HD Spectrum and the Mass of the Neutron

1 Introduction

Recall that the energy levels of a hydrogenic atom are proportional to the reduced mass m of the electron-nucleus system. This is almost the electron mass, but not precisely. The mass of the nucleus makes a small correction, and in the case of hydrogen and deuterium ("heavy hydrogen") the corrections are different. Thus the energy levels are slightly different in hydrogen and deuterium, and their emission spectra are slightly different. It takes high quality optics to be able to resolve this difference. In this experiment you will try to see it and measure it. If you can do that, you will have measured the mass of the deuterium nucleus, and hence, roughly speaking, the mass of the neutron (which is the extra mass in deuterium).

The theory of the effect is very simple. Every wavelength in the emission spectrum is inversely proportional to m, i.e.

l = A/m
(1)
where A is independent of m. Thus, if m should change by a small amount dm, the wavelength l would change by

dl = Ad(1/m)=lmd(1/m).
(2)
But in going from hydrogen to deuterium, i.e., from a proton as nucleus (m_p) to the deuteron as nucleus (m_d) we have

d(1/m)=(1/m_e+1/m_p)-(1/m_e+1/m_d)=1/m_p-1/m_d,
(3)
and, to excellent approximation, m » m_e, i.e., the reduced mass is the electron mass. Thus

dl
l
» m_e
m_p
æ
è 1- m_p
m_d
ö
ø
(4)
or, finally,

m_p/m_d=1-(m_p/m_e)(dl/l)
(5)
The ratio of the proton mass to the electron mass is well known to be about 1836. Thus we just choose a line l and measure the HD splitting dl.

2 Experiment

The source will be essentially a Balmer tube, but with both hydrogen and deuterium gas in it. The only experimental challenge is to get good resolution in order to measure the separation accurately. Notice that you could also use a Michelson interferometer to do this if you used a filter to look at just one color, say red. The red Balmer lines from H and D would form fringes with slightly different periodicities. Be sure to estimate the uncertainty in your determination of m_p/m_d, due mainly to the uncertainty in determining dl.

File translated from T_EX by T_TH, version 3.00.
On 27 Aug 2001, 17:37.