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Explain and derive an expression of Reduced mass.

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In our earlier discussion we have assumed that the nucleus(a proton in case of hydrogen atom) remains at rest. With this assumption the values of the Rydberg constant R and the ionization energy of hydrogen predicted by Bohr’s analysis are within 0.1% of the measured values. 

Rather the proton and electron both revolve in circular orbits about their common centre of mass. We can take the motion of the nucleus into account simply by replacing the mass of electron m by the reduced mass µ of the electron and nucleus.

where M = mass of nucleus. The reduced mass can also be written as,

Now, when M > > m, m/M → 0 or  µ → m

For ordinary hydrogen we let M = 1836.2 m. Substituting in equation (i), we get µ = 0.99946 m when this value is used instead of the electron mass m in the Bohr equations, the predicted values are well within 0.1% of the measured values. 

The concept of reduced mass has other applications. A positron has the same rest mass as an electron but a charge +e. A positronium atom consists of an electron and a positron, each with mass m, in orbit around their common centre of mass. This structure lasts only about 10–6 s before two particles annihilate (combine) one another and disappear, but this is enough time to study the positronium spectrum. The reduced mass is m/2,so the energy levels and photon frequencies have exactly half the values for the simple Bohr model with infinite proton mass.

Now, let us prove why m is replaced by the reduced mass µ when motion of nucleus(proton) is also to be considered. In figure both the nucleus(mass = M, charge = e) and electron (mass = m, charge =e ) revolve about their centre of mass(CM) with same angular velocity (ω) but different linear speeds. Let r1 and r2 be the distance of CM from proton and electron. Let r be the distance between the proton and the electron. Then

Applying the Bohr model to positronium. The electron and the positron revolve about their common centre of mass, which is located midway between them because they have equal mass

Centripetal force to the electron is provided by the electrostatic force. So,

The expression for En without considering the motion of proton is

i.e.,m is replaced byµ while considering the motion of proton.

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