describe the use of a diffraction grating to determine the wavelength of light (the structure and use of the spectrometer are not included)

Diffraction Grating – Determining the Wavelength of Light

Contents

• Assessment Objectives (AO) link
• Practical skills link (Paper 3/5)
• Cross‑topic connections
• 1. Theory of a Diffraction Grating
• 2. Experimental Determination of a Wavelength
• 3. Worked Example (including uncertainty)
• 4. Practical Considerations
• 5. Sources of Error & Mitigation
• 6. Typical Grating Specifications
• 7. Summary

Assessment Objectives (AO) Mapping

AO1 – Knowledge & Understanding: Identify the structure of a diffraction grating and state the grating equation.

AO2 – Application of Knowledge: Use the grating equation to calculate wavelength from measured angles, including propagation of uncertainties.

AO3 – Analysis & Evaluation: Analyse experimental data, evaluate sources of error and suggest improvements.

Practical Skills (Paper 3/5) – What you will do in the lab

• Set‑up a spectrometer with a transmission or reflection grating.
• Align a narrow, collimated beam so that it is incident normal to the grating surface.
• Read diffraction angles from the calibrated circular scale (both left‑ and right‑hand sides).
• Record data in a structured table, calculate λ and its uncertainty, and compare with known spectral lines.
• Complete a written evaluation that discusses systematic and random errors – a key component of Paper 5.

Cross‑topic Connections (Cambridge 9702)

• 8.3 – Wave interference: The grating produces multiple‑slit interference; the same path‑difference principle underlies both double‑slit and grating patterns.
• 8.5 – Polarisation: When a polariser is placed before the grating, only the component of the electric field parallel to the slits contributes to the diffracted orders.
• 22.1 – Photon energy: Once λ is known, the photon energy can be obtained via E = hc/λ, linking optics to quantum physics.

1. Theory of a Diffraction Grating

1.1 What a diffraction grating is

A diffraction grating is a plate that contains a very large number (typically > 10⁴) of equally spaced, parallel slits or reflective rulings. The spacing between adjacent slits is called the grating spacing d (the reciprocal of the line density).

1.2 Derivation of the grating equation (path‑difference method)

1. Consider two adjacent slits, S₁ and S₂, illuminated by monochromatic light of wavelength λ.
2. For a point P on a screen (or on the angular scale) at an angle θ measured from the normal, the extra distance travelled by the wave from S₂ compared with S₁ is Δ = d sin θ.
3. Constructive interference (bright order) occurs when this path‑difference is an integer multiple of the wavelength:
   Δ = n λ → d sin θ = n λ, where n = 0, ±1, ±2,…
4. The same relation holds for transmission and reflection gratings; the sign of θ indicates the side of the normal.

1.3 Grating equation – quick reference

2. Experimental Determination of a Wavelength

2.1 Aim / hypothesis

“If the diffraction angle θ for a known order n is measured accurately, then the wavelength λ of the incident light can be calculated using the grating equation.”

2.2 Apparatus (brief list)

• Transmission or reflection diffraction grating (known line density)
• Spectrometer with a circular angular scale (or a calibrated protractor + screen)
• Narrow slit + collimating lens (or a low‑power laser)
• Reading aid (e.g., a pointer or sight‑line) to avoid parallax
• Reference spectral line (e.g., a mercury or sodium lamp) for verification

2.3 Procedure (step‑by‑step)

1. Set‑up: Mount the grating on the spectrometer so that the incident beam strikes it at normal incidence (i = 0°). Verify normality by reflecting the beam from a plane mirror placed at the grating position – the reflected beam should retrace the incident path.
2. Alignment: Adjust the slit and collimating lens until a sharp, well‑defined beam passes through the centre of the grating.
3. Measurement: Rotate the telescope (or sight‑line) until the first‑order (or chosen order) diffracted spot is centred. Record the angle θ₁ on the left side and θ₂ on the right side. Repeat for at least three separate readings.
4. Data handling: For each reading compute the mean angle θ̄ = (θ₁ + θ₂)/2. Convert the line density to spacing d = 1/(lines mm⁻¹) × 10⁻³ m.
5. Calculation: Use λ = (d sin θ̄)/n. Propagate uncertainties (see Section 2.5).
6. Verification: Compare the obtained λ with a known value for the source; calculate the percentage error.
7. Evaluation: Discuss the size and origin of the error, referring to the table of sources of error.

