In 1927 T. E. Phipps and J. B. Taylor of the University of Illinois reported an important experiment similar to the Stern-Gerlach experiment but using hydrogen atoms instead of silver. This was done because hydrogen is the simplest atom, and the separation of the atomic beam in the inhomogeneous magnetic field would allow a clearer interpretation. The atomic hydrogen beam was produced in a discharge tube having a temperature of 663 K. The highly collimated beam passed along the x direction through an inhomogeneous field (of length 3 cm) having an average gradient of 1240 T/m along the z direction. If the magnetic moment of the hydrogen atom is 1 Bohr magneton, what is the separation of the atomic beam?
Strategy The force can be found from the potential energy of Equation (7.31).
V_{B}=-\mu_{z} B=+\mu_{ B } m_{\ell} B (7.31)
F_{z}=-\frac{d V}{d z}=\mu_{z} \frac{d B}{d z}The acceleration of the hydrogen atom along the z direction is a_{z}=F_{z} / m. The separation of the atom along the z direction due to this acceleration is d=a_{z} t^{2} / 2. The time that the atom spends within the inhomogeneous field is t=\Delta x / v_{x} where Δx is the length of the inhomogeneous field, and v_{x} is the constant speed of the atom within the field. The separation d is therefore found from
d=\frac{1}{2} a_{z} t^{2}=\frac{1}{2}\left(\frac{F_{z}}{m}\right) t^{2}=\frac{1}{2 m}\left(\mu_{z} \frac{d B}{d z}\right)\left(\frac{\Delta x}{v_{x}}\right)^{2}We know all the values needed to determine d except the speed v_{x}, but we do know the temperature of the hydrogen gas. The average energy of the atoms collimated along the x direction is \frac{1}{2} m\left\langle v_{x}^{2}\right\rangle=\frac{3}{2} k T.
