THE SPECTRUM OF HYDROGEN 247 



An account of the evidence for the molecular origin of the 

 secondary spectrum has been left to this point for the sake of 

 clearness. The assumption was made in the first place because 

 the Balmer series was known to come from the hydrogen atom, 

 and the total dissimilarity of the two spectra denoted a distinct 

 source. There is a very beautiful method for settling such 

 questions quantitatively, the theory of which has been discussed 

 very fully by Lord Rayleigh. The radiating particles in a gas 

 are moving with velocities governed by the Maxwell distribution 

 law. The light that they emit will, therefore, not be mono- 

 chromatic to an external observer, but will be spread over an 

 appreciable wave-length range by the Doppler effect. The 

 average number of particles with any given velocity in the line 

 of sight can be worked out from Maxwell's law, and the intensity 

 of the light they emit is clearly proportional to their number. 

 Every spectrum line, then, must have an intensity maximum, on 

 either side of which the intensity falls off according to a definite 

 law, which is actually 



L = L X e-^' 



where I, is the intensity at the maximum, I, the intensity at a 

 wave-length distance x from the central maximum, and <fe is a 

 constant. Now the mean velocity of the molecules of a gas, at 

 a given temperature, is inversely proportional to the square root 

 of the molecular weight, and so the magnitude of the Doppler 

 effect, and the corresponding broadening of the line, are func- 

 tions of the molecular weight of the radiating particles, and 

 their mean temperature. It is usual to express the breadth of 

 a line in terms of the distance on either side of the central 

 maximum in which the intensity drops to half its maximum 

 value, i.e. in the above formula it is the value of a; which makes 

 I^/I„=o-5. It is known as the " half-width" of the line. 

 Rayleigh arrived at the expression 



3X. = \ xk / I. 



V M 



where S\ is the half-width of a line of wave-length X, T is the 

 absolute temperature,^ M is the molecular weight of the radiating 

 particles, and ife is a constant. 



This means that the mass of the particles responsible for a 

 spectrum can be determined by measuring the half-widths of its 

 lines. Such measurements are not easy to accomplish. The 

 half-widths of the lines due to the hydrogen atom, where M has 

 its minimum value, are only of the order of 0-05 A, which is well 



^ It has been shown by Michelson that the mean temperature of the 

 radiating particles in a vacuum tube is no higher than that of the walls of the 

 tube. 



