Dispersion

(last edited June 14, 2013)

Dr. Larry Bortner

 Purpose

To examine how the index of refraction of the material of a prism depends on the wavelength of light. To determine what type of glass constitutes the prism.

 Background

 

 Refraction

The change in direction of a light ray as it passes from one medium to another is called refraction. This bending when a ray enters a medium of different optical density is due to a difference in velocities on opposite sides of the interface. A light beam composed of more than one wavelength (almost all light sources except for lasers) is separated or dispersed upon refraction, thus the component colors can be seen individually. One optical apparatus which is made especially for viewing this phenomenon is a prism spectrometer, with a prism being the component that disperses the light.

 

Upon striking a prism, the beam of light is usually bent twice, once as it enters and once when it leaves (Fig. 1). We call the total angle θ through which the ray is bent from its original direction the angle of deviation.

 

Figure 1.jpg

Figure 1 Two step refraction (air to glass, glass to air) of two different colors of light through an equilateral prism. The angles shown are the angles of deviation.

 

 Snell’s Law

We can derive a quantitative expression for the change in direction that happens when light refracts as it goes from one material to another. Suppose we have a light wave from a distant source (All the rays are parallel.) going from air to glass as in Fig. 2. The speed of light in air is c to at least three significant figures and it slows down to the speed v in glass. The index of refraction n is defined as the ratio of the speed of light in vacuum to that in the material: n ≡ c/v.

 

Figure 2 Diagram for derivation of Snell’s Law.

 

During the time Δt of bending, one end of the wavefront (ray 2) travels the distance cΔt in air while the other end (ray 1) travels vΔt in the glass. From Figure 2,

 

(1)                     

 

Eq. 1 is a form of Snell’s Law. Snell’s Law is reversible since the same argument would hold if we were to switch directions and go from glass to air. Another conclusion drawn from Snell’s Law is that the higher the index of refraction, the greater the deviation.

 Slowing Down Light

Light is an electromagnetic wave and when it encounters a charged element (ion, electron), the charged element experiences oscillating forces and it vibrates accordingly. A harmonically moving charge generates its own electromagnetic wave; it radiates. We regard this as the source of the scattered electromagnetic wave. The combination of this re-radiated wave and the incident wave proceeds through the material at a speed slower than that in a vacuum, and this speed depends upon the natural frequencies of oscillation of the charges in the medium. The light wave behaves like an elastic wave, transferring momentum and energy to charged elements of the medium having one or more resonant frequencies. If the light’s frequency is the same as the natural vibrational frequency of some of the charged elements of the medium, strong absorption of the light occurs.

 Dispersion: Variation of n with λ

As the index of refraction is easily measured directly and is closely related to the speed, we can study the variation of the speed with the wavelength (or frequency) by measuring the index of refraction as a function of the wavelength (or frequency). With visible light, we call it normal dispersion when the medium’s index of refraction decreases gradually as the wavelength increases. The graphical display (as in Fig. 3) of n vs. λ is called a dispersion curve. The strong absorption occurs (as mentioned) at the natural frequency of oscillation of some charged element in the material.

 

Figure 3.jpg

Figure 3 Typical dispersion curve.

 

 Angle of Minimum Deviation

The angle of deviation can vary from some minimum to a maximum of 90°.  It is the minimum angle that is particular to a color, as all colors will deviate to 90°. This minimum is what is measured and used to calculate the index of refraction. The minimum deviation occurs when the beam of light traverses the prism in a direction perpendicular to the plane bisecting the angle α formed by the two refracting surfaces. The angle of incidence is then equal to the angle of emergence. For an equilateral prism (α is 60°), this is the same as saying that the incident beam is perpendicular to the back face.

 Derivation of the Dispersion Equation

The index of refraction n can be found from the prism angle α and the angle of minimum deviation θmd (measured in the laboratory). Referring to Fig. 4, the derivation of the functional relationship of these three quantities is as follows:

 

Figure 4b.jpg

Figure 4 Angles used in derivation of dispersion equation.

 

From similar triangles,


1.   θ12

2.   γ12

3.   θ2=α/2 (with angle β)

From vertical angles,

4.   θin11

5.   γ23

From the geometry of the situation,

6.    

From Snell’s Law,

7.    

Then

8.    

And

                                      (2)

This is the dispersion equation. Again, for the equilateral prisms used in this experiment, α is 60°.

