Fix canonical single-particle order $(1\uparrow ,1\downarrow ,2\uparrow ,2\downarrow )$ and write the occupation ket as $|n_1{n}_2{n}_3{n}_4⟩$. The six states in the printed order are $|1010⟩$, $|1001⟩$, $|0110⟩$, $|0101⟩$, $|1100⟩$, $|0011⟩$. The last state is double occupancy on site 2.

For a fermionic raising and lowering operator acting on mode $j$, the sign is $(-1{)}^{{\sum }_{k<j}{n}_k}$, and annihilation requires that mode occupied while creation requires it empty. Starting from the second basis state, the hop that reaches the last state isIt therefore contributes $H_{6,2}=-t$. Starting from the third state instead,because the occupied $2\uparrow$ mode precedes $2\downarrow$. The Hamiltonian's leading $-t$ makes $H_{6,3}=+t$. Finally $n_{2\uparrow }{n}_{2\downarrow }|0011⟩=|0011⟩$, so $H_{6,6}=U$. No other displayed basis state reaches $|0011⟩$ in one allowed hop. These signs verify the three nonzero entries without assigning distinguishable-particle hopping signs to fermions.

Expand the field operator in orthonormal localized orbitals, $\hat{\psi }(\mathbf{r})={\sum }_i{\varphi }_i(\mathbf{r})c_i$. For a one-electron continuum Hamiltonian $h_0={\hat{p}}^2/(2m^{\ast })+V(\mathbf{r})$, projection givesThe coefficient in the displayed lattice Hamiltonian is $-t_{12}$. Thus the projected hopping matrix element and sign convention requireOrthogonality states $\int {\varphi }_1^{\ast }{\varphi }_2=0$; it does not imply $\int {\varphi }_1^{\ast }V{\varphi }_2=0$ for a spatially varying potential. If the off-diagonal potential is assumed zero, for example when it is a constant, the hopping coefficient is $h_{12}=K_{12}$. This is the kinetic integral requested by the paper, but it equals $-t_{12}$ under the paper's own leading-minus convention. A convention writing the hopping term with a plus sign would instead call $h_{12}$ itself $t_{12}$.

The printed positive equality is not a consequence of the stated orthogonality assumption and the displayed sign convention. Orthogonality does not diagonalize a varying potential gives an explicit counterexample to the underlying inference. The general projected relation above states the supported result; setting $V_{12}=0$ still leaves $-t_{12}=K_{12}$ in the question's convention. Rephasing an orbital changes both $h_{12}$ and the coefficient of the corresponding operator together; it does not remove this algebraic sign mismatch in a fixed convention. The spectrum in the remaining two-site calculation depends on $|t{|}^2$, so those parts are unaffected.

The first term assigns energy ${ϵ}_i$ to occupying the localized orbital at site $i$. The hopping term moves an electron between neighbouring sites, producing kinetic delocalization and band dispersion; Hermitian conjugate hops make the Hamiltonian Hermitian. The final term costs $U>0$ only when both spin states occupy the same local orbital. This is a Hubbard model with an on-site Coulomb penalty, not a classical interaction between magnetic dipoles.

At one electron per site and large $U$, a real hop is energetically costly, but a virtual hop into a doubly occupied intermediate state is possible. Opposite-spin states can exploit these virtual processes and lower their energy, while parallel spins cannot occupy the same one-orbital site. The resulting singlet–triplet splitting produces antiferromagnetic Hubbard kinetic exchange.

There is a terminology qualification. The paper calls this overlap-mediated process “direct exchange” because the hopping is directly between neighbouring magnetic orbitals. In the stricter distinction between direct Coulomb exchange versus kinetic exchange, the displayed Hamiltonian omits an explicit intersite Coulomb exchange integral. The antiferromagnetic $t^2/U$ mechanism derived below is kinetic exchange; a genuine intersite direct exchange term is separate and can compete with it. This distinction also separates the model from ligand-mediated superexchange in part f.

Experimental tests of BCS theory compare several properties of the same superconducting state. In the conventional weak-coupling setting, phonon-mediated attraction pairs opposite momenta and spins, and the BCS quasiparticle spectrum is $E_k=\sqrt{{\xi }_k^2+{\Delta }^2(T)}$. Agreement with a single energy scale and its temperature dependence is more discriminating than observing zero resistance alone.

