A Dynamical Study of Fusion Hindrance with Nakajima-Zwanzig Projection Method

A new framework is proposed for the study of collisions between very heavy ions which lead to the synthesis of Super-Heavy Elements (SHE), to address the fusion hindrance phenomenon. The dynamics of the reaction is studied in terms of collective degrees of freedom undergoing relaxation processes with different time scales. The Nakajima-Zwanzig projection operator method is employed to eliminate fast variable and derive a dynamical equation for the reduced system with only slow variables. There, the time evolution operator is renormalised and an inhomogeneous term appears, which represents a propagation of the given initial distribution. The term results in a slip to the initial values of the slow variables. We expect that gives a dynamical origin of parameter"injection point $s$"introduced by Swiatecki et al in order to reproduce absolute values of measured cross sections for SHE. Formula for the slip is given in terms of physical parameters of the system, which confirms the results recently obtained with a Langevin equation.


I. INTRODUCTION
Finding the limit of existence of nuclei is one of the challenging research programs in nuclear physics. It gives the limit of chemical elements, with which all the matter of the world is made. Within the Liquid Drop Model (LDM), the limit of the atomic charge Z is around 100, where the fission barrier becomes negligibly small. Beyond this limit, nuclei only owe their existence to the extra-stability provided by the shell correction energy due to quantum-mechanical structure effects in finite many-body Fermion systems. They are called superheavy elements (SHE). Their quests in nature turned out unsuccessful, and then the prediction of their existence is to be justified by synthesis with nuclear reactions, but the attempts face a challenge due to the extremely low cross sections, at the order of the picobarn or even smaller for the heaviest elements produced by fusion-evaporation reactions [1]. The reasons are twofold: First, fragility of the compound nuclei (CN) formed, that are supported by a very low fission barrier, and thus, undergo fission before cooling down through neutron evaporation. Second, the fusion process is hindered with respect to models developed for the fusion of light nuclei. Fusion hindrance is observed in systems with Z 1 · Z 2 being larger than 1 600 [2]. There is no commonly accepted explanation of the origin of the latter, while the former is quantitatively well described by the conventional statistical theory of decay [3], with some ambiguities in physical parameters such as nuclear masses [4].
An unveiling of the origin of the hindrance and its correct description is necessary to improve the predictive power of models describing the whole reaction leading to the synthesis of SHE. In all models, the fusion process consists in a sequence of two steps, i.e., overcoming of the Coulomb barrier and then formation of the CN, starting from the di-nucleus configuration of the hard contact of projectile and target nuclei. The necessity for the latter process is due to the fact that the contact configuration in heavy systems locates outside of the conditional saddle point, and thus, the system has to overcome the saddle point, after overcoming the Coulomb barrier [5][6][7][8][9][10][11][12]. In other words, the fusion probability is given by the product of the contact and the formation probabilities. The former factor is often called the capture probability or cross-section, while the latter is called formation factor or hindrance factor as in the Fusion-by-Diffusion model (FBD) [10,11].
The dynamics of the latter process cannot be simply described by Newtonian mechanics, or its quantum version. For, at the hard contact of two nuclei, the di-nucleus system is internally excited. Most of the kinetic energy carried in by the incident channel is supposed to be already transformed into those of nucleonic motions, i.e., to be dissipated into heat. Therefore, the collective motions of the system are to be described by dissipation-fluctuation dynamics like in fission decay [13][14][15][16][17][18]. However, first cross sections calculated with the Langevin model [19][20][21] as well as the Fusion-by-Diffusion model [10,11], turned out to be still too large, by a few orders of magnitude, compared with the measured cross sections in so-called cold fusion path for SHE. In order to reproduce the absolute value of the measured cross sections, an arbitrary "injection point parameter s" was introduced in the latter model, shifting the contact configuration outward by about 2.0 fm. Such a shift makes the hindrance factor stronger and thus, the cross section even smaller by a few order of magnitude as desired [10,11]. A systematics over several systems exhibited a regular behaviour [22,23]. A similar analysis was conducted on the so-called hot fusion path with an injection point parameter found to be incident-channel dependent or incident-energy dependent for systematic reproduction of the experiments [24][25][26]. Those phenomenological results suggest that something is missing in theories of fusion mechanism hitherto developed.
