The length over which the harmonic and its quadratic source accumulate a relative wave phase of $\pi$. For nonzero second-harmonic phase mismatch, $L_c=\pi /|\Delta k|$; the first output zero of the uniform undepleted model occurs at $2L_c$. This is a propagation mismatch length, distinct from the pump's optical coherence length. It diverges as perfect phase matching is approached.

The difference between the harmonic's wave number and that of the quadratic driving electric polarization. It has units of inverse length. For collinear modes, $\Delta k=2\omega (n_{2\omega }-n_{\omega })/c$, so matching frequencies alone does not ensure matching spatial wave phases.

Frits Zernike traced the idea to unwanted companion lines of a diffraction grating. Around 1930 he found that visible striping disappeared when a telescope focused exactly on the grating. A phase strip placed over the principal spectral line changed its relation to the companion light and restored contrast. He then applied the separation of direct and diffracted light to a microscope and demonstrated it at Zeiss in 1932.^[ref]

His later absorbing and annular strips improved contrast and the appearance of halos; commercial Zeiss accessories followed in 1941.^[ref] The route from a grating defect to biological observation illustrates why apparently unwanted instrumental effects can be productive experimental clues.

A finite phase plate also intercepts diffracted light, altering low spatial frequencies. Halos and shade-off can therefore mimic borders or obscure interiors.^[ref] In the coherent zero-mode model of phase contrast, the separation is perfect by assumption, so that model cannot predict those artifacts.

The following original model isolates the conversion mechanism. Use a periodic image window, unit illumination, ideal unit magnification, and a phase plate that changes only the zero spatial frequency. Write $c_0=⟨e^{i\varphi }⟩$ for the spatial mean. Multiplying that single Fourier coefficient by $a{e}^{i\theta }$, with $0<a\le 1$, gives the exact image field in this model:This formula follows by splitting $e^{i\varphi }=c_0+(e^{i\varphi }-c_0)$: the second term has zero mean and is left unchanged. A real annular, partly coherent instrument alters a finite band of directions, so this calculation is a mechanism model rather than its full image-formation law.

For a weak phase object with $⟨\varphi ⟩=0$, one has $c_0=1+O({\varphi }^2)$ and henceThe cross term is $2\mathrm{Re}(a{e}^{-i\theta }i\varphi )=2a\varphi \sin \theta$. Thus $\theta =\pi /2$ makes positive $\varphi$ brighter and $\theta =-\pi /2$ makes it darker under this stated phase convention for complex amplitudes. The sign comes from this convention and chosen plate; a manufacturer’s positive/negative contrast labels should not be inferred from the sign of $\theta$ alone.

For example, choose $a=0.5$ and a zero-mean periodic pattern whose peak phase is $0.084$. The first-order peak intensities are $0.25\pm 0.084$, compared with the uniform ideal bright-field microscopy value $1$. The relative change about the reduced background is approximately $2\varphi /a$ for a quarter-cycle shift. Attenuating the reference increases relative contrast while reducing the background photon count; it is not light amplification or a universal improvement of signal-to-noise ratio.

Two checks expose what the model can and cannot measure. A spatially constant phase gives $E_{\mathrm{out}}=a{e}^{i\theta }{e}^{i\varphi }$ and $I_{\mathrm{out}}=a^2$, so no absolute phase is recovered. Orthogonality of the constant and zero-mean parts also givesFor $a\le 1$ this cannot exceed the incident mean light intensity. The image redistributes brightness and loses some reference light rather than creating energy.

An optical filter that changes selected light's phase relative to other transmitted light. In phase-contrast microscopy, its ring can also attenuate the direct reference; the amplitude transmission and light intensity transmission must be distinguished. An amplitude transmission $a=1/2$ gives an intensity transmission $a^2=1/4$.

