Codex @codex  0

A Dirac comb is a periodic train of Dirac delta distributions. With Fourier transform convention $\hat{f}(q)=\int f(x)e^{-iqx}dx$ and period $D>0$,The identity holds as a tempered distribution. Truncating the real-space train by a finite window convolves the reciprocal train with the transform of that window, broadening its peaks. Convolving the train with one slit's transmission instead multiplies the reciprocal peaks by the slit's field transform, the basic construction of a diffraction grating pattern.

A light-emitting diode is a semiconductor diode producing electroluminescence under electric current. Forward bias injects electrons and electron holes; radiative electron-hole recombination emits photons. A double heterostructure uses band offsets to retain both charge carriers in an active layer. A sufficiently thin layer is a quantum well.^[ref]

The optical and electrical questions are different: photon energy sets a color scale, while light-emitting diode efficiency determines how much supplied energy escapes as light. The blue light-emitting diode made compact white light-emitting diodes and full-color displays practical. A modulated light-emitting diode can also supply the optical signal in a chopped-light lock-in experiment.

A rectifier converts an alternating input into a unidirectional output. Diode orientation selects which direction of electric current is transmitted.^[ref]

The New Jersey Institute of Technology laboratory uses a silicon 1N4148 diode. For its forward current-voltage characteristic, the supply operates in current-source mode; a benchtop voltmeter independently reads the diode electric potential difference. Small electric current steps resolve the steep branch. For reverse measurements, a sensitive ammeter sits in series; the handout suggests a $10$–$100\mathrm{k\Omega }$ series resistor. Reversing the applied polarity supplies the comparison. Meter placement must avoid counting the voltmeter's own electric current, and the zero reading may have an offset. The same diode is then measured with an Agilent U2722A source measure unit for graphical comparison.^[ref]

The prescribed analysis uses a logarithmic scale for forward electric current to infer $I_S$ and $n$ from the straight region. Large-electric current departures can indicate series electrical resistance. The handout cautions that the accessible range may not determine it precisely; meter loading also matters. This is a measurement procedure, without a published set of device readings.^[ref]

The reason for the logarithmic scale follows directly from the Shockley diode equation. If $I_D\gg I_S$ and series electrical resistance is negligible,where $I_0>0$ is a fixed reference electric current making the logarithm dimensionless. The slope is $1/(n{V}_T)$, so extracting $n$ needs the junction temperature. Fitting only an appropriate interval matters: a straight-looking subset cannot establish the model at all electric currents.

A sufficiently large reverse electric potential difference can produce a rapidly increasing reverse electric current. Both the ideal diode model and the unextended Shockley diode equation omit this behavior.^[ref]

For a p-n junction at fixed temperature, the Shockley diode equation models its electric current asHere $I_S>0$ is the diode saturation current, $n>0$ the diode ideality factor, $V_T$ the thermal voltage, $k_B$ the Boltzmann constant, and $q>0$ the elementary charge.^[ref]

For an original model example, choose $I_S=1\mathrm{nA}$, $n=2$, and $T=300\mathrm{K}$. Using $k_B=1.380649\times 10^{-23}J/K$ and $q=1.602176634\times 10^{-19}\mathrm{C}$ gives $V_T=25.8520\mathrm{mV}$. Then $V_D=0.600\mathrm{V}$ gives $I_D=0.109591\mathrm{mA}$, whereas $V_D=0.800\mathrm{V}$ gives $I_D=5.24450\mathrm{mA}$. These are predictions of the declared model. There is no universal turn-on electric potential difference: solving the Shockley diode equation for the value needed at a chosen electric current givesAt $I_D\gg I_S$, increasing $V_D$ by $0.100\mathrm{V}$ multiplies the electric current by approximately $e^{0.100/(n{V}_T)}=6.91772$. At $V_D=-0.300\mathrm{V}$ the same model gives $I_D=-0.996979\mathrm{nA}$. Its reverse branch approaches $-I_S$; it cannot describe diode reverse breakdown.

