Two projectile protons \( P_1 \) and \( P_2 \), both with spin up (along the \( +z \)-direction), are scattered from another fixed target proton \( T \) with spin up at rest in the \( xy \)-plane, as shown in the figure. They scatter one at a time. The nuclear interaction potential between both the projectiles and the target proton is \( \hat{\lambda} \vec{L} \cdot \vec{S} \), where \( \vec{L} \) is the orbital angular momentum of the system with respect to the target, \( \vec{S} \) is the spin angular momentum of the system, and \( \lambda \) is a negative constant in appropriate units. Which one of the following is correct?
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When dealing with spin-orbit interaction potentials like \( \hat{\lambda} \mathbf{L} \cdot \mathbf{S} \), the spin-alignment of the particles often causes them to scatter in the same direction due to the minimization of the total spin.
\( P_1 \) will be scattered in the \( +y \) direction (upward) and \( P_2 \) will be scattered in the \( -y \) direction (downward)
\( P_1 \) will be scattered in the \( +y \) direction (upward) and \( P_2 \) will be scattered in the \( +y \) direction (upward)
\( P_1 \) will be scattered in the \( -y \) direction (downward) and \( P_2 \) will be scattered in the \( +y \) direction (upward)
\( P_1 \) will be scattered in the \( -y \) direction (downward) and \( P_2 \) will be scattered in the \( -y \) direction (downward)
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The Correct Option isB
Solution and Explanation
In this problem, the protons \( P_1 \) and \( P_2 \) interact via a nuclear potential of the form \( \hat{\lambda} \mathbf{L} \cdot \mathbf{S} \), where \( \mathbf{L} \) is the orbital angular momentum and \( \mathbf{S} \) is the spin angular momentum.
1. Understanding the Potential:
The interaction term \( \hat{\lambda} \mathbf{L} \cdot \mathbf{S} \) implies that the energy of the system depends on the relative orientation of the orbital and spin angular momenta. Since \( \lambda < 0 \), the system energetically favors configurations where \( \mathbf{L} \) and \( \mathbf{S} \) are anti-parallel. This means the total energy is minimized when the spins are aligned in such a way that their dot product with orbital angular momentum is negative.
2. Effect of the Interaction:
The presence of the spin-orbit interaction causes a correlation between the spin direction and the angular distribution of the scattered protons. If the initial spins of both protons are aligned (e.g., in the \( +z \)-direction), the potential encourages a scattering configuration that aligns the angular momentum in such a way as to minimize the energy — typically resulting in both particles scattering in the same direction.
3. Scattering Directions:
As a result of the spin-orbit coupling and the alignment of spins, both protons \( P_1 \) and \( P_2 \) scatter in the same direction, which is identified as the \( +y \)-direction in this setup. This directional preference is due to conservation of total angular momentum and the attractive nature of the interaction when spins and orbital angular momenta are appropriately aligned.
