Study of Pre-equilibrium Fission Based on Diffusion Model

Transcription

1 Commun. Theor. Phys. (Beijing, China) 45 (2006) pp c International Academic Publishers Vol. 45, No. 2, February 15, 2006 Study of Pre-equilibrium Fission Based on Diffusion Model SUN Xiao-Jun 1,2 and ZHANG Jing-Shang 2 1 College of Physics and Information Technology, Guangxi Normal University, Guilin , China 2 China Institute of Atomic Energy, P.O. Box 275(41), Beijing , China (Received April 27, 2005) Abstract In terms of numerical method of Smoluchowski equation the behavior of fission process in diffusion model has been described and analyzed, including the reliance upon time, as well as the deformation parameters at several nuclear temperatures in this paper. The fission rates and the residual probabilities inside the saddle point are calculated for fissile nucleus n+ 238 U reaction and un-fissile nucleus p+ 208 Pb reaction. The results indicate that there really exists a transient fission process, which means that the pre-equilibrium fission should be taken into account for the fissile nucleus at the high temperature. Oppositely, the pre-equilibrium fission could be neglected for the un-fissile nucleus. In the certain case the overshooting phenomenon of the fission rates will occur, which is mainly determined by the diffusive current at the saddle point. The higher the temperature is, the more obvious the overshooting phenomenon is. However, the emissions of the light particles accompanying the diffusion process may weaken or vanish the overshooting phenomenon. PACS numbers: Jj, Ge Key words: pre-equilibrium fission, diffusion model, fissile nucleus, un-fissile nucleus 1 Introduction The experimental measurements indicate that the preequilibrium fission would occur when the excitation energies are high enough. In these cases, the equilibrium theory [1] cannot describe this kind of reaction mechanism. The pre-equilibrium fission, now, is described by diffusion model. In this model the fission process can be viewed as a diffusion process over a fission barrier. In 1940 Kramers derived the fission width in a quasi-stationary approximation through solving Fokker Planck equation. [2] However, there really is a transient process prior to the quasistationary. Indeed, the experimental data indicate that the number of light particles evaporated in a heavy-ion induced reaction evidently exceeds the expectations based on the equilibrium statistical model. [3 5] So the research of the transient process by means of the diffusion model is necessary. In order to explain the experimental data and to study nuclear dissipation mechanism, the transient behaviors have been described by solving Fokker Planck equation numerically only for the fissile nucleus. [6 8] However, if the friction coefficient of the compound nucleus is so large that the equilibration in momentum space attains very rapidly, then the Fokker Planck equation is translated into Smoluchowski equation, which describes the diffusive process only in deformation space. [9] In terms of the Van Kampen method, [10] Smoluchowski equation could be solved analytically. But this analytical method requires an auxiliary potential, so there are some unphysical features as mentioned in Ref. [10]. For instance, the curvatures of the fission barrier and the fission well are dependent on nuclear temperature T. Therefore, this approach implies that the shape of fission potential V (x) is changed with nuclear temperature. In order to avoid these shortcomings mentioned above, the fission potential, which is independent of the nuclear temperature, is obtained by simulating the features of various target nuclei. The fission potentials, such as fissile nucleus 238 U and un-fissile nucleus 208 Pb, are given in this paper, which are composed of three harmonic-oscillator potentials jointing smoothly with each other. However, it is very difficult to find the analytical solution for this kind of fission potential. Hence the numerical solution, which is introduced in Ref. [11], is employed to solve the Smoluchowski