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Keep the cubic translations above, but attach an atomic basis with A at fractional crystal coordinates $(0,0,0)$ and B at $(u,0,0)$. Stipulate real, angle-independent relative scattering amplitudes $f_A=1$ and $f_B=0.6$. Neglect wave absorption, repeated crystal scattering and thermal motion. This is a point-scatterer model under the Born approximation, not a claim about the scattering strengths of named elements. Its crystal basis structure factor isHere $J$ is a dimensionless squared motif amplitude. Comparing it with measured integrated wave intensities requires the experimental geometry and detector response as well. The algebra follows directly by multiplying $F$ by its complex conjugate; the cross term records wave interference between the two sites.

With $u=1/2$, reflections with even $h$ have $J=2.56$, whereas odd $h$ gives $J=0.16$, a ratio of $16:1$. With $u=1/4$, the first four $(h00)$ values instead are $1.36,0.16,1.36,2.56$. Their geometrically possible positions are unchanged, although their strengths change. For equal stipulated amplitudes $f_A=f_B=1$ and $u=1/2$, the odd-$h$ amplitude vanishes exactly: an example of a systematic absence. Different motif weights can thus expose a repeat that equal weights conceal.

The same formula also proves an ambiguity: $u=1/4$ and $u=3/4$ give identical $J_{hkl}$ for every integer $h,k,l$, because $\cos (2\pi h(1-u))=\cos (2\pi hu)$. Yet $F_{100}$ changes from $1+0.6i$ to $1-0.6i$, reversing its wave phase. This particular ambiguity is inversion of the motif; it does not prove that these two simple motifs are distinct chemical substances. The conclusion is conditional on the stipulated real scattering amplitudes.

For $V(r)=A{e}^{-\kappa r}/r$, with real $A$ and $\kappa >0$, the first Born approximation gives the scattering amplitudeIndeed, orienting the polar axis along the momentum transfer gives the angular integral $4\pi \sin (qr)/(qr)$, and the remaining integral is ${\int }_0^{\infty }{e}^{-\kappa r}\sin (qr)dr=q/(q^2+{\kappa }^2)$. The differential scattering cross-section is $|f_B(q){|}^2$.

The limit $\kappa \to 0^{+}$ at fixed nonzero $q$ produces the Rutherford differential cross-section for a Coulomb potential. The unscreened interaction has long range, so this limiting calculation by itself does not establish the validity of a naive Born approximation to the full Rutherford scattering state. The exact Rutherford scattering amplitude has an additional phase, while its modulus agrees with the Rutherford differential cross-section.

For a weak potential $u=ϵq$, the first Born approximation in the KdV Schrödinger convention is

Using $V=k_0^2(n^2-1)$, the scalar Lippmann-Schwinger equation is $\psi ={\psi }_i+{\int }_D{G}_{k_0}V\psi$. The Born approximation replaces the total field inside the integral by the known incident field:For a unit plane wave, its far-field pattern is $(4\pi {)}^{-1}{\int }_{D}V({\mathbf{r}}^{\prime })e^{i{k}_0({\hat{\mathbf{r}}}_0-\hat{\mathbf{r}})\cdot {\mathbf{r}}^{\prime }}d{\mathbf{r}}^{\prime }$. Thus each incident direction samples a shifted sphere in the Fourier transform of the potential.

For real $h\in C_c^{\infty }(D)$, the displayed real compactly supported contrast has Fourier transform $\hat{V}(\mathbf{\kappa })=[|\mathbf{\kappa }{|}^4-4({\mathbf{k}}_i\cdot \mathbf{\kappa }{)}^2]\hat{h}(\mathbf{\kappa })$. This vanishes on the shifted Ewald sphere because $|\mathbf{\kappa }{|}^2+2{\mathbf{k}}_i\cdot \mathbf{\kappa }=0$ there. Thus nonzero contrasts can have zero fixed-frequency, fixed-incidence Born approximation data. A small amplitude preserves weak contrast and a positive refractive index.

