Scattering amplitude

Crossing symmetry relates scattering amplitudes by analytic continuation exchanging incoming and outgoing particles. A symmetric four-point scalar amplitude has equivalent descriptions under permutations of the Mandelstam variables.

Relativistic scattering cross-section

With relativistically normalized external states, a two-particle initial state has differential cross-section

d σ = ∣ M ∣^{2} d Φ_{n} / F

, where

d Φ_{n}

is the Lorentz-invariant phase-space measure and

F

is the invariant flux factor. For identical unobserved final particles, divide the full labeled phase-space integral by their permutation multiplicity.

Inclusive electron-positron annihilation into hadrons ( $σ_{h} (s)$ )

At leading electromagnetic order and with negligible electron mass, the photon-channel relativistic cross-section is

σ_{h} = 4 π^{2} α ρ_{h} = 16 π^{3} α^{2} ρ_{J} / s

, in the stated hadronic photon spectral density and hadronic electromagnetic-current spectral density conventions. Initial electron and positron spins are averaged; final hadronic states are summed inclusively. Electroweak channels beyond photon exchange are separate contributions.

Hadronic R ratio ( $R (s) = σ_{h} (s) / [4 π α^{2} / (3 s)]$ )

The hadronic R ratio normalizes the photon-channel hadronic relativistic cross-section to the leading massless muon-pair value. The free active-quark result is

R = N_{c} \sum_{f} Q_{f}^{2}

. It depends on flavor charges as well as the number of active flavors; it is not simply

N_{c} N_{f}

Elastic scattering from a quartic scalar contact interaction ( $d σ_{event} / d Ω = λ^{2} / (128 π^{2} s)$ )

The factorial-normalized quartic vertex gives an angle-independent tree amplitude of magnitude

∣ λ ∣

. For equal incoming and outgoing masses, the momentum ratio in the two-body cross-section cancels. The event density for identical outgoing particles over a full sphere is

λ^{2} / (128 π^{2} s)

. On a hemisphere representing each event once, the density is

λ^{2} / (64 π^{2} s)

. Both conventions give total event cross-section

λ^{2} / (32 π s)

Invariant flux factor ( $F = 4 (p \cdot q)^{2} - m_{1}^{2} m_{2}^{2}$ )

For two incident particles in relativistic normalization, this is

2 κ (s, m_{1}^{2}, m_{2}^{2})

, where

κ

is the Källén function. For equal masses it is

2 s (s - 4 m^{2})

Lorentz-invariant phase-space measure ( $d Φ_{n} (P) = (2 π)^{4} δ^{4} (P - \sum_{i} p_{i}) \prod_{i} \frac{d ^{3} p _{i}}{( 2 π ) ^{3} 2 E _{i}}$ )

The on-shell measure

d^{3} p / (2 E)

is Lorentz invariant. The product measure together with the four-dimensional Dirac delta distribution enforces four-momentum conservation. For equal masses in two-body scattering,

d Φ_{2} = 1 - 4 m^{2} / s d Ω/ (32 π^{2})

Relativistic two-body phase space ( $\frac{d Φ _{2}}{d Ω} = \frac{k}{16 π ^{2} s}$ )

For two outgoing particles with centre-of-mass momentum magnitude

k

and total energy

s

, integrating the four-momentum delta function gives

d Φ_{2} / d Ω = k / (16 π^{2} s)

. A factor

1/2!

for identical outgoing particles is inserted when counting events over the full phase space, not into this labelled kinematic measure itself.

Positive-energy on-shell delta function identity ( $\frac{δ ( ω - E )}{E} = 2 δ (E^{2} - ω^{2}), ω > 0$ )

For

E > 0

and the positive-energy branch, the root Jacobian of the Dirac delta function gives this identity. In a massless collinear parton approximation, it produces

δ (x - ξ) / (P \cdot q)

Two-body decay phase space ( $Γ = \frac{∣ k ∣}{8 π m _{A}^{2}} ∣ M ∣^{2}$ )

This total decay-width formula applies in the parent's rest frame when the squared scattering amplitude is independent of direction and the daughters are distinct. For identical daughters an additional symmetry factor is needed.

Identical-particle factor in a final-state phase-space integral

When the full Lorentz-invariant phase-space measure treats

n

identical final particles as labeled, each physical final configuration is counted

n!

times. The unlabelled relativistic scattering cross-section therefore includes

1/ n!

. For two identical scalars, this divides the full-angle result by two; equivalently, integrate over a region containing only one representative of each exchanged pair.

