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Atomic Structure 141

demonstrated in Sec. 8.6.) If mis the electron mass and Mthe nuclear mass, then m

is given by

Reduced mass m (4.22)

The quantity mis called the reduced massof the electron because its value is less

than m.

To take into account the motion of the nucleus in the hydrogen atom, then, all we

need do is replace the electron with a particle of mass m. The energy levels of the

atom then become

En  (4.23)

Owing to motion of the nucleus, all the energy levels of hydrogen are changed by the

fraction

0.99945

This represents an increase of 0.055 percent because the energies En, being smaller in

absolute value, are therefore less negative.

The use of Eq. (4.23) in place of (4.15) removes a small but definite discrepancy

between the predicted wavelengths of the spectral lines of hydrogen and the measured

ones. The value of the Rydberg constant Rto eight significant figures without correct-

ing for nuclear motion is 1.0973731  107 m^1 ; the correction lowers it to 1.0967758

 107 m^1.

The notion of reduced mass played an important part in the discovery of deuterium,

a variety of hydrogen whose atomic mass is almost exactly double that of ordinary

hydrogen because its nucleus contains a neutron as well as a proton. About one

hydrogen atom in 6000 is a deuterium atom. Because of the greater nuclear mass, the

spectral lines of deuterium are all shifted slightly to wavelengths shorter than the

corresponding ones of ordinary hydrogen. Thus the Hline of deuterium, which arises

from a transition from the n3 to the n2 energy level, occurs at a wavelength of

656.1 nm, whereas the Hline of hydrogen occurs at 656.3 nm. This difference in

wavelength was responsible for the identification of deuterium in 1932 by the

American chemist Harold Urey.

Example 4.6

A positronium“atom” is a system that consists of a positron and an electron that orbit each

other. Compare the wavelengths of the spectral lines of positronium with those of ordinary

hydrogen.

Solution

Here the two particles have the same mass m, so the reduced mass is

m  

m



2

m^2



2 m

mM



mM

M



Mm

m



m

E 1



n^2

m



m

1



n^2

me^4



8 ^20 h^2

Energy levels

corrected for

nuclear motion

mM



mM

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