Nuclear recoil

Movement of an atomic nucleus after it receives momentum, for example in a collision. A finite nuclear mass requires sharing kinetic energy as well as momentum; the fixed-target approximation neglects this motion.

The relative light or charge yield of a nuclear recoil compared with an electron recoil at the same deposited energy in a specified material and field configuration. It connects an electron-equivalent calibration scale with a recoil-energy scale. It can depend on energy and measurement conditions; assuming a constant or extrapolating below calibration changes light-WIMP interpretations. It does not mean energy conservation fails: energy can enter channels not measured by that readout.

Nuclear-recoil kinematics of WIMP scattering ( $E_{R, m a x} = 2 μ_{χ N}^{2} v^{2} / m_{N}$ )

 0  0

For nonrelativistic elastic scattering of a WIMP of mass

m_{χ}

from a stationary nucleus of mass

m_{N}

, the reduced mass is

μ_{χ N} = m_{χ} m_{N} / (m_{χ} + m_{N})

. In the centre-of-momentum description, a scattering angle

ϑ

gives momentum transfer squared

q^{2} = 2 μ_{χ N}^{2} v^{2} (1 - cos ϑ)

. The target kinetic energy is

E_{R} = q^{2} / (2 m_{N})

, giving the displayed maximum at backscattering. In SI units, using masses in kilograms and velocity in metres per second gives joules. Low recoil energy and a falling recoil spectrum make threshold and response calibration important for dark-matter direct detection.

Nuclear elastic electron scattering kinematics ( $E^{'} = E / [1 + 2 E sin^{2} (θ /2) / (M c^{2})]$ )

 0  0

An elastic scattering event leaves the nuclear internal quantum state unchanged while transferring energy to its motion. Let a target of mass

M

initially be at rest. Neglect the electron rest mass, but retain relativistic target recoil. Conservation of momentum and conservation of relativistic energy give

ℏ^{2} c^{2} q^{2} = E^{2} + E^{'2} - 2 E E^{'} cos θ, (M c^{2} + E - E^{'})^{2} = M^{2} c^{4} + ℏ^{2} c^{2} q^{2} .

(1)

Eliminating

q

yields

M c^{2} (E - E^{'}) = E E^{'} (1 - cos θ)

and hence

E^{'} = \frac{E}{1 + \frac{2 E}{M c ^{2}} s i n ^{2} ( θ /2 )}, T_{R} = E - E^{'} .

(2)

The static relation

q = 2 E sin (θ /2) / (ℏ c)

follows only when

E / (M c^{2})

is negligible. For small recoil,

T_{R} ≃ ℏ^{2} q^{2} / (2 M)

; this last approximation is nonrelativistic for the recoiling nucleus, not for the incident electron. The invariant spacelike transfer, expressed in energy-squared units, is

Q^{2} = ℏ^{2} c^{2} q^{2} - (E - E^{'})^{2} = 4 E E^{'} sin^{2} (θ /2),

(3)

Q / (ℏ c)

and the spatial inverse length

q

coincide only in the static limit. Recoil kinematics alone does not correct the static differential scattering cross-section: a finite-mass target also changes its kinematic prefactor and the appropriate current matrix elements.