Neutron scattering practical

The sample under test is placed against a sample holder and held in place with aluminium foil (this is invisible to neutrons). An appature is set on the beam slit. The beam shutter is remotely opened and the data obtained. Pictures of the LOQ "pit" are in appendix 2 as well as a schematic of the LOQ instrument. The specifications[8] of LOQ are also in the appendix (appendix 3).

It is normal on a small angle neutron scattering (SANS) apparatus to measure both the sample as a SANS and TRANS (transmission). The blank is first measured (typically the sample which has been unchanged) to obtain a reference to be subtracted from the sample.

The samples are then altered and subjected to a fixed exposure to the beam and the intensities measured.

The instrument at ISIS used for this type of experiment is LOQ (Low Q). By plotting Q vs. I(Q)4 (intensity of Q), a typical graph (diagram 15) is produced.

Normalised graphs

Diagram 15.

 

Scattering intensities by computer simulation.

A computer simulation program to calculate both the exact and approximate lines for the scattering from a sphere for a given radius (R). It will also output the real numbers (Q4 values) for both the exact and approximate forms of scattering. The program is called ISHP.

ISHP requires a maximum value of Q (Qmax), a number of points to plot over and the sphere radius. This radius is in Angstroms, though it is expressed as if it was a reasonable (viewable!) number (i.e. 160 instead of 160 x 10-10).

The initial values of Q are calculated by first taking the entered Qmax and dividing this by the number of points to be plotted. This number is the fed into a loop and the values of Q calculated by using ((loop - 1) + ½) (loop is simply a counter inside the program). While this will not give the precise intensity for I(0), the value calculated will be very close.

For an exact number for I(0) for any given r, equation (31) is used.

equation 31

31

However, the number generated from (31) will not be greatly different to that calculated by the method outlined above.

The program then moves on to calculate (in order) exact I(Q), exact I(Q)4, approximate I(Q) and approximate I(Q)4 for the equations I(Q) vs. Q, ln I(Q) vs. Q2, log I(Q) vs. Q and log I(Q) vs. log I(Q).

For examination of the Guinier (low Q) region, the graphs of ln I(Q) vs. Q2 should be used. Log I(Q) vs. Q is used for low intensity I(Q) values while the log log graph examines the power law / fractal scattering.

The exact scattering is calculated using (32). On entry to the routine, Q1 equals Q, R0 is the sphere radius.

equation 32

32

If the Q4 value is required (32a) is multiplied by Q14.

Again, for the approximate form, the same values are passed into the routine.

equation 32a

32a

A comparison plot is given in diagram 16.

Diagram 16

Diagram 16. Comparitive plot of exact and approximate values

If plotted together, the approximate for can be seen to cut through the exact form in the forward (lower Q value) oscillation. Nearer to the convergence of the lines the approximate form will cut almost through the centre of the oscillations. At higher Q values (³ 2), the approximate form is of little use as it has already reached zero before the exact form.

What is the use for this?

In this form, its use is that we can see that the approximate that can be used for finding (roughly) were the oscillations reach zero far faster than calculating the exact scattering.

A far more useful method for seeing the scattering can be performed using a normalised plot over a given Q range.

Normalised plots are very simple to calculate as it is simply the ratio of I(Q)/I(0) i.e. the intensity at a given Q over the intensity when Q = 0 (or 0.5 in the program data is used). This gives a value between 0 and 1 for all intensities.

If a series of data for r = 5 to 160 (each new value of r is double the previous value) is treated in the normalised manner and taken over a range of Q = 0 to 2, then a diagram (such as diagram 15) is obtained.

The most interesting points about this point is that the intensities, no matter the value of R, are the same for each oscillation and the start of the oscillations (working from R = 5 backwards) is half that of the original (the first value for R = 5 is 0.9 while R = 10 is 0.45 and so on).

The intensity (when plotted as a non-normalised data set) is related to the radius by (33)

equation 33

33

If intensity is taken as the ratio of two intensities over R (for ease, R is taken as 5 and 100), then the n factor can be calculated.

equation equation 34

34

n comes out to be 106 for R = 10. If R = 20, then the difference will be 20 n6. It is plain to see that the intensity of scattering is very dependant on R.

The relationship between I(Q) and R.

If we have two intensities, I1 and I2, with two corresponding radii, r1 and r2, it can be shown that there exists a relationship of

equation 35

35

The proof of (35) can be derived from the following relationships and equations :

equation 36

36

Intensities can be represented as the average of the function F(Q) where F is (effectively) the mass of the sphere.

equation 37

37

equation 38

38

Combining (37) and (38) gives

equation equation equation equation 39

39

A simple proof

I1 and I2 are two intensities with two corresponding radii of r1 = 5Å and r2 = 100Å. By placing these into (35) we obtain

equation equation equation equation equation

The 10n is derived from taking logs of (35) thus

equation equation

If r2/r1 = 10,

equation equation equation