Diffraction Grating Experiment: Procedure, d sin θ = nλ, and Spectra Analysis (H2 Physics Paper 4)

$$d\sin\theta = n\lambda$$

Where:

• $d$ = Slit spacing (distance between lines)
• $\theta$ = Angle from normal
• $n$ = Order number $(0, \pm 1, \pm 2...)$
• $\lambda$ = Wavelength

Why CDs and DVDs Work

CD specifications: Track pitch $\pu{1.6 µm}$ $≈ 625 lines/mm$ per Ecma International ECMA-130.
DVD specifications: Track pitch $\pu{0.74 µm}$ $≈ 1350 lines/mm$ per Ecma International ECMA-267.

Building Your Spectrometer

Method 1: Simple Hand-Held

Materials:

• Old CD/DVD (remove metal layer)
• Cardboard tube (paper towel roll)
• Razor blade for slit
• Black paper/tape
• Protractor

Construction:

1. Cut narrow slit ($\pu{0.5 mm}$) in cardboard
2. Mount CD piece at 45° angle
3. Look through CD at slit
4. See spectrum on either side

Method 2: Measurement Setup

For quantitative work:

[Light Source] → [Slit] → [Grating] → [Screen/Eye]
| | | |
     0cm         20cm       30cm     Measure angle

Key improvements:

• Fixed positions for consistency
• Rotating grating mount
• Degree scale for angles
• Dark box to reduce stray light

Method 3: Smartphone Spectrometer

Modern approach:

1. 3D print or cardboard housing
2. DVD piece as grating
3. Phone camera as detector
4. Apps for analysis

Benefits: Digital recording, intensity plots, easy sharing

Measuring Known Sources

LED Wavelengths

Single-color LEDs are nearly monochromatic when you use the wavelength stated on the datasheet.

1. Read the dominant $\lambda$ from the datasheet (manufacturers list it for each color bin).
2. Aim LED at slit.
3. View through grating.
4. Measure angle to first order ($n=1$).
5. Calculate: $\lambda = d\sin\theta$ and compare with the datasheet spec.

White LED Spectrum

The US Department of Energy notes that white LEDs are typically a blue pump plus a yellow phosphor conversion layer; you should see a blue peak (≈$\pu{450 nm}$) and a broad yellow band with reduced blue-green intensity (DOE SSL LED basics).

Calculating Grating Spacing

Using Known Wavelength

With laser pointer (known $\lambda$):

1. Measure angles for multiple orders
2. Plot $\sin\theta$ vs $n$
3. Gradient = $\dfrac{\lambda}{d}$
4. Calculate $d$

Example with red laser (manufacturer-rated $\pu{650 nm}$ diode):

• First order at 25.4°
• $d = \dfrac{650 \times 10^{-9}}{\sin(25.4°)} = 1.52 \times 10^{-6}$ m
• Lines/mm = $\dfrac{1}{1.52 \times 10^{-3}} = 658$

Cross-Verification

Test your calculated $d$ value:

1. Use different laser color
2. Predict angle using your $d$
3. Measure actual angle
4. Should agree within 2%

Analyzing Gas Discharge Spectra

Hydrogen Spectrum

The quantum mechanics showcase:

Visible lines (NIST Atomic Spectra Database H I values):

• Red: $\pu{656.28 nm}$ (H-alpha)
• Blue-green: $\pu{486.13 nm}$ (H-beta)
• Blue: $\pu{434.05 nm}$ (H-gamma)
• Violet: $\pu{410.17 nm}$ (H-delta)

What you'll observe:

• Discrete lines (not continuous)
• Exact wavelengths match theory
• Balmer series demonstration

Mercury Vapor (Fluorescent Lights)

Characteristic lines (NIST Atomic Spectra Database Hg I strong lines):

• Violet: $\pu{404.66 nm}$
• Blue: $\pu{435.83 nm}$
• Green: $\pu{546.07 nm}$ (brightest)
• Yellow doublet: $\pu{576.96 nm}$, $\pu{579.07 nm}$

Environmental note: These narrow lines are why older fluorescent lamps distort colour rendering.

Need more lab write-ups to reinforce these spectroscopy techniques? Browse our H2 Physics practicals hub for diffraction grids, double-slit, and capacitor investigations you can adapt for promos and the H2 practical paper.

Sodium Street Lamps

The classic doublet (NIST Atomic Spectra Database Na I D-lines):

• $\pu{588.995 nm}$ and $\pu{589.592 nm}$
• Usually appears as single orange line
• Resolving needs $R \gtrsim 1000$

Experimental Procedures

Calibration First

1. Use laser pointer (known wavelength)
2. Find grating constant $d$
3. Check with second laser color
4. Now measure unknowns

For Each Light Source

1. Ensure stable setup (no movement)
2. Align carefully (perpendicular incidence)
3. Measure both sides (±$n$ orders)
4. Average for accuracy
5. Record order number clearly

Data Collection Table

†: 2nd-order measurement is less accurate at large diffraction angles. At those angles the diffracted beam is weaker, more dispersed, and can overlap with other orders, all of which reduce wavelength-measurement precision

• National MagLab diffraction grating primer
• Optometrics Buyer's Guide
• Diffraction grating efficiency (Wikipedia)