equation. The Smoluchowski equation does have a simple scaling property. So the Smoluchowski equation can be rewritten in a new form, in which the friction coefficient is not included with this time scaling commutation. [12] For n+ 238 U reaction, its fission rates tend to a steady value if the nuclear temperature is relatively low, while in the case of high nuclear temperature, the fission rates firstly tend to a very high value then decrease and turn to a steady value. These abnormal behaviors are described as the overshooting phenomenon of the fission rates in transient process. For p Pb = 209 Bi reaction in the medial and high energies of the projectile proton, the fission behavior is also studied, and the information is useful to studying the nuclear material on target physics of the accelerator driven sub-critical system (ADS). The Smoluchowski equation, which does not comprise the friction coefficient, is presented in Sec. 2. In Sec. 3 the fission potentials consisting of three harmonic-oscillator potentials and their parameters of target nucleus (such The project supported by National Natural Science Foundation of China under Grant No

2 326 SUN Xiao-Jun and ZHANG Jing-Shang Vol. 45 as 238 U and 208 Pb) are given. In Sec. 4, the fission rate, the diffusive current at the saddle point, and the time to reach the quasi-stationary state are discussed. In Sec. 5, the calculated residual probability inside the saddle point in various cases is given, and the homology and discrepancy between fissile nucleus and un-fissile nucleus are discussed. Conclusions and prospects are elaborated in the last section. 2 Smoluchowski Equation and Its Numerical Solution Nuclear fission can be viewed as a diffusion process over the fission barrier, and described by means of the Fokker Planck equation. [2] When the friction coefficient is so large that equilibration in momentum space attains very rapidly, the Fokker Planck equation is translated into the Smoluchowski equation, which describes the diffusive process only in deformation space. [9] Thus Smoluchowski equation reads P (x, t) = β 1 t x [K(x)P (x, t)] + β 2 ɛ 2 P (x, t). (1) x2 Here, P (x, t) refers to the probability of the system at deformation coordinate x and time t; K(x) = (1/µ)(dV (x)/dx), where V (x) is fission potential; and ɛ = βt /µ, where T is the nuclear temperature, β is the viscosity coefficient (which gives the coupling strength between the fission degree of freedom and the rest of the system as a heat bath), µ is the reduced mass. [10,12] As is well known, the first term on the right-hand side of Eq. (1) represents the drifting term, and the second term is the diffusive term. The Smoluchowski equation has a simple scaling property with the friction coefficient β, so new time τ (in units of MeV 2) ) is defined as τ = t β h 2 = c2 t β( hc) 2. (2) Thus, equation (1) has the form as follows: P (x, τ) = [ du(x) ] τ x dx P (x, τ) + D 2 P (x, τ). (3) x2 Here, the fission potential in new time scaling becomes the form U(x) = [( hc) 2 /µc 2 ]V (x) (in units of MeV 2 fm 2 ), and the new diffusive coefficient is D = [( hc) 2 /µc 2 ]T (in units of MeV 2 fm 2 ). Neither of them is relevant to the viscosity coefficient β, which only holds in new time unit of τ. The initial probability distribution takes a normalized Gaussian form as 1 ) P (x, τ = 0) = exp ( x2 2πσ0 2σ0 2, (4) where σ 0 is the initial distribution width, which weakly influences the solutions as mentioned in Ref. [12]. Using the method introduced in Ref. [11], equation (3) is written into a difference equation with a tri-diagonal form, so the numerical solution could be easily obtained. 