Atomic Spectra: MDCAT Physics notes
Atomic Spectra MDCAT notes: Bohr model, hydrogen spectral series and their regions, excitation, X-ray production, characteristic lines and lasers.
Atomic spectra
Atoms of a gas, when excited, emit radiation only at certain sharp wavelengths, giving a line spectrum. Each element has its own set of lines, which act like a fingerprint. Bright lines on a dark background form an emission spectrum; dark lines on a continuous background form an absorption spectrum. Hot solids give a continuous spectrum; molecules give band spectra.
Bohr's model of hydrogen
- Electrons move in certain circular orbits without radiating (stationary states).
- Angular momentum is quantized: $mvr=\dfrac{nh}{2\pi}$ (Bohr's postulate).
- A photon is emitted or absorbed only when the electron jumps between levels: $hf=E_n-E_p$.
For hydrogen: $r_n=n^2r_1$ with $r_1=0.053$ nm; $E_n=-\dfrac{13.6}{n^2}$ eV; speed in the first orbit $v_1\approx2.19\times10^6$ m s$^{-1}$ (about $c/137$). Lines close together come from levels whose energy difference is small.
Excitation: absorbing energy lifts an electron to a higher level; the accelerating potential needed is the excitation potential. Removing the electron completely needs the ionization potential (13.6 V for hydrogen).
Hydrogen spectral series
$$\frac{1}{\lambda}=R_H\left(\frac{1}{p^2}-\frac{1}{n^2}\right),\qquad R_H=1.097\times10^7\ \mathrm{m^{-1}}$$
| Series | Lower level $p$ | Region |
|---|---|---|
| Lyman | 1 | Ultraviolet |
| Balmer | 2 | Visible |
| Paschen | 3 | Infrared |
| Brackett | 4 | Infrared |
| Pfund | 5 | Infrared |
So the hydrogen spectrum has lines in the UV, visible and IR. The longest wavelength of a series comes from the smallest jump (e.g. Paschen $4\to3$, about $1.87\times10^{-6}$ m). For $2\to1$: $\dfrac1\lambda=\dfrac{3R_H}{4}$, so $\lambda=\dfrac{4}{3R_H}$.
X-rays
X-rays are high-energy, invisible photons of wavelength much shorter than visible light. In a Coolidge tube, fast electrons strike a heavy metal target (usually tungsten, which has a high melting point). X-ray production is the reverse of the photoelectric effect: electron energy becomes photon energy.
- Continuous X-rays (bremsstrahlung): electrons decelerated by nuclei lose any amount of energy. The hardest photon (shortest wavelength) comes from an electron losing all its energy at once: $hf_{max}=eV$. This is part of the continuous spectrum.
- Characteristic X-rays: an inner-shell vacancy is filled by an outer electron. $K_\alpha$ (L to K) is the most probable, so most intense; $K_\beta$ (M to K) has more energy and shorter wavelength than $K_\alpha$. M-series lines have much lower energy (longer wavelength) than K lines.
Bones absorb X-rays more than flesh (calcium has a higher atomic number), so they appear lighter on film. X-rays and gamma rays differ mainly in origin: X-rays from electron transitions, gamma rays from the nucleus.
Lasers
An atom can emit spontaneously or by stimulated emission, when an incoming photon triggers an identical photon. Lasers use stimulated emission (with population inversion and metastable states) to give coherent, monochromatic, intense light.
Key formulas
- $mvr=nh/2\pi$; $E_n=-13.6/n^2$ eV; $r_n=0.053n^2$ nm
- $1/\lambda=R_H(1/p^2-1/n^2)$
- $\lambda_{min}=hc/eV$ for continuous X-rays
Common MDCAT traps
- Lyman is UV, Balmer is visible; Paschen, Brackett and Pfund are all infrared.
- The hardest X-ray photon belongs to the continuous spectrum limit, not a characteristic line.
- Among K lines, $K_\alpha$ has the longest wavelength; an M line is longer still.
- Excitation potential lifts an electron; ionization potential removes it.
- X-rays have shorter, not longer, wavelength than visible light.
Quick revision
- Gaseous atoms give line spectra.
- Bohr quantized angular momentum.
- First-orbit speed $2.19\times10^6$ m s$^{-1}$.
- Tungsten target in X-ray tubes.
- Laser: stimulated emission.
- Gamma rays have shorter wavelengths than typical X-rays.