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Chapter 10

Experiment 8:The Spectral Nature of Light

10.1 Introduction: The Bohr Hydrogen Atom

Historical AsideThe 19th century was a golden age of science. We could explain everything we observed.Near the turn of the century that began to change. The arrival of the microscopeallowed us to see random motion of wheat grains. Light emitted from hot objects hadspectrum different than expected; indeed the expected spectrum was unreasonable.Light emitted or absorbed by atomic gases had sharp lines instead of continuity.

In this lab we will resolve some of the reasons for the confusion surrounding the state ofscience in nineteen hundred. These explanations involve accepting that light has a particle orquantum nature as well as the wave nature that our interference experiments have verified.These explanations involve an atomic nature for materials instead of the continuous, twocharged fluid model suggested by Benjamin Franklin. These explanations even involve stablequantum states in atoms where the accelerating electron does not radiate EM waves asexpected.

An atom has a diameter of about 10−10 m and is made up of a small (about 10−14 m),heavy, and positively charged nucleus, surrounded by orbiting electrons. The nucleus consistsof protons (positive electric charge) and neutrons (no electric charge) bound together bynuclear forces.

In a neutral atom there are as many negatively charged electrons as there are protons.Each electron occupies one of the many possible well defined atomic orbit states. Eachparticular orbit state has an associated total energy (kinetic + potential). Electrons “jump-ing” from one orbit with less energy to one with more (binding) energy will release discretequantities of energy in the form of light. The spectrum of quantized frequencies observedin the light emitted by excited atoms is a direct measurement of the allowed transitions of

141

CHAPTER 10: EXPERIMENT 8

M

e

-e

me

r

Figure 10.1: Schematic force diagram for the nucleus and electron in a hydrogen atom.

electrons among different orbits in the atom. The emission spectrum of a given atom is aunique signature of this atom. Even different isotopes in an element family have differentspectra.

Although the existence of discrete line spectra emitted by atoms had been known for along time, it was Niels Bohr who first calculated the spectrum of the hydrogen atom. This isthe simplest atomic system and consists of one proton as nucleus and one orbiting electron.The electron describes a circular orbit in the electric field of the proton; the force acting onthe electron is

− e2

4πε0 r2 = Fe = −me ar = −mev2

r. (10.1)

From Equation (10.1) the kinetic energy of the electron is

KE = 12me v

2 = e2

8πε0 r. (10.2)

The electrostatic potential due to the proton’s positive charge at a distance r is

V = e

4πε0 r(10.3)

and the potential energy, Ue = (−e)V of the electron is

Ue = − e2

4πε0 r= −2KE (10.4)

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The total energy of the electron-proton system (of the atom) is

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0Hydrogen Energy Levels

En

erg

y (

eV

)

Lyman

Series

Balmer

Series

Paschen

Series

Brackett

Series

Ser

ies

Lim

it

Ser

ies

Lim

it Ser

ies

Lim

it

Ser

ies

Lim

it

Figure 10.2: This figure shows an energy leveldiagram for hydrogen along with some of the possi-ble transitions. An infinite number of energy levelsare crowded between n = 6 and the escape energyn =∞.

E = KE + Ue = KE − 2KE

= − e2

8πε0 r(10.5)

Bohr quantized the value of r bypostulating that the electronic orbitmust be a standing wave with wave-length λ, consequently the length of thecircle describing the electron’s trajec-tory must be the product of an integerand λ,

2πrn = nλ (10.6)

where n = 1, 2, 3, . . .The French physicist Louis de

Broglie postulated that one can assigna wavelength λ to any object havinga momentum so that for the electronp = mev; λ = h

p= h

mev, where h is

Planck’s constant. Combining this withEquation (10.6) we see that

2π rn = nλ = nh

me v. (10.7)

We eliminate the electron’s velocityfrom Equation (10.2) and Equation (10.7) to find

e2

8πε0 r= 1

2me v2 = 1

2me

(nh

2πme r

)2

= (nh)2

8π2me r2 (10.8)

and1r

=(e

nh

)2 πme

ε0. (10.9)

CheckpointWhat did Bohr postulate in calculating the energy spectrum of the hydrogen atom?