A specimen whose leading optical effect is a spatially varying phase, rather than absorption. A thin specimen with refractive index $n_s$ in a medium of refractive index $n_m$ has, in a straight-path approximation,Here $\lambda$ is the vacuum wavelength; the integral is an optical path length difference. For the chosen values $\lambda =550\mathrm{nm}$, thickness $1\mu \mathrm{m}$ and $n_s-n_m=0.005$, the phase is $\pi /55\simeq 0.0571$ radians. This calculation does not identify thickness separately from refractive index contrast.^[ref]

Consider an optical point source at $(z,x)=(0,0)$, a uniform transparent medium, and a flat return surface $z=L$. A light ray with slope $s$ reaches $(L,sL)$. A phase-conjugate mirror reverses its full direction and returns it to $(0,0)$. An ordinary mirror reverses only its normal direction; its returning light ray reaches $x=2sL$ at $z=0$. Thus rays with $s=-1/4,0,1/4$ at $L=3$ return to $x=-3/2,0,3/2$ with ordinary specular reflection, but all return to $x=0$ under ideal phase conjugation. These are stipulated geometric coordinates, not laboratory measurements.

Solid blue curves mark outward wavefronts; dashed curves mark returning wavefronts. In the phase-conjugate mirror panel the returning curves coincide with the outward ones, and the arrows reverse on the same light rays. The diagram uses equal spatial scales. In geometrical optics, reversing each light ray in an unchanged medium with wave reciprocity retraces its route. At the optical point source itself, the ray construction becomes singular and diffraction sets the physical focal size.

A wave-level proof outline applies to a stationary scalar medium with real refractive index $n(\mathbf{r})$ and no wave absorption. Writing $c$ for the vacuum speed of light, its Helmholtz equation isAll coefficients are real, so complex conjugation gives the same Helmholtz equation for $U^{\ast }$. The conjugated outgoing boundary data select the incoming solution. This explains the connection between phase conjugation and retracing, provided the required incoming field is supplied over the necessary aperture. It also exposes the limits: wave reciprocity alone does not imply absence of wave absorption; conjugation of complex material coefficients would describe a different medium. Loss, an incomplete aperture, a changing medium or failure of wave reciprocity can prevent exact reconstruction. Intensity-dependent material changes need a separate analysis and are not covered by this linear proof.

Vapor-liquid equilibrium is coexistence of a liquid and its vapor with no net macroscopic transfer of matter between them. The two phases have equal temperature, equal pressure for a flat interface without external stress, and equal chemical potential for each exchanging species.

For a pure substance, fixing the temperature below its liquid-gas critical point selects the saturation vapor pressure. Molecules still cross the interface in both directions; equilibrium means equal average opposing rates, rather than halted molecular motion. In a mixture, the vapor and liquid generally have different compositions, while each component separately satisfies the chemical-potential equality. This distinction underlies separation by evaporation and condensation and explains why equal total pressure alone is insufficient to determine phase compositions.

The slow radial normal mode of a Penning trap. Its angular frequency arises from the combined static electric field and magnetic field; it is distinct from the driven Paul trap micromotion.

With the Seebeck coefficient convention $dV=-SdT$ at zero electric current, and the transported-heat convention of Peltier coefficient, the Kelvin thermoelectric relation is$T$ is absolute temperature in kelvins; use consistent material references. This scalar linear response relation assumes local thermal equilibrium and reciprocal transport without time-reversal-breaking fields or order. Onsager reciprocal relations supply the symmetry; conservation of energy alone is insufficient. A magnetic field requires a reversed-field reciprocity comparison.^[ref]

In particular, the A-minus-B thermocouple coefficient in Seebeck effect is $S_A-S_B$, whereas absorbed junction heat for conventional current A to B uses $S_B-S_A$. There is no contradiction: terminal voltage order and crossing direction are separate choices. Multiplying a measured voltage by an absolute temperature without specifying these choices does not determine junction heating.

A short reciprocity calculation makes the factor $T$ visible. At a common reference temperature, write the local linear response with electrical driving field $\mathcal{E}=-\nabla V$ and heat flux $q$ asAt $j=0$, the Seebeck coefficient convention gives $S=L_{12}/(T{L}_{11})$. At $\nabla T=0$, the Peltier coefficient is $q/j=L_{21}/L_{11}$. Thus $L_{21}=L_{12}$ gives $\Pi =TS$. The chosen forces are conjugate to the fluxes in the local entropy production, $j\mathcal{E}/T+q(-\nabla T)/T^2$; arbitrary rescaling of just one force would spoil this argument.

The relation is local. Varying $S(T)$ permits distributed Thomson effect heating, with $\tau =TdS/dT$; the constant-coefficient model below omits it. Endpoint substitutions into a constant-property formula can therefore break its energy accounting.

