Collision Fraction Calculation
This problem asks for the proportion of incoming spheres that collide with a fixed sphere G, given constraints on their trajectories.
- Fixed Sphere G: Has a radius of $b$.
- Shot Spheres: These spheres are identical to G, meaning they also have a radius of $b$. Their velocities are parallel to each other.
- Trajectory Constraint: The centers of the shot spheres are confined within an imaginary cylinder of radius $a$. We are given that $b \ll a$, meaning $b$ is significantly smaller than $a$.
Hit Condition Analysis
A collision between a shot sphere and the fixed sphere G occurs if the distance between their centers is less than or equal to the sum of their radii.
- The radius of the fixed sphere G is $b$.
- The radius of each identical shot sphere is also $b$.
- The sum of their radii is $b + b = 2b$.
- Therefore, for a hit to occur, the center of a shot sphere must pass within a distance of $2b$ from the center of the fixed sphere G.
Area Calculations for Probability
To find the fraction of spheres that hit G, we compare the "effective target area" to the total area where the centers of the shot spheres can land. We analyze this in a 2D plane perpendicular to the parallel velocities.
- Effective Target Area: This is the area representing the successful collision condition. It's a circle centered on G with a radius equal to the distance required for a hit, which is $2b$.
$ \text{Area}_{\text{target}} = \pi \times (\text{radius})^2 = \pi \times (2b)^2 = 4\pi b^2 $. - Total Possible Area: This is the area defined by the cylinder's radius $a$, within which the centers of the shot spheres land. It's a circle with radius $a$.
$ \text{Area}_{\text{total}} = \pi \times (\text{radius})^2 = \pi a^2 $.
Fraction of Collisions Calculation
The fraction of spheres that hit G is the ratio of the effective target area to the total possible area available for the centers of the shot spheres.
Fraction = $ \frac{\text{Area}_{\text{target}}}{\text{Area}_{\text{total}}} $
Substituting the calculated areas:
Fraction = $ \frac{4\pi b^2}{\pi a^2} $
After simplification, the fraction is:
Fraction = $ \frac{4b^2}{a^2} $
Important Questions from Scattering Theory
- A phase shift of $30^\circ$ is observed when a beam of particles of energy $0.1$ MeV is scattered by a target. When the beam energy is changed, the observed phase shift is $60^\circ$. Assuming that only s-wave scattering is relevant and that the cross-section does not change with energy, the beam energy is
- 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 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 - In the partial wave expansion, the differential scattering cross-section is given by
$$ \frac{d\sigma}{d(\cos\theta)} = \left| \sum_l (2l+1)e^{i\delta_l} \sin\delta_l P_l(\cos\theta) \right|^2 $$
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 - The range of the inter-atomic potential in gaseous hydrogen is approximately $5\text{ \AA}$. In thermal equilibrium, the maximum temperature for which the atom-atom scattering is dominantly $s$-wave, is