3 Fission Potentials and Their Parameters Based on the diffusion model, the fission process is depicted mainly by potential barrier height, potential well curvature, and potential barrier curvature, as shown in the Kramers formula. [2] These three parameters are primary physical quantities of the fission potential used in the diffusion model. In this paper, the fission potential is composed of three harmonic-oscillator potentials, which connect smoothly with each other. Their parameters simulate the features of various target nuclei for both of fissile nucleus and un-fissile nucleus. Fission is then described as a diffusion process out of the shallow well into the deep well of the bistable potential V (x). The fission potential reads µc 2 ( hω 1 ) 2 2( hc) 2 x 2, x a, V (x) = µc2 ( hω f ) 2 2( hc) 2 (x b) 2 + V f, a x c, µc 2 ( hω 2 ) 2 2( hc) 2 (x d) 2 V 0, c x. Here, hω 1 is the curvature of the interior well; hω f is the fission barrier curvature; hω 2 is the curvature of the exterior well; V f is the fission barrier height and V 0 is the depth of the exterior well, and the bottom position of the interior well is at the original point, as shown in Fig. 1. Fig. 1 The fission potential. (5) Physically, there is no exterior potential well, so that hω 2 and V 0 have no physical significance. Since the condition P (x ±, t) = 0 must hold in the numerical method, then the exterior potential well should be so deep to avoid negative current at the saddle point. To do so in this way the probability keeps conservation in the fission process. In Eq. (5) d denotes the bottom position of the exterior well, b is the position of the saddle point, a and c are the two points to joint the three harmonic-oscillator potentials smoothly each other. From the smoothly connecting conditions, the four parameters of a,b, c, d could

3 No. 2 Study of Pre-equilibrium Fission Based on Diffusion Model 327 be determined by the five parameters of hω 1, hω f, hω 2, V f, and V 0. After five parameters of the fission potential mentioned above are given, there are still other parameters, such as the reduced mass µ and the nuclear temperature T (T = E /a ), where E is excitation energy, and a stands for level density parameter) of the nuclear system. In order to compare the physical figures of the fissile nucleus 238 U and the un-fissile nucleus 208 Pb, the common parameters are hω 1 = 1.0 MeV, hω 2 = 1.0 MeV, V 0 = 100 MeV. For n+ 238 U nuclear reaction, the reduced mass is µ = 57.9m (m is a nucleon mass), the fission barrier height V f = 5.9 MeV, the fission barrier curvature hω f = 0.6 MeV, and the level density parameter a = MeV 1, which is used by Chinese Nuclear Data Center. [13] For p Pb = 209 Bi reaction, because the fission behavior is at medial and high energy of the projectile proton, then the symmetrical fission is taken into account with the reduced mass µ = 52.25m, the fission barrier height V f = MeV (see the result of Ref. [14]), the curvature of the fission barrier hω f = MeV, and the level density parameter a = MeV 1. [13] 4 Fission Rate and Quasi-stationary Time by [8,10] The residual probability inside the saddle point is given P s (τ) = b P (x, τ)dx. (6) Let λ t f (t) denote the fission rate in t time scaling as defined in Refs. [8] and [10], and λ τ f (τ) is the fission rate in τ time scaling used in this paper. Then there is a relation of them, λ t f (t) = dp s(t)/dt P s (t) = 1 dp s (τ)/dτ h 2 = 1 β P s (τ) h 2 β λτ f (τ), (7) where λ τ f (τ) is independent of the friction coefficient β, and λ t f (t) is proportional to β 1. Let τ tr denote the time to reach the quasi-stationary state, which is defined in our calculation by the following two conditions: (a) the time when λ τ f (τ) has approached its steady value with sufficient accuracy of 5%, and (b) the time when P s (τ) reaches to the value 1/e. This definition is intended to include the transient process. Fig. 2 The fission rates of the different target nuclei at the different temperatures as indicated. Figure 2 shows the fission rates λ τ f (τ) as function of τ at the different nuclear temperatures (or excitation energy) as indicated for fissile nucleus 238 U and the unfissile nucleus 208 Pb, where figure 2(a) shows the case of T < V f, and figure 2(b) shows the case of T > V f for target nucleus 238 U. In the case of high temperature and low temperature, the rates λ τ f (τ) are very different from each other in the transient process. At low temperature (T < V f /2), λ τ f (τ) increases slowly, and reaches its steady value when τ is large enough. However, at high temperature (T > V f /2), λ τ f (τ) increases at a swoop, and decreases slowly to the steady value. This is the so-called overshooting phenomenon of the fission rate λ τ f (τ). The higher the temperature is, the more evident the overshooting phenomenon is. For un-fissile nucleus 208 Pb, the rates λ τ f (τ) are similar to the feature of fissile nucleus 238 U at

