Astronomy 217

 
 

Prof. Andrew W. Steiner

 
 
Oct. 4, 2021
 
 

TA James Ternullo

Last Time

  • Wave Theory of Light
  • Photoelectric Effect
  • Kirchoff's Laws

Today

  • Light and Atoms

Orbiting Electron

  • To model the atom, one can equate the electrical force of the atomic nucleus on an electron with the centripetal force, creating an election orbital $$ F_e = \frac{K_e q_1 q_2}{r^2} = - \frac{m v^2}{r} = F_{\mathrm{cen}} $$
  • Substituting the values of \( K_e \) and q $$ \frac{(Ze)(-e)}{4 \pi \varepsilon_0 r^2} = - \frac{m_e v^2}{r} $$
  • This gives an expression for the Kinetic Energy $$ K = \frac{1}{2} m_e v^2 = \frac{Z e^2}{8 \pi \varepsilon_0 r} $$

Orbital Energy

  • As we showed for gravity, one can integrate a central force to calculate a potential energy, \( U_E \) $$ U_E = \int_{R}^{\infty} \frac{1}{4 \pi \varepsilon_0} \frac{q}{r^2}~dr = - \frac{Z e^2}{4 \pi \varepsilon_0 r} $$
  • Given this potential energy and the kinetic energy, the total energy of the electron is $$ E = K + U = \frac{Z e^2}{8 \pi \varepsilon_0 r} - \frac{Z e^2}{4 \pi \varepsilon_0 r} = - \frac{Z e^2}{8 \pi \varepsilon_0}{r} $$
  • As in the case of the circular gravitational orbit, the energy is negative, with \( - \left< U \right> = 2 \left< K \right> = - 2 \left< E \right> \)

Bohr's Insight

  • While these expressions for the energy are true, they are also classical expressions and, like those for gravity, continuous. All values of E, or equivalently r, are valid.
  • Bohr, recognizing the discrete nature of the spectra, and following the quantum nature of light upon which Einstein’s explanation of the photoelectric effect and Planck’s derivation of the Blackbody radiation curve depend, proposed that the electron’s motion was also quantized.
  • Specifically, the orbital angular momentum can only have prescribed values $$ L = \frac{n h}{2 \pi} = n \hbar = m_e v r $$ where \( h = 6.626 \times 10^{-34}~\mathrm{J}~ \mathrm{s} \) is Planck's constant

Bohr's Atom

  • Returning to the equality of electrostatic and centripetal forces $$ \frac{(Ze)(-e)}{4 \pi \varepsilon_0 r^2} = - \frac{m_e v^2}{r} $$
  • Using Bohr's insight, the velocity is \( v = n \hbar / (m_e r) \) $$ \frac{Z e^2}{4 \pi \varepsilon_0 r} = m_e v^2 = \frac{n^2 \hbar^2}{m_e r^2} $$
  • This yields a radius that depends on n and Z $$ r_n = \frac{4 \pi \varepsilon_0 \hbar^2}{Z e^2 m_e} n^2 $$

Quantized Energy

  • Quantization of the angular momentum and radius naturally leads to quantization of the orbital energy $$ E_n = K + U = - \frac{Z e^2}{8 \pi \varepsilon_0} \frac{1}{r_n} = - \frac{Z e^2}{8 \pi \varepsilon_0} \frac{Z e^2 M_e}{4 \pi \varepsilon_0 \hbar^2} \frac{1}{n^2} $$
  • This can be written in terms of the "electron volt", the energy gained by an electron crossing a 1 Volt potential difference $$ E_n = - \left( \frac{e^2}{4 \pi \varepsilon_0} \right)^2 \frac{m_e}{2 \hbar^2} \frac{Z^2}{n^2} = - \frac{m_e c^2 \alpha^2}{2} \frac{Z^2}{n^2} = -13.6 \frac{Z^2}{n^2}~\mathrm{eV} $$ where \( \alpha \equiv e^2/(4 \pi \varepsilon_0 \hbar c) \approx 1/137 \)
  • Thus the eV provides a natural energy scale for the electron's orbital energy