2.4 Data‑recording template

2.5 Uncertainty propagation

For the calculation λ = (d sin θ)/n the dominant uncertainties are those in θ and, if applicable, in d. Using standard propagation of independent errors:

\[

\frac{\Delta \lambda}{\lambda}= \sqrt{\left(\frac{\Delta d}{d}\right)^{2}+\left(\frac{\Delta\theta\;\cos\theta}{\sin\theta}\right)^{2}}

\]

where:

• Δd – manufacturer tolerance or calibration error (often ±1 % of d).
• Δθ – reading uncertainty (typically ±0.2° for a spectrometer; convert to radians when using the formula).

Calculate Δλ and report the result as λ ± Δλ (e.g., 571 ± 4 nm).

3. Worked Example (including uncertainty)

Given:

• Grating line density = 600 lines mm⁻¹ → d = 1/(600) mm = 1.667 × 10⁻⁶ m
• First‑order (n = 1) bright spot measured: θ₁ = 20.2°, θ₂ = 19.8°
• Instrumental angle uncertainty = ±0.2° (one‑sigma)
• Line‑density tolerance = ±1 %

Step 1 – Mean angle:

\[

\bar θ = \frac{20.2° + 19.8°}{2}=20.0°

\]

Step 2 – Convert to radians for the uncertainty term:

\[

Δθ = 0.2° = 0.2\times\frac{\pi}{180}=3.49\times10^{-3}\,\text{rad}

\]

Step 3 – Calculate λ:

\[

λ = \frac{d\sin\bar θ}{n}= \frac{1.667\times10^{-6}\,\text{m}\times\sin20.0°}{1}=5.71\times10^{-7}\,\text{m}=571\,\text{nm}

\]

Step 4 – Propagate uncertainties:

\[

\frac{Δd}{d}=0.01,\qquad

\frac{Δθ\cos\bar θ}{\sin\bar θ}= \frac{3.49\times10^{-3}\times\cos20°}{\sin20°}=0.0091

\]

\[

\frac{Δλ}{λ}= \sqrt{(0.01)^{2}+(0.0091)^{2}}=0.0135\;(1.35\%)

\]

\[

Δλ = 0.0135\times571\,\text{nm}=7.7\,\text{nm}

\]

Result: λ = 571 ± 8 nm (rounded to one significant figure in the uncertainty).

Comparing with the known green‑yellow sodium D‑line (589 nm) gives a percentage error of ≈ 3 %, which is acceptable for a routine AS‑level experiment.

4. Practical Considerations

• Order selection & resolution: Higher orders increase angular dispersion (Δθ/Δλ) and thus improve resolution, but may cause overlap of different wavelengths (order confusion).
• Line density (d): Finer gratings (≥ 1200 lines mm⁻¹) give larger θ for a given λ, useful for precise measurements; coarse gratings (≈ 300 lines mm⁻¹) are easier to align and give wider angular separation for teaching demonstrations.
• Normal incidence: Any tilt of the grating changes the effective incident angle i, modifying the equation to d(sin θ + sin i) = nλ. Keep i ≈ 0°.
• Angle measurement technique: Read the circular scale from directly above, align the pointer with the index line, and record both left‑ and right‑hand angles.
• Wavelength range: Use UV‑grade gratings (often fused silica) for λ < 350 nm; standard glass gratings are suitable for the visible range (400–700 nm).
• Beam quality: A narrow slit (≈ 0.1 mm) and good collimation reduce spot broadening and improve angular precision.

5. Sources of Error and Mitigation

6. Typical Grating Specifications (Cambridge 9702 labs)

7. Summary – Key Points to Remember

• The grating equation d sin θ = nλ links the measured diffraction angle to the wavelength.
• Accurate λ determination requires:

  • Correct knowledge of the grating spacing d.
  • Normal incidence (i ≈ 0°) and precise alignment.
  • Accurate angle measurement (average left/right, minimise parallax).
  • Appropriate choice of order n and line density to avoid overlap while maximising dispersion.

• Propagate uncertainties from d and θ to obtain a realistic error estimate for λ.
• Evaluate results against known spectral lines; discuss systematic and random errors – a vital skill for Paper 5.