 Cauchy’s equation

Fine. We can find the index of refraction of light of a particular wavelength once we know its angle of minimum deviation through a prism. How do we relate that to the wavelength itself? That’s the whole point of the experiment. Deriving the functional dependence of the index of refraction with wavelength is complex and involved and, to coin a phrase, “… beyond the scope of this course.” However, an empirical relationship exists for normal dispersion called Cauchy’s equation,

 

(3)                     

 

where B and C are material-dependent constants. Note that the larger the value of C, the more dispersive the medium. Table 1 includes values for some glasses offered by Edmund Industrial Optics.

 

Glass

B

C (μm2)

Fused Silica

1.4580

0.00354

Schott Borofloat

1.4720

0.00376

Corning Pyrex

1.4740

0.00379

BK7

1.5170

0.00422

borosilicate crown

1.5035

0.00434

hard crown

1.5043

0.00455

K5

1.5220

0.00459

B270

1.5230

0.00468

SK11

1.5460

0.00470

Schott Zerodur

1.5420

0.00505

BaK4

1.5690

0.00531

Sapphire

1.7680

0.00557

medium barium crown

1.5593

0.00572

dense barium crown

1.5969

0.00572

light barium crown

1.5646

0.00649

SSKN8

1.6180

0.00650

light flint

1.5542

0.00710

BaF11

1.6670

0.00723

BaF10

1.6700

0.00743

BaFN10

1.6700

0.00745

BaF13

1.6690

0.00778

dense flint

1.5961

0.00880

F2

1.6200

0.00892

LaSFN30

1.8030

0.00906

heavy flint

1.6221

0.00970

SF2

1.6480

0.01001

SF5/FDS

1.6730

0.01091

SF8

1.6890

0.01156

SF18

1.7220

0.01290

SF10

1.7280

0.01342

FD10

1.7280

0.01347

LaSFN9

1.8500

0.01382

SF14

1.7620

0.01506

SF11/SFL11

1.7850

0.01593

SF6

1.8050

0.01660

SF57

1.8470

0.01864

heaviest flint

1.8332

0.01972

Table 1 Cauchy’s equation coefficients for some typical optical glasses.

Procedure

You need the following items:

*        spectrometer

*        prism

*        magnifier

*        metal frame and cloth hood to shield the prism from light not coming through the collimator

*        power supply with sodium lamp

*        spectral tube power supply

*        spectral tubes of hydrogen, helium and mercury

*        lab jack (to adjust height of power supplies)

*        flashlight with incandescent and fluorescent bulbs (shared)

 

Figure 5.jpg

Figure 5 Spectrometer with working parts. Not shown is the vernier window on the opposite side.

 

1.     Turn on the sodium light. It takes a while to warm up.

To save time and frustration, the spectrometers are adjusted through Step 9. Take the following steps only if instructed to do so by your TA.

 

Focus Adjustment


2.   With no prism on the prism table, focus the telescope at infinity (something at one end of the room or through the window, not through the collimator). You may have to loosen the telescope rotation lock screw to freely move the telescope.

3.   While looking through the telescope, slide the eyepiece in and out until the cross hairs come into sharp focus. Do not use the focusing knob to do this. There is a locking adjustment around the eyepiece that you can loosen. When you tighten it up again, make sure the cross hairs are aligned so that one is totally horizontal and the other is vertical. (If it’s rotated a little, it’s hard to align the spectral lines.)

4.   If necessary, repeat steps 2 & 3 until the distant object and the cross hairs can be put in sharp focus at the same time.

5.   Check to see that the collimator slit is partially open. Adjust, if necessary.

6.   View the collimator slit through the telescope (the slit can be illuminated with room light). Focus the collimator (not the telescope) until the slit comes into sharp focus.

 

Note: θmd measurements should always be made with the cross hairs aligned on the fixed edge of the collimator slit. This enables you to adjust the slit's width for optimal line visibility at any time during data collection. This way, your angle measurement is independent of the width of the slit.

 

7.   Lock the telescope rotation lock screw. Use the telescope rotation fine adjustment to align the vertical cross hair with the fixed edge of the slit. (The fine adjust does not work unless the rotation lock screw is set.)

Zeroing the Reference Angle

8.   Loosen the table base lock screw and rotate the table so that the right window vernier reads about 0°. Retighten the lock screw.

9.   Using the magnifier and the table base fine adjust, set the vernier to read 0° 0'. This is your reference angle. Setting this to zero means that the vernier reading for a particular spectral line is directly the angle of minimum deviation. Do not move the table after you have set this angle.