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Table 10.1: Details about the photon interactions with hydrogen atoms for several ‘lines’.Most humans can see the first 3-4 Balmer lines.

Series Name nl lower level nu upper level Wavelength (nm)1 2 121.61 3 102.6

Lyman 1 4 97.01 5 94.91 6 94.01 ∞ 91.22 3 656.32 4 486.1

Balmer 2 5 434.12 6 410.22 7 397.02 ∞ 365.03 4 1875.13 5 1281.8

Paschen 3 6 1093.83 7 1005.03 8 954.63 ∞ 822.0

CheckpointWhat does quantization mean? Name five quantities that are quantized in nature.

Now we substitute this into Equation (10.5) to determine the atom’s energy to be

En = − e2

8πε0 rn= − e2

8πε0

(e

nh

)2 πme

ε0= −me

2

(e2

2hε0

)2 1n2 = −EI

1n2 (10.10)

where the ‘ionization energy’ for hydrogen is found to be

EI = me

2

(e2

2hε0

)2

= 13.605 eV. (10.11)

The radius rn of the orbiting electron and the binding energy En are quantized; they canassume only discrete values! The quantity n is called the principal quantum number. The

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Figure 10.3: The spectrum of white light as viewed using a grating instrument like theone in this experiment. The different orders identified by the order number m, are shownseparated vertically for clarity. In actuality they would overlap.

radius rn increases as n2. (See Equation (10.9).) The energy values are negative because weare calculating a binding energy; (one must supply energy in order to achieve an isolatedelectron and an isolated proton at rest). Figure 10.2 shows the energy of stationary statesand their associated quantum numbers. The upper level marked n = ∞, corresponds to astate in which the electron is completely removed from the atom; the atom is ionized andthe ionization energy is required to achieve this when n = 1.

CheckpointWhy does the binding energy of an atomic electron have a negative value?

10.2 The Apparatus

An atom radiates when, after being excited, it makes a transition from one state with theprinciple quantum number n, and consequently with energy En, to a state m with lowerenergy Em. The quantum energy carried by the photon is

hν = Eγ = Eu − El = EI

(1n2

l− 1n2

u

). (10.12)

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Figure 10.4: A sketch of our grating spectrometer. Our gas is ionized to provide the sourceof light. Lens A collimates the light into a beam the size of the grating. Lens B focuses thelight onto the cross-hair (see inset) and the eyepiece allows us to see the spectral lines. TheVernier angular scale allows us to determine the wavelength of the line at the cross-hair.

The wave speed, the wavelength, and the frequency of oscillation are related by c = λν, sowe can find the wavelength of the emitted photon to be

1λ

= ν

c= Eγhc

= Ry

(1n2

l− 1n2

u

)(10.13)

where we have defined the ‘Rydberg’ to be Ry = EIhc

= 0.0109737916 nm−1 . Equation (10.13)is Rydberg’s law and was the first relation that successfully described hydrogen’s emissionand absorption spectrum. In our experiment we will wish to verify Rydberg’s law and tomeasure the Rydberg constant using our measured wavelengths,

Ry = 1

λ(

1n2

l− 1

n2u

) . (10.14)

Historical AsideRydberg discovered this law empirically simply by examining the emitted wavelengthssome time before Bohr discovered this theoretical basis.

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Hydrogen has thousands of spectral lines from the ultra-violet to high-frequency radiowaves, but the only four which are visible to humans are the lowest energy Balmer seriesphotons having nl = 2 in the de-excited atom and nu = 3, 4, 5, 6. The nu = 6 transitionappears very dim and many students cannot see it; therefore we will only view the first three.

We must keep in mind that Bohr’s model above applies only to hydrogen and to hydrogen-like atoms. To understand more complex atoms we must incorporate the revolutionary theoryof quantum mechanics. Even quantum mechanics applied to these complex systems cannotbe solved exactly, but we have developed approximate solutions that agree quite well withexperiment. Additionally, quantum mechanics can explain the subtle variations in hydrogen’sspectrum that are visible in the data you will gather today.

