Atomic Physics Domain Guide

Every spectral line is a subtraction. The atom does not emit its energy levels; it emits the difference between two of them, and that difference is the photon. Once that is automatic, the Balmer, Lyman, and Paschen series stop being lists to memorise and become the obvious consequence of which level the atom falls to.

Scaling beats computing. Hydrogen-like energies go as $Z^{2}/n^{2}$ and radii as $n^{2}/Z$; well energies go as $1/L^{2}$; X-ray cutoffs go as $1/V$; decay populations halve on a fixed clock. The exam builds items around these proportionalities precisely because they can be answered in one step, and a candidate who reaches for constants instead of ratios runs out of time rather than out of knowledge.

The nucleus and the solid are the same quantum story at different scales. Confinement produces discrete levels in an atom, bands in a crystal where the levels of $10^{23}$ atoms smear together, and shell structure in nuclei. Recognising that a band gap and an atomic level spacing are the same kind of object turns condensed matter from a separate subject into a corollary.

ETS lists atomic physics among the content areas of the GRE Physics Test, and our learning model additionally groups condensed-matter and nuclear material into this canonical domain — that grouping is our own reading of the outline, chosen because these specialised topics share a formalism, not because ETS labels them together. Practically, this is the block where breadth beats depth: a wide, shallow command of many standard results is worth far more than mastery of any one of them.

The Bohr model remains the exam's workhorse for one-electron systems. Its results are $E_n = -13.6Z^{2}/n^{2}\ \mathrm{eV}$, $r_n = n^{2}a_0/Z$ with $a_0 = 0.0529\ \mathrm{nm}$, and the emitted photon energy $\Delta E = 13.6Z^{2}\left(\dfrac{1}{n_f^{2}}-\dfrac{1}{n_i^{2}}\right)\ \mathrm{eV}$, equivalently the Rydberg formula $\dfrac{1}{\lambda} = R_\infty Z^{2}\left(\dfrac{1}{n_f^{2}}-\dfrac{1}{n_i^{2}}\right)$ with $R_\infty = 1.097\times10^{7}\ \mathrm{m^{-1}}$. Transitions ending at $n_f = 1$ form the ultraviolet Lyman series, at $n_f = 2$ the visible Balmer series, and at $n_f = 3$ the infrared Paschen series. The ionisation energy of ground-state hydrogen is $13.6\ \mathrm{eV}$, which is the single most useful number in the domain.

Quantum numbers organise multi-electron atoms even though the Bohr energies no longer apply. The set $(n, \ell, m_\ell, m_s)$ has $\ell = 0,\ldots,n-1$, $m_\ell = -\ell,\ldots,+\ell$, and $m_s = \pm\tfrac{1}{2}$; the Pauli exclusion principle forbids two electrons from sharing all four, so a subshell $\ell$ holds $2(2\ell+1)$ electrons and a shell $n$ holds $2n^{2}$. Electric-dipole transitions obey the selection rule $\Delta\ell = \pm1$, which is why not every energetically allowed transition actually appears in a spectrum. In a magnetic field the $m_\ell$ degeneracy splits with spacing $\mu_BB$ where $\mu_B = 5.79\times10^{-5}\ \mathrm{eV/T}$ — the Zeeman effect — and spin-orbit coupling produces fine structure at the scale $\alpha^{2} \approx (1/137)^{2}$ relative to the gross level spacing.

X-rays connect atomic structure to crystals. Electrons accelerated through a potential $V$ strike a target; the continuous bremsstrahlung spectrum they produce cuts off sharply at $\lambda_{\min} = hc/eV$, because no photon can carry more than the electron's whole kinetic energy. Superposed on it are characteristic lines from inner-shell vacancies, whose frequencies follow Moseley's law $\sqrt{f} \propto (Z-1)$ for the $K_\alpha$ line — historically the evidence that atomic number, not atomic weight, orders the elements. Sent into a crystal, X-rays reflect constructively when Bragg's condition $2d\sin\theta = m\lambda$ is met, where $\theta$ is measured from the crystal plane rather than from its normal, and $d$ is the interplanar spacing.

Condensed matter on this exam is a small set of qualitative facts with a few numbers attached. Solids are classified by their band structure: metals have a partly filled band and conduct at any temperature, with resistivity rising as temperature increases because lattice vibrations scatter electrons more; semiconductors have a gap of order an electronvolt and conduct better when heated because more carriers are thermally excited across it; insulators have a gap far above $k_BT$. The free-electron model gives a Fermi energy $E_F = \dfrac{\hbar^{2}}{2m}(3\pi^{2}n)^{2/3}$ of a few electronvolts for typical metals, far above $k_BT \approx 0.025\ \mathrm{eV}$ at room temperature, which is why the electron gas is strongly degenerate and contributes far less to heat capacity than classical equipartition would predict. Lattice vibrations supply the rest, giving the Dulong-Petit value $3R$ per mole at high temperature and a $T^{3}$ dependence at low temperature.

Nuclear physics closes the domain with conservation laws and one exponential. Decays conserve nucleon number and charge: alpha decay lowers $A$ by 4 and $Z$ by 2; beta-minus decay converts a neutron into a proton, raising $Z$ by 1 at fixed $A$ and emitting an electron and an antineutrino; beta-plus decay does the reverse; gamma decay changes neither. The binding energy comes from the mass defect, $B = \left[Zm_p + Nm_n - m_{\text{nucleus}}\right]c^{2}$, with $1\ \mathrm{u} = 931.5\ \mathrm{MeV}/c^{2}$ and a binding energy per nucleon that rises steeply to about $8.8\ \mathrm{MeV}$ near iron and then falls slowly — hence energy release from fusing light nuclei and from fissioning heavy ones. Decay itself is $N(t) = N_0e^{-\lambda t}$, but on a timed test the fastest route is almost always to count half-lives: after $k$ half-lives a fraction $2^{-k}$ remains.