First attempts to explain the shift of the injection parameter to dynamical process from the di-nucleus to a mono-nucleus shape were related to the fast disappearance of the neck degree of freedom that affects the injection point and gives rise to an additional hindrance [27][28][29][30][31][32]. Later, combining of the elimination of both neck degree of freedom and momentum ones in case of strong friction lead to the incident-energy dependence of the initial slip, thus explaining the behaviour of the injection point parameter, systematically both for the cold and the hot fusion paths [33]. This is encouraging for theoretical predictions on on-going and/or future experiments for synthesis of heavier elements with various incident channels.
More generally, the elimination of fast variables in a multidimensional system to reduce the model to few slow variables is addressed in several seminal articles [34,35] and textbooks [36][37][38]. It always leads to a slip of the initial condition of the slow variables [39][40][41][42][43][44]. The purpose of the present article is to propose another framework for this problem.
In contrast with Ref. [33] which is based on the Langevin formalism [45], we shall rather use here the so-called Fokker-Planck [46,47] formalism which is based on partial differential equations describing the time evolution of the probability density function in the phase space.
To look at the problem from two viewpoints often provides deeper physical understanding of the dynamics. We, thus, expect that the present approach sheds another light on the problem how the fast neck degree affects the motion of the slow variables and gives rise to a new additional hindrance.
There are various Fokker-Planck type equations to study the diffusion of collective degrees of freedom dragged by potential forces. In this study, we shall consider special forms describing the Brownian motion in an external field. When applied to position distributions, it is better known as Smoluchowski equation [48], and, when it also takes into account the conjugated momenta or velocity, it is called Klein-Kramers (K-K) equation [13,49]. The former is an approximation of the latter in the high viscosity limit.
Applied to collective degrees of freedom describing the shape of the colliding nuclei from contact to compound shape, dragged by a macroscopic potential map based on the LDM, we aim to give a global base-line theory of fusion of heavy systems. The fate of the system is either fusion (CN formation) or to re-separation (quasi-fission decay). Since the neck degree is distinctly (by factor several to one order of magnitude) faster than the other degrees of freedom [28][29][30][31][32], the neck rapidly reaches the equilibrium point, or the equilibrium distribution at the very beginning of the dynamical processes.
In order to rigorously eliminate the fast variable, we employ Nakajima-Zwanzig projection operator method [50,51] and derive a dynamical equation only for the slow variables and see how the initial slip arises. In Section 2, the derivation of the reduced dynamical equation is given in the case of Smoluchowski equation with the fast (neck) and the slow (radial and mass-asymmetry) coordinates. For simplicity and facilitation of analytic calculations, we assume the potential (LDM energy surface) to be approximated by multi-dimensional parabola (Taylor expansion at the saddle point up to the second order) and the friction tensor for the collective degrees to be coordinate-independent, i.e., be constant. The coupling between the fast and the slow variables is supposed to be weak. Perturbative approximation with respect to the coupling enables us to obtain a reduced Fokker-Planck equation with simple analytic expressions for the evolution operator and the inhomogeneous term. The latter term is mostly neglected in the scientific literature with a few exceptions [39][40][41][42][43][44], but turns out to play a key role in the present subject, giving rise to the slip of the initial point and thus, an additional hindrance. In Section 3, we summarise the results, including those for K-K equation, which is equivalent to full Langevin equation.

II. ELIMINATION OF FAST VARIABLES BY NAKAJIMA-ZWANZIG PROJEC-TION METHOD AND INITIAL SLIP
As mentioned in the introduction, the elimination of fast variables with the Nakajima-Zwanzig (N-Z) projection method [50,51] is general and can be applied to either K-K or Smoluchowski equation. Here, we shall only consider the Smoluchowski equation for general N-dimensional collective coordinates which describe shapes of the di-nucleus system. As an illustration, we eliminate the neck variable to derive a dynamical equation for the system with only slow variables and investigate possible effects of the eliminated variable to the motion of the slow ones. N-Z method is amenable to analytic calculations practically for cases that a coupling is separable with respect to fast and slow variables, like in Caldeira-Leggett model [52]. In the present model, however, the coupling is in Smoluchowski type, as given below in 2.1. Nevertheless, it turns out to be effectively bi-linear, helped by the quadratic potential, etc. It, thus, is feasible to calculate the projection, i.e., the integration (trace) over the fast variable analytically, which results in a simple dynamical equation for the reduced system, as shown in detail in 2.2.