The change in incident and outgoing electron wavefunctions caused by the target Coulomb potential. It can invalidate a plane wave treatment in the Born approximation, especially for a large target atomic number.

In the Born approximation, scattering from identical atomic potentials translated to lattice sites factors into one atomic form factor times a finite sum of phases over the crystal sites.

Take $N$ equally spaced scattering sites at $x_j=jd$ on a line, a normally incident plane wave, equal site scattering amplitudes, and observation far enough away that outgoing directions are effectively parallel. Retain only one scattering event per amplitude, as in the Born approximation. For angle $\varphi$ from the outward normal, write the adjacent-site quantum phase difference as $\delta =kd\sin \varphi$, with $k=2\pi /\lambda$; reversing its overall sign leaves the intensity unchanged. Summing a geometric series givesAt $\delta =2\pi m$ use the continuous limit $I/I(0)=1$, rather than treating $0/0$ as a missing peak. The integer $m$ is the diffraction order: $d\sin \varphi =m\lambda$. This is a surface-row spacing, not the plane spacing in Bragg's law. The plotted intensity is normalized to its central value; it is not an integrated detection probability or an absolute collector electric current.

At $100\mathrm{V}$, $\lambda \simeq 0.12264\mathrm{nm}$ and the positive first-order angle is $\arcsin (\lambda /d)\simeq 29.4^{\circ }$. At $400\mathrm{V}$, the wavelength halves and this angle becomes about $14.2^{\circ }$. Increasing $N$ narrows each peak through more selective interference, while increasing momentum brings a fixed-order peak nearer the normal. Unequal site amplitudes, multiple scattering, a three-dimensional crystal lattice structure factor, and finite detector acceptance can change this simple picture; it is an explanatory model with hypothetical inputs.

An inverse scattering problem reconstructs a medium or obstacle from measured scattered waves. Under the Born approximation, fixed-frequency far-field data sample the Fourier transform of the scattering potential on a restricted frequency surface.

For an ultrarelativistic electron scattered elastically by a static, spherical, spin-zero atomic nucleus, a plane-wave Born approximation retains a single electromagnetic interaction. The Fourier transform of its Coulomb potential is proportional to $ZeF(q)/q^2$. Relative to the same point charge, the scattering amplitude is therefore multiplied by $F(q)$ and the differential scattering cross-section by $|F(q){|}^2$:Here $E\gg m_e{c}^2$ is the incident total electron energy, $\theta$ the scattering angle, $d\Omega$ a solid angle element, and $\alpha$ the fine-structure constant. The cosine factor already accounts for the projectile's spin; a spinless projectile would give a different point-charge factor. The formula neglects nuclear recoil, electron rest-mass corrections, radiative corrections and Coulomb distortion in electron scattering. The latter can be substantial at large $Z\alpha$: electron waves then require treatment in the target's Coulomb potential, rather than undistorted plane waves.

Nonzero target spin can introduce magnetic and additional electric contributions, so division by this point-charge expression does not generally isolate one charge form factor for a proton or deuteron. Even for spin zero, finite energy resolution, recoil and Coulomb distortion in electron scattering can change a measured minimum. A zero of the uniform-sphere model is a clear illustration of amplitude cancellation, not a universal prediction of a zero experimental counting rate.

Fermi's golden rule converts a weak transition matrix element and the available density of final states into a transition rate. For a static perturbation $W$ and a continuum,When the matrix element is nearly constant on the energy shell, this becomes $(2\pi /\hslash )|W_{fi}{|}^2\rho (E_i)$. The energy delta function is to be integrated, rather than interpreted as a finite rate to one exactly resonant discrete level. The observation time must resolve a smooth final-state continuum while remaining short enough that first-order depletion is small. A periodically driven perturbation instead gives the corresponding absorption and emission energy shells.

