To solve this problem, we need to determine the maximum temperature for which the atom-atom scattering in hydrogen gas is dominantly $s$-wave. This is generally determined by the condition where the thermal de Broglie wavelength \(\lambda_T\) is comparable to or larger than the range of the potential. Let's go through this step-by-step:
- First, the thermal de Broglie wavelength is given by the formula:
- where \(h\) is the Planck's constant, \(m\) is the mass of the hydrogen atom, \(k_B\) is the Boltzmann constant, and \(T\) is the temperature in Kelvin.
- The range of the inter-atomic potential is given as \(5\text{ \AA}\), which is \(5 \times 10^{-10}\text{ m}\).
- For dominantly $s$-wave scattering, we need \(\lambda_T \geq 5 \times 10^{-10}\text{ m}\). Let's substitute and solve for \(T\):
- Rearranging for \(T\), we get:
- Knowing the constants:
- Planck's constant, \(h = 6.626 \times 10^{-34}\text{ Js}\)
- Mass of hydrogen atom, \(m = 1.67 \times 10^{-27}\text{ kg}\)
- Boltzmann constant, \(k_B = 1.38 \times 10^{-23}\text{ J/K}\)
- After substituting these values and solving, we find that \(T \approx 1\text{ K}\).
Therefore, the correct option is \(1\text{ K}\) as it satisfies the condition for $s$-wave scattering.
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 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?
