QMD is the quantum extension of the classical molecular dynamics model and is widely used to analyze various aspects of heavy ion reactions, especially for many-body processes, and in particular the formation of complex fragments. In the previous section, we mentioned several similar and dissimilar points between Binary Cascade and QMD. There are three major differences between them:

The definition of a participant particle,
The potential term in the Hamiltonian, and
Participant-participant interactions.

At first, we will explain how they are each treated in QMD. The entire nucleons in the target and projectile nucleus are considered as participant particles in the QMD model. Therefore each nucleon has its own wave function, however the total wave function of a system is still assumed as the direct product of them. The potential terms of the Hamiltonian in QMD are calculated from the entire relation of particles in the system, in other words, it can be regarded as self-generating from the system configuration. On the contrary to Binary Cascade which tracks the participant particles sequentially, all particles in the system are tracked simultaneously in QMD. Along with the time evolution of the system, its potential is also dynamically changed. As there is no criterion between participant particle and others in QMD, participant-participant scatterings are naturally included. Therefore QMD accomplishes more detailed treatments of the above three points, however with a cost of computing performance.

Equations of Motion

The basic assumption of QMD is that each nucleon state is represented by a Gaussian wave function of width \(L\),

\[\varphi_i(\mathbf{r}) \equiv \frac{1}{(2\pi L)^{3/4} } \exp \left( -\frac{(r-r_i)^2}{4L} + \frac{i}{\hbar} r\cdot p_i \right)\]

where \(r_i\) and \(p_i\) represent the center values of position and momentum of the \(i^{\rm th}\) particle. The total wave function is assumed to be a direct product of them,

\[\Psi (\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) \equiv \prod\limits_i\varphi_i(r_i)\ .\]

Equations of the motion of particle derived on the basis of the time dependent variation principle as

\[\dot r_i = \frac{\partial H}{\partial p_i}\,,\quad \dot p_i = -\frac{\partial H}{\partial r_i}\]

where \(H\) is the Hamiltonian which consists particle energy including mass energy and the energy of the two-body interaction.

However, further details in the prescription of QMD differ from author to author and JAERI QMD (JQMD) [eal95, eal99] is selected as a basis for our model. In this model, the Hamiltonian is

\[H=\sum\limits_i\sqrt{m_i^2 +p_i^2 } + \hat{V}\]

A Skyrme type interaction, a Coulomb interaction, and a symmetry term are included in the effective Potential (\(\hat{V}\)). The relativistic form of the energy expression is introduced in the Hamiltonian. The interaction term is a function of the squared spatial distance:

\[R_{ij} = (R_i-R_j)^2\]

This is not a Lorentz scalar. In Relativistic QMD (RQMD) [SStockerG89], they are replaced by the squared transverse four-dimensional distance,

\[-q_{Tij}^2 =-q_{ij}^2 +\frac{(q_{ij}\cdot p_{ij})^2}{p_{ij}^2}\]

where \(q_{ij}\) is the four-dimensional distance and \(p_{ij}\) is the sum of the four momentum. In JQMD they change the argument by the squared distance in center of mass system of the two particles,

\[\tilde{R} = R^2_{ij} + \gamma^2_{ij}(R_{ij}\cdot\beta_{ij})^2\]

with

\[\beta_{ij} =\frac{p_i +p_j }{E_i +E_j }\,,\quad \gamma_{ij} =\frac{1}{\sqrt{1-\beta_{ij}}}\]

As a result of this, the interaction term in also depends on momentum.

Recently R-JQMD, the Lorentz covariant version of JQMD, has been proposed [MNMS09]. The covariant version of Hamiltonian is

\[H_C =\sum\limits_i\sqrt{p_i^2 + m_i^2 + 2m_iV_i}\]

where \(V_i\) is the effective potential felt by the \(i^{\rm th}\) particle.

With on-mass-shell constraints and a simple form of the “time fixations” constraint, the entire particle has the same time coordinate. They justified the latter assumption with the following argument “In high-energy reactions, two-body collisions are dominant; the purpose of the Lorentz-covariant formalism is only to describe relatively low energy phenomena between particles in a fast-moving medium” [MNMS09].

