For equal masses $M$ of the same liquid with constant specific heat $c_P$, adiabatic constant-pressure mixing gives final temperature $T_f=(T_1+T_2)/2$ by enthalpy conservation. The entropy change is $M{c}_P\ln (T_f/T_1)+M{c}_P\ln (T_f/T_2)=2M{c}_P\ln [(T_1+T_2)/(2\sqrt{T_1{T}_2})]$. The arithmetic-geometric mean inequality proves nonnegativity, with equality only for equal initial temperatures. There is no separate compositional mixing entropy for identical fluids.

An ideal chain with endpoint charges $+q$ and $-q$ has coupling energy $-q\mathbf{E}\cdot \mathbf{R}$ in a uniform electric field, when $\mathbf{R}$ points from negative to positive charge. Completing the square in its Gaussian end-to-end distribution gives the displayed mean extension. The force conjugate to end separation is $qE$, even though the two endpoints experience opposite forces; adding their magnitudes would double-count the work. The zero-field rms length is $a\sqrt{N}$, and the rms of a single Cartesian component is $a\sqrt{N/3}$. Direct endpoint interactions and charge-induced changes of solvent quality are excluded from this model.

For a unit-trace two-dimensional reduced state of a bipartite pure state, determinant $D$ gives eigenvalues $(1\pm \sqrt{1-4D})/2$. Their binary entropy quantifies entanglement. Natural logarithms give a maximum $\ln 2$; base-two logarithms give one bit.

A plot of Gibbs free energy along a specified reaction coordinate under stated conditions. A barrier must be measured from the appropriate starting minimum.

The range of energy within one electronic energy band, obtained by varying its wavevector. This differs from an interband band gap.

For real $|k|<1$, define $E(\varphi ,k)={\int }_0^{\varphi }\sqrt{1-k^2{\sin }^{2}x}dx$ and $F(\varphi ,k)={\int }_0^{\varphi }(1-k^2{\sin }^{2}x{)}^{-1/2}dx$. Smoothness of the integrands permits differentiation under the integral sign. Subtracting the two integrands gives $E-F=-k^2{\int }_0^{\varphi }{\sin }^{2}x/\sqrt{1-k^2{\sin }^{2}x}dx$, which is $k{\partial }_{k}E$. This proves the displayed identity for $k\ne 0$. At $k=0$, both sides extend continuously with value zero. The variable $k$ is the modulus; derivatives with respect to the parameter $m=k^2$ instead obey ${\partial }_{m}E=(E-F)/(2m)$, with removable value $-\frac{1}{2}{\int }_0^{\varphi }{\sin }^{2}xdx$ at zero.

Let a radius-$R$ circle have center $c\in {\mathbb{R}}^2$, and apply the real affine map $y=Mx+b$. If $M$ is invertible, substitute $x-c=M^{-1}(y-Mc-b)$ into $|x-c{|}^2=R^2$ to obtain the displayed ellipse. Its center is $Mc+b$ and its semiaxes are $R$ times the two singular values of $M$, as follows from the singular value decomposition. A rotation or reflection preserves both semiaxes and hence gives another circle. If $M$ has rank one, the image is a segment; rank zero gives one point. This real-plane description also handles maps that stretch real and imaginary coordinates differently and are not holomorphic complex maps.

The Steiner inellipse of a nondegenerate triangle is the unique ellipse of largest area contained in the triangle. Its center is the centroid, and it touches each side at its midpoint.

cryo-electron microscopy images specimens at cryogenic temperature. In biological single-particle analysis, many weak projections of vitrified particles are combined to estimate a three-dimensional scattering electrostatic potential map; specimen heterogeneity and radiation damage limit the recoverable detail.

dose weighting combines image information with weights that depend on accumulated exposure and spatial frequency. Frames recorded after fine-detail damage may still supply useful lower-frequency information.

single-particle analysis reconstructs a three-dimensional structure from many two-dimensional images of individual particles in different orientations. Classification, alignment and assumptions about symmetry or structural sameness determine which information can be combined.

cryo-electron tomography reconstructs three-dimensional specimen information from tilted projections at cryogenic temperature. Radiation exposure and incomplete angular coverage limit resolution and interpretation.

subtomogram averaging aligns and combines repeated structures extracted from tomographic volumes. Structural equivalence, orientation coverage and independent validation determine which details are reproducible.

DNA-selective electron-microscopy staining combined with tomography for reconstructing chromatin ultrastructure in biological cells.^[ref]

Reconstruction of three-dimensional structure from electron-microscopy views at different tilts. Preparation, sampling and angular coverage affect the reconstruction.

Negative-stain electron microscopy makes a small specimen visible by surrounding it with an electron-scattering stain. Regions containing more stain scatter more electrons out of the accepted transmitted beam, so the specimen can appear relatively bright against a darker background in bright-field transmission electron microscopy.

A typical preparation deposits a dilute sample onto a supported grid, removes excess liquid, applies stain and removes excess stain before examination in a transmission electron microscope.^[ref] The resulting outline is useful for checking particle size, aggregation and gross shape. Drying, flattening and stain distribution can change the appearance, so an apparent cavity or fine boundary need not reproduce the hydrated specimen's structure.

Crystallographic electron channeling is orientation-dependent propagation of electrons in a crystal, where the spatially ordered electrostatic potential changes how the beam travels relative to atomic planes or columns. Changing crystal orientation can therefore change the collected signal even without changing chemical composition.

A small specimen tilt at fixed beam energy is a useful test: brightness changes correlated with orientation suggest a channeling or electron diffraction contribution. This matters when interpreting backscattered-electron atomic-number contrast or atomic-scale chemical maps. A bright region is not by itself proof of a different atomic number.

Hydrocarbon contamination in electron microscopy is carbonaceous material present on, or deposited onto, a specimen during examination. Material from specimen preparation or the chamber can accumulate in an irradiated region and alter the measured electron scattering or chemical signal.

A darkening patch that follows repeated scanning can therefore be a measurement-induced deposit rather than a pre-existing structure. Compare a repeatedly exposed field with a fresh neighboring field, and repeat at lower exposure if possible. Cleaning and suitable specimen handling reduce contamination, while such controls distinguish it from genuine specimen features or other forms of electron beam damage.

A specimen change caused by electron irradiation, including atomic displacement, ionization-driven chemical changes, heating or contamination. Compare low-dose repeated observations and unexposed regions to test whether measured nanostructure is being altered by its measurement.

In Auger recombination an electron-hole pair annihilates while transferring its energy to another electron or hole rather than to an emitted photon. The hot recipient subsequently loses energy, commonly to phonons. In a simple nondegenerate carrier-density model the rate is $R_A=C_n{n}^{2}p+C_{p}n{p}^2$, with electron and hole number densities $n,p$ and material-dependent coefficients. This makes the process especially important at high carrier injection. It is the reverse type of energy transfer to impact ionization, although detailed thresholds and matrix elements depend on the bands.