 Prism Mounting

10. Mount the prism at the center of the prism table. If necessary, loosen the (long) table lock screw which fastens the prism table to the base and adjust the height so that light from the collimator is striking the prism, then lightly tighten the screw. Recall that the prism orientation for the minimum deviation is where one of the sides is roughly perpendicular to the incident light beam, as in Fig. 4. The frosted side of the prism should be to the right as you’re facing the light source.

 Minimum Deviation Adjustment

11. Set up the spectrometer to view light from the sodium lamp. (See Fig. 6.) Based on Figure 4, estimate the direction the dispersed light from the collimator will exit the prism. With your bare eye, look at the prism along this direction to find the image of the collimator's slit. You should see a series of brightly-colored lines, a yellow line dominating.

 

 

Figure 6.jpg

Figure 6 The prism spectrometer trained on a sodium lamp. The prism as shown is oriented for minimum deviation of refracted rays.

 

12. If the prism is in exactly the right orientation to provide the angle of minimum deviation, the series of colored lines move to the right as a whole, whether you rotate the prism clockwise or counterclockwise. As you view the lines, rotate the prism back and forth (touching only the edges and not the faces of the prism) until you are satisfied that the orientation is where the lines bounce, or change direction.

13. Now view the lines through the telescope.

a.   Rotate the prism back and forth slightly to fine-tune the exact orientation that puts the lines at their extreme position (the telescope's cross hairs make a convenient reference mark).

b.   Since the position of the prism for minimum deviation is a slowly varying function of the wavelength, it is not necessary to reset the minimum deviation for each line (color). Once the prism is set, this orientation should not be altered for the duration of the experiment.

c.   Lightly tighten the prism holder.

Checkpoint! Have the TA check your setup before you proceed further.

 Measure θmd

14. Measure θmd for the strong yellow line in the sodium spectrum:

a.   Loosen the telescope rotation lock screw and rotate the telescope so that its cross hairs are near the fixed-edge side of the slit's image.

b.   Lightly tighten the telescope rotation lock screw.

c.   Use the telescope fine adjust knob to carefully align the cross hairs and the fixed-edge side of the slit's image.

d.   Find the value of θmd for this color using the vernier scale.

i. Identify the color and record the wavelength of the line from Table 2.

ii.Record the two numbers from the spectrometer that indicate this angle, the nearest, lowest half-degree and the minute.

 Reading the Angle

The angle θmd between the moveable telescope and the fixed collimator is determined using the vernier scale as in Figure 7. The bottom degree plate is graduated in units of 0.5° or 30' (30 arc minutes). The top set of numbers, or the vernier scale, provides a further resolution of 1' and has a range of 30'. To get the angle, you need to record two numbers, the nearest, lowest half-degree and the minute.  Follow the procedure below:

 

Figure 7 The half degree-minute vernier on the spectrometer.

 


 

 

A.    Find the Zero. Locate where the zero mark of the vernier scale aligns with the degree plate.

B.    Record the degrees to the nearest, lowest 0.5°. When the zero of the top scale is between two lines on the degree plate, use the smaller value. In Fig. 7, the zero is between the lines 155° 30' and 156° 00'. 155° 30' is the smaller of the two, so you record the degrees as 155.5.

C.    Record the minutes. Use the magnifying glass to determine the line on the vernier scale that aligns most closely with any line on the degree plate. In Fig. 7, this is 13', so you write down 13 as the minutes.

 

15.  Based on your use of the spectrometer to measure θmd , estimate u{θmin}. You can assume u{α}= 0.

Checkpoint! Have the TA check your measurements before you proceed further.

16.  Switch to the other power supply and make necessary height adjustments.

a.   Repeat procedure 14 for...

i. the four lines of hydrogen,

ii.the twelve lines of helium,

Checkpoint! Have the TA check your measurements before you proceed further.

iii.   …and the eight listed mercury lines.

All members of the group should participate in the data collection. Everybody should find several lines and read the two numbers that determine the angle for that color.

b.   Insert the proper tubes when necessary.

c.   You must measure angles for all of the lines listed in Table 2.

