Question

A muon is a particle very similar to an electron. It has the same charge but its mass is \(1.88 \cdot 10^{-28} \mathrm{~kg}\). a) Calculate the reduced mass for a hydrogen-like muonic atom consisting of a single proton and a muon. b) Calculate the ionization energy for such an atom, assuming the muon starts off in its ground state.

Short answer

Answer: The reduced mass of the muonic atom is \(1.671 \cdot 10^{-28} \mathrm{~kg}\) and the ionization energy is \(2.194 \cdot 10^{-17} \mathrm{~J}\).

Step-by-step solution

Find the reduced mass of the muonic atom

To find the reduced mass of the muonic atom, we will use the formula for reduced mass, which is given by: $$ \mu = \frac{m_1 m_2}{m_1 + m_2} $$ Here, \(m_1\) and \(m_2\) are the masses of the two particles in the system. In this case, \(m_1\) will be the mass of the proton and \(m_2\) will be the mass of the muon. The mass of a proton is approximately \(1.67 \cdot 10^{-27} \mathrm{~kg}\), and the mass of the muon is given as \(1.88 \cdot 10^{-28} \mathrm{~kg}\).

Calculate the reduced mass

Now, we can plug the values for the proton and muon masses into the formula for the reduced mass: $$ \mu = \frac{(1.67 \cdot 10^{-27} \mathrm{~kg})(1.88 \cdot 10^{-28} \mathrm{~kg})}{(1.67 \cdot 10^{-27} \mathrm{~kg}) + (1.88 \cdot 10^{-28} \mathrm{~kg})} = 1.671 \cdot 10^{-28} \mathrm{~kg} $$ So, the reduced mass of the muonic atom is \(1.671 \cdot 10^{-28} \mathrm{~kg}\).

Calculate the ionization energy using the Rydberg formula

To calculate the ionization energy of the muonic atom, we can use the Rydberg formula for hydrogen-like atoms, which is: $$ E_n = -\frac{Z^2e^4m_e}{2(4\pi\epsilon_0)^2n^2h^2} \cdot \frac{\mu}{m_e} $$ Here, - \(E_n\) is the energy of the nth energy level, - \(Z\) is the atomic number (for hydrogen, \(Z=1\)), - \(e\) is the charge of the electron (\(1.602 \cdot 10^{-19} \mathrm{~C}\)), - \(m_e\) is the mass of an electron (\(9.11 \cdot 10^{-31} \mathrm{~kg}\)), - \(4\pi\epsilon_0\) is a constant (\(8.99 \cdot 10^9 \mathrm{ ~N m^2 C^{-2}}\)), - \(n\) is the principal quantum number, - \(h\) is the Planck's constant (\(6.626 \cdot 10^{-34} \mathrm{~J s}\)), and - \(\mu\) is the reduced mass of the system. For the ground state, \(n=1\), and considering the substitution of the electron mass with reduced mass, we can plug the values into the Rydberg formula to find the ionization energy.

Calculate the ionization energy for the ground state

Now, we can plug the values into the Rydberg formula to find the ionization energy for the ground state: $$ E_1 = -\frac{1^2(1.602 \cdot 10^{-19} \mathrm{~C})^4(1.671 \cdot 10^{-28} \mathrm{~kg})}{2(8.99 \cdot 10^9 \mathrm{ ~N m^2 C^{-2}})^2(1)(6.626 \cdot 10^{-34} \mathrm{~J s})^2} = -2.194 \cdot 10^{-17} \mathrm{~J} $$ The ionization energy for the muonic atom is \(2.194 \cdot 10^{-17} \mathrm{~J}\). Since it is negative, we must take the absolute value of the energy to find the ionization energy, which is \(2.194 \cdot 10^{-17} \mathrm{~J}\).

Key concepts

Reduced Mass Formula

When delving into the realm of atomic physics and specifically muonic atoms, the concept of reduced mass becomes central. In a two-particle system, like a muon and a proton forming muonic hydrogen, the movements of the particles are interdependent. If we try to describe the motion of one particle in response to the other, things can get quite complicated due to their individual masses. This complexity is significantly simplified by introducing the idea of a single body moving under the influence of a given force, which is where the reduced mass comes in.

The reduced mass formula is expressed mathematically as
\(\frac{m_1 m_2}{m_1 + m_2}\) where \(m_1\) and \(m_2\) are the masses of the two interacting particles. In the case of a muonic hydrogen-like atom, this formula helps us understand the effective mass that plays a role in the atom's internal dynamic and energetic interactions. The reason behind the use of the reduced mass is that it accounts for the contribution of both particles' masses to their motion around their common center of mass. It ensures that the computed energy levels of the system reflect the unique dynamics of the muonic atom, which are different than those seen in a typical hydrogen atom due to the larger mass of the muon compared to the electron.

Rydberg Formula for Hydrogen-like Atoms

The Rydberg formula is an essential tool in understanding the energy levels of hydrogen-like atoms, in which a single electron orbits a nucleus. What distinguishes a hydrogen-like atom is its possession of only one electron, which makes its mathematical treatment somewhat similar to the hydrogen atom. Although initially formulated for hydrogen, the Rydberg formula can be extended to such one-electron systems, albeit with a specific adjustment - incorporating the reduced mass.

The generalized Rydberg formula takes the form
\(-\frac{Z^2 e^4 m_e}{2(4\tildepi\tildeepsilon_0)^2 n^2 h^2} \)
enhanced by the factor \(\frac{mu}{m_e}\) to account for different masses involved, where \(mu\) is the reduced mass of the system, \(m_e\) is the regular mass of an electron, and \(Z\) represents the atomic number. For a muonic atom, calculating the ionization energy requires plugging in the reduced mass of the muon-proton system into this formula. Such calculations allow scientists to predict the respective energy when the muon is removed from the atom - a phenomenon that is central to understanding the behavior of muonic atoms.

Principal Quantum Number

At the core of quantized energy levels in an atom lies the principal quantum number, symbolized as \(n\). This number is integral in determining the atom's electron configurations and their corresponding energy states. In a nutshell, the principal quantum number defines the size of the electron's orbit, or in quantum terms, the probability cloud around the nucleus.

Values of \(n\) are positive integers (1, 2, 3, ...), with each representing a different electron shell. As \(n\) increases, the electron's energy also increases, which means it is located further from the nucleus. In the case of the ground state, the electron or muon is in the lowest energy level, which corresponds to \(n=1\). When calculating ionization energies using the Rydberg formula for a muonic atom, setting the principal quantum number to reflect the ground state reveals the minimum amount of energy necessary to free the muon from the atom. Understanding the principal quantum number is crucial for students as it helps predict electron behavior and interaction energies within atoms.

Muon Mass

Characterizing a muon's properties is a pivotal step in understanding muonic atoms. The muon mass is significantly greater than that of its cousin, the electron, at approximately 200 times more massive. In numbers, the mass of a muon is around \(1.88 \times 10^{-28}\) kilograms, compared to the electron's mass of \(9.11 \times 10^{-31}\) kilograms. This difference in mass has profound implications for the behavior and binding energy of muonic atoms.

Physics observes that particles with greater mass, like muons, will orbit closer to the nucleus when taking the place of an electron in a hydrogen-like atom. Such tight orbits lead to energy levels that diverge from those predicted by models based solely on electronic hydrogen atoms. The greater mass of the muon also affects the calculation of the reduced mass in the atom, thus influencing the energy levels as determined by the Rydberg formula. As a consequence, muonic atoms exhibit different spectra and ionization energies, a fact which must be taken into account when studying atomic systems that include muons.