The Plane Diffraction Grating

Andy Chmilenko, 20310799 Instructor: Jeff Gardiner Section 1

(Dated: 2:30 pm Tuesday November 19, 2013)

I. PURPOSE

The purpose of this lab is to observe the diffraction patterns from plane diffraction gratings of two types, transmission gratings and reflection gratings, and confirm the relation between the slit widths, spacing and the diffraction pattern observed. Using these observations and theory we will also use the observed diffraction pattern from a reflection grating to determine the wavelengths of spectral lines from a Helium source.

II. ANALYSIS

A. The Transmission Grating

Order Spectral  θ_{L}  θ_{r}  θ (rad)  λ  %
Line  (± 1′ deg) (± 1′ deg) (± 4.11 × 10^{−4} rad) () Deviation
1^{st}  Hg Blue  119^{∘}38′ 150^{∘}12′  0.2667  %
Hg Green  115^{∘}39′ 154^{∘}10′  0.3361  %
Hg Yellow I 114^{∘}30′ 155^{∘}21′  0.3565  %
Hg Yellow II 114^{∘}26′ 155^{∘}24′  0.3575  %
Hg Red  111^{∘}04′ 158^{∘}40′  0.4154  %
2^{nd}  Hg Blue  103^{∘}05′ 166^{∘}39′  0.5547  %
Hg Green  93^{∘}31′  176^{∘}10′  0.7213  %
Hg Yellow I 90 90^{∘}35′  179^{∘}04′  0.7722  %
Hg Yellow II 90^{∘}22′  179^{∘}15′  0.7757  %
Hg Red  –  –  –  –  –

TABLE I: Left and right angle measurments of the first and second order diffraction patterns for Mercury Blue, Green, Yellow I&II, and Helium Red spectral lines with calculated wavelengths

Sample Calculations for d using row 1 of Table I

d = \frac{|d_{L}−d_{R}|}{2}
d = \frac{|119^{∘}38′−150^{∘}12′|}{2}
d = \frac{|2.08800−2.62148|}{2}
d = 0.2667rad

Sample Calculations for Δd

Δd = \sqrt{Δd_{L}^{2} + Δd_{R}^{2}}
Δd = \sqrt{(1′)^{2} + (1′)^{2}}

Δd = ±4.11 × 10^{−4}rad

Sample Calculations for Calculated Diffraction Grating d_{1} using Row 3 of Table I using λ 5770Å

λ = \frac{d}{m} · sin θ

Sample Calculations for % deviation Diffraction Angle using Row 1 of Table I

\begin{matrix}%_{deviation} = \frac{|1.15−1.16|}{1.15} × 100% \\ %_{deviation} = 0.9% \end{matrix}

White Light Source and Single Slit Diffraction

The white light diffraction pattern has a dark central band, with fringes that had rainbow like patterns on each. This is because each wavelength of light bends at different amounts when going through the single slit.

The Minima are proportional to:

sin α_{n} = n · \frac{λ}{b}  (1)

So red light, will have a smaller angle of diffraction compared to a larger wavelength colour like blue. This process is similar to refraction in transparent mediums of different indices of refraction.

B. The Double Slit

b_{L}(± 0.0005cm) b_{r}(± 0.0005cm) b(± 0.0007cm)

Slit 1

3.1338  3.1467  0.0129
3.1329  3.1249  0.0080
3.1377  3.1463  0.0086

Slit 2

3.1668  3.1739  0.0071
3.1600  3.1526  0.0074
3.1662  3.1732  0.0070

TABLE II: Measurements for the left and right bounds respectively of the the two slit widths for the double slit using the travelling microscope

Sample Calculations for b using row 1 of Table II

b = |b_{L} − b_{R}|
b = |3.1338 − 3.1467|
b = 0.0129

Sample Calculations for Δb

Δb = \sqrt{Δb_{L}^{2} + Δb_{R}^{2}}

Δb = \sqrt{0.0005^{2} + 0.0005cm^{2}} Δb = ±0.0007cm

Sample Calculations for \overline{b}

\overline{b} = ^{\frac{∑_{i}^{n} b_{i}}{n}}

\overline{b} = \frac{0.0080+0.0086+0.00710.0074+0.0070}{5}

\overline{b} = 0.0076 × 10^{−2}m

(Note: Outlier value 0.0129 cm was not used in the calculation for \overline{b})