 

Light Source

Line Color

Wavelength (Å)

Sodium

Yellow

5893

Hydrogen

Red

6563

 

Blue-Green

4861

 

Blue (purple)

4340

 

Violet (faint)

4102

Helium

Red (strong)

7065

 

Red (strong)

6678

 

Yellow (strong)

5876

 

Green

5048

 

Green

5015

 

Blue-Green(strong)

4922

 

Blue (strong)

4713

 

Deep Blue

4471

 

Blue-Violet

4388

 

Violet

4026

 

Violet

3964

 

Violet

3889

Mercury

Red

6907

 

Yellow

5790

 

Yellow

5770

 

Green

5461

 

Blue-Green

4916

 

Blue

4358

 

Violet

4078

 

Violet

4047

Table 2 Wavelengths of spectral lines for elements used in this lab, sorted according to the element.

 

Light Source

Line Color

Wavelength (Å)

Helium

Red (strong)

7065

Mercury

Red

6907

Helium

Red (strong)

6678

Hydrogen

Red

6563

Sodium

Yellow

5893

Helium

Yellow (strong)

5876

Mercury

Yellow

5790

Mercury

Yellow

5770

Mercury

Green

5461

Helium

Green

5048

Helium

Green

5015

Helium

Blue-Green(strong)

4922

Mercury

Blue-Green

4916

Hydrogen

Blue-Green

4861

Helium

Blue (strong)

4713

Helium

Deep Blue

4471

Helium

Blue-Violet

4388

Mercury

Blue

4358

Hydrogen

Blue (purple)

4340

Hydrogen

Violet (faint)

4102

Mercury

Violet

4078

Mercury

Violet

4047

Helium

Violet

4026

Helium

Violet

3964

Helium

Violet

3889

Table 3 Spectral lines sorted according to wavelength.

 

Hints:

*        Opening up the slit allows more light to come through, increasing the intensity and allowing you to better see the dimmer lines. However, some of the lines are so close to each other that a wider slit washes out the distinction. In particular, the two yellow lines of mercury and the two green lines of helium need a very narrow slit in order to distinguish between the two.

*        Also, repositioning the lamp can increase the brightness.

*        You can also use the blackout cloth to shield the prism from unwanted light. See Fig. 8.

*        It is important that you identify the lines correctly. If Cauchy’s equation (Eq. 3) holds, the index of refraction decreases as the wavelength increases. Also, n is a continuous function of λ, as well as θmd. This means that the smallest angle you measure is for the longest wavelength and the largest angle is for the shortest wavelength. If you were to add another column in Table 3 and put the measured angle with the wavelength, the angles would go from the smallest to the largest, each one being larger than the previous one.

 

 


17.  Observe the spectrum when the slit is illuminated by white light from outside or from a yellow incandescent bulb. Which color is deviated the most?

18.  Observe the light of a fluorescent tube. Which one of the four tested elements contributes strongly to the light from a fluorescent tube?

 

Dispersion- using blackout cloth.jpg

 

Figure 8 Arranging the blackout cloth around the spectrometer.

Analysis

The top of the spreadsheet should look something like this (The neon line is for demonstration purposes only. Don’t use it.):

 

α(°)=

u{θmd} (min) =

α(rad)=

u{θmd} (rad) =

For λ=5893Å of sodium, u{n}=

θmd

θmd

Element

color

λ ()

°

 & min

radians

λ2(μm-2)

n

Ne

blue

4792

53.0

17

0.9300

4.355

1.6705

 

 

1.     Trigonometric functions in Excel require the angle to be in radians. Our measurements are in degrees and minutes. There is an Excel function called (Are you ready for this?) RADIANS that does the conversion from degrees. Calculate each angle in radians. There are 60 minutes in a degree; be sure to convert the total angle (degrees plus minutes) to radians.

2.     Calculate λ-2 in μm-2. 104 Å = 1 μm.

3.     Calculate n from Eq. (2).

4.     Calculate u{n} for the sodium line. The expression is

 

(4)                     

 

where θmd and α are in radians. To 1 s.f., it’s the same for all wavelengths used here.

5.     Plot n vs. λ-2, with error bars. Is this a straight line as expected?

6.     Do a least squares fit with errors to fit the data to Cauchy’s equation. You should get the quantities B, u{B}, C, and u{C}. To within experimental uncertainty, does it appear that the prism used in this experiment is made from any of the glasses listed in Table 1? Keep in mind that the slope C is more indicative of the material than the intercept B. What type of glass provides the best match to your data?

 Questions

1.     Calculate the velocity of light in the glass prism from your values of the index of refraction for the sodium yellow line and from the known value of the speed of light in a vacuum.

2.     How is a rainbow formed? Give details.