4 328 SUN Xiao-Jun and ZHANG Jing-Shang Vol. 45 low temperature, as shown in Fig. 2(c). The time span of the un-fissile nucleus is greatly longer than that of the fissile nucleus, as shown in Fig. 2 or Tables 1 and 2 in Sec. 5. These properties of un-fissile nucleus are tightly relevant to its fission potential, whose barrier height is much higher, and fission barrier curvature is less than that of fissile nucleus, so the probability of the pre-equilibrium fission is relatively small than that of fissile nucleus. Rewriting Eq. (3) into continuity equation form, the fission probability current is obtained as J(x, τ) = du(x) P (x, τ) P (x, τ) D, (8) dx x where the first term is the drifting current, mainly decided by the fission potential, and the second term is the diffusive current, which is proportional to the diffusive coefficient and the negative gradient of probability. Integrating Eq. (3) from to b in x space, and using Eqs. (5) (8), the probability current at the saddle point is written as J b (τ) = P s(τ) P (b, τ) = D. (9) τ x Thus the fission rate can be written as λ τ f (τ) = J b(τ) P s (τ). (10) Because the change of P s (τ) is very small at τ τ tr, the fission rate λ τ f (τ) is primarily decided by the change of current J b (τ), as shown in Fig. 3. The current J b (τ) versus τ at various T as indi- Fig. 3 cated. Figure 3 shows the current J b (τ) with τ at T = 1 6 MeV for fissile nucleus n U reaction. At the high temperature, the current J b (τ) firstly increases at one swoop and forms a steep slope, and then decreases very slowly with increasing τ. This steep slope is familiar in the overshooting of λ τ f (τ) as shown in Fig. 2(b). The higher the temperature is, the steeper the slope is. At low temperature, the slope is gradually weakened or vanishes, thus there is no overshooting phenomenon. Fig. 4 The fission rates λ τ b (τ) at different scission points for 238 U. For the aim of comparison, in this paper we take b (= b + x) as the so-called scission point beyond the saddle point. One can find that the transient behaviors at these points are quite different from that at the saddle point. The overshooting phenomenon at the scission point b weakens and eventually disappears as shown in Fig. 4. Therefore, the overshooting phenomenon is dependent on the position of the selected fission point. [15] This kind of phenomenon mentioned above is caused by the currents J b (τ) at the scission point, which consist of both the drifting current and the diffusive currents, not only of the diffusive current as defined in Eq. (9). Although the overshooting phenomenon at the scission points is not observed, it cannot deny the fact that the overshooting phenomenon occurs at the saddle point. As is well known, in the liquid drop model, the fission process is the competition between the surface tense and the Coulomb force. The scission point is the feature only in the liquid drop model. Moreover, there are diverges about the position of the scission point at the fission potential surface, when the shell correction and the pairing correction are or are not taken into account. [16] On the other hand, in the diffusion model the fission process is caused by the stochastic forces acted on the Brown particles, so the scission point is an obscure quantity in the diffusion model, in which the saddle point is an only commonly recognized physical quantity. Thus, it is reasonable to depict the fission point only at the saddle point, as used in Kramers formula [2] and the diffusion model as shown in Refs. [6] [8], and [10]. In addition, the negative current at the saddle point may occur when the fission process is depicted by Langevin equation. As is well known, the average of the stochastic forces with white noise in Langevin equation over time is the friction coefficient. Although the Langevin