Sample Calculations for Δ\overline{b}

Δ\overline{b} = ±\frac{Δb}{\sqrt{n}}
Δ\overline{b} = ±\frac{0.0007}{\sqrt{5}}
Δ\overline{b} = ±0.0003cm

Sample Calculations for d to b relation

\begin{matrix}d : b = 0.0076 : 0.0276 \\ d : b = 1 : 3.6 \end{matrix}

The slit separation to the slit width ratio for the double slit pair is approximately 4.

d_{L}(± 0.0005cm) d_{r}(± 0.0005cm) d(± 0.0007cm)

Left Edge to Left Edge

3.4137  3.4418  0.0281
3.4146  3.4415  0.0269
3.4136  3.4424  0.0288

Right Edge to Right Edge

3.4363  3.4092  0.0271
3.4373  3.4104  0.0269
3.4378  3.4102  0.0276

TABLE III: Measurements for the left and right bounds respectively of the the two slit separation using the travelling microscope

Sample Calculations for d using row 1 of Table III

d = |d_{L} − d_{R}|

d = |3.4137 − 3.4418|
d = 0.0281

Sample Calculations for Δd

Δd = \sqrt{Δd_{L}^{2} + Δd_{R}^{2}} Δd = \sqrt{0.0005^{2} + 0.0005cm^{2}}

Δd = ±0.0007cm

Sample Calculations for \overline{d}

\overline{d} = ^{\frac{∑_{i}^{n} d_{i}}{n}}

\overline{d} = \frac{0.0281+0.0269+0.0288+0.0271+0.0269+0.0276}{6} \overline{d} = 0.0276 × 10^{−2}m

Sample Calculations for Δ\overline{d}

Δ\overline{d} = ±\frac{Δd}{\sqrt{n}}
Δ\overline{d} = ±\frac{0.0007}{\sqrt{6}}
Δ\overline{d} = ±0.0003cm

Order of  M_{L}  M_{r}  M (×10^{−2} rad) Calculated Diffraction  %
Maximum (± 1′ deg) (± 1′ deg) (± 4.11 × 10^{−4} rad) Angle(×10^{−2} rad) Deviation
1  190^{∘}56′ 191^{∘}13′  0.247  0.213  16%
2  190^{∘}49′ 191^{∘}19′  0.436  0.427  2%
3  190^{∘}44′ 191^{∘}27′  0.625  0.640  2%
4  –  –  –  –  –
5  190^{∘}28′ 191^{∘}41′  1.062  1.067  0.5%
6  190^{∘}20′ 191^{∘}48′  1.280  1.280  0%
7  190^{∘}15′ 191^{∘}56′  1.469  1.494  1.67%

TABLE IV: Measurements for the left and right bounds respectively of orders of diffraction minima for Sodium light and a single slit, as well as calculated quantities such as diffraction angle and deviation from measured diffraction angle.

Sample Calculations for M using row 1 of Table IV

M = \frac{|M_{L}−M_{R}|}{2}
M = \frac{|190^{∘}56′−191^{∘}13′|}{2}
M = \frac{|3.33242−3.33736|}{2}
M = 0.247 × 10^{−2}rad

Sample Calculations for ΔM

ΔM = \sqrt{ΔM_{L}^{2} + ΔM_{R}^{2}}

ΔM = \sqrt{(1′)^{2} + (1′)^{2}}

ΔM = ±4.11 × 10^{−4}rad

Sample Calculations for Calculated Diffraction Angle using Row 1 of Table IV using λ 5890Å

α_{m} = \frac{mλ}{d}
α_{1} = \frac{1·5890×10^{−10}}{0.0276×10^{−2}}
α_{1} = 0.213 × 10^{−2}rad

