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Optical diffraction and defocus calculations

Optical diffractometer

For the present calibration grating, spacing = 0.167 mm, and radius of 1st diffraction spot = 3.05 mm (you need to check this in case the optical components have been moved)

Camera constant = [grating spacing (mm) x diffraction repeat (mm) /magnification] x 106 to convert to nm.

NB: The diffraction repeat is measured in mm, but really corresponds to reciprocal mm, so that bigger spacings on the image correspond to smaller distances on the diffraction pattern.

For these values, and a magnification of 40,000, the camera constant CC = 12.73 (the repeat spacing in nm on a film at x40,000 giving rise to a 1 mm repeat on the optical diffraction pattern). For a magnification of 30,000, the camera constant CC = 17.

For a feature at radius R mm on the optical diffraction pattern, the corresponding spacing on the image is CC/R nm.

Procedure for determining defocus

Measure the radius of the first CTF minimum on the optical diffraction pattern of a drift-free area of carbon film on your negative. Note that if the defocus is less than 3-400 nm, it will be very difficult to see the minimum. Using a PC in the lab, run the CTF program (type 'ctf'). Check that the parameters are correct for your image (voltage, magnification, calibration of optical diffractometer) by selecting option p. Option c allows you to calculate the defocus by entering the position of the first minimum in mm. This also gives you the position of the first zero in nm. To see what spacings will be visible at a given defocus, use option g to plot the ctf function on the screen, either as amplitude or intensity.

Measurement of digital images and calculated diffraction patterns

Scanning films

The full field that can be captured in one image is 768x512 pixels. For diffraction calculations, you can only use 512x512 (or square areas of 256, 128, 64,..). The magnification of the image on the CCD camera is determined by the setting of the camera lens: with the lens barrel fully extended, 512 pixels covers 5.5 mm, giving 10.8um/pixel. This is the maximum resolution we can get with the CCD camera setup. 20 um/pixel is good enough, but the camera is less accurate in one direction, so it is best to scan at this resolution and later interpolate the digital image to twice the pixel size. If you just want to see features on the diffraction pattern, and do not require the maximum possible resolution, you should set the lens so that 512 pixels covers about 11 mm. This will give you 4 times the area, and make the diffraction pattern stronger. Measure the field size by capturing the image of an eyepiece graticule. Note that focussing is done by adjusting the height of the camera on the copy stand, once the lens zoom setting is chosen. Do not change the lens position once you have chosen the zoom (field size)! The lens aperture should be set around F5.6. If the contrast is low, you may need to try other settings.

Measurement of diffraction spacings

Semper commands

par 1 size 512 [set the size of the display partition to 5122]

live [capture the video image]

ps ln dis:1 100 [calculate the power spectrum (diffraction pattern) of the screen image (display:1) on a logarithmic scale (more convenient for display) and put the result in file number 100]

dis 100 [display result; only half is shown as the other half is the same]

xwires line [puts a cursor on the screen; click on center and then on point you want to measure]

type r [To get the length of the line you have just measured, in pixels]

you can also try:

section 100 101 [This calculates the radial average of your diffraction pattern, making weak features easier to see on a linear plot]

dis 101 [shows the radial plot]

Camera constant CC = width of square field in mm x 106 / EM magnification

Spacing in nm = CC/tu, where tu is the number of transform units (pixels on the diffraction pattern) from the center to the position being measured.

For a field of 5.5 mm and a magnification of 40,000, CC = 137.5 nm.