Electron orbiting with speed $v$, radius $r$: current $I = \frac{ev}{2\pi r}$
$$\mu = IA = \frac{ev}{2\pi r} \cdot \pi r^2 = \frac{evr}{2}$$Bohr quantization: $mvr = \frac{nh}{2\pi}$ $\Rightarrow$ $vr = \frac{nh}{2\pi m}$
$$\mu = \frac{e}{2} \cdot \frac{nh}{2\pi m} = \boxed{\frac{neh}{4\pi m}}$$An electron revolving in a circular orbit constitutes a current loop. Current $I = e/T = ev/(2\pi r)$ (charge per unit time). Magnetic moment $\mu = IA = \frac{ev}{2\pi r}\cdot\pi r^2 = \frac{evr}{2} = \frac{eL}{2m}$ where $L = mvr$ = angular momentum. So $\mu = \frac{e}{2m}L$. Since $L$ is quantized ($L = n\hbar$): $\mu_n = \frac{e\hbar}{2m}\cdot n = n\mu_B$ where $\mu_B = e\hbar/(2m)$ = Bohr magneton.
$\mu_B = \frac{e\hbar}{2m_e} = \frac{eh}{4\pi m_e} = 9.274\times10^{-24}$ J/T. Smallest unit of magnetic moment for an electron. Magnetic moment of electron in $n$th Bohr orbit: $\mu_n = n\mu_B$. Spin magnetic moment of electron: $\mu_s = g_s\mu_B\sqrt{s(s+1)} \approx \mu_B$ (where $g_s \approx 2$, $s = 1/2$). Nuclear magneton: $\mu_N = e\hbar/(2m_p) = 5.05\times10^{-27}$ J/T (proton mass replaces electron mass, so 1836 times smaller).
Radius: $r_n = n^2 a_0$ where $a_0 = 0.529$ Å. Energy: $E_n = -13.6/n^2$ eV. Velocity: $v_n = v_1/n$ where $v_1 = 2.18\times10^6$ m/s. Angular momentum: $L_n = n\hbar$. Magnetic moment: $\mu_n = n\mu_B$. For hydrogen-like ions ($Z > 1$): $r_n = n^2 a_0/Z$, $E_n = -13.6Z^2/n^2$ eV, $v_n = Zv_1/n$. Energy of photon emitted: $h\nu = E_i - E_f = 13.6(1/n_f^2 - 1/n_i^2)$ eV... wait: $h\nu = E_i - E_f$ for $n_i > n_f$.
Principal quantum number $n$: energy and size. $n = 1, 2, 3...$. Azimuthal $l$: shape of orbital. $l = 0, 1, ..., n-1$. Magnetic $m_l$: orientation. $m_l = -l, ..., 0, ..., +l$. Spin $m_s$: $\pm 1/2$. Total orbitals in shell $n$: $n^2$. Total electrons: $2n^2$. Pauli exclusion: no two electrons have same four quantum numbers. Hunds rule: half-fill before pairing. Aufbau: fill lowest energy first.
de Broglie: $\lambda = h/p = h/(mv)$. Bohr condition $mvr = n\hbar$ is equivalent to $2\pi r = n\lambda$ (integer wavelengths fit in orbit). For $n$th orbit: $n$ complete waves fit around circumference. This gives a physical meaning to Bohr quantization: standing wave condition for electron wave. Davisson-Germer (1927): confirmed wave nature of electrons by diffraction experiment.
Rydberg formula: $\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$ where $R_H = 1.097\times10^7$ m$^{-1}$. Series: Lyman ($n_f=1$, UV), Balmer ($n_f=2$, visible: $H_\alpha=656$ nm red, $H_\beta=486$ nm blue-green), Paschen ($n_f=3$, IR), Brackett ($n_f=4$), Pfund ($n_f=5$). Ionization energy of hydrogen from ground state: 13.6 eV.
$N = N_0 e^{-\lambda t}$. Half-life $t_{1/2} = 0.693/\lambda$. Activity $A = \lambda N$. Alpha decay: $^A_Z X \rightarrow ^{A-4}_{Z-2}Y + ^4_2He$. Beta minus: neutron $\rightarrow$ proton + $e^-$ + $\bar{\nu}_e$. Beta plus: proton $\rightarrow$ neutron + $e^+$ + $\nu_e$. Gamma: no change in $A$ or $Z$ (energy release from excited nucleus). Binding energy: $BE = (Zm_p + Nm_n - M_{nucleus})c^2$. Mass defect $\Delta m = Zm_p + Nm_n - M$.
Fission: heavy nucleus splits ($^{235}_{92}U + n \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3n + \sim200$ MeV). Chain reaction: each fission releases 2–3 neutrons. Critical mass: minimum for sustained chain reaction. Reactor: controlled fission ($k=1$). Bomb: uncontrolled ($k>1$). Fusion: light nuclei combine ($^2H + ^3H \rightarrow ^4He + n + 17.6$ MeV). Requires $\sim10^7$ K. Sun powered by pp chain. Hydrogen bomb: uncontrolled fusion. ITER (fusion reactor): under construction in France.