The superconducting isotope effect tests the lattice's role. Replacing one isotope by another changes ionic masses while leaving the chemical valence nearly unchanged. Phonon frequencies scale approximately as $M^{-1/2}$, so a weak-coupling phonon mechanism predicts $T_c\propto M^{-\alpha }$ with $\alpha \simeq 1/2$ when other parameters stay fixed. The shift was measured in mercury before the microscopic pairing theory was developed.^[ref] Deviations from $1/2$ can result from Coulomb, anharmonic and strong-coupling effects; the isotope observation supports lattice-mediated pairing without uniquely proving every assumption of the simplest model.

Thermodynamic measurements test the gapped excitations and condensation energy. An isotropic fully gapped superconductor has exponentially small low-temperature electronic heat capacity, rather than the normal linear law. At the transition the weak-coupling BCS thermodynamic ratios areHere $C_n$ means the electronic contribution after subtracting the lattice background. The gap closes at $T_c$, and the same framework predicts the temperature dependence of penetration depth and thermodynamic critical field. These quantities were quantitative comparisons in the original theory.^[ref]

Electron tunnelling measures the excitation spectrum more directly. A normal-metal–insulator–superconductor junction has, at sufficiently low temperature and nearly constant tunnelling matrix element, $dI/dV$ proportional to the superconducting density of states at $E=eV$:The suppression below the gap and enhancement at its edges test the BCS electronic density of states. Giaever's thin oxide-barrier junction measurements established such energy-gap structure.^[ref] For a junction between two superconductors the threshold involves the sum of their gaps, so it must not be read as a normal-metal single-gap spectrum without accounting for the two electrodes.

Nuclear spin-lattice relaxation supplies a phase-coherence-sensitive test. In the conventional s-wave case, the Hebel-Slichter coherence peak just below $T_c$ is followed by a falling rate at lower temperature. Density of states alone is not enough: the BCS coherence factors enter the transition matrix elements. The contrast between nuclear relaxation and ultrasonic attenuation was explained using the paired state, unlike a simple two-fluid population model.^[ref]

Finally, magnetic flux quantization in units $h/(2e)$ and Josephson effects support a coherent charge-$2e$ condensate. Those observations establish pairing and phase coherence more broadly; their existence alone does not specify a phonon mechanism or an isotropic weak-coupling gap. The converging isotope, spectroscopic, thermodynamic and coherence-factor tests support conventional BCS superconductivity, with explicit limits to its simplest universal ratios.

A Wannier-Mott exciton is a bound conduction-band electron and valence-band hole whose envelope extends over many lattice cells. That large radius permits an effective mass description and dielectric screening of the electron–hole attraction. With electron and hole masses $m_e^{\ast },m_h^{\ast }$, define reduced mass $\mu =m_e^{\ast }{m}_h^{\ast }/(m_e^{\ast }+m_h^{\ast })$ and center-of-mass mass $M=m_e^{\ast }+m_h^{\ast }$. The bulk relative-motion Hamiltonian is hydrogenic with an appropriate relative permittivity ${ϵ}_r$:A semiconductor quantum well confines the electron and hole perpendicular to the plane while leaving approximately free in-plane center-of-mass motion. If the well width is small compared with the bulk exciton Bohr radius, their relative motion becomes quasi-two-dimensional. The ideal zero-width, same-dielectric limit isIts ground-state binding is $4R_X$, compared with $R_X$ in three dimensions, and the ground envelope decays on the scale $a_X/2$. Stronger binding follows because confinement prevents the pair from separating in the third direction.

For a finite-width quantum-well exciton, average the Coulomb attraction over the normalized electron and hole confinement envelopes:This smooths the strict planar singularity and changes the binding. The ideal factor four is a useful reference, not a universal value for every real well. Finite barriers, dielectric mismatch, several confinement subbands and nonparabolic valence bands can require a coupled treatment.

Optical absorption creates a nearly zero in-plane center-of-mass wavevector exciton. For the lowest confinement subbands, its line lies approximately atThe discrete bound-state peaks occur below the free electron–hole continuum. Dipole-allowed envelope states have nonzero relative wavefunction at zero separation; the absorption strength also contains the confinement overlap $|\int {\chi }_e^{\ast }(z){\chi }_h(z)dz{|}^2$. Confinement can increase the overlap and the binding, making excitonic optical structure more prominent. A perpendicular electric field shifts and separates the confinement envelopes, changing the transition energy and usually reducing their optical overlap: this is the quantum-confined Stark effect.

