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A muon is an unstable elementary particl...

A `muon` is an unstable elementary particle whose mass is `207 m_e` and whose charge is `either^+ e` or `-_e`. A negative muon `(mu-1)` can be captured by a nucleus to form a muonic atom.
(a) A proton captures a `mu^-`. Find the radius of the first Bohr orbit of this atom.
(b) Find the ionization energy of the atom.

Text Solution

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Hence `m = 207m_(e) and M = 1836m_(e)`:so the reduced mass is
`mu = (M m)/(m + M) = ((207m_(e)) (1836m_(e)))/(207m_(e)) + (1836m_(e)) = 186 m_(e)`
According to equation ` lambda = h//p` , the orbid radius corresponding to `n = 1 is r_(1) = (h^(2) epsilon_(0))/(pi m_(e) e^(2)) = 5.29 xx 10^(-11)m`
Hence , the redius `r'` that corresponding to the reduced mass `mu` is
` r'_(1) = ((m)/(mu)) r_(1) = ((m_(e))/(1836m_(e))) r_(1) = 2.85 xx 10^(-13) m`
The muon is `186` times closer to the proton than an electron would be in the same orbit.
b. We have `E_(1) = ((mu)/(m)) e_(1) = 186 xx (-13.6) e V`
`implies E_(1) = - 2.53 xx 10^(3) e V = - 2.53 ke V`.
The ionization energy is , therefore , `2.53 ke V. 186` times that for an ordinary hydrogen atom.
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