Momentum transfer ( $Q$ )

The momentum transfer in a scattering process is the difference between the incoming and outgoing wavevectors, commonly

Q = k - k^{'}

up to a sign convention and a factor of

ℏ

Elastic scattering

Elastic scattering preserves the total kinetic energy. For one particle scattered by a fixed target, the incoming and outgoing wavevectors therefore have equal magnitudes.

Diffraction

 2  0

Diffraction is interference produced when waves scatter from spatial structure. A periodic structure concentrates the scattered intensity at reciprocal-lattice conditions.

Crystal scattering

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.

Bravais lattice

 2  0

A three-dimensional Bravais lattice is

Λ = {n_{1} a_{1} + n_{2} a_{2} + n_{3} a_{3} : n_{i} \in Z},

where the primitive vectors

a_{1}, a_{2}, a_{3}

are linearly independent.

Face-centered cubic lattice

A face-centered cubic lattice has lattice points at every corner and face center of a conventional cube. One primitive cell has one quarter of the conventional cube's volume.

Lattice point

A lattice point is a vector

l = n_{1} a_{1} + n_{2} a_{2} + n_{3} a_{3}

whose coefficients in a chosen primitive basis are integers.

Reciprocal lattice ( $Λ^{*}$ )

The reciprocal lattice is

Λ^{*} = {q : q \cdot l \in 2 π Z for every l \in Λ} .

Ω = a_{1} \cdot (a_{2} \times a_{3})

, its primitive vectors are

b_{1} = 2 π \frac{a _{2} \times a _{3}}{Ω}, b_{2} = 2 π \frac{a _{3} \times a _{1}}{Ω}, b_{3} = 2 π \frac{a _{1} \times a _{2}}{Ω},

(2)

and satisfy

a_{i} \cdot b_{j} = 2 π δ_{ij}

Wigner-Seitz cell

The Wigner--Seitz cell of a lattice point consists of points at least as close to it as to every other lattice point. Its faces lie on perpendicular bisectors to neighbouring lattice points.

Reciprocal lattice and Wigner-Seitz cell of the unit triangular lattice

For direct primitive vectors

(1, 0)

and

(- 1/2, 3 /2)

, reciprocal primitive vectors are

(2 π, \frac{2 π}{3}), (0, \frac{4 π}{3}) .

The reciprocal Wigner--Seitz cell is a regular hexagon of area

8 π^{2} / 3

and circumradius

4 π /3

Body-centered cubic lattice

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.

Reciprocal lattice of the 2023 Cambridge body-centered cubic basis

For

a_{1} = \frac{a}{2} (1, 1, 1), a_{2} = \frac{a}{2} (1, - 1, 1), a_{3} = a (0, 0, 1),

a reciprocal basis is

b_{1} = \frac{2 π}{a} (1, 1, 0), b_{2} = \frac{2 π}{a} (1, - 1, 0), b_{3} = \frac{2 π}{a} (- 1, 0, 1) .

(2)

Thus

Λ^{*} = (2 π / a) {(h, k, l) \in Z^{3} : h + k + l even}

and its shortest nonzero vectors have length

22 π / a

Crystal lattice structure factor

For a finite set

S

of crystal sites, the lattice structure factor at momentum transfer

Q

Δ (Q) = \sum_{l \in S} e^{i Q \cdot l} .

It multiplies the single-atom scattering amplitude.

Reciprocal-lattice peaks of a finite crystal

For

S = {\sum_{i} l_{i} a_{i} : - L_{i} /2 \leq l_{i} \leq L_{i} /2}

and

α_{i} = Q \cdot a_{i}

Δ (Q) = \prod_{i = 1}^{3} \frac{s i n (( L _{i} + 1 ) α _{i} /2 )}{s i n ( α _{i} /2 )} .

For large

L_{i}

, each factor is sharply peaked when

α_{i} \in 2 π Z

, exactly the condition

Q \in Λ^{*}

Elastic Bragg scattering condition

For elastic scattering with

k - k^{'} = q \in Λ^{*}

and

∣ k ∣ = ∣ k^{'} ∣ = k

2 k \cdot q = ∣ q ∣^{2}, ∣ q ∣ = 2 k sin \frac{θ}{2} .

The smallest nonzero diffraction angle is determined by the shortest reciprocal-lattice vector.

Bragg scattering

Bragg scattering is coherent elastic scattering from a periodic lattice, enhanced when the wavevector transfer is a reciprocal lattice vector. Finite crystals have narrow rather than infinitely sharp reciprocal-space peaks.

Bound-state pole of the scattering amplitude