Neutron Scattering

Then equation (5 .21) can be converted into a sum :
N
(A(o)A(T)) = lim 1 AjAj+n . (5 .22)
N-cc N j--1
In optical correlation spectroscopy sums like (5 .22) are calculated from the photodetector
signal by special purpose computers .
per time interval, i .e . the light intensity .
In this case A is the number of photons detected
It is easy to see that the autocorrelation function has the following properties
(A(0)A(T)) < (A(0)A(0)) - (A 2 ) (5 .23)
lim OO
(A(0)A(T)) = (A) 2 . (5 .24)
Figure 5 .9 shows a simulation of data of a light scattering experiment . Such data could
arise e .g . from scattering of polystyrene spheres in an aqueous dispersion .
The correlation function usually decays following a simple exponential law :
(A( 0)A(T)) = (A) 2 + ((A 2 ) - (A)2)
exp ( -T/TT)
(5 .25)
where Tr is the correlation time of the system . In general, also more complicated decays,
e .g . involving multiple characteristic times, are possible . But the decay always takes places
between the limits given by (5 .23) and (5 .24) .
Alternatively, one can consider the fluctuations 6A(t) = A(t) - (A), i .e . the deviations of
the observable from its average . For its autocorrelation function follows :
(SA(o)6A(t)) = (A(o)A(T)) - (A)2
= (6A2) exp ( -TI?T)
. (5 .26)
The general result is that the fluctuation autocorrelation function decays starting from
the variance of the observable, (6A2) = (A2) - 2 (A) to zero .
The time-dependent autocorrelation function describes the temporal fluctuation behaviour
of the system . In the case presented here of a polymer colloid the characteristic
time is directly connected to the diffusion constant : Tr-1 = DQ2 .
5-1 2