For elastic scattering from a weak localized potential $W(\mathbf{r})$, use box-normalized free states $⟨\mathbf{r}|\mathbf{k}⟩={\mathcal{V}}^{-1/2}{e}^{i\mathbf{k}\cdot \mathbf{r}}$. ThenThe number of final states is $\mathcal{V}{k}_f^2\mathrm{d}{k}_f\mathrm{d}\Omega /(2\pi {)}^3$. With $E_f={\hslash }^2{k}_f^2/(2m)$, the density per energy and solid angle is $\mathcal{V}m{k}_f/[(2\pi {)}^3{\hslash }^2]$. Dividing the resulting golden-rule rate by the incident single-particle flux $j_i=\hslash k_i/(m\mathcal{V})$ givesThis is the golden-rule derivation of the Born cross-section. For the elastic problem $k_f=k_i$, so the first Born scattering amplitude and cross-section areThe box volume cancels, as a physical cross-section must. The minus sign is fixed by the outgoing-wave scattering convention; the golden-rule rate alone determines only the modulus.

For example, $W(r)=g{e}^{-\kappa r}/r$ has $\tilde{W}=4\pi g/(q^2+{\kappa }^2)$, hence $f_B=-2mg/[{\hslash }^2(q^2+{\kappa }^2)]$, with $q=2k\sin (\theta /2)$. Replacing the exact scattering state by the incident plane wave is the essential Born approximation. It can fail for strong scattering, low-energy resonances and situations requiring repeated scattering; a small perturbation and a suitable continuum are separate conditions.^[ref]

For elastic scattering by the Yukawa potential, take $\kappa >0$ and use the first Born approximation in the plane-wave normalizationChoose the polar axis along the momentum transfer $\hslash \mathbf{q}$. The angular integral isThe radial volume element cancels the remaining powers of $r$, leavingFor real $A$, the differential scattering cross-section is thereforeThis also has a finite forward limit when $\kappa >0$. The Born amplitude for a Yukawa potential provides a screened route to the Coulomb potential by taking $\kappa \to 0^{+}$ at fixed $q>0$.

For an alpha particle of charge $2e$ scattering from a heavy nucleus of charge $Ze$, set $A=2Z{e}^2/(4\pi {ϵ}_0)$. Neglect recoil, so that the reduced mass is approximately the projectile mass, $E={\hslash }^2{k}^2/(2m)$, and elastic geometry givesThe Coulomb-limit scattering amplitude becomeswhich gives the Rutherford differential cross-sectionThe unscreened Coulomb potential has long range, so taking this limit does not justify the Born approximation for its full scattering state, particularly for a strongly charged target. The exact Rutherford scattering amplitude differs by a phase and has the same modulus. Thus the resulting differential scattering cross-section remains applicable to the ideal nonrelativistic Coulomb problem beyond the weak-potential regime.

To relate the detector counts to the atomic number, first compute the number of target atoms per illuminated area. With molar mass $M=0.197\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$, the number density integrated through the foil isThe detector's transverse area is already its area normal to the scattered ray. Its small solid angle isThere is no extra $\cos 45^{\circ }$ factor: the foil is traversed normally and the detector area is transverse to its line of sight.

For a thin foil with negligible multiple scattering and energy loss, the detected fraction is $f=\mathcal{N}(d\sigma /d\Omega )\Delta \Omega$. Thus the measurements implyUsing $E=7.68\times 10^{6}e\mathrm{J}=1.23047\times 10^{-12}\mathrm{J}$ in the Rutherford differential cross-section gives

The stated measurements do not reproduce gold's atomic number 79 in this model. The values and units above agree with the original PDF. Substituting $Z=79$ instead would predict $f\simeq 3.17\times 10^{-7}$ with the other inputs unchanged, rather than $3.7\times 10^{-7}$. No stated correction accounts for that discrepancy; forcing the answer to 79 would require changing the data or adding an experimental correction not specified in the problem.

