Optical Spectroscopy with Grating and Prism PDF

Preview

Summary

An explanation of the different types of spectroscopy , grating and prism, that can be used and then introduces the mathematical theory behind their use and how to actually go about collecting data. ...

Description

LABORATORY INSTRUCTIONS FOR PHYSICS 277

DIFFRACTION GRATING AND DIRECT WAVELENGTH MEASUREMENT Introduction The diffraction grating is one of the most important optical instruments. It spite of being a very simple device, it allows for quick, and highly precise, determination of wavelength of light. The object of this experiment is to measure the wavelengths of selected atomic spectral lines using grating spectrometer. In contrast to the prism spectrometer used in the previous experiment - the grating version provides a direct measure of wavelengths and does not require "known sources" for equipment calibration. Theory The diffraction grating consists of a very large number of fine, equally spaced parallel slits. There are two types of diffraction gratings: the reflecting type and the transmitting type. The lines of the reflection grating are ruled on a polished metal surface; the incident light is reflected from the unruled portions, which act as a set of narrow mirrors. The lines of the transmission grating are ruled on glass; the unruled portions of the glass act as narrow windows letting the light through. Gratings usually have about 1,000 lines per millimeter. The principles of diffraction and interference can be used to explain the measurement of wavelength in the diffraction grating. Let the broken line in Fig.1 represent a magnified portion of a transmission diffraction grating. Let a beam of parallel monochromatic light impinge upon the grating from the left. By the Huygens' principle, the light spreads out in every direction from the narrow apertures of the grating, each one of them behaving as a separate new source of light. The envelope of the secondary wavelets determines the position of the advancing resultant wave. In the figure are seen the instantaneous positions of several successive wavelets after they have advanced beyond the grating. Lines, drawn tangent to these wavelets, represent the new wave fronts (surfaces of common phase).

1

One of these wave fronts, tangent to the wavelets which have all advanced the same distance from the slits, is parallel to the original wave front and corresponds to the light traveling straight ahead, i.e. in the original direction. A converging lens placed in the path of these rays would form the central image, or zero order spectrum. Another wave front is tangent to wavelengths whose distances from adjacent slits differ by one wavelength. This wave front advances in the direction 1 and forms the first order spectrum. The next recognizable wave front is tangent to wavelets whose distances from adjacent slits differ by two wavelengths. This wave front advances in the direction 2 and forms the secondorder spectrum. Spectra of higher orders will be formed at correspondingly greater angles. The directions at which the wave fronts (and the actually observed light beams), travel behind the grating depend on the light wavelength. Simple geometrical arguments can be used to show that the relation between those parameters can be written as d sin θ = nλ ,

(1)

where: λ is the wavelength of the light, n is the order of the spectrum (e.g. n=1 for the side image closest to the center, etc.), d is the distance between the lines on the grating, and θ is the angle by which the particular beam deviates from the original propagation direction of the incident light. As long as the incident light is monochromatic, θ has a well defined value for each n, and a converging lens behind the grating produces a single image in each order of the spectrum. However, when a multi-chromatic light is used, there will be as many images in each order as there are wavelengths represented in the source, the diffracting angle of each being given by the formula (1) (Every time one plugs in a different wavelength value the answer for the theta changes). The spectrometer used in this experiment does not differ from the one used for studying of hydrogen spectrum, except that the glass prism is to be replaced by a grating. Like before, the function of the collimator is to render parallel the rays of light coming from the source and to direct them to the dispersion element. The diffracted image of the collimator slit is viewed with the lunette whose angular position can be read from the protractor circle. Since the relation between the wavelength and the angle at which spectral image is observed is now perfectly known, an absolute value of the wavelength can be easily determined if the grating constant (d) is known. No calibration with "known source" is needed.

2

Experimental Aim: Measure the wavelength of optical light. Apparatus -

Spectrometer Diffraction grating(s) Helium discharge tube with power supply Sodium lamp with power supply (optional) Other light sources, e.g. mercury discharge tube, etc (optional).

How to setup your spectrometer: -

Adjust the eyepiece of the lunette for clear vision of the crosshairs.

-

Adjust the lunette (telescope) length to obtain clear images of distant objects behind the window. (Those objects are convenient sources of parallel beams. After this adjustment we are sure that parallel beams will give sharp images.)