Quantum wells tune both the continuum threshold and the bound-state energy; an optical line shift is not automatically a change in binding alone. The hydrogenic envelope model requires a Wannier-sized state, rather than an exciton localized to one lattice cell.

Landau quasiparticles describe the long-lived low-energy excitations of an interacting normal Fermi liquid. Starting from a free Fermi gas, imagine switching on interactions without changing the phase or destroying the one-to-one labelling of low-energy states. The resulting excitation carries the quantum numbers of a dressed particle but has a renormalized energy and interactions with other excitations. Its microscopic wavefunction need not resemble a bare electron state. The occupied momenta define a Fermi surface, and the quasiparticle energy measured from the chemical potential vanishes on that surface.

The Landau quasiparticle energy functional expresses the change near equilibrium in terms of occupation changes:Differentiation gives the energy of one quasiparticle in the distorted distribution. This is a phenomenological expansion around the true interacting ground state, not the assertion that its total energy equals that of independent particles. For an isotropic system, spin-symmetric and spin-antisymmetric components of $f$ are expanded in Legendre polynomials. Multiplication by the appropriate Fermi-surface density of states defines dimensionless Landau parameters $F_l^s,F_l^a$. The original formulation relates a change in the distribution to the energy shift.^[ref]

Pauli blocking makes the low-energy excitations sharp. The phase space for a quasiparticle to scatter while creating another particle–hole pair shrinks as the Fermi surface is approached, giving the regular three-dimensional low-energy Fermi-liquid quasiparticle lifetimeThus at zero temperature its width is small compared with its excitation energy. The single-particle spectral function has a pole with quasiparticle residue $Z$, generally less than one, and an incoherent remainder. In a conventional Fermi liquid the occupation distribution has a discontinuity of height $Z$ at the Fermi surface, rather than a unit free-particle step.

For an isotropic three-dimensional system, define $N^{\ast }(0)$ to include both spins. The quasiparticle density of states determines the low-temperature electronic heat capacity, $C_V/V=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$. In the usual Landau-parameter normalization, the spin response is modified by $1/(1+F_0^a)$ and the compressibility by $1/(1+F_0^s)$. In a Galilean-invariant continuum, $m^{\ast }/m=1+F_1^s/3$; that particular relation must not be imposed unchanged on a crystal band. A denominator approaching zero signals a possible instability of the assumed normal state.

The same interaction functional and a kinetic equation describe collective zero sound when the excitation frequency exceeds the collision rate. Ordinary first sound is the collision-dominated hydrodynamic limit. The theory organizes thermodynamics, response and transport through a small set of measured effective parameters while explaining why interactions coexist with long-lived fermionic excitations. It is not applicable without qualification in phases where the Fermi-liquid quasiparticle pole disappears or a gap opens over the relevant Fermi surface.

The central limitation is that a Hartree-Fock electron gas includes exchange but omits the correlation energy that is not representable by one determinant. Opposite-spin electrons can lower their repulsion by avoiding one another in a correlated unpolarized state. Hartree-Fock misses that avoidance while retaining the exchange advantage of aligned spins, so it need not place a magnetic instability at a reliable interaction strength. The relevant energy differences between competing spin states can be much smaller than their total energies. Correlation calculations of the electron gas explicitly address this competition.^[ref]

There are further model restrictions. A constant contact coupling discards the momentum dependence of the Coulomb interaction and the resulting nontrivial screening. Real materials have a lattice, band structure and material-dependent effective interactions. Spatial magnetic textures or other competing states are excluded by a comparison restricted to uniform Fermi spheres. Even before those changes, part c's two endpoints do not find the continuous polarization onset.

The symbol $r_s$ also needs care. Here it is an energy ratio proportional to $\gamma n^{1/3}$. The usual Dimensionless Wigner-Seitz radius of a three-dimensional Coulomb electron gas is $r_s^{\mathrm{WS}}=[3/(4\pi n){]}^{1/3}/a_B$, proportional to $n^{-1/3}$. They are not the same density parameter. A quoted experimental value of the latter cannot be compared numerically to $1.175$ without a model-specific mapping. The contact interaction, missing correlations, restricted variational class and parameter convention prevent the number $1.175$ from being a quantitative experimental prediction. The paper supplies no particular experimental threshold, so none is invented here.