The Grating Spectrometer

Figure 10.5: A sketch of the apparatus (sideview) illustrating how to view the spectrum ofa hot tungsten filament.

As already discussed in the fourth labora-tory exercise, gratings are used to analyzethe frequency spectrum of light. Light of agiven wavelength, λ, will be deflected by agrating with a line spacing constant d, by anangle θ, according to the relation

±mλ = d sin θ with m = 0, 1, . . . (10.15)

where ±m is the order number.Figure 10.3 shows how a grating spec-

trometer diffracts visible light in groups ofintensity distributions as a function of θ,and each of these groups corresponds to adifferent order m. Previously, our experience with diffraction gratings was limited toillumination by monochromatic lasers and we saw the spots into which the laser wasdiffracted. Previously we saw spots because we were using only one color (or wavelength,λ) of light. Today, we are illuminating the gratings with many colors simultaneously andeach color will excite every order of the gratings. Because of this it is important that wenot allow ourselves to become overwhelmed by what the spectrometer does with our light.Adjacent distributions do overlap at the larger orders; third order violet occurs for smallerθ than second order red.

Figure 10.4 shows the spectrometer used for this experiment. It consists of a circular tablewith a diffraction grating at its center and an angular Vernier scale along its circumference.Two pipes are mounted on the table to keep out room light and these pipes intersect in theregion of the grating. The mounting of the first pipe is permanent. It is designed to carryparallel light from an external source to shine upon the grating.

The second pipe is mounted in such a way as to be able to pivot about the center ofthe circular table. This rotation makes it possible to separate the light, collecting only whathas been diffracted through some chosen angel, θ. If this pipe is set to an angle such as

147


θ = 10◦, then whatever light the grating diffracts through an angle of 10◦ will travel downthe length of the pipe. It will be focused by the adjustable lens (B) and the eyepiece. A pairof cross-hairs identify the center of the pipe and, together with the vernier scale located onthe equipment, allow the accurate measurement of the angle, θ.

CheckpointDraw the path of two parallel light rays diverging from the spectrometer slit andentering the observer’s eye.

CheckpointWhat is the physical significance of the orderm of the spectrum produced by a grating?

Figure 10.6: Focusing system of the eye.Most of the refraction occurs at the first sur-face, where the cornea C separates a liquidwith index of refraction n1 from the air. Thelens L, which has a refractive index varyingfrom 1.42 in the center to 1.37 at the edge,is focused by tension at the edges. The mainvolume contains a jelly like ‘vitreous humour’(n = 1.34). The image is formed on theretina. Illumination levels are controlled bythe aperture of the iris.

The grating constant d is engraved onthe frame holding the grating. Slit S, inFigure 10.4 defines the light source; the slitis parallel to the lines of the grating sothe diffracted light will form vertical linesinstead of spots. Lens A is separated fromthe slit by a distance equal to its focallength, f , and consequently it projects abeam of parallel light on to the grating.Such a collimated beam utilizes most of thegrating so that the spectra are quite sharp.Lens B focuses the light diffracted by thegrating into a focal plane which contains thecross-hair. Both positions of lens B and thecross-hair can be adjusted in order to focusthe image better. First focus the cross-hairand then focus the slit line.

The optics of the human eye is shownschematically in Figure 10.6. The optimumfocal distance of our eyes is about 25 cm.In order to observe an image at a closerdistance, a lens called an eyepiece may beused as indicated in Figure 10.7. The spec-trometer has an eyepiece with focal lengthof 2.5 cm. Each student will have to adjustthe position of the of lens B to see a sharp

image the spectral fringe. The placement of the cross-hairs should then be adjusted so thatthe two are in focus. A small light bulb controlled by a push-button switch allows the cross-

148


hairs to be illuminated. To determine the axis (θ = 0) of the spectrometer, place the tablelight in front of the spectrometer slit, S, remove the diffraction grating and look throughthe pivoting arm of the spectrometer directly at the light source. Adjust the position of theslit, of lens B, and of the cross-hair in order to see a sharp, white image (order m = 0).When a sharp vertical image of the slit is centered on the cross-hair, you can read the anglecorresponding to the axis of the spectrometer; use the Vernier scale as shown in the appendix.All measurements of θ that you will make must be with respect to the spectrometer axisso subtract this θ0 from all remaining angle measurements. Ga3 can do this for you if yousimply incorporate the subtraction into the column formula. Don’t forget to record the unitsand experimental error for your measurements.

