In an independent-electron crystal with two spin states and no spin splitting, each completely filled energy band contains two electrons per primitive cell. An odd electron count per primitive cell therefore cannot fill an integer number of such bands. Enlarging the actual primitive cell, breaking the relevant degeneracy or including correlated-electron physics changes the premises. Even filling is necessary in this model but not sufficient: bands can overlap.

Moving a surface adsorbate between lateral positions by coupling it to a nearby scanning-probe tip and moving the tip parallel to the surface. Pulling, pushing and sliding depend on the competing tip–adsorbate and substrate interactions.

Transferring an adsorbate between substrate and probe tip, allowing pickup, lateral repositioning and deposition at a selected site. Its feasibility depends on the tip and substrate binding landscapes and on the controlled excitation.

A deliberately assembled xenon-atom surface pattern illustrating controlled atomic manipulation. Don Eigler and Erhard Schweizer reported individual xenon positioning on single-crystal nickel at about $4\mathrm{K}$ in their 1990 paper.^[ref]

2016 paper: Public original PDF, physical page 779. Student-answer availability for this paper remains unverified.

Historical Part III Atmospheric Physics papers. Availability of complete worked student solutions has not been established; advertised answer years are skipped when identified. Only solutions are reproduced here, with the original public paper linked from each paper.

A scanning transmission electron microscopy mode collecting electrons scattered to sufficiently large angles. It often gives approximately monotonic atomic-number and mass-thickness contrast, with detector angles and multiple scattering still important.

An expression of the form $a_0+{\sum }_{j=1}^m{a}_j{x}_j$, with fixed coefficients $a_j$. It contains a constant term and terms of degree one, without products of variables. It represents an affine function once the scalar domain is specified.

2016 paper: Public original PDF, physical page 783. Student-answer availability for this paper remains unverified.

A saturated ascending air parcel releases latent heat as water vapor condenses, offsetting part of its expansion cooling. Its temperature decreases less steeply than the dry adiabatic lapse rate, with the gradient depending on temperature, pressure, moisture and whether condensate remains in the parcel. It is not a universal constant equal to the typical environmental lapse rate.

The elastic stiffness satisfies $c_{ijkl}{a}_i{p}_j{a}_k{p}_l>0$ for every nonzero polarization $a$ and propagation vector $p$. It makes the elastic acoustic tensor positive definite, ensuring positive squared wave speeds. Positive strain energy is a sufficient stronger condition in ordinary linear elasticity.

Jeu de taquin slides entries of a skew Young tableau to remove an inner corner. Place a hole at a chosen inner corner, move the smaller of the neighboring entries immediately right and below into the hole, and repeat until the hole reaches the outer boundary. For a semistandard Young tableau with weakly increasing rows and strictly increasing columns, break equal-entry ties by moving the entry below.

Each slide preserves the tableau inequalities. Repeated inward slides rectify a skew tableau to a straight shape; the rectified tableau is independent of the order of inner-corner choices. The procedure connects skew tableaux with Schur functions and the Robinson-Schensted-Knuth correspondence. Its tie convention is essential when entries repeat.

The largest square of cells in the upper-left corner of a Young diagram. For an integer partition $\lambda$, its side length is $d=\max \{i:{\lambda }_i\ge i\}$.

For an even periodic classical chain with spin length $S$ and $J>0$, each bond attains the negative largest singular value of $\mathrm{diag}(1,1,\Delta )$. For $\Delta >1$ the two minimizing configurations are alternating $z$ spins. For $|\Delta |<1$ they are planar Néel states with arbitrary in-plane angle. For $\Delta <-1$ the two configurations have uniform $z$ spins. At $\Delta =1$ the Néel direction is arbitrary on a sphere; at $\Delta =-1$ a staggered transverse component and uniform longitudinal component parameterize another spherical degeneracy. These are classical configuration manifolds, not a claim about finite quantum-chain degeneracy.

An X-ray diffractometer measures scattering intensity as a function of sample orientation, scattering angle or photon energy. Its essential elements are an incident X-ray beam, a mounted sample, controlled angular geometry and a detector recording the outgoing beam or pattern.

A monochromatic angle scan changes which reciprocal-lattice vectors satisfy the elastic diffraction condition. A position-sensitive detector can instead record many outgoing directions at once, while sample rotation brings further reflections into view. Converting the counts into structural information requires the wavelength, angular calibration, sample alignment and intensity corrections. A single crystal and a powder produce different sampling patterns. The recorded intensities contain both peak-position information about the lattice and structure-factor information about the atomic basis.

A diffraction geometry in which the incident and diffracted rays make equal angles with the specimen surface. The selected reciprocal-lattice vector is normal to that surface.

Reflection diffraction from planes inclined to the specimen surface. Incident and exit angles relative to the surface differ; both must correspond to rays on the accessible side of an opaque specimen.

A geometry in which the diffracted beam traverses the specimen and exits another surface. Absorption and thickness affect whether it is observable.

Diffraction from an oriented single crystal. A reflection requires both an allowed nonzero structure factor and access to its reciprocal-lattice direction and Bragg angle.