Matrix element for direct reactions

In summary, the matrix element for direct reactions is a quantity that describes the probability amplitude for a specific interaction to occur. It is calculated by summing over all possible intermediate states and integrating over the coordinates and momenta of the particles involved in the reaction. Various factors such as energy, momentum, and interaction type can affect its value. The matrix element is directly related to the cross section of a direct reaction, and can be experimentally measured by comparing theoretical predictions to precise and accurate measurements of the particles' properties.
  • #1
Silversonic
130
1
I apologise since I already have a question on this board, but I've been stuck for a good few hours understanding exactly how this has been done. The differential cross section for a direct reaction from [itex]\alpha[/itex] to [itex]\beta[/itex] is given by

[itex] \frac{d\sigma}{d\Omega} = f(k,k')|T_{\beta \alpha}|^2 [/itex]

f(k,k') is just a constant dependent on the initial and final momentums. For a one-nucleon transfer, e.g. A(B,C)D where B may be a deuteron and C a proton, [itex] T_{\alpha \beta} [/itex] is given by

[itex] T_{\alpha \beta} \int e^{-i\mathbf{k}\cdot\mathbf{r_{\beta}}} \int \psi^*_C \psi^*_D V \psi_A \psi_B d\tau e^{i\mathbf{k}\cdot\mathbf{r_{\alpha}}} d^3\mathbf{r_{\alpha}} d^3\mathbf{r_{\beta}} [/itex]

[itex] \alpha [/itex] represents the relative vector distance between incident particle A and target B
[itex] \beta [/itex] ^^ projectile C and residual nucleus D

V is the potential responsible for the direct reaction process.

[itex] d\tau [/itex] is the integral over all internal coordinates.

For reference Bertulani (nuclear physics in a nutshell) chapter 11.1 and Wong (intro to nuclear) chapter 8.3 focus on this.

There are three approximations then used:
(1) [itex] r_{\alpha} = r_{\beta} = r [/itex] because of the short range of [itex] V [/itex].
(2) [itex] V = V_0\delta(r-R) [/itex] because the interaction is short ranged, but below a certain distance the formation of a compound nucleus is more favourable.
(3) For a (d,p) reaction, which is what we're considering, the neutron is transferred to a single particle state (M=0) of B. i.e. [itex] \psi_D = \psi_B \phi_n = \psi_B f(r) Y_{L0}(\theta,\phi) [/itex]

What I'm then confused with is that I'm told these approximations are used to obtain

[itex] T_{\alpha \beta} = V_0\int e^{i (\mathbf{k - k'}) \cdot r} Y_{L0}(\theta,\phi)^*\delta(r-R) d^3\mathbf{r} [/itex]

I think some factors were left out because we're interesting in really the angular dependence of the differential cross section. I've tried getting to this myself, but I'm a bit confused as to what is a function of what variables for integration, and I also have no idea how [itex] Y_{L0} [/itex] makes its way outside of the integral for [itex] d\tau [/itex]. This is as far as I can get subbing in and using the approximations above;

[itex] T_{\alpha \beta} = V_0 \int e^{i (\mathbf{k - k'})\cdot\mathbf{r}} \int \psi^*_C \psi^*_D \delta(r-R) \psi_A \psi_B d\tau d^3\mathbf{r} [/itex]

So I guess all I need is

[itex] \int \psi^*_C \psi^*_D \psi_A \psi_B d\tau = Y_{L0}(\theta,\phi)[/itex]

But all I can show is;

[itex] \int \psi^*_C \psi^*_D \psi_A \psi_B = \int \psi^*_C \psi^*_B f(r')Y_{L0}(\theta',\phi') \psi_A \psi_B d\tau [/itex]

Then that's really as far as I can get. I don't understand how this results in [itex] Y_{L0} [/itex] when we're integrating over [itex] d\tau [/itex] - all internal coordinates of those wavefunctions. Anyone think they know?
 
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  • #2
In order to get the expression you provided, we need to make a few more assumptions. The first is that the wave functions for particles A and B are the same (this implies that they are in the same state). Secondly, we need to assume that the wave function for the particle C is the same as the wave function for particle D. With these two assumptions, we can then reduce the integral over dτ to just an integration over the coordinates of particle C (or equivalently, particle D). When doing this, we can replace the wave functions of particles A and B with the single wave function they both have, leading to the integral depending only on the wave function of C/D and the delta function. Finally, since we are integrating over the coordinates of C/D, we can pull the angular part of the wave function out, leaving us with the expression you provided. I hope this helps!
 

1. What is the matrix element for direct reactions?

The matrix element for direct reactions is a quantity that describes the probability amplitude for a specific interaction to occur. It takes into account the initial and final states of the particles involved in the reaction and their interaction potential.

2. How is the matrix element calculated?

The matrix element is calculated by summing over all possible intermediate states and integrating over the coordinates and momenta of the particles involved in the reaction. This can be a complex mathematical process and often requires advanced techniques such as perturbation theory or Feynman diagrams.

3. What factors can affect the value of the matrix element?

The value of the matrix element can be affected by various factors such as the energy and momentum of the incoming particles, the type of interaction (e.g. strong, weak, electromagnetic), the spin and isospin of the particles, and the shape of the interaction potential.

4. How does the matrix element relate to cross sections in direct reactions?

The matrix element is directly related to the cross section of a direct reaction. The cross section is a measure of the probability of a reaction occurring, and the matrix element is one of the key factors that determines this probability. A larger matrix element generally corresponds to a larger cross section.

5. Can the matrix element be experimentally measured?

Yes, the matrix element can be experimentally measured by studying the properties of the particles before and after the reaction and comparing them to theoretical predictions. This requires precise and accurate measurements of the particles' energies, momenta, and other properties, as well as a good understanding of the interaction potential.

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