3.1 Optical Model (OM)

The simplest calculation that can be performed with FRESCO is the standard OM calculation. Here, the interaction between the projectile and target is described in terms of a complex potential, whose imaginary part accounts for the loss of flux in the elastic channel going to any other channels.

The elastic differential cross section is evaluated through the expression¹

$$\frac{d\sigma}{d\Omega}(\theta)=\vert f(\theta)\vert^{2}\,,$$ (1)

where $f(\theta)$ is the scattering amplitude. The quantity $v\vert f(\theta)\vert^{2}/R^{2}$ represents the flux of particles elastically scattered by the target at angle $\theta$, with v the asymptotic velocity of the outgoing particles. This is obtained from the asymptotic expression of the scattering wave function:

$$\Psi(\mathbf{K},\mathbf{R})\rightarrow e^{i\mathbf{K}\cdot\mathbf{R}}+f(\theta)\frac{e^{iKR}}{R}\,,$$ (2)

where K denotes the incident linear momentum of the projectile in the center of mass system and R is the relative coordinate between projectile and target.

This in turn is calculated by solving the Schrödinger equation for the complex potential U(R):

$$\left[\frac{\hbar^{2}}{2\mu}\nabla^{2}+U(R)-E\right]\Psi(\mathbf{K},\mathbf{R})=0$$ (3)

where $\mu$ is the reduced mass and E is the energy in the center of mass system: $E=\hbar^{2}K^{2}/2\mu$. This equation has a solution with the form of a plane wave with relative momentum K plus outgoing scattered waves:

$$\Psi(\mathbf{K},\mathbf{R})=e^{i\mathbf{K}\cdot\mathbf{R}}+\chi^{scat}(\mathbf{K},\mathbf{R})$$ (4)

where $\chi^{(+)}(\mathbf{K},\mathbf{R})$ represents the set of scattered waves. Notice that in absence of the target (U(R)=0) there are not such scattered waves and $\Psi(\mathbf{K},\mathbf{R})=e^{i\mathbf{K}\cdot\mathbf{R}}$. Commonly, the wave function $\Psi(\mathbf{K},\mathbf{R})$ is decomposed in partial waves, in order to separate the angular and radial parts:

$$\Psi(\mathbf{K},\mathbf{R})=\frac{1}{KR}\sum_{L}(2L+1)i^{L}\chi_{L}(K,R)P_{L}(\cos\theta),$$ (5)

where L is the orbital angular momentum between the projectile and target and $\theta$ is the angle between K and R.

Replacing this solution into the Schrödinger equation (3) we get the following equation for the radial functions $\chi_{L}(K,R)$

$$\left[\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{dR^{2}}-\frac{\hbar^{2}}{2\mu}\frac{L(L+1)}{R^{2}}-U(R)+E\right]\chi_{L}(K,R)=0.$$ (6)

Asymptotically, the radial functions behave as

$$\chi_{L}(K,R)\rightarrow\frac{1}{2}i[H_{L}(KR)^{*}-S_{L}H_{L}(KR)].$$

Here, we distinguish two cases:

• U(R) is a purely short range potential (i.e., decays faster than 1/R). In this case $H_{L}(\rho)\rightarrow\rho h^{(+)}(\rho)$, where h_L⁽⁺⁾ is a spherical Hankel function of the first kind [1]
• U(R) contains a long range part. In this case, $H_{L}(\rho)$ are the so called Coulomb wave functions.

The coefficients S_L are the scattering matrix (or S-matrix) elements. They are related to the nuclear phase-shifts, $\delta_{L}$, by:

$$S_{L}=e^{2i\delta_{L}}$$ (7)

These are very important quantities as all the information of the effect that the target produces on the scattering wave function (and hence in the observables) is contained in these coefficients. Hence it is possible to write the all the scattering observables in terms of the S-matrix elements. Notice that in the absence of target S_L=1 for all partial waves. Even in the presence of a point Coulomb interaction, these coefficients remains equal to one, as the Coulomb phase-shifts are already included in the Coulomb wave functions H_L(KR). It is also important to note that, if only real potentials are involved, the S-matrices verify |S_L|=1 and the phase shifts $\delta_{L}$ are real numbers. This expresses the conservation of flux of particles (i.e., the number of scattered particles equals the number of incident particles). On the contrary, if the scattering potential contains an imaginary part, then |S_L|<1 and $\delta_{L}$ become complex numbers. In this case, the outgoing flux of particles is less than the incoming flux, indicating that part of the incident flux leaves the elastic channel and goes to other channels.

Then, the calculation of the elastic cross section involves the following steps:

• Integration of the differential equation (6) for each value of L. The integration is carried out starting from R=0 and solving the differential equation in steps of $\Delta R$ up to a certain maximum value R_m.
• For R large enough, the solution (5) obeys the asymptotic form (2). Thus, choosing the value of R_m large enough it is possible to identify the scattering amplitude $f(\theta$) comparison of (2) and the solution (5) for R=R_m.
  
  $$f(\theta)=f_{C}(\theta)+f'(\theta)\,\,,$$
  
  
  
  

  $$f'(\theta)=\frac{1}{2iK}\sum_{L}(2L+1)e^{2i\sigma_{L}}(S_{L}-1)P_{L}(\cos\theta)\,.$$ (8)

  
  

• Finally, the differential elastic cross section is evaluated according to Eq. (1).

It results obvious that the particular choice of $\Delta R$ and R_m depends on each particular problem. If only short-ranged potentials are involved (eg. neutron scattering) the value of R_m should be chosen outside of the range of the projectile-target interaction. However, in general U(R) has both Coulomb and nuclear parts. The Coulomb part has a long (``infinite'') range that requires special treatment. Actually, in this case the asymptotic solution of (3) does not behave as planes waves and so expression (2) is not strictly valid. In particular, planes waves should be replaced by the so called Coulomb functions, which are the solutions of the Schrodinger equation in presence of the Coulomb potential alone. For our purposes, the important point to remind is that in that when long-range interactions are present, the radial equations have to be solved up to larger distances in order to ``reach'' their asymptotic form.

Concerning the radial step, $\Delta R$, its choice will depend mainly on the diffusiveness of the potentials. Very abrupt or sharp potentials will normally require an smaller step.

Finally, a maximum value of L, denoted (L_max) has to be chosen. In principle, the sum in (5) goes to infinity. In practice, convergence of the scattering observables is achieved for finite values of L_max. Within a semiclassical picture, the orbital angular momentum can be related to the impact parameter, b, and the incident linear momentum by $Kb\approx L+1/2$. This means that, for a fixed incident energy, large impact parameters correspond also to large values of the orbital angular momentum. If only short-ranged potentials are present, these values of L do not feel the potential of the target. In presence of the Coulomb potential, these large L values are commonly identified with trajectories that explore uniquely the Coulomb part of the projectile-target interaction.

The parameters discussed above are controlled by the following variables in fresco:

-

rmatch: Matching radius (R_m in the discussion above)

-

hcm: Radial step

-

lmax: Maximum partial wave. Fresco also allows fixing the minimum partial wave, through the variable LMIN. However, in normal calculations this will be set to zero.

In the pure optical model approach only the ground state of the projectile and targets are considered explicitly. Thus, a OM calculation only provides the elastic scattering cross section. The loss of flux effect from the elastic channel to the excluded channels (excitations, rearrangement reactions, etc) is assumed to be included in the imaginary part of the optical potential.