8.2.5 The Diffraction Grating

• A diffraction grating is a plate on which there is a very large number of parallel, identical, close-spaced slits
• When monochromatic light is incident on a grating, a pattern of narrow bright fringes is produced on a screen

Diagram of diffraction grating used to obtain a fringe pattern

• The angles at which the maxima of intensity (constructive interference) are produced can be deduced by the diffraction grating equation

Diffraction grating equation for the angle of bright fringes

• Exam questions sometime state the lines per m (or per mm, per nm etc.) on the grating which is represented by the symbol N
• d can be calculated from N using the equation

Angular Separation

• The angular separation of each maxima is calculated by rearranging the grating equation to make θ the subject
• The angle θ is taken from the centre meaning the higher orders are at greater angles

Angular separation

• The angular separation between two angles is found by subtracting the smaller angle from the larger one
• The angular separation between the first and second maxima n₁ and n₂ is θ₂ – θ₁

Orders of Maxima

• The maximum angle to see orders of maxima is when the beam is at right angles to the diffraction grating
  • This means θ = 90^o and sin θ = 1

• The highest order of maxima visible is therefore calculated by the equation:

• Note that since n must be an integer, if the value is a decimal it must be rounded down
  • E.g If n is calculated as 2.7 then n = 2 is the highest order visible

Method

• The wavelength of light can be determined by rearranging the grating equation to make the wavelength λ the subject
• The value of θ, the angle to the specific order of maximum measured from the centre, can be calculated through trigonometry
• The distance from the grating to the screen is marked as D
• The distance between the centre and the order of maxima (e.g. n = 2 in the diagram) on the screen is labelled as h - the fringe spacing
• Measure both these values with a ruler
• This makes a right-angled triangle with the angle θ as the ratio of the h/D = tanθ

The wavelength of light is calculated by the angle to the order of maximum

• Remember to find the inverse of tan to find θ = tan^-1(h/D)
• This value of θ can then be substituted back into the diffraction grating equation to find the value of the wavelength (with the corresponding order n)

Improving experiment and reducing uncertainties

• The fringe spacing can be subjective depending on its intensity on the screen. Take multiple measurements of h (between 3-8) and finding the average
• Use a Vernier scale to record h, in order to reduce percentage uncertainty
• Reduce the uncertainty in h by measuring across all fringes and dividing by the number of fringes
• Increase the grating to screen distance D to increase the fringe separation (although this may decrease the intensity of light reaching the screen)
• Conduct the experiment in a darkened room, so the fringes are clearer
• Use grating with more lines per mm, so values of h are greater to lower percentage uncertainty