def compute_cutoff_beta(T, frac=0.99):
"""Returns the velocity for which the fraction frac of a beam emitted from a thermionic
cathode with temperature T move more slowly in the longitudinal (z) direction.
Arguments:
T (float) : temperature of the cathode in K
frac (Optional[float]) : Fraction of beam with vz < cutoff. Defaults to 0.99.
Returns:
beta_cutoff (float) : cutoff velocity divided by c.
"""
sigma = np.sqrt(2*kb_J*T/m) # effective sigma for half-Gaussian distribution
multiplier = erfinv(frac) # the multiplier on sigma accounting for frac of the distribution
return multiplier*sigma/c
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