5 No. 2 Study of Pre-equilibrium Fission Based on Diffusion Model 329 equation is equivalent to the Fokker Planck equation, the fission process described by the Fokker Planck equation is the time-averaged behaviors of the stochastic force. So there is always no negative current at the saddle point in the solution of Fokker Planck equation, neither in the Smoluchowski equation. Figure 5 shows the quasi-stationary time τ tr as a function of different temperature for fissile nucleus n+ 238 U reaction and un-fissile nucleus p Pb reaction. At the low temperature for the fissile nucleus 238 U as shown in Fig. 5(a), τ tr decreases with increasing of the temperature, but almost has a steady value when the temperature is so high that the overshooting phenomenon appears. For the un-fissile nucleus 208 Pb as shown in Fig. 5(b), τ tr decreases with increasing of the temperature, as the same as the behaviors of the fissile nucleus 238 U without overshooting regime. The values τ tr of the un-fissile nucleus are larger obviously than those of the fissile nucleus at the same temperature, as indicated in Tables 1 and 2. Fig. 5 The quasi-stationary time τ tr as a function of temperature T for different target nuclei as indicated. Table 1 The values τ tr, P s(τ) and 1 P s(τ) for fissile nucleus 238 U at the different temperature as listed. T (MeV) E (MeV) V f /T τ tr (MeV 2 ) P s(τ tr) 1 P s(τ tr) Table 2 The values τ tr, P s(τ), and 1 P s(τ) for un-fissile nucleus 208 Pb at the different excitation energies as listed. E (MeV) T (MeV) V f /T τ tr (MeV 2 ) P s(τ tr) 1 P s(τ tr)

6 330 SUN Xiao-Jun and ZHANG Jing-Shang Vol Residual Probability Inside Saddle Point The residual probability P s (τ) inside the saddle point is given by Eq. (6). The curves P s (τ) as shown in Fig. 6 for both the fissile nucleus n U reaction and the unfissile nucleus p Pb reaction, respectively, are given. There are several features such as (i) There exists a very short time τ p to reach the equilibrium in momentum space. For time τ with 0 τ τ p, very little probability escapes over the fission barrier, i.e. P s (τ) 1. (ii) At the time τ > τ p, log 10 P s (τ) is almost in a linear form with τ. The higher the temperature is, the steeper the line of log 10 P s (τ) is. This fact indicates that the probability escapes over the fission barrier is rapid, when the temperature is high. (iii) The time to reach the steady values of the unfissile nucleus p Pb reaction is much longer than that of the fissile nucleus n U reaction, since the doubly magic target nucleus 208 Pb is very stable, and the fission is not easy to take place. respectively. P s (τ tr ) stands for the probabilities without fission, while 1 P s (τ tr ) refers to the probabilities undergoing fission, respectively. The results of Table 1 show that the contribution of the pre-equilibrium fission is very small (< 1%) below 100 MeV, so it could be neglected in the calculation of the nuclear reactions. However, the contribution of the pre-equilibrium fission is obvious, even over 15% at 1000 MeV, so the pre-equilibrium fission process should be considered properly. The probabilities P s (τ tr ) and 1 P s (τ tr ) for proton induced un-fissile nucleus p Pb fission are given in Table 2. The results of Table 2 show that fission probability is small in the transient process even at high excitation energies. For the structure material 208 Pb, however, the ratio of the high fission barrier to nuclear temperature is over 2, so the overshooting phenomenon does not occur (see Fig. 2(c)). Although the quasi-stationary time τ tr of 208 Pb is longer than that of 238 U, the value λ τ f (τ tr) of 208 Pb is much smaller than that of 238 U. At the high excitation energy (for example, 1000 MeV), the contribution of the pre-equilibrium for the target nucleus 208 Pb is only 1.4%, so it could be omitted in model calculations, as