At fixed total density, the unpolarized and fully polarized candidate energies arePolarization increases the kinetic energy because one spin species must occupy a larger Fermi sphere, but gains negative exchange energy. The fully polarized candidate wins this endpoint comparison ifUsing the defined $r_s$ givesThus the specified two-state comparison indeed predicts a polarized state below the unpolarized state's energy for $r_s>1.175$.

The endpoint and continuous polarization thresholds of a contact electron gas are different. To check the actual onset within the same uniform Hartree-Fock model, allow $P=(n_{\uparrow }-n_{\downarrow })/n$ to vary continuously. In units $2A(n/2{)}^{5/3}$,The unpolarized state loses local stability already at $r_s=10/9\simeq 1.111$. Full polarization becomes the minimizing endpoint only when $f^{\prime }(1)\le 0$, or $r_s\ge (5/6)2^{2/3}\simeq 1.323$. Between those values the optimum is partly polarized. Consequently $1.175$ is the requested endpoint crossing, not the exact onset of spontaneous polarization after all spin populations are allowed.

For the unpolarized gas write $x=n/2$. The positive direct Hartree energy before background cancellation and the kinetic energy areTheir ratio is thereforeThis compares the separate direct and kinetic terms. The net Hartree-plus-background contribution after the stipulated cancellation is zero, so it would be incorrect to interpret $r_s$ as that residual contribution divided by $T$. The exchange-energy magnitude in the unpolarized state is $E_H/2$, not $E_H$.

Use normalized plane-wave spin orbitals $V^{-1/2}{e}^{i\mathbf{k}\cdot \mathbf{r}}{\chi }_{\sigma }$ and fill a separate Fermi sphere for each spin species. Each spin state counts once, soIts kinetic energy per volume isThe spin-independent contact interaction has uncancelled direct energy $E_H/V=\gamma n^2/2$, with $n=n_{\uparrow }+n_{\downarrow }$. By the imposed background assumption this direct contribution cancels. For the Fock term the contact delta function sets the two spatial positions equal; its spin factors then select equal spins. The local one-body density matrix for each spin has diagonal value $n_{\sigma }$, givingCombining these results gives the contact-interaction Hartree-Fock spin energyHere $\gamma$ has units of energy times volume. The result is the model with the stated background subtraction; an isolated contact-interacting gas without that subtraction retains its direct contribution.

An exciton is an electron-hole bound state and behaves approximately as a boson when the gas is dilute compared with the inverse exciton volume. A condensation experiment needs a long enough lifetime for cooling, high phase-space density and evidence of spontaneous coherence; excessive density causes ionization or a Mott transition instead.

Bulk Cu₂O offers long-lived excitons, but heating, losses and Auger recombination complicate interpretation. Spectral narrowing or anomalous transport alone is insufficient: early claims required reinterpretation. Coupled quantum wells separate electron and hole layers, increasing lifetime and enabling cold indirect-exciton gases, though disorder, trapping and nonequilibrium flow must be distinguished from condensation.^[ref]

In a semiconductor microcavity, coherent strong coupling mixes an exciton with a light cavity photon to form polaritons with very small effective mass. Nonresonant pumping supplies particles without imposing the condensate's phase. The 2006 CdTe experiment reported a threshold, strong low-momentum occupation, increased temporal coherence and long-range spatial coherence. These combined signatures support an exciton-polariton condensate; a threshold or bright emission alone does not distinguish it from photon lasing. Check that strong coupling persists above threshold. Finite lifetime and pumping make this an open, driven system, so thermalization must be demonstrated rather than inferred merely from the word condensate.^[ref]

In mixed-valence manganites, mobile $e_g$ electrons hop between sites whose localized $t_{2g}$ core spins are coupled to them by strong Hund's coupling. Projecting onto local spin-aligned states gives the double-exchange mechanismAlignment maximizes the hopping bandwidth and lowers carrier kinetic energy, favouring ferromagnetism. This involves real charge transfer, unlike insulating superexchange. Heating disorders core spins and reduces coherent transport; a magnetic field aligns spins, increasing mobility and reducing resistivity. The large negative response near the magnetic transition is colossal magnetoresistance.

