0.8 t 0.8 0.6 ~: 0.4 / i"/ I 0.2 c0=O.8Me v /j /P t 0.4 l// 0.2 0.0 ~ 0 / 5 10 ~: ( M e V ) 15 20 Fig. 2a + b. Transmission coefficient u? as a function of the energy e of the emitted neutron for l = 0 and angular velocity co of the nucleus. Case (a) corresponds to co=0 and case (b) to hco = 0.8 MeV. The dashed and fully drawn curves correspond to emission fiom a spherical and deformed nucleus, respectively coefficient increases with increasing energy of the particle. The emission from the spherical nucleus is favored with respect to the deformed one. This effect is particularly strong in the energy range (2 10MeV). On the other hand, the emission probability is not strongly affected by the rotation of the nucleus. A close inspection shows that for h e ) = 0.8 MeV the emission is slightly favored for e < 4 MeV and the reverse is true for higher energy. The situation changes quantitatively when l # 0, as can be seen in Fig. 3 which shows cases corresponding to 1 = 4. The spherical nucleus does not emit particles below some particle emission threshold whereas the particle can escape from the nucleus for any energy of the particle in the deformed case. These effects may have a sizable influence on the calculation of the emission widths (1). The effect of the nuclear rotation is quantitatively important (~ = 0 for the spherical case and # ~- 0.27 for the deformed case at = 10 MeV). The difference becomes smaller for increasing emission energy e. In Fig. 4 we show the dependence of # on l. F o r an emission energy e = 10 MeV the rotation of the nucleus again does not affect the emission rate sizably. For both // I= 4 h 0.0 5 10 ~: ( M e V ) 15 20 Fig. 3a + b. Same as Fig. 2 for I = 4. See discussion in the text co = 0 and co r 0 the higher I values in the one. This is expected gated along its axis rotation axis. tail of the emission rates extend to deformed case than in the spherical since the deformed nucleus is elonof symmetry perpendicular to the 5. Conclusions and discussion We have worked out a simple semi-classical model which allows to calculate the transmission coefficients which enter the emission widths (1) of a particle from a rotating nucleus with an arbitrarily deformed surface. This expression may be used in actual dynamical calculations which take care of prefission particle emission. Our preliminary numerical results show that deformation effects indeed strongly affect the transmission coefficients. On the other hand, the effect of the nuclear rotation is found to be weak, even for large rotation velocity. Our evaporation model is similar in spirit to the Richardson theory describing the emission of electrons from a hot metal. In our case, an additional complication is due to the deformed shape of the emitting body and to its rotation in space. 404 1.0 ' 9 ' ~ I " 9 9 * I ' ' 9 9 I ' ~ ' " ! (o) -~ 0.8 ' ' " " I ~ " ' " ( ,p)=C.exp ~,=o a=lOMeV - \~mm V(2')-col; , (54) where the normalisation factor C is given by the ratio of the level densities PR and +plof the daughter and m o t h e r nucleus times the factor ~ as in the Weisskopf formula (see Eq. (1)) 0.6 V 2S~ + 1 DR(E~, IR) 27rh p(E*, I) 0.4 c = 0.2 Using the Fermi gas formula for PR and p, Aleshin obtains .... !, 0.0 0 1.0 .... 2 '''' 4 .... 6 P~ = e -s/T, 8 i(h) 10 12 ~=O.8MeV l , , , , l , , ' ' l ' ' ' ' l ' ' ' ' i ' ' ' ' (b) a=lOMeV 0.8 , , 0 2 \ 0.6 v 0.4 0.2 0.0 4 , 6 8 , 10 12 I . , , , Fig. 4a + b. Transmission coefficient w as a function of the angular momentum I of the emitted neutron for fixed e. Case (a) corresponds to co = 0 and case (b) to he) = 0.8 MeV So far, we only calculated the average transmission factor #"(e~, la) for neutrons assuming a constant distribution of the neutrons prior to emission. In the next step, we shall i m p r o v e this calculation by using a m o r e realistic phase space distribution of the neutrons and p e r f o r m analogous calculations for protons and c~-particles. We also will calculate the angular distribution of the emitted particles assuming that the decaying nuclei are partially or completely polarized. Finally, it is tempting to investigate the effect of the q u a n t u m corrections in a m o r e systematic way than as done in chapt. 3, including the effect of shell corrections. Of course, there is also the question whether nucl e o n - n u c l e o n collisions will have a noticeable effect on the emission rate. It is difficult to foresee the o u t c o m e of such an investigation which one could base on the Bethe-Goldstone equation at finite temperature. Finally, we would like to c o m m e n t on the relation between our work and the one of Aleshin [10]: In Aleshin's work the distribution of the nucleons inside the nucleus is described as a classical gas of particles m o v i n g in the average potential V(U) at a given temperature T. In our notation, Aleshin's distribution has the form (see 1st of the references [10]) (55) (56) P where S is the (empirical) separation energy of the emitted particle from the m o t h e r nucleus in its ground state. This has to be c o m p a r e d with the F e r m i - D i r a c distributions (2) and (3) in our theory: In the classical limit, our distribution f ( 2 ' , fi') becomes equivalent to the one of Aleshin with the difference that we obtain the chemical potential ~t a t finite temperature T instead of the negative separation energy ( - S) (see eqs. (56) and (54)). If the temperature T is small, the chemical potential is almost equal to the negative separation energy S from the ground state and, furthermore, the ratio ~T ~ S is large c o m p a r e d to 1. In this case, our theory becomes identical to the one of Aleshin. As the temperature rises, the absolute value of the chemical potential becomes smaller. When ~T becomes comparable to 1, the F e r m i - D i r a c distribution will yield a different emission probability from the classical one. In a later paper we shall investigate the differences in detail as a function of the temperature. One of us (J.R.) acknowledges a grant for three months from the Commission of the European Community DG XII under contract no. ERBCIPACT-93-176 which allowed him to stay and to perform the present work at the Institute of Physics, M. Curie-Sktodowska University in Lublin, Poland. K.D. acknowledges gratefully the support during a two week stay in Lublin by the Marie Curie Sktodowska University and support by the BMFT. References 1. D. Hilscher, H. Rossner: Ann. Phys. Fr. 17 (1992) 471 2. V. Weisskopf: Phys. Rev. 52 (1937) 295; E. Strumberger, K. Dietrich, K. Pomorski: Nucl. Phys. A529 (1991) 522 3. D. Wilmore, P.E. Hodgson: Nucl. Phys. 55 (1964) 673; F.G. Perey: Phys. Rev. 131 (1963) 745; J.R. Huizenga, G. Igo: Nucl. Phys. 29 (1962) 462 4. D.L. Hill, J.A. Wheeler: Phys. Rev. 89 (1953) 1102 5. A. Bohr, B.R. Mottelsson: "Nuclear Structure", Vol. I, Benjamin, New York (1969) 6. T. Dossing: 1977, Licentiat thesis University of Copenhagen 7. M. Blann: Phys. Rev. C21 (1980) 1770 8. N.N. 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