Separate the entire-system atom index into a cell translation $\mathbf{R}$ and an index $p$ within a single primitive unit cell. The potential becomesThe same $u_p$ is used in every translated cell because the atomic basis repeats. Normalize plane waves in volume $\Omega$ by $⟨\mathbf{x}|\mathbf{k}⟩={\Omega }^{-1/2}{e}^{i\mathbf{k}\cdot \mathbf{x}}$. For $\mathbf{G}={\mathbf{k}}^{\prime }-\mathbf{k}$, substitution and $\mathbf{s}=\mathbf{x}-\mathbf{R}-{\mathbf{r}}_p$ giveFor a real static scalar potential, define the atomic scattering factor in the positive-exponent convention of the question byThe second bracket in the matrix element is $S(\mathbf{G}{)}^{\ast }$. At a reciprocal lattice vector every term of the first bracket is one, giving the number of cells $N_c$. Therefore the stated transition-rate rule, equivalently the first Born approximation and Fermi's golden rule up to the final-state kinematic factors, yieldsThe harmless complex conjugation records the distinction between the source's positive-exponent Fourier transform and the usual positive-exponent incoming plane wave. Reversing every Fourier sign gives the equivalent amplitude convention. Only the cell-translation sum ranges over the full crystal; the crystal basis structure factor contains the atoms of one cell.

The differential scattering cross-section is scattered rate per unit solid angle divided by incident flux. For the outgoing scattering solution, the incident plane wave has current magnitude $\hslash k/m$, while the outgoing spherical wave has radial current $(\hslash k/m)|f(\hat{x}){|}^2/r^2$. ThereforeThe incoming plane-wave normalization and outgoing radiation condition select the physical scattering state.

The time-independent Schrödinger equation is $[-{\hslash }^2{\nabla }^2/(2m)+V]\psi =E\psi$, equivalently $({\nabla }^2+k^2)\psi =(2m/{\hslash }^2)V\psi$. The given outgoing Helmholtz fundamental solution has source $-4\pi \delta$. Multiplying it by the source and adding the incident homogeneous solution givesApplying ${\nabla }^2+k^2$ verifies its equation and sign. At large $r$, $|x-x^{\prime }|=r-\hat{x}\cdot x^{\prime }+O(r^{-1})$ and $|x-x^{\prime }{|}^{-1}=r^{-1}+O(r^{-2})$ over the potential's support. ConsequentlyThe Born approximation replaces $\psi (x^{\prime })$ by the incident wave, givingIt requires weak wave distortion. A sufficient smallness condition from the integral equation is $(m/2\pi {\hslash }^2){\sup }_x\int |V(x^{\prime })|/|x-x^{\prime }|d^3{x}^{\prime }\ll 1$; for a bounded potential of range $R$ this is of order $m|V_0|R^2/{\hslash }^2\ll 1$. At high energy a small accumulated phase and weak deflection can also justify the approximation, but small potential relative to energy alone is not sufficient for an arbitrarily long interaction range.

For the exponentially screened potential, angular integration in its Fourier transform givesIt is not compactly supported, but for $\mu >0$ its decay gives the same scattering asymptotics and convergent integrals. Since $q=2k\sin (\theta /2)$,Thus the Born amplitude in this limit is independent of $\hslash$ when expressed using $E$ and angle; its squared magnitude is the Rutherford cross-section. This is a fixed-nonzero-angle limit of the Born formula. It does not establish uniform validity in the forward direction or preserve the short-range outgoing-wave asymptotics for the unscreened Coulomb potential.

For $p^2+q^2>1$, $m=i\sqrt{p^2+q^2-1}$ and the Weyl plane-wave representation contains evanescent waves. Their amplitudes decay exponentially with the distance from the scattering region. Excluding them removes high transverse spatial frequencies, including subwavelength information. Near-field measurements can in principle detect them, but back-propagating their decay exponentially amplifies measurement errors.