performed in the model calculation codes used in the accelerator driven sub-critical system (ADS). [17 19] 6 Conclusions and Prospects Fig. 6 The residual probability P s(τ) inside the saddle point as a function of time τ at the different temperatures as indicated. The probabilities of P s (τ tr ) and 1 P s (τ tr ) for neutron induced fissile nucleus n+ 238 U fission with the temperatures of 1, 2, 3, 4, 5, and 6 MeV, are given in Table 1, The contribution of the pre-equilibrium fission and the fission rates, and the transient process are analyzed for the fissile and un-fissile nucleus in this paper with the numerical solution of the Smoluchowski equation. The results show that there really exists pre-equilibrium fission process, of which the contribution must be considered at the high excitation energies for the fissile nucleus, but might be neglected at low excitation energies. Because of the high fission barrier of the un-fissile nucleus, the contribution of the pre-equilibrium also might be neglected in model calculations, as that in the codes used in ADS. [17 19] There is no overshooting phenomenon at V f /2 > T, but at T V f /2 for the fissile nucleus, the overshooting phenomenon exists obviously. The overshooting phenomenon is caused by the diffusive current at the saddle point. The research mentioned above is performed only qualitatively because the light particle emissions are absent. Once the emissions of the light particles are taken into account in the diffusive process, the emissive particles take away a part of the excitation energies from the mother nucleus, and decrease the nuclear temperature of its daughter nucleus. So the light particle emissions reduce the fission probability prior to equilibrium state. Therefore, the overshooting phenomenon may weaken or vanish gradually. After consideration of the emissions of the light particles, one can understand the reason why the number

7 No. 2 Study of Pre-equilibrium Fission Based on Diffusion Model 331 of the emitted particles from a heavy-ion induced reaction prior to fission considerably exceeds the expectations only based on the statistical model. The diffusion model including the light particle emissions could give the explanation quantitatively. The study of the fission process accompanied by the light particle emissions needs to set up the coupled Smoluchowski equation, which may really describe the fission process. The coupled Smoluchowski equations have been constructed and used in some papers such as Refs. [15] and [20], which focused on the influence of the shell correction and the angular momentum effect on the light particle emissions. References [1] N. Bohr and J.A. Wheeler, Phys. Rev. 56 (1939) 426. [2] H.A. Kramers, Physica 7 (1940) 284. [3] A. Gavron, A. Gayer, et al., Phys. Rev. Lett. 47 (1981) [4] D.J. Hindle, R.J. Charity, et al., Phys. Rev. Lett. 52 (1984) 986. [5] E. Holub, D. Hilscher, et al., Phys. Rev. C 28 (1983) 252. [6] Wu Xi-Zhen, Zhuo Yi-Zhong, et al., High Energy Phys. Nucl. Phys. (Beijing, China) 7 (1983) 716. [7] Wu Xi-Zhen, Zhuo Yi-Zhong, et al., Commun. Theor. Phys. (Beijing, China) 1 (1982) 769. [8] P. Grangé, Li Jun-Qing, and H.A. Weidenmüller, Phys. Rev. C 27 (1983) [9] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [10] H.A. Weidenmüller and Zhang Jing-Shang, Phys. Rev. C 29 (1984) 879. [11] Zhang Jing-Shang and Georg Wolschin, Z. Phys. A 311 (1983) 177. [12] Sun Xiao-Jun, Zhang Jing-Shang, et al., Commun. Theor. Phys. (Beijing, China) 43 (2005) [13] Zhang Jing-Shang, CNDC, private communication (2005). [14] Zhou Jin-Feng, Shen Qing-Biao, Sun Xiu-Quan, Nuclear Technology 127 (1999) 113. [15] Lu Zhong-Dao, Zhang Jing-Shang, et al., Phys. Rev. C 42 (1990) 707. [16] A.J. Sierk and J.R. Nix, Phys. Rev. C 21 (1980) 982. [17] K.K. Gudima, et al., Nucl. Phys. A 401 (1983) 329. [18] A. Dementyev, et al., Nucl. Phys. Suppl. 70 (1990) 486; Radiation Measurements 30 (1999) 553. [19] A.S. Botvina, et al., Nucl. Phys. A 475 (1987) 663. [20] Ye Wei, High Energy Phys. Nucl. Phys. (Beijing, China) 27 (2003) 798; 28 (2004) 181.