Figure 10.7: Simple lens used as a mag-nifier. An object at O close to the eye canbe focused by the eye to appear to be muchlarger and at a more distant point O’.

Figure 10.8: A sketch of the apparatus(side view) illustrating how to view the spec-trum of an ionized gas.

The Light Sources

In this laboratory you will observe the spectrum of one hot filament and four gases: H, He,Ne, and Hg. The light source consists of a glass tube which contains one of these gasesat a pressure of about 20-30 cmHg. A 60Hz voltage of a few kilovolts is applied across thegas in the tube via electrodes at each end. The non-uniform electric field between the twoelectrodes will excite the gas in the tube.

An electron will remain in an excited state for a time on the order of 10−8 s before jumpingdown to a state with higher binding energy. The excited state, with a lifetime τ ∼ 10−8 s,has then decayed into a lower energy state. In order to prolong the usefulness of these tubesdo not keep the light source on any longer than necessary.

CheckpointHow might you excite an atom and how long does it stay excited?

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10.2.1 The Spectrum of White Light

The electromagnetic spectrum emitted by the hot filament of the table lamp is a continuousspectrum whose relative intensity depends upon its temperature. The shape of the spectrumof a black body is shown in Figure 10.9. It was Planck who first calculated the shape of thisspectrum. Planck introduced a constant h, now known as Planck’s constant, to reproducemathematically the observed distribution of this spectrum.

Observe the visible part of the electromagnetic spectrum emitted by your incandescentlamp. Look at the spectrum associated with the order numbers m = 0, 1, and 2. Observethe different colors associated with different frequencies or wavelengths.

0 0.5 1 1.5 2 2.5 30

2000

4000

6000

8000

10000

12000

14000

Wavelength (microns)

Watt

s/(m

2 n

m) Rayleigh-Jeans

Law

T=3000 K

Planck

Law

T=5000 K

Planck

Law

T=4000 K

Planck Law, T=3000 K

Figure 10.9: Blackbody Radiation Spectrum. Planck’s energy density of blackbody radiationat various temperatures as a function of wavelength. Note that the wavelength at which thecurve is a maximum decreases with temperature.

Measure the spectral sensitivity of your eye by observing the highest frequency your eyecan discern. Use the order m = 1 to make these measurements. Measure the angle θ withrespect to the spectrometer axis (don’t forget to subtract off θ0) and use Equation (10.15)to calculate the corresponding λ. Don’t forget your units and errors. Now measure thelowest frequency your eye can see using the same method. Note the characteristics of thefilament’s spectrum in your Data. Are there holes in the spectrum or is it continuous?Humans can see 400 nm < λ < 700 nm so be sure your measured wavelengths are reasonablebefore continuing. Once you understand how to use the spectrometer, double-check your

150


knowledge by repeating the exercise on the other side of zero (for m = −1).

10.2.2 The Hydrogen Spectrum

Start Vernier Software’s Ga3 graphical analysis program. A suitable Ga3 configuration filecan be downloaded from the lab’s website at

http://groups.physics.northwestern.edu/lab/spectral.html

The wavelength is calculated as you enter the order m and the diffracted angle θ usingEquation (10.15) λ = d

msin θ; since the grating specifies lines/inch, this becomes

25.4 * 10^6 * sin(“Angle” - θ0) / (lines * “Order”)

in Ga3’s calculation equation. Another calculated column “Rydberg” with units “1/nm” andformula

1 / (“Wavelength” * (1/4 - 1/“Upper”^2))

is your measured Rydberg constant and is plotted on the y-axis. Equation (10.14) has beenused as this column’s equation. Use the controls at the bottom left to tell Ga3 about yourθ0 and lines/inch.

