Thermal Velocity Effects in Electron Guns

A method is presented for determining the effect of the transverse velocities of electron emission on the flow in accelerated electron streams. Complete expressions are derived giving the degree and character of the beam spread in a Pierce-type electron gun resulting from the transverse velocity distribution, and an expression is derived for the image magnification in the pin-hole camera beam analyzer described in a companion paper. The theory clearly shows the increase in effective (transverse) beam temperature caused by beam convergence.