Sample Calculations for % deviation Diffraction Angle using Row 1 of Table IV

\begin{matrix}%_{deviation} = \frac{|0.213−0.247|}{0.213} × 100% \\ %_{deviation} = 16% \end{matrix}

Laser

Order Fringe separation (2nt) Spacing (t)
(cm ± 0.05cm)  (cm ± 0.05cm)
1  0.95  0.475
2  1.80  0.450
3  2.70  0.450
4  –  –
5  4.60  0.460
6  5.50  0.458
7  6.40  0.457
8  7.30  0.456

TABLE V: The fringe separation from the left order to the equivalent right order maximum if a He-Ne Laser incident on a double slit with a distance D = 198.5 cm between the slit and the screen.

Sample Calculations for t using row 1 of Table V

2nt = F ringeSeparation

t = \frac{F ringeSeparation}{2n}
t = \frac{0.95}{2·1}
t = 0.475
Sample Calculations for \overline{t}
\overline{t} = ^{\frac{∑_{i}^{n} t_{i}}{n}}

\overline{t} = \frac{0.475+0.450+0.450+0.460+0.458+0.457+0.456}{7} \overline{t} = 0.458 × 10^{−2}m

Sample Calculations for Δ\overline{t}
Δ\overline{t} = ±\frac{Δt}{\sqrt{n}}
Δ\overline{t} = ±\frac{0.05}{\sqrt{7}}
Δ\overline{t} = ±0.02cm

Sample Calculations for λ using D = 198.5 ± 0.5cm and t = 0.458 ± 0.02cm

d · sin α_{m} = mλ

λ = d · \frac{t}{D}

λ = 0.0276 × 10^{−2} · \frac{0.458×10^{−2}}{198.5×10^{−2}}

λ = 636.8nm

Sample Calculations for Δλ

Δλ = λ · \sqrt{(\frac{δt}{t})^{2} + (\frac{δD}{D})^{2} + (\frac{δd}{d})^{2}} Δλ = 636.8 · \sqrt{(\frac{0.02}{0.458})^{2} + (\frac{0.5}{198.5})^{2} + (\frac{0.0003}{0.0276})^{2}}

Δλ = ±28.7nm

Sample Calculations for % deviation of λ with the accepted value of 632.8 nm

\begin{matrix}%_{deviation} = \frac{|632.8−636.8|}{632.8} × 100% \\ %_{deviation} = 0.6% \end{matrix}

III. CONCLUSION

For single slit diffraction, when the slit width was varied it was observed that the diffraction pattern changed proportions. As the slit was bade larger, the diffraction patter became squished with the fringes and spaces between becoming smaller and the pattern sharper. Conversely as the slit was made smaller the pattern stretches out and the fringes and the spaces become thicker and more fuzzy.

Using a fixed slit, it was found that the slit width, b, to be 0.0513 ± 0.0004 cm, and using this value we were able to calculate and and compare the Diffraction angle of 5890Ålight with our measurements (Table I), and we had mixed results between 0.9% and 52% deviation. The diffraction pattern was fuzzy and rather dim which made it difficult to discern the minima and make accurate measurements.

Using the double slit slide, we were able to measure the slit widths b, and the slit separation d, to be 0.0076 ± 0.0003 cm and 0.0276 ± 0.0003 cm respectively. The ratio of the slit widths to the slit separation was found to be 3.6, which is approximately 4. Using these values we were able to calculate the diffraction angle for specific orders and compare them to our measured angles for these diffraction angles (Table 5). Our results were in rather good agreement only deviating between 0% and 16%. In observations the 4th order maximum for the double slit pattern was not visible, this is due to the single slit envelope having a minimum at the corresponding position where there is a maximum for the double slit.

Using the same double slit pattern, using a laser and measuring the distance between maximum fringes to find the constant maximum separation and knowing the distance from the slits to the screen we were able to calculate the wavelength of the light from the He-Ne laser to be 636.8 ± 28.7 nm. This only differs by 0.6% of the accepted value of 632.8 nm.