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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)

here

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

Thanks in advance to anyone helping

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