# Beam separation distance of silver atoms in Stern gerlach experiment

The Sge experiment is well known and the principle is well understood. I am asked the question to find the distance $$d$$ between the 2 beams after they are separated at the end of the magnets in the in.h magnet field in $$z$$ direction (with a gradient).

My problem is to prove that this distance between 2 beams $$d$$ is as follows $$d = \frac{a^2 e \hbar}{6 m_e k_B T} \frac{\partial B}{\partial z}$$

where $$T$$ is the temperature that the silver Ag 107 atoms are cooked in the oven. $$k_B$$ is probably the Boltzmann constant (is not specified in the problem.) And $$a$$ is the length of the magnet.

The Energy of all silver atoms can be taken to be the same at the value of $$E_{kin} = \frac{3}{2}k_B T$$

Picture for reference (it is in german)

Question 2 is, how would $$d$$ change for other silver isotopes?

Thanks in advance to anyone helping