$cot(kr_0 + \delta) \approx -\frac{\gamma}{k}$
where $\delta$ is the phase shift, $k$ the wave number and $(-\gamma)$ the logarithmic derivative of the deuteron ground state wave function, the phase shift is
$\delta \approx -\frac{k}{y} - kr_0$
Scattering Phase Shift Approximation
Given Information
The problem involves the scattering of neutrons by protons at very low energy. The relationship provided is:
$ cot(kr_0 + \delta) \approx -\frac{\gamma}{k} $
Where:
- $r_0$ = range of the nuclear potential
- $k$ = wave number
- $\delta$ = phase shift
- $(-\gamma)$ = logarithmic derivative parameter
We need to find the approximation for the phase shift $\delta$. We assume $\gamma$ corresponds to $y$ in the options for consistency.
Derivation using Approximation
Let's evaluate the validity of the proposed phase shift approximation $\delta \approx -\frac{k}{y} - kr_0$. This approximation implies:
$ kr_0 + \delta \approx -\frac{k}{y} $
Substitute this expression for $kr_0 + \delta$ into the left side of the given equation:
$ cot\left(-\frac{k}{y}\right) \approx -\frac{\gamma}{k} $
For small values of the argument (as implied by very low energy, $k \to 0$), we use the Taylor expansion of $cot(x)$ around $x=0$: $cot(x) \approx \frac{1}{x} - \frac{x}{3}$.
Applying this expansion to the left side:
$ cot\left(-\frac{k}{y}\right) \approx \frac{1}{-k/y} - \frac{-k/y}{3} $
$ cot\left(-\frac{k}{y}\right) \approx -\frac{y}{k} + \frac{k}{3y} $
Now, let's assume $y = \gamma$ based on the options provided. The equation becomes:
$ -\frac{\gamma}{k} + \frac{k}{3\gamma} \approx -\frac{\gamma}{k} $
For the condition of very low energy ($k \to 0$), the term $\frac{k}{3\gamma}$ is negligible compared to $-\frac{\gamma}{k}$, provided $\gamma$ is finite and non-zero.
Therefore, the approximation holds true: $-\frac{\gamma}{k} \approx -\frac{\gamma}{k}$.
Conclusion
The approximation $\delta \approx -\frac{k}{y} - kr_0$ is consistent with the given scattering relation $cot(kr_0 + \delta) \approx -\frac{\gamma}{k}$ under the condition of very low energy.
Important Questions from Scattering Cross Section Phase Shift Method
- A particle is scattered from a potential $ V(\vec{r}) = g\delta^3(\vec{r}) $, where $ g $ is a positive constant. Using the first Born approximation, the angular $ (\theta, \phi) $ dependence of differential scattering cross section $ \frac{d\sigma}{d\Omega} $ is
Consider the potential $U(r)$ defined as $$U(r) = -U_0 \frac{e^{- \alpha r}}{r}$$ where $ \alpha$ and $U_0$ are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with $U(r)$ for a (wave-vector) momentum transfer $q$ and $ \alpha \to 0$, is proportional to
(Useful integral: $ \int_0^{ \infty} \sin(qr)e^{- \alpha r} dr = \frac{q}{ \alpha^2+q^2}$)
- The scattering of particles by a potential can be analyzed by Born approximation. In particular, if the scattered wave is replaced by an appropriate plane wave, the corresponding Born approximation is known as the first Born approximation. Such an approximation is valid for
- Consider an elastic scattering of particles in $l = 0$ states. If the corresponding phase shift $\delta_0$ is $90^\circ$ and the magnitude of the incident wave vector is equal to $\sqrt{2}\pi$ fm$^{-1}$ then the total scattering cross section in units of fm$^2$ is ________.
- Protons and $\alpha$-particles of equal initial momenta are scattered off a gold foil in a Rutherford scattering experiment. The scattering cross sections for proton on gold and $\alpha$-particle on gold are $\sigma_p$ and $\sigma_\alpha$ respectively. The ratio $\sigma_\alpha/\sigma_p$ is _________
