Question

The muon is a subatomic particle with the same charge as an electron but with a mass that is $207$times greater: $m_e=207m_e$Physicists think of muons as "heavy electrons," However, the muon is not a stable particle; it decays with a half-life of $1.5\mu \mathrm{s}$into an electron plus two neutrinos. Muons from cosmic rays are sometimes "captured" by the nuclei of the atoms in a solid. A captured muon orbits this nucleus, like an electron, until it decays. Because the muon is often captured into an excited orbit $\left(\right(n>1)$, its presence can be detected by observing the photons emitted in transitions such as $2\to 1$and $3\to 1$.

Consider a muon captured by a carbon nucleus $(Z=6)$. Because of its long mass, the muon orbits well inside the electron cloud and is not affected by the electrons. Thus, the muon "sees" the full nuclear charge $2$and acts like the electron in a hydrogen like ion.

a. What is the orbital radius and speed of a muon in the $n=1$ground state? Note that the mass of a muon differs from the mass of an electron.

b. What is the wavelength of the $2\to 1$muon transition?

c. Is the photon emitted in the $2\to 1$transition infrared, visible, ultraviolet, or $\mathrm{x}$ray?

d. How many orbits will the muon complete during $1.5\mu$s? Is this a sufficiently large number that the Bohr model "makes sense, " even though the muon is not stable?

Short answer

(a)$r_1=42.59\mathrm{fm};v_1=1.31·{10}^7\frac{\mathrm{m}}{\mathrm{s}}$

(b) $\lambda =1.63·{10}^{-11}=16.3\mathrm{pm}$

(c) The photon that is emitted is a$x$-ray photon.

(d) $N=7.34·{10}^{13}$

Step-by-step solution

Step:1 Given information

(a)the orbital radius and speed of a muon in the $n=1$ground state? Note that the mass of a muon differs from the mass of an electron.

Calculation

We can begin with an expression for both the radius and muon.

$$\begin{gathered} a_{\mu }=\frac{4\pi {ϵ}_0{\hslash }^2}{m_{\mu }{e}^2} \\ {a}_B=\frac{4\pi {ϵ}_0{\hslash }^2}{m_e{e}^2} \\ {a}_B{m}_e=\frac{4\pi {ϵ}_0{\hslash }^2}{e^2} \\ {a}_{\mu }=\frac{m_e{a}_B}{m_{\mu }} \\ {a}_{\mu }=r_{1}Z \\ {r}_1=\frac{a_{\mu }}{Z} \\ {r}_1=\frac{m_e{a}_B}{m_{\mu }Z} \\ {r}_1=\frac{0.0529·{10}^{-9}}{207·6} \end{gathered}$$

$$\begin{gathered} r_1=42.59\mathrm{fm} \\ 2r\pi =n\lambda =n\frac{h}{mv} \end{gathered}$$

$$\begin{gathered} mvr=n\frac{h}{2\pi }=n\hslash \\ v=\frac{n\hslash }{mr} \\ {v}_1=\frac{\hslash }{m_{\mu }{r}_1} \\ {v}_1=\frac{1.05·{10}^{-34}}{207·9.11·{10}^{-31}·42.59·{10}^{-15}} \\ {v}_1=1.31·{10}^7\frac{\mathrm{m}}{\mathrm{s}} \end{gathered}$$

Given information

the $2\to 1$muon transition?

Calculation

We can now use the given equations to express the energy of stationary states:

$$\begin{gathered} E_n=-\frac{Z^2}{n^2}\frac{1}{4\pi {ϵ}_0}\frac{e^2}{2a_{\mu }} \\ {E}_1=-\frac{Z^2}{1^2}\frac{1}{4\pi {ϵ}_0}\frac{e^2}{2r_{1}Z} \\ {E}_1=-\frac{36}{1^2}\frac{1}{4\pi ·8.85·{10}^{-12}}\frac{{\left(1.6·{10}^{-19}\right)}^2}{2·42.59·{10}^{-15}·6} \\ {E}_1=-101567.73\mathrm{eV} \\ {E}_2=-\frac{36}{2^2}\frac{1}{4\pi ·8.85·{10}^{-12}}\frac{{\left(1.6·{10}^{-19}\right)}^2}{2·42.59·{10}^{-15}·6} \\ {E}_2=-25391.93\mathrm{eV} \\ {E}_{ph}=E_2-E_1 \\ {E}_{ph}=-25391.93+101567.73 \\ {E}_{ph}=76175.8\mathrm{eV} \\ \lambda =\frac{hc}{E_{ph}} \\ \lambda =\frac{6.63·{10}^{-34}·3·{10}^8}{76175.8·1.6·{10}^{-19}} \\ \lambda =1.63·{10}^{-11}=16.3\mathrm{pm} \end{gathered}$$

Step:5 Given information

the $2\to 1$transition infrared, visible, ultraviolet, or $\mathrm{x}$ ray?

Step: 6: Explanation

The photon that is emitted is an x-ray photon.

Given information

Is the muon complete during $1.5\mu$s? Is this a sufficiently large enough number that the Bohr model "makes sense, " even though the muon is not stable?

Step:8 Calculation

(d).The ratio of the muon's half-life to its orbital period can be used. :

$$\begin{gathered} N=\frac{\Delta t}{T} \\ T=\frac{2r_1\pi }{v_1} \\ N=\frac{\Delta t{v}_1}{2r_1\pi } \\ N=\frac{1.5·{10}^{-6}·1.31·{10}^7}{2·42.59·{10}^{-15·\pi }} \\ N=7.34·{10}^{13} \end{gathered}$$

This is large number that it makes sense to use the Bohr model