For propagating data, the Fourier transfer satisfies $|{\mathbf{k}}_s-{\mathbf{k}}_i|\le 2k$. One illumination still samples only the Ewald sphere, so this frequency bound does not imply access to every mode inside that ball. The lack of a volumetric spectrum is the separate nonuniqueness described in the previous part.

The Born approximation requires the incident field to remain a good approximation inside the scatterer. A useful sufficient smallness estimate for a weak, extended inclusion is small accumulated phase $kD|n-1|\ll 1$, alongside weak reflection and multiple scattering. Small pointwise contrast alone can fail for a thick inclusion or a resonance. The scalar Helmholtz and background-medium assumptions, and the extra linearization of $n^2-1$, also belong to the model's validity conditions.

Numerically, a finite aperture windows the measured fields and limits angular and Fourier resolution. Discrete sampling must satisfy the Nyquist–Shannon sampling theorem for the retained transverse bandwidth; spacings $\Delta x,\Delta y\lesssim \pi /K_{\max }$ avoid its simplest aliasing. Finite discrete Fourier transforms also introduce periodic continuation, truncation, interpolation and window errors. Noise, unresolved evanescent modes and near-grazing modes with $m\simeq 0$ limit reliable recovery. Refining the grid cannot by itself fill the missing three-dimensional Fourier data.

The nullspace in the Fourier diffraction theorem already prevents uniqueness, and the rapid decay of evanescent high-frequency information prevents stable recovery. Both make this an ill-posed inverse problem.

Let $A_{\pm }$ be the linear Born approximation operators mapping a real contrast supported in a known region $D$ to the measured fields on the two observation planes. For data $d_{\pm }$, a suitable Tikhonov reconstruction from two-plane scattering data minimizesA Sobolev norm penalty can instead express a smoothness preference. Reality, known support and $n=1-V/k>0$ can be imposed as constraints. For the unconstrained complex Hilbert-space problem with the displayed quadratic penalty, write $A=(A_{+},A_{-})$ and $d=(d_{+},d_{-})$. Its Tikhonov normal equation isTikhonov regularization selects a stable preferred estimate; it does not establish that the original two-plane data uniquely determine the true contrast.

Translate the scattering solution for the atom at the origin by $\mathbf{a}$. Its incident wave becomes $e^{i\mathbf{k}\cdot (\mathbf{r}-\mathbf{a})}$; multiplying the whole solution by $e^{i\mathbf{k}\cdot \mathbf{a}}$ restores the prescribed incident wave. At large $r$,Thus the outgoing amplitude acquires the phase $e^{i(\mathbf{k}-{\mathbf{k}}^{\prime })\cdot \mathbf{a}}$, with ${\mathbf{k}}^{\prime }=k\hat{\mathbf{r}}$.

In the single-scattering approximation, amplitudes from lattice sites add coherently. If $\mathbf{Q}=\mathbf{k}-{\mathbf{k}}^{\prime }$,For a rectangular block of a Bravais lattice with $N_j$ sites along primitive vectors ${\mathbf{a}}_j$ and $N=N_1{N}_2{N}_3$, the geometric series givesEach ratio has its limiting value $N_j$ when its argument is a multiple of $2\pi$. Thus the peaks have height $N$ and widths of order $1/N_j$ in those phase variables, at $\mathbf{Q}\cdot {\mathbf{a}}_j\in 2\pi \mathbb{Z}$, exactly the reciprocal lattice vectors.

With the incident-wave normalization used here, the Born approximation isFor the stated attractive three-dimensional delta potential,independent of scattering angle. This is the formal first Born amplitude, not a claim that an unregularized three-dimensional delta interaction defines an exact scattering Hamiltonian.