Observe and record as many spectral lines as you can see in the spectrum of hydrogen,using a hydrogen gas tube. Make sure that the light source is properly aligned with thespectrometer slit. At 0◦ the red and blue will merge into a bright sort of tan at properalignment (the same color as the directly viewed hydrogen tube). Hold the table as youmove the eyepiece or you will mess up your careful alignment and need to repeat it beforecontinuing. Enter the angle into Ga3 (θ), the diffraction order (m), and the upper electronshell’s principal quantum number (red⇒ nu = 3, cyan⇒ nu = 4, violet⇒ nu = 5). Comparethe wavelength of the lines you observe with the value shown in Table 10.1. Which series oflines do you observe? Table 10.1 also gives the principal quantum numbers involved in theobserved transitions.

Hints for measuring the wavelengths of spectral lines:

1. Align and center the light source to make the lines as bright as possible.

2. Optimize the focus for each measurement using lens B. Make sure the diffraction gratingis aligned perpendicular to the incident, collimated beam (a peg on the slide’s bottomshould fit into a groove on the mount.)

3. Measure each color using all visible orders on both sides of the optical axis. Each orderyields a measured value of the Rydberg constant. Note correlations between Rydbergconstants gathered using one color and contrast these with constants gathered usingother colors.

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4. Use some intuition in matching the measured wavelengths to the chart of known lines.Even with careful measurement procedures your measured wavelengths may easily varyas much as a couple dozen nanometers from those in the chart. Also the chart onlylists the brighter lines; other dim lines may also be present.

Rescale your y-axis to be (0, 0.012 nm-1) and decide whether the result is a constant plusexperimental errors. If all of the points were taken from the same distribution, then it isreasonable to compute their statistics as described in Section 2.6. Ga3 can compute thestatistics for you; do you remember how? Specify your best measurement of the Rydbergconstant and its tolerance.

CheckpointWhich one of the hydrogen series describes light in the visible spectrum?

CheckpointWhat is the difference between the electromagnetic spectrum produced by anincandescent object and the spectrum generated by excited atoms?

10.2.3 The Spectrum of He, Ne, and Hg

The He gas tube is labeled while the tubes with Ne and Hg are only marked with colored tape.Record the wavelength of 5 to 10 spectral lines of each one of these elements. Measure thesewavelengths only once each rather than once for each visible order and note their units anderrors. Notice that the spectrum of these gases is much more complicated, with many morelines than the one of hydrogen. This is due to the larger number of electrons orbiting thenuclei of these elements, the large number of electron-electron interactions, and consequentlythe larger number of possible transitions corresponding to spectral lines. Table 10.2 gives thewavelengths and the relative intensity of the line spectra of these elements. By comparingyour observed values with those on the table you will be able to tell which gas is enclosed ina tube marked with a given color. Chemists find the spectroscopic uniqueness of atoms andmolecules quite useful in identifying unknown substances.

10.3 Analysis

Discuss your graph of measured Rydberg constants. What trends do you see in the graph?Bohr’s model and quantum mechanics predicts that all of the Rydberg constants are thesame; is it conceivable that your data agrees with this within your errors? Use the strategyin Section 2.9.1 to compare your measurement of Ry to that measured by others on page 146.

152


Table 10.2: Several known lines for hydrogen (H), helium (He), neon (Ne), and mercury(Hg). The units are nm and items marked with asterisks (*) are very dim. These may beused to identify these substances in chemicals when they are burned in a flame.

Hydrogen Helium Neon Mercury656.28 706.5 724.5 734.5∗656.27 667.8 717.0 690.5486.1 587.5 703.2 615.6434.0 501.5 692.9 579.0410.4 447.1 640.2 576.9∗

388.8 614.3 546.0588.1 435.8585.2 404.6∗540.0535.0479.0472.5457.6

Does your Rydberg constant agree with everyone else’s? Disagree? How well do youknow? What do you observe about the environment of the hydrogen atoms that might affectthe electrons’ energies? What does this all mean versus Bohr’s model?

Now discuss the three more complex spectra. Take a few minutes and try to fit helium’sspectrum to Bohr’s model (Equation (10.10)). Do all of the lines fit or do we need a bettertheory? Does it seem like we might have better luck with neon or mercury?

