Question

The Born's approximation is applicable for:

• A. High energy, low atomic number for scatterer
• B. Low energy, low atomic number for scatterer
• C. High energy, high atomic number for scatterer
• D. Low energy, high atomic number for scatterer

Hint:

Born Approximation Rule of Thumb: Treat it as a weak interaction condition. - High Energy means the particle flies by too fast to be heavily distorted. - Low Atomic Number (\(Z\)) means the scattering target's electric field is weak enough to be treated as a small perturbation.

Answer 1

The Correct Option is A

Concept: In quantum scattering theory, the First Born Approximation is a perturbation technique used to calculate the scattering amplitude. It treats the scattering potential \(V(\mathbf{r})\) as a weak perturbation acting on an incoming free particle plane wave. For this approximation to yield accurate and physically valid results, the incident wave must be only minimally distorted by the presence of the scattering potential.

Step 1: Analyzing the mathematical validity condition.
Let \(V_0\) be the characteristic depth or strength of the scattering potential well, and let \(a\) represent the effective spatial range (radius) of the potential. The general validity criterion for the Born approximation requires that the potential must be small compared to a characteristic kinetic energy scale. Specifically, at high incident energies, the condition for validity takes the following form: \[ \frac{|V_0| a}{\hbar v} \ll 1 \] where \(v\) represents the magnitude of the velocity of the incoming particle.

Step 2: Interpreting the condition in terms of particle energy.
From the inequality established above, we can see that:
• As the incident velocity \(v\) increases, the fraction \(\frac{|V_0| a}{\hbar v}\) becomes significantly smaller.
• High velocity implies a very large kinetic energy (\(E = \frac{1}{2}mv^2\)).
• Physically, if the incident particle travels at a very high energy, it passes through the scattering region quickly. As a result, the potential has very little time to cause any major, non-linear distortions to the wavefunction. This makes perturbation theory highly accurate.

Step 3: Interpreting the condition in terms of the scatterer's properties.
Now let us analyze the strength of the potential \(V_0\):
• The strength of an atomic scattering field is directly determined by the total electrical charge of its nucleus, which is proportional to its atomic number (\(Z\)).
• A high atomic number \(Z\) creates a very strong, deep potential well \(V_0\), which violates the weak perturbation assumption (\(|V_0| \rightarrow \text{large}\)).
• Conversely, a low atomic number (\(Z\)) ensures that the scattering potential remains weak and shallow. This satisfies the required condition that \(|V_0|\) must be small. Combining these insights, the Born approximation is highly valid and effective when the incident particle has high energy and the target scatterer has a low atomic number. This aligns with option (1).