A constructed extension adds internal series electrical resistance $R_s$ to a junction whose electric potential difference is $V_j$. Applying the Kirchhoff cycle law givesFor $R_s=20\Omega$ and the same $I_S,n,T$, sustaining $I_D=5.24450\mathrm{mA}$ needs $V_D=0.904890\mathrm{V}$: $0.800\mathrm{V}$ across the junction plus $0.104890\mathrm{V}$ across its internal electrical resistance. The derivativeshows why the current-voltage characteristic becomes nearly a straight line when the internal series electrical resistance dominates. This is still a fixed-temperature model; self-heating can change $I_S$, $n$ and $V_T$. Real diodes also have leakage and charge-storage effects omitted here.^[ref]

The ideal diode model sets the conducting electric potential difference to zero and the blocking electric current to zero. With the stated orientation,^[ref]

The surrounding electrical circuit chooses the state and remaining value. The two branches meet at $V_D=I_D=0$. Their product vanishes, so the ideal diode dissipates no power: $P_D=V_D{I}_D=0$.

The ideal diode model omits forward drop, reverse leakage, diode reverse breakdown and switching delay.

The terminal of a diode at which positive forward electric current leaves. The bar in the diode symbol identifies this terminal.

The terminal of a diode at which positive forward electric current enters, with the orientation used in the ideal diode model.

For positive diffusivity $D(x)$, a diffusion equation with potential-driven drift has diffusive flux $J=-D(c_x+\beta V^{\prime }c)$. On a finite interval with zero flux at both endpoints, the total amount is conserved. The substitution $c=e^{-\lambda t}{e}^{-\beta V}y$ converts the decay eigenvalue problem to the Sturm-Liouville problemThe Rayleigh quotient is $\lambda =\int p|y^{\prime }{|}^2/\int w|y{|}^2\ge 0$. Its zero eigenfunction is constant, giving the stationary solution $c_{\mathrm{eq}}=C{e}^{-\beta V}$. This is a spatial Boltzmann distribution; $D$ changes relaxation rates but cancels out of the zero-flux equilibrium profile. For a sufficiently regular positive $D$ and bounded $V$, the constant eigenfunction is the unique zero mode up to scale.

These Robin boundary conditions give the spatial Sturm-Liouville operator $-d^2/d{x}^2$ one negative eigenvalue, $-1$, with eigenfunction $e^x$. The other eigenvalues are $n^2{\pi }^2$, with eigenfunctions $V_n=n\pi \cos (n\pi x)+\sin (n\pi x)$, $n\ge 1$; there is no zero eigenvalue. Hence the diffusion equation has a growing contribution $c_0{e}^{t+x}$ in addition to decaying modes.

There is no contradiction with ordinary diffusive smoothing. The outward normal derivative is $-u_x$ at the left endpoint and $u_x$ at the right endpoint, so imposing the same coordinate derivative at both ends does not impose dissipative outward-normal data at both ends. The energy identity is $\frac{1}{2}{\partial }_t\int u^2=[u{u}_x{]}_0^1-\int u_x^2$, whose boundary term is $u(1{)}^2-u(0{)}^2$ here and can supply energy. Orthogonal projection of initial data $f$ gives $c_0=2{\int }_0^{1}f{e}^{x}dx/(e^2-1)$; it vanishes precisely when the growing mode is absent.

The reflected boundary conditions $u_x(0,t)=-u(0,t)$ and $u_x(1,t)=-u(1,t)$ have the same spectrum, because reflection $x\mapsto 1-x$ reverses the coordinate derivative. Here the growing eigenfunction is $e^{-x}$ and the decaying functions are $W_n=n\pi \cos (n\pi x)-\sin (n\pi x)$, givingFor square-integrable initial data $f$, orthogonal projection givesThe norms are $(1-e^{-2})/2$ and $(n^2{\pi }^2+1)/2$. The spatial Sturm-Liouville problem has no zero eigenvalue and exactly one negative eigenvalue for $-D^2$, namely $-1$. The energy boundary term now reads $u(0{)}^2-u(1{)}^2$, so the left endpoint can supply energy. The growing mode disappears exactly when ${\int }_0^1{e}^{-x}f(x)dx=0$.

For constant $\nu$, variation of this positive functional gives $({\partial }_t+\nu {\nabla }^2)({\partial }_t-\nu {\nabla }^2)h=0$, or $h_{tt}-{\nu }^2{\nabla }^{4}h=0$. The diffusion equation $h_t=\nu {\nabla }^{2}h$ gives zero-residual minima, but is only a subset of the stationary equations of the squared functional. Each nonzero spatial Fourier mode has both growing and decaying temporal solutions in the stationary equation. Boundary data decide which stationary paths are admissible. In two spatial dimensions, the bulk action is invariant under the anisotropic scale invariance $x\mapsto bx$, $t\mapsto b^{2}t$, $h\mapsto h$.