The 60-degree scattering angle is twice the Bragg angle, so $\theta =30^{\circ }$. First-order elastic Bragg scattering condition gives $2d\sin \theta =\lambda$, henceHigher orders would give $d=n\lambda$; the smallest, usual first-order spacing is the likely answer.

Write the observation point as $r\hat{r}$, put $q=k_0(\hat{r}-{\hat{r}}_0)$, and take ${\psi }_0(r\hat{r})=e^{i{k}_{0}r{\hat{r}}_0\cdot \hat{r}}$. For a bounded medium, the far-field approximation for an outgoing source usesFor finite-distance accuracy require $r\gg \ell$ and $k_0{\ell }^2/r\ll 1$ as well as being in the radiation region. Define the Fourier transform sample $S(q)={\int }_{D}V(r^{\prime })e^{-iq\cdot r^{\prime }}{d}^3{r}^{\prime }$ and the known factorThen the required total Born approximation for scalar wave scattering and total Rytov approximation areThe first Born approximation far-field pattern is $S(q)/(4\pi )$. In both formulas the incident field is retained; an outgoing scattered term alone would not be a total-field answer.

Let $\delta {\psi }_B={\psi }_1^{(B)}-{\psi }_i$ be the first-order scattered correction in the Born approximation for scalar wave scattering, using the paper's Green function convention. Comparing its outgoing integral with the preceding logarithmic correction givesTherefore the exact algebraic relation between the two first-order approximations isIf $V=O(\epsilon )$, then ${\chi }_1=O(\epsilon )$ on a controlled fixed region, andThus Born and Rytov agree to first order in the scattering potential, but differ as finite approximations. The Rytov approximation exponentiates the first logarithmic correction; the Born approximation adds the first field correction. The exponential's higher powers are not a calculation of all higher multiple-scattering terms in the Born series. This distinction explains why a smooth, appreciable wave phase accumulation can be represented more naturally by the Rytov approximation even when the corresponding linear field expansion is inaccurate.

The reciprocal lattice is ${\Lambda }^{\ast }=\{G:G\cdot \ell \in 2\pi \mathbb{Z}\text{ for every }\ell \in \Lambda \}$. With oriented primitive-cell volume $v_c=a_1\cdot (a_2\times a_3)$, its basis is $b_1=2\pi (a_2\times a_3)/v_c$ and the cyclic variants; thus $b_i\cdot a_j=2\pi {\delta }_{ij}$.

The total potential is $V(x)={\sum }_{\ell }U(x-\ell )$. Using the Fourier transform convention $\tilde{U}(q)=\int e^{-iq\cdot x}U(x)d^{3}x$, translation gives $\tilde{V}(q)=\tilde{U}(q)\Delta (q)$, withThe given Born approximation therefore supplies the amplitude $\boxed{-m\Delta (q)\tilde{U}(q)/(2\pi {\hslash }^2)}$. A finite geometric series gives, when the denominators are nonzero,At reciprocal-lattice vectors each phase is one, so $|\Delta |=N_1{N}_2{N}_3$, the number of atoms. For large finite crystals these narrow peaks represent coherent Bragg scattering, with peak intensity proportional to the squared number of atoms; their finite width reflects the finite crystal size.

For the specified face-centered cubic lattice, $v_c=a^3/4$ andThe reciprocal lattice have coordinates $(2\pi /a)(h,k,l)$ with $h,k,l$ all even or all odd. The two shortest nonzero length families are $G_1=2\pi \sqrt{3}/a$ for $(111)$ and $G_2=4\pi /a$ for $(200)$. Elastic scattering satisfies $|q|=2|k|\sin (\theta /2)$. The assumed incident magnitude exceeds $G_2/2$, so both families are kinematically accessible, andThis is the angular prediction for suitable crystal orientations, or for a powder containing those orientations. A fixed single-crystal orientation must additionally satisfy the vector Elastic Bragg scattering condition $2k\cdot G+|G{|}^2=0$; the magnitude bound alone cannot guarantee both peaks for every incident direction.