Neutron Scattering

10.5 Back-Scattering Spectrometer
It follows from Fig .6) and eq . (12) that for a given divergence
of the neutron beam a crystal will achieve the optimum
resolution in the modulus of k_ for the case of backscattering,
i .e . for 0 = 90° . This is the basic idea for the
backscattering-( ;r )-spectrometer which realizes this optimum
deflection angle both at the monochromator and the
analyzer . For the case of backscattering one may rewrite
eq. (12) to :
Aḵ div
_4k 1 -1 -
(AO)2
4, j d , cos(AO / 2) 8
(15)
Fig. 16) Bragg-reflection in the case of
near baclcscattering
Assuming that the divergence of the beam is determined by a neutron guide, one gets from eq .
(9) and for the isotope 58Ni
AO,,; -
2 (4zpb )
k
(16)
Inserting AO,,,; into eq. (15) delivers ° o ~dN - 510-5 and with eq. (12) an energy resolution
of AE = i AkN; 2 - E / k2 - 2 .4-10 -7 eV. This contribution to the energy resolution is thus independent
of the selected energy . Even with a triple axis spectrometer at a cold source this extreme
value of AE is out of question . An additional contribution to the variance of k results
from primary extinction, i .e . the fact that a final number of lattice planes contributes to the
Bragg-reflection, only. Perfect crystals are used in order to maximise this number . This second
variance is expressed in Fig . 5) by the thickness ofthe bisecting plane . The primary extinction
is proportional to the number of unit cells per volume NZ and the absolute value of the structure
factor FG, and inversely proportional to G2. For perfect crystals like e.g . Si the additional
variance is of about the saine order as that due to the divergence of the k-vector . The maximum
error for the energy results then from the sum of both contributions, i.e . adding the extinction
in eq. (15) (without derivation).
AE Ak (AO)2
=2(
E -ZCk
8 +
0 div
16 ZNZF, 1
G Z (17)
Yet, how is it now still possible to vary the incident energy at such a spectrometer being restricted
to the deflection angles 0=90' ? To this end one needs - according to eq. (1) - a
variation of the lattice parameter or the reciprocal lattice vector !2 . This can be achieved by
heating the monochromator crystal or simply by moving the crystal (periodically) parallel to
the direction of G with a velocity v,, . The change of energy (in the laboratoty system, Doppler
effect) is then
AE= 2E- vo / v (18)
9- 1 4