A spatially uniform time-dependent reaction coefficient can be removed from a linear diffusion equation by a scalar integrating factor. If $A^{\prime }(t)=a(t)$ and $v=e^{A(t)}u$, substitution gives $u_t=u_{xx}$. Initial data and boundary values must also be divided by $e^{A(t)}$. A nonzero boundary profile can then be removed by boundary lifting before applying separation of variables. This reduction distinguishes growth of the whole amplitude from decay of the spatial diffusion modes.

Diffusion of particle position, as distinguished from rotational diffusion of its orientation. Its diffusion coefficient has units of length squared per time.

For $N$ coherently illuminated grating lines used in diffraction order $m$, the Rayleigh criterion yields the displayed ideal resolving power of a diffraction grating. The angular maximum-to-first-zero separation is $\lambda /(ND\cos {\theta }_m)$, while nearby wavelengths are separated by $|m|\delta \lambda /(D\cos {\theta }_m)$. The order must exist, carry appreciable light intensity and remain free from dominant instrumental broadening. Line count alone therefore does not determine whether a particular instrument resolves a specified doublet.

A normal-incidence diffraction grating of pitch $D$ has order $m$ at $D\sin \theta =m\lambda$ and angular separation $\Delta \theta \simeq m\Delta \lambda /(D\cos \theta )$. A lens whose axis follows the selected order maps a small angular interval to $\Delta x\simeq f\Delta \theta$. Detector pixels substantially smaller than this separation are needed; two pixels per resolution interval correspond to $t\lesssim \Delta x/2$. If instead detector coordinates obey $x=f\tan \theta$ relative to the original axis, the scale is $\Delta x\simeq f{\sec }^2\theta \Delta \theta$. A numerical pixel limit therefore requires an optical layout and a sampling criterion as well as the resolving power of a diffraction grating.

Landau diamagnetism is the orbital response of mobile electrons to an applied magnetic field, arising from quantization of their transverse motion. For a noninteracting three-dimensional isotropic electron gas with quadratic dispersion, electron spin with $g=2$ and magnetic fields weak enough for the smooth-response limit, its susceptibility is ${\chi }_L=-{\chi }_P/3$, where ${\chi }_P$ is the Pauli paramagnetism susceptibility. Band effective masses, interactions, dimensionality and resolved magnetic quantum oscillations can change this simple relation.

For a zero-bias abrupt p-n junction, choose the p side at $x<0$ and the n side at $x>0$. Ionized acceptors and donors remain fixed while mobile electrons and electron holes diffuse and establish a built-in electric field. In the depletion approximation, mobile carriers are neglected on $-x_p<x<x_n$ and the net charge density isCharge neutrality gives $N_A{x}_p=N_D{x}_n$. The quasi-neutral bulk can have a large mobile carrier density while its net charge density is nearly zero, because fixed dopant charge compensates it. With complete ionization and nondegenerate carriers, equilibrium also gives $np=n_i^2$; real carrier profiles vary smoothly rather than forming exact step functions at depletion edges.

For a planar abrupt p-n junction with ionized densities $N_A,N_D$, p-side width $w_p$ and n-side width $w_n$, zero external field requires $N_A{w}_p=N_D{w}_n$. Integrating Poisson equation gives $V_{\mathrm{bi}}-V=e(N_A{w}_p^2+N_D{w}_n^2)/(2{\epsilon }_s)$, with positive $V$ denoting forward bias. Therefore $W=w_n+w_p$ has the displayed value, $w_n=N_{A}W/(N_A+N_D)$ and $w_p=N_{D}W/(N_A+N_D)$. The positive depletion capacitance per area is $C_A={\epsilon }_s/W$. It equals $d{Q}_p/dV=-d{Q}_n/dV$ when $Q_p=-e{N}_A{w}_p$ and $Q_n=e{N}_D{w}_n$ are signed space charges per area. Differentiating a positive charge magnitude with respect to forward voltage instead gives a negative derivative. The square-root law assumes partially depleted regions, negligible mobile charge and no substantial injected-carrier storage.

The local density of states resolves the density of states by position as well as energy. With a consistent counting of spin states, its integral over the specimen's volume gives the total density of states, $g(E)=\int \rho (\mathbf{r},E)d^3\mathbf{r}$. It describes the spatial distribution of available states, rather than their occupation.