A. Projectors
The equation is where indexes (i, j) denote all the fast and the slow variables, while indexes (α, β) used later only denote slow variables, i.e., the radial and mass-asymmetry coordinates. V denotes LDM potential, while µ is the inverse of the friction tensor γ. T is the temperature of the system.
Smoluchowskian is rewritten as follows, with the premise of perturbative approximation with respect to the coupling between the fast and the slow coordinates, where where µ f represents the µ f f . For the sake of simplicity, we have only considered a single fast variable here. This could be easily generalised to any number of fast variables.
Assuming that the fast variable quickly converges towards an equilibrium distribution φ 0 (q f ), we introduce the projection operators [50,51], Since q f becomes rapidly in equilibrium, non-equilibrium component disappears rapidly, and thus, life time of Q space is short, compared with time scale of the slow variables.
It is easy to show that projected distributions P w and Qw satisfy ∂ ∂t P w = P LP w + P LQw, which is rewritten into a closed equation for P w, So far, no approximation is made and therefore, this equation is equivalent to the original one, but is amenable to perturbative approximation.
The last term of Eq. The second term of Eq. (11) exhibits a memory effect, which systematically appears when some variables are eliminated [34][35][36][37]. The method has been generally discussed in the field of statistical mechanics, but their interests are mostly to what dynamics the reduced system obeys. We, however, are also interested in the inhomogeneous term in Eq. (11), which carries the Q space component in the initial distribution.

B. Renormalised Smoluchowski operator and initial slip
To facilitate the calculations, we first diagonalise the coefficient matrix of the potential, although it is quite close to be diagonal in case of Two-Center-Parameterization (TCP) of di-nucleus system [53,54]. Thus, the potential is as follows, then, Note that L f and L s are operators solely in the fast and the slow variables, respectively, while L 1 is the coupling between them.
In the following calculations, we use the basic properties of L f : Naturally, distribution functions are supposed to be zero at the boundaries of q f variable.
For the equilibrium distribution φ 0 (q f ), we have that The eigenvalues and eigenfunctions are Here, H n (x) are the Hermite polynomials.
Hereafter, we calculate the terms in r.h.s. of Eq. (11) at the lowest order with respect to the coupling L 1 between the fast and the slow variables. For the first term, we simply have that P LP w = φ 0 L s · w( q α , t), with w( q α , t) = dq f w( q i , t).
This is just the Smoluchowski operator for the (N − 1) slow variables only.
The second term is already at the second order with respect to L 1 , due to the pre-and post-factors of the propagator. Therefore, we approximate the full Smoluchowski operator L in the propagator by L 0 , so that, e QL·t ≃ e QL 0 ·t . Noticing that QL s and QL f commute, . Thus, with the properties of L f given above, the memory kernel is governed by the factor e −µ f ·c f ·(t−t ′ ) . Thus, the duration of the memory effect is determined by the relaxation time of the fast neck degree. In the t ′ integration, we use Laplace approximation, taken into account that the integrand is slowly varying function, Then, the second term is rewritten as follows, The first two terms of Eq. (11) are summed up into This means that the elimination of the fast variable results in the renormalisation of Smoluchowski operator for the slow variables with µ eff αβ : Now, we analyse the last term in Eq (11) which is most important in the present study.
Using a perturbative expansion of the propagator e QL·t = e Q(L 0 +L 1 )·t with respect to L 1 , the third term is given as follows, The zeroth order term is simply zero, and the first order term which does not vanish is, Since e QL f ·t = e L f ·t , and e QLs·t · e QL f ·t · Qw(0) = e Ls·t · e L f ·t · Qw(0), where w(q i , 0) = w(q f , 0)w(q α , 0) is assumed. And denotes the solution of 1-D Smoluchowski equation, which is well known [34][35][36][37]: with where q f 0 is the initial value of the fast variable. Finally, integrated over q f , the term I 1 becomes which denotes α-th component of the vector B(t) T . It clearly shows that this term rapidly diminishes with the time scale of the fast variable, as expected.
Eventually, Eq. (11) turns out to be The question is what the last term in r.h.s. of Eq. (31), i.e., the inhomogeneous term, gives rise to. As usual, a formal solution is written down, with This means that the reduced system for only slow variables is described by the renormalised Smoluchowski operator L eff with the t-dependent "initial distribution". The function w eff ( q, t) is equal to w( q, 0) at t = 0, while at t ≫ 1/(µ f · c f ), i.e., after the equilibration of the fast variable, where we use again Laplace approximation in the time integration to calculate the 2nd term in Eq. (33).B denotes a constant shift vector whose component is given explicitly, µ f q f 0 , and the initial value is effectively shifted as q eff In other words, after the beginning short time, the system evolves as if it has started with the slipped initial point. Now, it should be noted that the slip which gives the additional hindrance is given in terms of physical parameters of the system.
Actually, the slip −B R for the radial variable can be estimated for the system 209 Bi + 70 Zn, for example. In TCP of LDM, the initial point q f0 of the neck ε is 1.0 (non-dimensional) [53,54], and µ αf µ f is estimated with the one-body dissipation [55] and at the contact configuration with a relaxed neck, say, ε = 0.2. The slip in physical dimension is obtained with the nuclear radius constant r 0 = 1.1 fm. A preliminary value obtained is about 2.3fm. Considering the crude parabolic approximation of LDM, etc, the result well explains the phenomenological "injection point parameter s"( about 2fm) in FBD [10,11]. Analyses of system-dependence etc. will be given elsewhere.