Photons

  • An electron going from the \( n_2 \) orbit to a lower \( n_1 \) orbit becomes more bound by $$ \Delta E = E_{n_2} - E_{n_1} = 13.6~Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)~ \mathrm{eV} $$
  • Energy conservation requires that this much energy be emitted from the atom $$ \frac{\hbar c}{\lambda} = \Delta E = \frac{m_e c^2 \alpha^2}{2} Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) $$
  • This is called Rydberg's formula, and the "Rydberg constant is defined $$ R_H \equiv \frac{m_e c \alpha^2}{2 \hbar} $$

Hydrogenic Atoms

  • Bohr’s model for the Hydrogen atom was a tremendous success, providing a physical basis for the empirical relations of Balmer and Rydberg.
  • Photon are emitted as electrons make atomic transitions, descending from excited states toward the ground state, with each named series corresponding to different destination state.
  • Absorption of photons of the appropriate energies allow electrons to move to excited states.

Beyond Bohr's Atom

  • While wildly successful for mono-electron atoms, hydrogen or nearly fully ionized heavier atoms, Bohr’s model fails for multi-electron atoms.
  • Modern QM replaces orbitals with probability distributions.
  • Electrons are fermions, with half-integer spin, obeying the Pauli exclusion principle.
  • Bohr’s quantum number becomes the principal quantum number, but additional quantum numbers for orbital angular momentum and spin.

Emission Lines

  • Following Bohr’s model, emission lines can also be understood as atomic photo-de-excitation transitions, descending from excited states toward the ground state.
  • Photodeexcitation comes in two forms, spontaneous emission $$ X^{*} \rightarrow X + h \nu $$ and stimulated emission $$ X^{*} + h \nu \rightarrow X + h \nu + h \nu $$
  • Stimulated emission is the mechanism on which lasers and masers are based.

Atomic Processes

  • Both forms of photodeexcitation require the atom to be in an excited state, X*.
  • One way to achieve this is photoexcitation, but that is a zero sum game. The photons emitted may not be the same as those absorbed, but the total energy is the same.
  • Another excitation process is collisional excitation, where a collision with a free electron or another atom leaves the atom in an excited state, $$ X + \frac{1}{2} m v^2 \rightarrow X^{*} + \frac{1}{2} m v^{\prime 2} $$ with the initial \( v \) larger than the final \( v^{\prime} \).
  • Collisional de-excitation $$ X^{*} + \frac{1}{2} m v^2 \rightarrow X + \frac{1}{2} m v^{\prime 2} $$ is also a possibility, with the final \( v^{\prime} \) larger than the initial \( v \).

Ionization

  • If sufficiently energetic, atomic transitions can excite an electron to energies above \( n_{\infty} \). This removes an electron, ionizing the atom.
  • Photoionization: $$ X + h \nu \rightarrow X^{+} + \frac{1}{2} m_e v_e^2 $$ requires \( h \nu > E_{\infty} - E_n + \frac{1}{2} m_e v_e^2 \) (For Hydrogen in the ground state, \( h \nu > 13.6~\mathrm{eV} \)
  • \( E_{\infty} - E_{1} \) is called the "ionization potential"
  • Collisional ionization: $$ X + \frac{1}{2} m v^2 \rightarrow X^{+} + \frac{1}{2} m v^{\prime 2} + \frac{1}{2} m_e v_e^2 $$ requires a similar energy exchange from the colliding atom or electron
  • The reverse process is called recombination: $$ X^{+} + \frac{1}{2} m v^{\prime 2} + \frac{1}{2} m_e v_e^2 \rightarrow X + h \nu $$

Thermal Velocity

  • A common source of collisional excitation is thermal motion of the gas particles.
  • In an ideal gas, the distribution of particle velocities is given by the Maxwell-Boltzmann distribution $$ F(v)~dv = 4 \pi \left( \frac{m}{2 \pi kT} \right) v^2 \exp \left( - \frac{m v^2}{2 k T} \right)~dv $$
  • Computing an average quantity requires integrating over the distribution.
  • For example, the mean velocity and energy are $$ \left< v \right> = \int v F(v)~dv = \left( \frac{8 k T}{\pi m} \right)^{1/2} \left< E \right> = \int E F(v)~dv = \frac{3}{2} k T $$