Advanced Techniques

Resolution and Resolving Power

Theoretical resolution: $$R = \frac{\lambda}{\Delta\lambda} = nN$$

Where $N$ = total number of lines illuminated

Testing resolution:

• Try to resolve sodium doublet
• Separation only 0.6 nm
• Need $R > 1000$
• Requires many grating lines

Blazed Gratings

Commercial gratings are "blazed":

• Angled grooves concentrate light
• Most intensity in one order
• Much brighter spectra
• Worth buying for serious work

Measuring Grating Defects

Real gratings aren't perfect:

1. Photograph spectrum
2. Plot intensity vs wavelength
3. Look for ghost lines
4. Indicate manufacturing quality

Uncertainty Analysis

Angle Measurement

Biggest error source:

• Protractor reading: $\pu{\delta\theta ≈ \pm 0.5 ^\circ}$
• At $\pu{30 ^\circ}$: $\dfrac{\delta(\sin\theta)}{\sin\theta} ≈ 1.5\%$ → $\pm$$\pu{9}$–$\pu{10 nm}$ uncertainty near $\pu{650 nm}$
• Improves with digital angle sensor or camera fitting ($\pu{0.1 ^\circ}$ resolution gives few-nanometre agreement)

Systematic Errors

Common issues:

1. Grating not perpendicular (shifts all angles)
2. Wide slit (broadens lines)
3. Stray light (false readings)
4. Temperature effects (minimal for gratings)

Improving Accuracy

1. Use higher orders when possible
2. Average multiple measurements
3. Measure complementary angles
4. Digital photography for angle analysis

Connecting to Atomic Physics

Energy Levels and Photons

Each spectral line represents: $$E = h f = \frac{hc}{\lambda}$$

For hydrogen red line (656nm): $$E = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{656 \times 10^{-9}} = 3.03 \times 10^{-19} \text{ J} = 1.89 \text{ eV}$$

This matches $n=3$ to $n=2$ transition perfectly!

Spectroscopy Applications

Your simple setup demonstrates:

• Element identification (forensics)
• Star composition analysis
• LED quality testing
• Laser wavelength verification

Common Exam Questions

Q1: "Why are higher orders dimmer?"

Key points:

• Light energy spread among orders
• Path differences increase
• Not all rays interfere constructively
• Blazed gratings optimize one order

Q2: "Calculate wavelength from measurements"

Given: $600$ lines/mm grating, first order at $\pu{22.5^\circ}$

Solution:

• $d = \dfrac{1}{600} = 1.67 \times 10^{-6} \space \pu{m}$
• $\lambda = d\sin\theta = 1.67 \times 10^{-6} \times \sin(\pu{22.5 ^\circ})$
• $\pu{\lambda = 639 \space nm}$ (red light)

Q3: "Why do CDs show colors but mirrors don't?"

Model answer:

• CDs have regular line spacing ($d ≈ \lambda$)
• Acts as diffraction grating
• Different wavelengths diffract at different angles
• Mirrors have no regular structure

Practical Applications

DIY Projects

1. Gem testing: Real vs synthetic by fluorescence
2. Plant health: Chlorophyll absorption spectra
3. Monitor calibration: Check RGB peaks
4. Chemistry: Flame test spectra

Industry Uses

• Quality control: LED manufacturing
• Environmental: Pollution monitoring
• Medical: Blood oxygen sensors
• Astronomy: Exoplanet detection

Building a Quantitative Spectrometer

Professional-Grade Features

For research-quality results:

1. Collimating lens (parallel light)
2. Focusing lens (sharp image)
3. Micrometer slit (adjustable width)
4. Rotation stage (precise angles)

Digital Detection

Replace eye with:

• Webcam (remove IR filter)
• DSLR in RAW mode
• Specialized sensor (linear CCD)
• Software for calibration

Wavelength Calibration

Using multiple known sources:

1. Plot pixel position vs wavelength
2. Fit polynomial curve
3. Interpolate unknown wavelengths
4. Achieve few-nanometre accuracy once the angle fit is tied to multiple reference lines

Your Laboratory Checklist

✓ Remove metallic layer from CD/DVD carefully
✓ Make slit narrow ($\pu{0.5 mm}$ or less)
✓ Work in darkened room for best contrast
✓ Calibrate with laser first
✓ Measure angles from perpendicular
✓ Check both positive and negative orders
✓ Calculate wavelengths using $d\sin\theta = n\lambda$
✓ Compare with published values

Master diffraction gratings and you hold the key to understanding atomic structure, analyzing starlight, and checking if that "ruby" is real. From forensic labs to semiconductor fabs, spectroscopy rules - and it all starts with the physics you're exploring here.

Sources

• CD track pitch $\pu{1.6 µm}$: IEC 60908 (summarised at https://en.wikipedia.org/wiki/Compact_disc)
• DVD/Blu-ray track pitch $\pu{0.74 µm}$/$\pu{0.32 µm}$: Blu-ray Disc Association physical format whitepaper (via https://en.wikipedia.org/wiki/Blu-ray#cite_note-BDF_physical_format_whitepaper-121)
• White LED emission mechanism (blue pump + phosphor): https://www.energy.gov/eere/ssl/led-basics
• Spectral lines (Hydrogen, Mercury, Sodium): NIST Atomic Spectra Database, https://physics.nist.gov/asd

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