C. Formation Probability with the initial shift: A New Dynamical Hindrance
Now we obtain explicit expressions of Formation Probability (Hindrance Factor), taking into the initial slip. The solution of Smoluchowski equation with parabolic potential and given initial conditions is known as, with Here note that the exponent is 2 · c · µ, not µ · c, and that the matrix Σ is symmetric, so Σ −1 as well.
The initial vector q 0 should be the slipped one due to the effective initial value given in Eq. (35) and the matrix µ should be the effective one given in Eq. (20), though it is supposed to be close to the original one.
We now consider the system of the two slow variables (R, α) after the elimination of the fast neck variable ε. The integration over mass-asymmetry α should give a kind of the solution of 1-D Smoluchowski equation for the radial coordinate R. Then, by integrating over R from minus infinity to zero, we obtain the formation probability at time t, with the mean trajectory being where the subscript 1 denotes the first component of the vector which is the product of the exponential matrix and the vector in two dimensions. The complementary error function is defined as The formation probability is explicitly obtained by taking the limit of the time t → ∞ [56,57]. Here we should remind that the origin is taken to be the saddle point of the which gives a simple approximate formula, Then, where the asymptotic expansion of the error function is used for large argument. If the second term in r.h.s of Eq. (41) for the expression ofR D which depends on α 0 is negligibly small, the above formula is the same as a simple 1-D problem. It, however, is worth noticing here that the initial point above is the slipped one given in Eq. (35), i.e., the saddle point height V R is higher by that like in FBD model.

III. CONCLUDING REMARKS
We consider the dynamical evolution of di-nucleus systems of very heavy ions as relaxation Ref. [33]. For intuition, we take two steps, depending on time scales: Firstly, we eliminate the whole momentum space, to obtain N-dimensional Smoluchowski equation with memory kernel and an inhomogeneous term. Next, we follow the neck elimination given in the present article. We end up with three inhomogeneous terms, which, in the lowest order, gives rise to a sum of two shifts from the momentum and the neck eliminations, respectively. The results coincide with those obtained in [33]. They will be published elsewhere, together with systematic analysis of incident-channel dependence. The present dynamical approach shall be widely developed not only to fusion, but also to quasi-fission etc. It will give a novel vista to various dynamical aspects in heavy-ion collisions as well as to quantitative predictions of synthesis cross sections of SHE.