Physics and Engineering of Radiation Detection

12.3. Neutron Spectroscopy 705

Ei Ef

kf

2θa

2θs

2θm

ki

Neutron Source Sample

Detector

Analyzer

Monochoromator

Figure 12.3.3: Principle of triple-axis neutron spectrometry.

scattering angle. The above equation can also be written in terms of wave vector by
using the identityki=2π/λi.

2 πn

ki

=2dmsinθm (12.3.13)

For first order diffraction (n= 1), this equation reduces to

ki=

π

dmsinθm

. (12.3.14)

Sincedmfor the monochromator is known a priori andθmis measured, the wave
vector can be determined from this relation. The second wave vector, that iskf,can
be determined in a similar manner using the analyzer (see Fig.12.3.3). The defining
equation forkfis

kf=

π

dasinθa

. (12.3.15)

wheredais the atomic plane spacing of the analyzer andθais the analyzer scatter-
ingangle. Onceweknowkiandkfwe can determineω. Additionally, using the
knowledge of sample’s scattering angleθswe can calculate the momentum transfer
Q ̄through the relation 12.3.11.
Let us now turn our attention to the energy resolution of the system. We first
note that it depends on the resolution of the neutron wavelength. This can be seen
by differentiating equation 12.3.3 on both sides, which gives

dE = −

h^2

mλ^3

dλ (12.3.16)

⇒δE ∝ λ−^3 δλ. (12.3.17)

Next we differentiate equation 12.3.12 (withn=1)onbothsidestoget

δλ=2dcosθδθ. (12.3.18)

Substitutingdλinto equation 12.3.17 gives

δE∝

d

λ^3

cosθδθ, (12.3.19)