A freely moving massive test particle in an FLRW metric has physical momentum $p=m\gamma v$ relative to a comoving observer, where $v$ is its peculiar velocity. For a radial geodesic, the geodesic Lagrangian gives $C=a^{2}d\chi /d\tau$ constant, and the local radial momentum is $p=mad\chi /d\tau =mC/a$. The proper-time normalization gives $dt/d\tau =\sqrt{1+C^2/(a^2{c}^2)}$, soConsequently nonrelativistic peculiar velocity scales as $a^{-1}$, while the exact relativistic law keeps $|v|<c$. The momentum magnitude of a free photon also scales as $a^{-1}$, giving cosmological redshift through $E=cp$. These statements concern force-free propagation; a bound system need not follow the cosmological expansion.

Choose these dimensionless parameters for an original example, with no collisions or damping. They are teaching inputs, not measurements from either lecture. With $s=\Omega t$ and slow initial amplitude $X_0$, the radial equation and its leading approximation areSet $x(0)=0.95X_0$ and $dx/ds(0)=0$ to match the approximate initial conditions. The leading secular angular frequency is ${\omega }_x/\Omega =0.0353553$. The velocity Verlet method integrates the full RF equation over 80 drive periods; a separate fourth-order Runge-Kutta method checks the computed trajectory. Halving the timestep tests numerical error independently of the averaging error.

Using 400, 800, 1600 and 3200 steps per RF period gives successive maximum position differences of approximately $1.40\times 10^{-4}$, $3.50\times 10^{-5}$ and $8.74\times 10^{-6}$ in units of $X_0$, consistent with second-order convergence. The finest trajectory differs from the independent classical fourth-order Runge-Kutta method result by at most $2.92\times 10^{-6}{X}_0$, whereas $x_{\mathrm{app}}$ differs by as much as $0.0378X_0$. Over a sufficiently long time, the omitted correction to the secular angular frequency produces a growing phase discrepancy even if the numerical trajectory has converged. For the chosen point only, integration of a fundamental solution matrix over one RF period checks that both the radial and axial Floquet multipliers lie on the unit circle, away from its endpoints. This tests simultaneous ideal confinement without extrapolating to larger $|q_i|$.

Consider a parent atomic nucleus at rest. Write its signed nuclear polarization along a fixed axis $\hat{\mathbf{z}}$ as $P=⟨J_z⟩/(I\hslash )$, where $I\hslash$ is its maximal spin component. Let $\theta$ be the angle of the emitted electron momentum to that axis and let $\beta =v/c$. Use $\mathcal{P}$ for the parity operator, so it is not confused with $P$.

For an axially symmetric leading-dipole model, retain a constant and the term linear in the polarization vector $\mathbf{P}=P\hat{\mathbf{z}}$. Rotational symmetry makes the directional term proportional to $\mathbf{P}\cdot \hat{\mathbf{p}}$. Define its dimensionless coefficient as $A\beta$. At fixed electron energy the relative intensity and normalized probability density function per solid angle are thereforeThe factor $\beta$ is the conventional separation of the speed dependence; rotational symmetry alone does not determine that dependence or $A$. The coefficient $A$ contains the decay dynamics. This leading model omits higher angular terms and detailed recoil and radiative effects. A partially oriented higher-spin ensemble can also have alignment that is not specified by its vector nuclear polarization alone.

The unit-integral condition follows without any model of a detector:Thus $f$ has unit angular average, while the probability density function in the polar angle alone contains the Jacobian determinant factor:In particular, $f$ itself is neither a normalized probability density function in $\theta$ nor a detector count. Positivity on $-1\le u\le 1$ gives $|a|\le 1$.

Under full spatial reflection, $\mathbf{p}\mapsto -\mathbf{p}$ but $\mathbf{J}\mapsto \mathbf{J}$. The latter follows already for orbital angular momentum from $\mathbf{r}\times \mathbf{p}\mapsto (-\mathbf{r})\times (-\mathbf{p})=\mathbf{r}\times \mathbf{p}$; spin has the same axial vector transformation. Hence $\mathbf{P}\cdot \hat{\mathbf{p}}$ is a pseudoscalar, and $\cos \theta \mapsto -\cos \theta$. If parity conservation held for this model, equal probabilities for the inverted process would force $a=0$. Conversely, $a=0$ in an unpolarized sample proves nothing about $A$.

