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The Diffraction Grating (CIE A Level Physics)

Revision Note

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The diffraction grating equation

  • A diffraction grating is a piece of optical equipment that creates a diffraction pattern when it diffracts monochromatic light into bright and dark fringes

bRavNlJQ_laser-diffraction-grating-set-up

A laser light is diffracted using a diffraction grating

  • A diffraction grating consists of a large number of very thin, equally spaced parallel slits carved into a glass plate

diffraction-grating

A diffraction grating consists of many parallel equally spaced slits cut into the glass plate

  • Just like for single and double-slit diffraction the regions where constructive interference occurs are also the regions of maximum intensity
  • Their location can be calculated using the diffraction grating equation

d sin open parentheses theta close parentheses space equals space n lambda

  • Where:
    • d = spacing between adjacent slits (m)
    • θ = angular separation between the order of maxima (degrees)
    • n = order of maxima (n = 0, 1, 2, 3...)
    • λ = wavelength of light source (m)

Slit spacing

  • Diffraction gratings come in different sizes
    • The sizes are determined by the number of lines per millimetre (lines / mm) or lines per m
    • This is represented by the symbol N
  • d can be calculated from N using the equation
    • If is given in terms of lines per mm then will be in mm
    • If is given in terms of lines per m then will be in m

d space equals space 1 over N

diffraction-grating-sizes

Diffraction gratings come in different sizes according to the number of lines per mm

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 of n are at greater angles

Angular separation, downloadable AS & A Level Physics revision notes

Angular separation increases as the order of maxima increases

  • 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 at n1 and n2 is θ2  θ1

Orders of maxima

  • The maximum angle of diffraction with which maxima can be seen is when the beam is at right angles to the diffraction grating
    • This means θ = 90o and sin θ = 1

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

n space equals space d over lambda

  • Since n is an integer number of maxima, if the value obtained is a decimal it must be rounded down to determine the highest-order visible
    • E.g If n is calculated as 2.7 then n = 2 is the highest-order visible

Worked example

An experiment was set up to investigate light passing through a diffraction grating with a slit spacing of 1.7 µm. The fringe pattern was observed on a screen. The wavelength of the light is 550 nm.

Worked Example: Diffraction Grating, downloadable AS & A Level Physics revision notes

Calculate the angle α between the two second-order lines.

Answer:

Step 1: List the known quantities

  • Order of maxima, n = 2
  • Diffraction slit spacing, d = 1.7 µm = 1.7 × 10–6 m
  • Wavelength, λ = 550 nm = 550 × 10–9 m

Step 3: Rearrange for β and substitute in the values

d sin open parentheses theta close parentheses space equals space n lambda

sin open parentheses beta close parentheses space equals space fraction numerator n lambda over denominator d end fraction

sin open parentheses beta close parentheses space equals space fraction numerator 2 space cross times space open parentheses 550 space cross times space 10 to the power of negative 9 end exponent close parentheses space over denominator 1.7 space cross times space 10 to the power of negative 6 end exponent end fraction space equals space 0.64706

beta space equals space sin to the power of negative 1 end exponent open parentheses 0.64706 close parentheses space equals space 40.32 to the power of degree

Step 4: Calculate α

  • β is the angle from the centre to the second-order line

alpha space equals space 2 beta space equals space 2 cross times 40.32 space equals space 81 degree space open parentheses 2 space straight s. straight f close parentheses

Examiner Tip

Take care that the angle θ is the correct angle taken from the centre and not the angle taken between two orders of maxima.

Determining the wavelength of light

Method

  • The wavelength of light can be determined by rearranging the grating equation to make the wavelength λ the subject

d sin open parentheses theta close parentheses space equals space n lambda space rightwards arrow space lambda space equals space fraction numerator d sin open parentheses theta close parentheses over denominator n end fraction

  • The value of θ, the angle to the specific order of maximum measured from the centre, can be calculated through trigonometry
  • Create a right angled triangle to determine the angle of diffraction, θ
    • 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 below) on the screen is labelled as h, the fringe spacing
  • Measure both of these values with a ruler
  • Obtain the ratio space tan space theta space equals opposite over adjacent space equals space h over D space

Order of maxima and wavelength

Wavelength of light setup, downloadable AS & A Level Physics revision notes

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

  • Calculate the inverse of tan to find theta space equals space tan to the power of negative 1 end exponent open parentheses h over D close parentheses
  • Substitute for θ back into the diffraction grating equation to find the value of the wavelength (with the corresponding order n)

Improving the 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 find 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

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Ashika

Author: Ashika

Expertise: Physics Project Lead

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.