Double exchange supplies the conduction–magnetism link, but its simplest spin-only model is insufficient for the full resistivity change. Lattice distortions, Jahn-Teller effect and carrier localization can amplify the effect; materials need not obey a universal resistivity-versus-magnetization law.^[ref][ref]

In weak-probe optical absorption, Fermi's golden rule creates an electron-hole pair, with negligible population depletion. Negligible photon momentum makes a direct band gap transition nearly vertical in crystal momentum:For allowed transitions between three-dimensional parabolic bands, the joint density of states yields $\alpha \propto \sqrt{\hslash \omega -E_g}/\omega$ above the gap. Selection rules and polarization enter the dipole matrix element. An indirect band gap needs an additional phonon to conserve crystal momentum, with phonon absorption/emission weights and shifted thresholds.

Electron-hole attraction produces Wannier excitons below the continuum, approximately $E_n=E_g-R^{\ast }/n^2$, with bright $s$-state oscillator strength decreasing as $n^{-3}$. It also enhances the continuum edge. Screening, disorder and phonons broaden these features; heavy/light valence bands can create distinct series. This linear-response picture includes Coulomb binding but excludes a resolved coherent photon–exciton anticrossing: that would require a strong light–matter coupling treatment. “Weak coupling” here is distinguished from neglecting the electron-hole interaction altogether.

Density functional theory replaces the interacting many-electron ground state wavefunction by its spatial density $n(\mathbf{r})$. For fixed interaction and a nondegenerate ground state, the Hohenberg-Kohn theorem fixes the external potential up to an additive constant. If two distinct potentials gave the same density, their strict variational energy inequalities would sum to the contradiction $E_0+E_0^{\prime }<E_0+E_0^{\prime }$.

The density functional variational principle minimizesover admissible densities of the correct particle number. The universal $F$ contains the kinetic and interaction energies. This is exact in principle; an explicit useful functional is the practical difficulty.^[ref]

The Kohn-Sham equations introduce noninteracting orbitals with the same density and decompose $F=T_s+E_H+E_{\mathrm{xc}}$. With $n={\sum }_i{f}_i|{\varphi }_i{|}^2$,The exchange-correlation functional includes exchange, correlation and the interacting kinetic-energy correction relative to $T_s$. Guess a density, construct the potentials, solve and occupy orbitals, then mix and iterate to self-consistent field convergence. The local density approximation uses $E_{\mathrm{xc}}^{\mathrm{LDA}}=\int n{ϵ}_{\mathrm{xc}}^{\mathrm{uniform}}(n)d^{3}r$; a generalized gradient approximation also depends on density gradients.

Kohn–Sham orbital energies are not generally excitation energies, and their sum double counts interactions. The Kohn-Sham energy double-counting correction iswith ion–ion energy added for a total crystal energy. Approximate functionals can miss self-interaction cancellation, band gaps and strong correlation; convergence alone does not prove the global ground state.^[ref]

In quasi-one-dimensional quantum transport, transverse confinement makes discrete subbands while motion along a quantum wire remains continuous. Only occupied propagating subbands conduct. For elastic scattering between equilibrated reservoirs, the Landauer formula isThe sum explicitly counts spin. A perfectly transmitting spin-degenerate subband gives $2e^2/h$ at low temperature; a quantum point contact shows steps as a gate opens additional subbands. The one-dimensional density of states cancels the carrier velocity in the current, giving a conductance quantum rather than a material-dependent velocity factor. Finite two-terminal contact resistance remains even without internal scattering.

Transport is ballistic transport for length $L\ll \ell$, where $\ell$ is the mean free path. In a multichannel wire with $\ell \ll L$ below its localization length, scattering reduces transmission and gives approximately Ohmic behaviour. If the phase coherence length exceeds the sample length, amplitudes interfere; weak localization and sample-dependent universal conductance fluctuations can appear. At greater lengths, generic coherent disordered wires undergo Anderson localization, with localization length of order the channel number times $\ell$, up to symmetry-dependent factors. Thermal averaging and dephasing smear the quantum features.

Strong interactions require more than independent-electron transmission. A clean single-channel interacting wire is described at low energy by a Luttinger liquid, with collective modes and possible spin-charge separation; impurities can generate temperature-dependent power laws. Ideal contacts to noninteracting leads can still fix the two-terminal conductance of a clean wire, so it must not automatically be multiplied by a bulk Luttinger parameter.

