Question

Particles called muons exist in cosmic rays and can be created in particle accelerators. Muons are very similar to electrons, having the same charge and spin, but they have a mass 207 times greater. When muons are captured by an atom, they orbit just like an electron but with a smaller radius, since the mass in\({{\bf{a}}_{\bf{B}}}{\bf{ = }}\frac{{{{\bf{h}}^{\bf{2}}}}}{{{\bf{4}}{{\bf{\pi }}^{\bf{2}}}{{\bf{m}}_{\bf{e}}}{\bf{kq}}_{\bf{e}}^{\bf{2}}}}{\bf{ = 0}}{\bf{.529 \times 1}}{{\bf{0}}^{{\bf{ - 10}}}}{\bf{\;m is 207}}{{\bf{m}}_{\bf{e}}}\)

(a) Calculate the radius of then = 1 orbit for a muon in a uranium ion\(\left (Z = 92).

(b) Compare this with the7.5 - fmradius of a uranium nucleus. Note that since the muon orbits inside the electron, it falls into a hydrogen-like orbit. Since your answer is less than the radius of the nucleus, you can see that the photons emitted as the muon falls into its lowest orbit can give information about the nucleus.

Short answer

a.The radius of the muon around the uranium nucleus is 2.78 fm.

b. The radius of muon is 2.78 fm is smaller than the radius of uranium nucleus, which is 7.5 fm.

Step-by-step solution

The radius of the muon around the uranium nucleus

(a)

Consider the formula for the radius of the electron atom is:

\({{\bf{r}}_{\bf{n}}}{\bf{ = }}\frac{{{{\bf{n}}^{\bf{2}}}{{\bf{a}}_{\bf{B}}}}}{{\bf{Z}}}\)

Here,\({{\rm{a}}_{\rm{B}}}\)is Bohr’s radius, and for electron, the Bohr’s radius is

\({a_B} = 0.529 \times {10^{ - 10}}\;{\rm{m}}\)

Consider the mass of muon is 207 times higher than electron, hence, the Bohr’s radius for muon is given by:

\(\begin{align}{}{a_{B\mu }} &= \frac{{{a_B}}}{{207}}\\ &= \frac{{0.529 \times {{10}^{ - 10}}\;{\rm{m}}}}{{207}}\\ &= 2.55 \times {10^{ - 13}}\;\;{\rm{m}}\end{align}\)

Hence, the radius of the muon around the uranium nucleus is

\(\begin{align}{}r &= \frac{{{n^2}{a_{B\mu }}}}{Z}\\ &= \frac{{{1^2}\left( {2.55 \times {{10}^{ - 13}}\;{\rm{m}}} \right)}}{{92}}\\ &= 2.78 \times {10^{ - 15}}\;{\rm{m}}\\ &= 2.78\;{\rm{fm}}\end{align}\)

Hence, the radius of the muon around the uranium nucleus is 2.78 fm.

Determine the radius of muon

(b)

Consider the radius of muon is 2.78 fm is smaller than the radius of uranium nucleus, which is 7.5 fm.