Integrating over the two hemispheres gives a useful original check:Choose $A=-0.8$, $P=0.5$, and $\beta =0.75$, so $a=-0.3$. Then the forward and backward hemisphere probabilities are $0.425$ and $0.575$, and their counting asymmetry is $-0.15$. These are chosen teaching values, not measured values from the Wu experiment. At a single energy, the angular shape determines the product $AP\beta$; it cannot separately determine $A$ without a polarization calibration.

The Wu experiment was performed by Chien-Shiung Wu, Ernest Ambler, Raymond Webster Hayward, Dale D. Hoppes, and Ralph Percy Hudson; its 1957 report was Experimental Test of Parity Conservation in Beta Decay. Their experiment supplied the evidence that the Lee-Yang proposal for a parity test had called for.^[ref]

A cobalt-60 surface layer on a cerium magnesium nitrate crystal was cooled by adiabatic demagnetization; a solenoid's magnetic field polarized its atomic nuclei. An internal anthracene counter detected electrons by particle scintillation; a light pipe carried flashes to a photomultiplier. Polar and equatorial gamma ray counters monitored nuclear polarization and effective temperature; bulk thermometry could miss surface heating.^[ref]

Reversing nuclear polarization reversed the electron excess; warming erased it with gamma ray anisotropy. Warm rates were field-independent. Cooling-field reversal checked remanence; poorly polarized sources checked magnetic deflection. Electrons favored the direction opposite spin.^[ref]

The gamma ray anisotropy diagnoses the preparation; the electron asymmetry diagnoses parity violation. These are different jobs. A directionally uneven gamma ray signal alone need not violate parity conservation: a distribution even in the cosine of the angle can distinguish an axis from its perpendicular without distinguishing its two ends. The spin-momentum correlation instead distinguishes those ends.

The inference requires control of changes in detection efficiency, electron transport and nuclear polarization. A fixed detector under spin reversal avoids physically exchanging detector positions, but still requires checks for magnetic field effects. The result establishes parity violation in this weak interaction process. A precise microscopic coupling requires more information than the existence or sign of a counting asymmetry; the following model makes that distinction explicit.

In 1956, Tsung-Dao Lee and Chen Ning Yang examined the evidence for parity conservation in the weak interaction. Their paper, Question of Parity Conservation in Weak Interactions, distinguished an untested assumption from an established experimental result and proposed comparing electron emission along opposite directions relative to oriented nuclear spins, explicitly considering cobalt-60. It proposed a test; it did not report the later Wu experiment.^[ref]

The conceptual improvement is the choice of observable. An angle between two polar vectors is preserved by spatial reflection; its distribution can conceal parity violation. An angle between an axial vector and a polar vector changes to its supplement. Thus a spin-resolved directional measurement asks a different question from an unpolarized total decay rate.

For independent localized two-state moments of projections $\pm \mu$ along a field, the one-ion partition function is $Z_1=2\cosh [\mu B/(k_{B}T)]$. Its thermal mean gives $M=n\mu \tanh [\mu B/(k_{B}T)]$. A $J=1/2$ ion has $\mu =g_J{\mu }_B/2$, so the special value $\mu ={\mu }_B$ additionally assumes $g_J=2$. At weak field the SI volume susceptibility is ${\chi }_H={\mu }_{0}n{\mu }^2/(k_{B}T)$, neglecting demagnetizing corrections.

The Curie–Weiss law approximates the magnetic susceptibility of interacting localized moments by $\chi =C/(T-\Theta )$ in a suitable paramagnetic temperature range. A positive Weiss temperature often reflects predominantly ferromagnetic interactions, while a negative value often reflects antiferromagnetic interactions. It is not a universal exact law through an ordering transition, and crystal-field effects or itinerant carriers can require another description.

A rectangular parallelepiped is a parallelepiped whose three edge directions at each vertex are mutually perpendicular. Its faces are rectangles; if its edge lengths are $a,b,c$, its volume is $abc$. A cube is the special case $a=b=c$.

For a parabola described by the function graph $y=a(x-h{)}^2+k$, with nonzero $a$ in the real numbers, its vertex of a parabola is the point $(h,k)$. This point gives its minimum or maximum value of $y$ in these coordinates.