$$ \frac{d\sigma}{d(\cos\theta)} = \left| \sum_l (2l+1)e^{i\delta_l} \sin\delta_l P_l(\cos\theta) \right|^2 $$
where $\theta$ is the scattering angle. For a certain neutron-nucleus scattering, it is found that the two lowest phase shifts $\delta_0$ and $\delta_1$ corresponding to $s$-wave and $p$-wave, respectively, satisfy $\delta_1 \approx \delta_0/2$. Assuming that the other phase shifts are negligibly small, the differential cross-section reaches its minimum for $\cos\theta$ equal to
Partial Wave Expansion Cross-Section Analysis
The differential scattering cross-section calculation involves the partial wave expansion formula:
$ \frac{d\sigma}{d(\cos\theta)} = \left| \sum_l (2l+1)e^{i\delta_l} \sin\delta_l P_l(\cos\theta) \right|^2 $For neutron-nucleus scattering, we consider the low energy approximation where only the $s$-wave ($\delta_0$) and $p$-wave ($\delta_1$) phase shifts are significant. Thus, $\delta_l \approx 0$ for $l \ge 2$. We are also given the condition $\delta_1 \approx \delta_0 / 2$.
Simplified Cross-Section Calculation
With only $l=0$ and $l=1$ terms contributing, and using $P_0(\cos\theta) = 1$ and $P_1(\cos\theta) = \cos\theta$, the cross-section simplifies to:
$ \frac{d\sigma}{d(\cos\theta)} \approx \left| (2(0)+1)e^{i\delta_0} \sin\delta_0 P_0(\cos\theta) + (2(1)+1)e^{i\delta_1} \sin\delta_1 P_1(\cos\theta) \right|^2 $ $ \frac{d\sigma}{d(\cos\theta)} \approx \left| e^{i\delta_0} \sin\delta_0 + 3e^{i\delta_1} \sin\delta_1 \cos\theta \right|^2 $Let $x = \cos\theta$. Expanding the squared magnitude gives a quadratic function of $x$:
$ \frac{d\sigma}{d(\cos\theta)} = \sin^2\delta_0 + 9\sin^2\delta_1 x^2 + 6 \sin\delta_0 \sin\delta_1 \cos(\delta_0 - \delta_1) x $Finding the Minimum Condition
The expression is a quadratic $f(x) = C_2 x^2 + C_1 x + C_0$, where $C_2 = 9\sin^2\delta_1$ and $C_1 = 6 \sin\delta_0 \sin\delta_1 \cos(\delta_0 - \delta_1)$. The minimum occurs where the derivative with respect to $x$ is zero:
$ \frac{df}{dx} = 2C_2 x + C_1 = 0 $ $ x = -\frac{C_1}{2C_2} = -\frac{6 \sin\delta_0 \sin\delta_1 \cos(\delta_0 - \delta_1)}{2 \times 9\sin^2\delta_1} $ $ \cos\theta = -\frac{\sin\delta_0 \cos(\delta_0 - \delta_1)}{3\sin\delta_1} $Applying Phase Shift Approximation
We use the given approximation $\delta_0 \approx 2\delta_1$. Substituting this into the expression for $\cos\theta$:
$ \cos\theta \approx -\frac{\sin(2\delta_1) \cos(2\delta_1 - \delta_1)}{3\sin\delta_1} $Using the identity $\sin(2\delta_1) = 2\sin\delta_1\cos\delta_1$:
$ \cos\theta \approx -\frac{(2\sin\delta_1 \cos\delta_1) \cos(\delta_1)}{3\sin\delta_1} $Assuming $\sin\delta_1 \neq 0$, we simplify to find the value of $\cos\theta$ at the minimum:
$ \cos\theta \approx -\frac{2\sin\delta_1 \cos^2\delta_1}{3\sin\delta_1} = -\frac{2}{3}\cos^2\delta_1 $Important Questions from Scattering Theory
- When a neutron of $1 \text{ keV}$ kinetic energy impinges on a $^{12}\text{C}$ target, the total scattering cross section is $1000 \text{ barns}$. The approximate value of the phase shift $\delta_0$ is
- A particle of energy $E$ scatters off a repulsive spherical potential
$V(r) = \begin{cases} V_0 & \text{for } r < a \\ 0 & \text{for } r \ge a \end{cases}$
where $V_0$ and $a$ are positive constants. In the low energy limit, the total scattering cross-section is $\sigma = 4\pi a^2 \left(\frac{1}{ka}\tanh ka - 1\right)^2$, where $k^2 = \frac{2m}{\hbar^2}(V_0 - E) > 0$. In the limit $V_0 \to \infty$ the ratio of $\sigma$ to the classical scattering cross-section off a sphere of radius $a$ is A particle of mass $m$ and energy $E > 0$, in one dimension is scattered by the potential shown below.
If the particle was moving from $x = -\infty$ to $x = \infty$, which of the following graphs gives the best qualitative representation of the wavefunction of this particle?
- Consider the potential
$$V(\vec{r}) = \sum_i V_0 a^3 \delta^{(3)}(\vec{r} - \vec{r}_i)$$
where $\vec{r}_i$ are the position vectors of the vertices of a cube of length $a$ centered at the origin and $V_0$ is a constant. If $V_0 a^2 \ll \frac{\hbar^2}{m}$, the total scattering cross-section, in the low-energy limit, is - A sphere G of radius $b$ is fixed mid-air and several spheres identical to the first one are shot at it with their velocities parallel to each other. If the shot spheres fall within an imaginary cylinder of radius $a \ (b \ll a)$ then the fraction of spheres that will hit G is
