Problem 6.128 — Born Approximation: Scattering Cross Section

Problem Statement

Using the Born approximation, find the differential scattering cross section for a Yukawa potential $V = (V_0/r)e^{-r/a}$.

Given Information

• All quantities, constants, and constraints stated in the problem above
• Physical constants used as needed (see Concepts section)

Physical Concepts & Formulas

This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.

• Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
• State the mathematical form of that law as it applies here
• Check dimensions at every step: both sides of an equation must have the same units

Step-by-Step Solution

Problem Statement

Using the Born approximation, find the differential scattering cross section for a Yukawa potential $V = (V_0/r)e^{-r/a}$.

Given Information

• All quantities, constants, and constraints stated in the problem above
• Physical constants used as needed (see Concepts section)

Physical Concepts & Formulas

This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.

• Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
• State the mathematical form of that law as it applies here
• Check dimensions at every step: both sides of an equation must have the same units

Step-by-Step Solution

Using the Born approximation, find the differential scattering cross section for a Yukawa potential $V = (V_0/r)e^{-r/a}$.

Solution

Born approximation for differential cross section:

$$\frac{d\sigma}{d\Omega} = \frac{m^2}{4\pi^2\hbar^4}|\tilde{V}(q)|^2$$

where $\mathbf{q} = \mathbf{k}’ – \mathbf{k}$, $q = 2k\sin(\theta/2)$, and $\tilde{V}(q)$ is the 3D Fourier transform of $V(r)$.

Fourier transform of Yukawa potential:

$$\tilde{V}(q) = \int V(r)e^{i\mathbf{q}\cdot\mathbf{r}}d^3r = 4\pi V_0/(q^2+1/a^2)$$

Differential cross section:

$$\frac{d\sigma}{d\Omega} = \frac{m^2}{4\pi^2\hbar^4}\left(\frac{4\pi V_0}{q^2+a^{-2}}\right)^2 = \frac{4m^2V_0^2}{\hbar^4(q^2+a^{-2})^2}$$

For $a\to\infty$ (Coulomb potential $V_0 = Ze^2/4\pi\varepsilon_0$), this becomes the Rutherford formula.

Worked Calculation

Substituting all given numerical values with their units into the derived formula:

$$\text{Numerical result} = \text{given expression substituted with values}$$

Answer

$$\boxed{\frac{d\sigma}{d\Omega} = \frac{m^2}{4\pi^2\hbar^4}\left(\frac{4\pi V_0}{q^2+a^{-2}}\right)^2 = \frac{4m^2V_0^2}{\hbar^4(q^2+a^{-2})^2}}$$

Physical Interpretation

The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.

Worked Calculation

Substituting all given numerical values with their units into the derived formula:

$$\text{Numerical result} = \text{given expression substituted with values}$$

Answer

$$\boxed{\boxed{\frac{d\sigma}{d\Omega} = \frac{m^2}{4\pi^2\hbar^4}\left(\frac{4\pi V_0}{q^2+a^{-2}}\right)^2 = \frac{4m^2V_0^2}{\hbar^4(q^2+a^{-2})^2}}}$$

Physical Interpretation

The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.