10.4 Conclusions

Is blackbody radiation fundamentally the same or different than gas discharge light? IsBohr’s model valid? Under what circumstances? Did you measure anything that you and/orothers might find of value in the future? Remember to communicate using complete sentencesand to define all symbols used in equations. Can you think of any applications for what wehave observed? What improvements might you suggest?

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10.4.1 Reference

Sections of this write up were taken from:Bruno Gobi, Physics Laboratory: Third Quarter, Northwestern University

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10.5 APPENDIX: The Vernier Principle

A vernier consists of a movable scale next to a stationary scale. The movable scale is markedoff in divisions that are slightly smaller than the divisions on the main scale. If you line upthe zero mark of the vernier with the zero mark of the fixed scale (see Figure 10.10), you willnotice that not all of the other divisions are aligned. However, if you examine the two scalesclosely, you should observe that the nth division of the movable scale exactly coincides withthe n− 1 division of the fixed scale. This implies that n− 1 divisions of the main scale hasthe same length as n divisions on the vernier scale; so each vernier division is n−1

nor 1− 1

n

times the length of a main scale division. Determining the quantity 1nis an important first

step in making any measurement with a vernier; 1nis called the “least count”.

Figure 10.10: An illustration of the Vernier scale.The position of the movable 0 indicates the coarsereading on the fixed scale. One additional digit ofprecision can be obtained accurately from the MovableScale digit that aligns best with a Fixed Scale number.

The least count gives an indica-tion of the precision of the particu-lar measuring device. For example,if we can read the main scale ofour device to the nearest degree,and the device has a least countof 1/4 (i.e. 4 vernier divisions isequal in length to 3 main scaledivisions) then we can make mea-surements with an instrumentallylimited error of ± 1/4 of a degree.Examine Figure 10.10. What isn? What is the least count of thisVernier scale? If each division ofthe main scale corresponds to the

one degree, what is the precision of the measurements we can make with this device? Awell-designed instrument will have accuracy approximately equal to precision; the abilityto provide faithful measurements (accuracy) should be about the same as the amount ofinformation conveyed in a reading (precision). An instrument with too few displayed digitsdefaults to having round-off be its greatest experimental error. Contrarily, having toomany displayed digits wastes money on useless precision and implies an accuracy whichit is incapable of providing. Keeping this in mind, we should expect that the experimentalerror associated with an instrument is usually about the same as one or two least significantdigits of its carefully read display. In the case of our spectrometers, we should expect 0.1◦errors in our angle measurements with reasonable care and attention to detail.

To read a measurement from a vernier such as the one in Figure 10.11, follow these steps:

1. Determine the least count of your device.

2. Look opposite the zero mark of the movable scale and read off what division of thefixed scale is at the zero mark. If the zero mark does not exactly align and is situatedbetween two divisions, then read off the value of the lower of these two divisions. InFigure 10.11, we read 46 degrees.

155


3. Examine each mark along the movable scale and determine which division (from zeroto n) is aligned exactly with a division of the main scale. Read off this number fromthe movable scale. In Figure 10.11, we can see that 3 on the movable scale is alignedwith a division on the main scale. So we must add 3 least counts of 3/10 of a degree tothe number read from the main scale earlier, 46. Thus our final answer is 46.3 degrees.If the zero mark (and simultaneously the n mark) of the movable scale had exactlycoincided with a division of the main scale, as in Figure 10.10, then we would add zeroleast counts and arrive at an answer of 0.0 degrees.

Figure 10.11: An example tutorial in reading the Vernier scale. The movable 0 is betweenthe fixed 46 − 47, so the reading is 46.x. The x is determined by noting that the movable 3matches the Fixed Scale numbers best (49 in this case); this makes x = 3 and our reading is46.3. We might also note that the movable 0 is indeed quite close to 46.3 to double-check ourresult.

Historical AsideAll of the above is brought to you courtesy of Pierre Vernier (1585-1638), a Frenchengineer who spent most of his life building fortifications, but who designed the Vernierscale in about 1630.

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