Question
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
The correct answer is
$1\text{ K}$
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.
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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 is scattered by a central potential $V(r) = V_0 r e^{-\mu r}$, where $V_0$ and $\mu$ are positive constants. If the momentum transfer $\vec{q}$ is such that $q = |\vec{q}| \gg \mu$, the scattering cross-section in the Born approximation, as $q \to \infty$, depends on $q$ as
[You may use $\int x^n e^{ax} dx = \frac{d^n}{da^n} \int e^{ax} dx$] - 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
- The differential scattering cross-section $\frac{d\sigma}{d\Omega}$ for the central potential $V(r) = \frac{\beta}{r}e^{-\mu r}$, where $\beta$ and $\mu$ are positive constants, is calculated in the first Born approximation. Its dependence on the scattering angle $\theta$ is proportional to ($A$ is a constant below.)
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