Neutron time-of-flight scattering is a form of inelastic neutron scattering. It can be pulsed or continuous. In the pulsed version, the incoming neutron beam is collineated and monochomatized by a neutron chopper, which cuts the incoming beam into short pulses of known velocity, direction, and time at which it leaves the chopper. The pulses scatter off the sample, and the scattered neutrons are detected by a screen of detectors. Each detector detects and measures the time at which neutrons arrive at the detector. For example, a double-disk neutron chopper consists of two rotating disks with ends notched, in parallel, separated at a given distance d {\displaystyle d} , and rotating at a given angular velocity ω {\displaystyle \omega } . Neutrons entering one notch could only exit out of the other notch if its velocity is v 1 = n ω d / 2 π {\displaystyle v_{1}=n\omega d/2\pi } for some n = 1 , 2 , 3 , … {\displaystyle n=1,2,3,\dots } . To make the output monochromatic, a cascade of several choppers is used. After each pulse departure event, many neutron arrival events occur at various angles and time-delays. By taking average over many pulses of the same velocity and direction, we obtain a time-delay function of form Δ t ( k ^ ) {\displaystyle \Delta t({\hat {k}})} , which denotes the average time-delay between the departure of the pulse and the arrival of scattered neutron at the detector at direction k ^ {\displaystyle {\hat {k}}} . Now, suppose input neutron has velocity v 1 {\displaystyle v_{1}} , and the distance between the chopper output and the sample is L 1 {\displaystyle L_{1}} , and the distance between the sample and the detector at direction k ^ {\displaystyle {\hat {k}}} is L 2 ( k ^ ) {\displaystyle L_{2}({\hat {k}})} , then the velocity of the post-scattering neutron is v 2 ( k ^ ) = L 2 ( k ^ ) Δ t ( k ^ ) − L 1 / v 1 {\displaystyle v_{2}({\hat {k}})={\frac {L_{2}({\hat {k}})}{\Delta t({\hat {k}})-L_{1}/v_{1}}}} . This then allows us to calculate the momentum and energy transferred by the neutron to the sample. Inverse geometry spectrometers are also possible. In this case, the final position and velocity are fixed, and the incident coordinates are varied. The neutron time-of-flight peak for a pulsed-source moderator can be modeled using the Ikeda-Carpenter function . The Ikeda-Carpenter function is:
ϕ ( v , t ) = α 2 { ( 1 − R ) ∗ ( α t 2 ) e − α t + 2 R α 2 β ( α − β ) 3 } {\displaystyle \phi (v,t)={\frac {\alpha }{2}}\{(1-R)*(\alpha t^{2})e^{-\alpha t}+2R{\frac {\alpha ^{2}\beta }{(\alpha -\beta )^{3}}}\}}
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