Thermal Equilibrium

  • If the rate of collisional excitation is sufficient, the populations of the excited states and even ionization states also obey the thermal temperature. This is termed local thermodynamic equilibrium (LTE).
  • In LTE, the relative populations obey the Boltzmann equation $$ \frac{n_2}{n_1} = \frac{g_2}{g_1} \exp \left( \frac{E_1 - E_2}{k T} \right) $$ (Note since \( E_1 - E_2 < 0 \), then \( n_2 < n_1 \)
  • LTE also applies ot the ionization states of an atom, the populations obey the Saha equation $$ \frac{n^{i+1} n_e}{n^i} = 2 \frac{Q^{i+1}}{Q^i} \left( \frac{m_e k T}{2 \pi \hbar^2} \right)^{3/2} \exp \left( \frac{E^{i}_1 - E^{i}_{\infty}}{kT} \right) $$ where the term in the exponential is the inverse of the ionization potential

Thermal Linewidth

  • The same thermal motion which can cause collisional excitation also results in doppler shift in the emitted light.
  • The combination of motion toward and away from the observer produces a velocity dispersion. $$ \left< \sigma_z \right> = \left( \frac{k T}{m} \right)^{1/2} \approx 100 \left( \frac{T}{1~{\mathrm{K}}} \right)^{1/2} \mu^{-1/2}~\mathrm{m}~\mathrm{s}^{-1} $$ where \( \mu \equiv m/m_p \) is the mean molecular mass
  • and $$ \frac{\Delta \lambda}{\lambda} \approx \frac{\sigma_z}{c} \approx 3 \times 10^{7} \left( \frac{T}{1~{\mathrm{K}}}\right)^{1/2} \mu^{-1/2} $$

Line Profile

  • Each process gives an intrinsic shape to the line called the line profile. For example, the thermal line profile is $$ \phi(\nu)~d\nu = \left( \frac{m c^2}{2 \pi kT} \right)^{1/2} \exp \left[ - \frac{m c^2}{2 \pi k T} \frac{ \left( \nu - \nu_0 \right)^2}{\nu_0^2} \right] \frac{d \nu}{\nu_0} $$
  • This is a Gaussian, centered on \( \nu_0 \) with the width given by the velocity dispersion $$ \left< \sigma_z \right> = \left( \frac{k T}{m} \right)^{1/2} $$

Intrinsic Linewidth

  • Even at zero temperature, the emission and absorption lines are not infinitesimally thin. As a result of the finite lifetime of each state, there is a finite uncertainty in the energy of the photon
  • Heisenberg’s uncertainty principle $$ \Delta x \cdot \Delta p \geq \hbar \leq \Delta E \cdot \Delta t \approx \Delta E \cdot \tau $$
  • Thus the intrinsic linewidth is inversely proportional to the lifetime, τ, or directly proportional to the decay probability, called the Einstein A coefficient, \( A_{nm} \) $$ \phi(\nu)~d\nu = \frac{\gamma_n}{(\nu - \nu_0)^2 + (\gamma_n/4 pi)^2} \frac{d \nu}{4 \pi^2} $$ where $$ \gamma_n = \sum_{n^{\prime} < n} A_{n n^{\prime}} $$
  • Then \( \Delta \lambda/\lambda \approx \gamma_n c / ( 4 \pi \lambda ) \)

Bulk Motions

  • Larger scale motions also affect line profiles.
  • For motions that can be resolved, in space or time, line shifts result.
  • For example, the Sun’s rotation results in a line shift between the approaching and receding side of the Sun.
  • However, the same motions for a distant star result in a line broadening as we can’t separate the sides.

Other Line Effects

  • Turbulent motions, unresolved at a distance, also result in line broadening. Because all species move at the same turbulent velocity, the linewidths are independent of the element (μ), unlike thermal broadening.
  • Rotational broadening is also independent of μ
  • Under high pressure conditions, the overlapping electric fields of nearby particles result in pressure broadening.
  • Magnetic Field coupling to the angular momentum quantum numbers can split otherwise degenerate states, resulting in Zeeman broadening (or splitting if it can be resolved), which is dependent on the magnetic field strength.
  • All of these effects allow spectra to teach us about the conditions of distant objects.