An organic semiconductor has conjugated molecular units with strong intramolecular bonding and comparatively weak intermolecular electronic overlap. Optical excitation commonly promotes a $\pi$ electron from an occupied molecular orbital toward an unoccupied one. Weak dielectric screening and narrow bands favour bound Frenkel excitons, sometimes mixed with intermolecular charge-transfer states, rather than immediate production of free carriers. The optical excitation energy is below the unbound electron–hole threshold by the exciton binding energy, with relaxation conventions specified when comparing measured gaps.

The electric-dipole interaction selects bright states through their transition dipoles, producing polarization-dependent absorption in oriented samples. From a singlet ground state, strong electric-dipole absorption normally preserves spin and reaches singlet excitons; direct triplet excitation is weak without spin mixing. The Franck-Condon principle treats excitation as vertical on the nuclear-motion timescale. For one displaced harmonic vibration at low temperature, the displaced-oscillator Franck-Condon progression has relative strengthsHere the Huang-Rhys factor $S$ measures the displacement and $E_{00}$ is the zero-vibration transition energy. Vibrational relaxation before emission produces a Stokes shift. Molecular packing and excitonic coupling shift or split bright features and change their intensities; disorder and vibrations broaden them. Consequently a universal parabolic-band absorption edge is inappropriate for a localized molecular excitation.^[ref]

An in-plane magnetic field lowers the Zeeman energy when moments align with it. It competes with antiferromagnetic interlayer exchange coupling; the layers generally cant or switch toward a parallel configuration and ultimately saturate if the field overcomes coupling and magnetic anisotropy. The high-resistance antiparallel state evolves toward the lower-resistance parallel state. The switching path and field scale depend on anisotropy, pinning and coupling strength, which are not numerically specified.

With opposite magnetic moments, a fixed global spin is the low-resistivity species in one ferromagnetic layer and the high-resistivity species in the other. Thus both spin channels have the same four-layer averaged resistivityEach channel's square resistance is $(a+b)/(4D^2)$, and the two channels run in parallel. ThereforeBoth channels encounter the strongly scattering magnetic layer; neither retains the uniformly favourable path available for parallel magnetizations.

Put $D=t_1+t_2$, $a=t_1{\rho }_{\mathrm{FM}}^L+t_2{\rho }_{\mathrm{NM}}$ and $b=t_1{\rho }_{\mathrm{FM}}^H+t_2{\rho }_{\mathrm{NM}}$. In the CIP thickness-averaged two-current model, assume spin is conserved and electron trajectories sample the layers, so each channel's effective resistivity is the thickness average: ${\bar{\rho }}_L=a/D$, ${\bar{\rho }}_H=b/D$. This uses averaged relaxation rates with comparable transport weights, neglecting additional interface resistance. Here ${\rho }_{\mathrm{NM}}$ is the per-spin-channel resistivity, twice the measured nonmagnetic bulk resistivity. The stated long mean free path motivates sampling more than one layer.

A four-layer period has thickness $2D$. For an in-plane square, $R_{\sigma }={\bar{\rho }}_{\sigma }L/(2DL)={\bar{\rho }}_{\sigma }/(2D)$, hence the two parallel spin conductances add:The square's side $L$ cancels. The layers are not being treated as four macroscopic resistors in series along the in-plane current; the averaging is over the traversing carriers' scattering histories.

The spacer electrons mediate interlayer exchange coupling. Spin-dependent reflection at the two magnetic interfaces changes their electronic energy as the spacer thickness varies. In the oscillatory spacer-mediated exchange picture, Fermi-surface spanning wavevectors produce terms such as $J(t_2)\propto \cos (Q{t}_2+\phi )$ with a decaying envelope; real spacers can have several periods.

Define the coupling energy per area by $E_{\mathrm{ex}}=-J(t_2){\hat{\mathbf{m}}}_1\cdot {\hat{\mathbf{m}}}_2$. $J>0$ favours parallel moments and $J<0$ antiparallel moments. The oscillation period and phase depend on spacer band structure and interface reflection; there is no universal monotone rule or numerical crossover thickness from $t_2$ alone. Roughness and nonuniform thickness average and weaken the coupling. This is conduction-electron-mediated exchange, not a requirement that the nonmagnetic spacer itself be ferromagnetic.