From this assumption, they get following equation of motion together with a big improvement in CPU performance.

\[\begin{split}\dot r_i &= \frac{p_i}{2p^0_i} + \sum\limits_j \frac{2m_j}{2p^0_j}\frac{\tilde V_j}{\partial p_i} \\ &= \frac{\partial}{\partial p_i} \sum\limits_j \sqrt{p_j^2 + m_j^2 + 2m_j\tilde V} \\ \dot p_i &= -\sum\limits_j \frac{2m_j}{2p^0_j}\frac{\tilde V_j}{\partial r_i} \\ &= \frac{\partial}{\partial r_i} \sum\limits_j \sqrt{p_j^2 + m_j^2 + 2m_j\tilde V}\end{split}\]

The \(i^{\rm th}\) particle has an effective mass of

\[m^*_i =\sqrt{m_i^2 +2m_i V_i}\ .\]

We follow their prescription and also use the same parameter values, such as the width of the Gaussian \(L\) = 2.0 fm\({}^2\) and so on.

Ion-ion Implementation

For the case of two body collisions and resonance decay, we used the same codes which the Binary Cascade uses in Geant4. However for the relativistic covariant kinematic case, the effective mass of \(i^{\rm th}\) particle depends on the one-particle effective potential, \(V_i\), which also depends on the momentum of the entire particle system. Therefore, in R-JQMD, all the effective masses are calculated iteratively for keeping energy conservation of the whole system. We track their treatment for this.

As already mentioned, the Binary cascade model creates detailed \(3r+3p\) dimensional nucleus at the beginning of each reaction. However, we could not use them in our QMD code, because they are not stable enough in time evolution. Also, a real ground state as an energy minimum state of the nucleus is not available in the framework of QMD, because it does not have fermionic properties. However, a reasonably stable “ground state” nucleus is required for the initial phase space distribution of nucleons in the QMD calculation. JQMD succeeded to create such a “ground state” nucleus. We also follow their prescription of generating the ground state nucleus. And “ground state” nuclei for target and projectile will be Lorentz-boosted (construct) to the center-of-mass system between them. By this Lorentz transformation, additional instabilities are introduced into both nuclei in the case of the non-covariant version.

The time evolution of the QMD system will be calculated until a certain time, typically 100 fm/\(c\). The \(\delta T\) of the evolution is 1 fm/\(c\). The user can modify both values from the Physics List of Geant4. After the termination of the time evolution, cluster identification is carried out in the phase space distribution of nucleons in the system. Each identified cluster is considered as a fragmented nucleus from the reaction and it usually has more energy than the ground state. Therefore, excitation energy of the nucleus is calculated and then the nucleus is passed on to other Geant4 models like Binary Cascade. However, unlike Binary Cascade which passes them to Precompound model and Excitation models by calling them inside of the model, the QMD model uses Excitation models directly. There are multiple choices of excitation model and one of them is the GEM model [Fur00] which JQMD and RJQMD use. The default excitation model is currently this GEM model.

Figure [fig:qmd-time] shows an example of time evolution of the reaction of 290 MeV/n ⁵⁶Fe ions bombarding a ²⁰⁸Pb target. Because of the small Lorentz factor (~1.3), the Lorentz contractions of both nuclei are not seen clearly.

Fig. 117 Time evolution of reaction of 290 MeV/n Fe on Pb in position space. Red and Blue circle represents neutron and proton respectively. Full scale of each panel is 50 fm.

Cross Sections

Nucleus-Nucleus (NN) cross section is not a fundamental component of either QMD or Binary Light Ions Cascade model. However without the cross section, no meaningful simulation beyond the study of the NN reaction itself can be done. In other words, Geant4 needs the cross section to decide where an NN reaction will happen in simulation geometry.

Many cross section formulae for NN collisions are included in Geant4, such as Tripathi[TCW97] and Tripathi Light System[TCW99], Shen[SWF+89], Kox[eal87] and Sihver[STS+93]. These are empirical and parameterized formulae with theoretical insights and give total reaction cross section of wide variety of combination of projectile and target nucleus in fast. These cross sections are also used in the sampling of impact parameter in the QMD model.

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