## Neutron Small Angle Scattering

The diffraction of light from a set of apertures (Young's experiment)

Most of the basic phenomena in small angle scattering can be understood from the much simpler experiment of diffraction from a set of apertures. For rectangular apertures (slits) the problem reduces to one dimension and the diffraction (scattering) pattern can be solved exactly. The scattering from a set of up to six slits is shown in the applet below (click here for information concerning applets; the filenames here are techniques/slits/slits.jar and techniques/slits/YoungsAppletJ.html). Below are some suggestions for examining the behaviour of the interference pattern, shown in the right hand panel.

(i) The initial setting shows the diffraction from a single slit. By varying the slit width you can verify that the diffraction gets wider as the slit gets narrower, which is an illustration of the inverse relation between width of the scattering and width of the object.

(ii) Change the number of slits to two to obtain interference between the light scattered at the two slits. The interference pattern for a number of slits is best viewed on a log scale (move right hand slider fully to the right). Note that the fringes move closer together as their separation is increased, also illustrating the inverse relation between scattering pattern and separation.

(iii) Set the slit width at about 8 μm and switch the number of slits between 2 and 1. Note that the overall intensity pattern is similar. What is being observed is an overall intensity contour determined by the scattering from a single slit. This intensity is then broken up by the strong interference between the slits but its overall contour does not change. This behaviour is common to all diffraction; the position of the stripes or spots depends on the arrangement of the slits or objects, but the intensity contour is determined by the structure of each individual slit or object.

(iv) Increase the number of slits. Note that the strong fringes are separated by subfringes and that there are n − 2 subfringes between each pair of strong fringes. This makes it easy to identify the number of slits giving rise to a given pattern.

(v) The diffraction pattern is sensitive to the wavelength. The best effect with white light (move bottom slider to far right) can be seen with 2-3 slits, separation about 2 mm and slit width about 2 μm. Since the single slit diffraction spreads longer wavelengths more than shorter wavelengths the progression of colours in each fringe is from the shorter blue to the longer red.

The same applet is used below, but with different settings, to illustrate what happens when the slits form a disordered array. This has interesting implications for small angle scattering, which is usually from a disordered arrangement of objects.

(i) The slit width is now fixed at 9 μm. Set the number of slits to 2 or more. Moving the disorder slider introduces disorder into the arrangement of the slits and hence changes in the diffraction pattern. Note that the pattern is always symmetrical about its centre even though the slits are not. This "centrosymmetry" is another characteristic feature of diffraction patterns and stems from the fact that the intensity in the pattern is the square of the amplitude.

(ii) Moving the right hand slider to the extreme right replaces the individual slit patterns by an average over 100 such patterns. Although this does not give a completely smooth average, it is sufficient to show up a feature very important in small angle scattering. This is that, except at very small angles, the average pattern of the disordered system starts to resemble the pattern from the individual slit. This can be seen approximately (there is some overall intensity variation) by setting the system to averaged maximum disorder and then varying the number of slits between 6 and 1. The effect of the number of slits, i.e. the interference between the different objects, is only retained close to the central maximum.

The diffraction of light from a two dimensional set of apertures

We now extend the one dimensional experiment demonstrated above to two dimensions. The scattering from a set of up to eleven vertical and horizontal apertures is shown in the applet below (click here for information concerning applets; the filenames here are techniques/2D/twoD.jar and techniques/2D/TwoDAppletJ.html). The left hand side is a schematic diagram of the optical arrangement except that the pattern is characteristic of Fraunhofer (far field) diffraction. The right hand side shows a circular cross section at right angles to the diffracted beam. Below are some suggestions for examining the behaviour of the interference pattern, shown in the right hand panel. The calculation is a large one. If it runs too slowly, you can speed it up by reducing the number of points for the calculation of the diffraction pattern (edit the html file), by setting the alias parameter to 0 (also in the html file), and/or by reducing the number of slits.

(i) The initial setting is for an array of 7 x 7 circular apertures. As for the one dimensional problem you can verify the inverse relation between aperture radius and aperture separation by combinations of different aperture sizes.

(ii) Change the number of apertures in the vertical and horizontal directions. Note that the diffraction spots are always broader along the direction of fewer apertures (all part of the inverse relation between scattering pattern and separation).

(iii) Set the slit width at the maximum value (5.5) and alter the number of vertical or horizontal slits. Note that the overall intensity pattern remains the same. What is being observed is an overall intensity contour determined by the scattering from a single aperture. This intensity is then broken up by the strong interference between the apertures but its overall contour does not change. This behaviour is common to all diffraction; the position of the stripes or spots depends on the arrangement of the objects, but the intensity contour is determined by the structure of each individual object.

(iv) Increase the number of apertures along either direction. Note that the strong fringes are separated by subfringes and that there are n − 2 subfringes between each pair of strong fringes. This makes it easy to identify the number of slits giving rise to a given pattern.

(v) The diffraction pattern is sensitive to the wavelength. Check that it varies in the direction that you expect.

(vi) The symmetry of the array can be altered using the shear slider. This varies the array between square and hexagonal. Note, however, that, owing to the finite extent of the array, only the positions of the spots have hexagonal symmetry. The shapes are determined by the separation of the spots along the two hexagonal directions, which is not the same.

(vii) The shape of the aperture can be changed from circular to square to rotated square using the button change aperture style. With square apertures the shape slider can be used to change the shape through rectangles of different aspect. By making the rectangle as long as possible and by reducing the slit size, you can approach the one dimensional slit pattern. Do this and follow the evolution of the pattern as you increase the number of vertical apertures from 1 upwards. Also verify that the intensity along the horizontal direction is more narrowly concentrated as the aperture width increases.

(viii) With the diamond aperture the shape, rotation slider rotates the aperture. The effect is more difficult to see but you should be able to verify that the intensity extends least along the diagonal directions of the square aperture.

To be continued.