Neutron Scattering

esults, where il is an irrelevant calibration factor and w, resp . w A are normalized distribution
functions . w", represents the spectrum of the sample as found in the scattering
function and wA takes account for the fact that the NSE spectrometers usually are operated
with a broad incoming wavelength distribution (O,\Fwxml,\ = 10 . . .20%) . Observing
the linear dependences of k, A and v and series expansion of the squares in the
expression for the double differential cross section 10 .2 and insertion into equation 11 .7
leads to :
yJ1
'Y J2
+
S , w wa dw dA
Il 2 [ ( (h/m )A-1 (h/mn) ,\-1 + ,\wl27r)] (Q ) (,\)
I = 1 1 + cos -
11 .3 .1 Symmetric Case
At the point of symmetry J1 = JZ = J it is possible to collect the A-dependend terms in
equation 11 .8 and to write them as series expansion for small w :
3
h27r .\- 1 + .\ -1 + ,\(m /h)w/27r ,.J --
11 .9
To see the salient features of the spectrometer signal more clearly the finite wavelength
distribution is temporarily ignored
The underbraced product has the unit "time", the integral in equation 11 .10 represents
the cosine Fourier transform of S(Q, w) with respect to w, the resulting function is called
intermediate scattering function, S(Q, t) s From equation 11 .10 it is further reckognizable
that the time parameter t = yJm,,
(hz 27r) depends on the third power of the wavelength
A (i .e . long wavelength very long Fourier times) . In addition t oc J, i .e . mainly
s Strictly this is only true for a S(Q, w) that is symmetric with respect to w, i.e . in the
classical approximation . For any practical problems however this is well fulfilled since the
minute energy transfers corresponding to the NSE time scale are very small compared to
kBT .
=
=
z
rl2 [S(Q) + f cos `yJh 27r
A
3 w) S(Q, w)dw]
2
--t
(S(Q) + S(Q, t)) (11 .10)
11-9