-

Place a helium discharge tube in front of the collimator slit, and adjust its lens position till a sharp image of the slit is observed through the lunette. This assures that the collimator produces parallel beams. Adjust the collimator slit width.

3

-

You need to place a diffraction grating in the center of the spectrometer table. The orientation of the grating should be normal to the path of the incident light. The diffraction gratting has a number of slits. Can you figure out the slit spacing?

-

You need to count the diffraction pattern for each wavelength and count the order. Which formula from above do you need? What is the lowest possible order? What is the image color for the lowest order diffraction?.

-

To record your data you need to use a vernier on the instrument to a reasonable precision. Which way is it better to turn the spectrometer. Left or right? Do you expect a symmetric pattern?

-

Can you think of any trick to increase the accuracy of your measurement? (Hint: see Fig.2).

-

Create a table for data recording in EXCEL. Calculate the wavelength value and its error for each color possible, and compare with the literature values. How large is the deviation from the literature? Does it agree within error with your measurements?

-

Go to the second order spectrum, and conduct similar set of measurement

Thoughts on Error analysis If you are a careful experimenter, the error in lunette position reading should be limited by the resolution of the protractor (0.1 deg). Uncertainty in the angle propagates into the wavelength result. Calculate how much wavelength error is associated with 0.1 degree uncertainty in the angle. Carry the calculation for two selected lines in first and, again, for the second order spectrum. Are the error bars identical? What makes them large? Are the numbers big enough to explain the deviations from the accepted values?

4

PRISM SPECTROMETER AND ANALYSIS OF SIMPLE ATOMIC SPECTRA Introduction Wavelength separation in a prism The index of refraction for glass varies somewhat with the wavelength of the incident radiation. Accordingly, when a multi-colored beam enters a glass prism, the variation of the refractive index with wavelength causes various colors to refract at different angles, and separate. After going through a prism the white light is dispersed into a rainbow of colors, with red deviating the least and the blue the most.

Fig.1 Light beam refraction in a glass prism. Determining the color composition of a light beam is in principle very easy; just let the beam go though a prism and see how it splits into different colors. However, precise determination of wavelengths represented in the light beam is a bit harder, primarily because there is no simple mathematical relation available, which would connect the wavelength ( λ ) with the deviation angle ( δ ) for the beam. This relation has to be established experimentally for any given prism, in order to make the prism-based spectrometer useful in measurements of wavelengths. Prism spectrometer

5

Fig.2 Prism spectrometer, a schematic view showing the essential components of the device. A typical spectrometer (see figure on the previous page) consists essentially of a collimator C, a telescope T, a turntable KLM (with a divided circular scale marked off in degrees), and platform N supporting a prism. The collimator is a tube with a slit S at one end and a positive lens at the other. The width of the slit and its position relative to the collimator lens are independently adjustable. With the slit positioned at the focal point of the collimator lens, the rays emerging from the collimator are parallel. When a prism is positioned on the platform, the beams of light from the collimator are refracted to the telescope and can be seen by the observer. For a polychromatic beam the observer sees images of the slit corresponding to each wavelength component at slightly different angles. This constitutes "the spectrum". The telescope and the turntable are mounted so that they can be independently rotated about a common axis that passes through the center of the divided circle. The collimator is generally fixed to the spectrometer so that its optical axis passes through the same center. The circular protractor scale is used to determine relative angular displacements between turntable and the telescope, and find the deviation of light beams caused by the prism. This information is later used for determination of wavelengths emitted by various sources. Light Sources There are many possibilities for production of light for spectroscopic investigations. The principal ones are the temperature radiation and all kinds of luminescence. In temperature radiation, atoms, molecules, or electrons in a solid body are excited to light emission by collisions with other atoms or molecules, the necessary energy being derived from kinetic energy of colliding particles. Therefore a high temperature is required. This kind of excitation is typical for tungsten filament in a light bulb. Atomic spectra of this kind can also be obtained, e.g. in flames. Luminescence includes all forms of light emission in which kinetic heat energy is not essential for the mechanism of excitation. A typical representative of this class is an

6

electroluminescence, i.e. light produced by all kinds of sparks, arcs, discharges, etc. Excitation in these cases results mostly from electron and ion collisions, that is, the kinetic energy of electrons or ions accelerated in an electric field is given up to the atoms or molecules of the gas present and causes light emission. Generally, spectrum produced by a source depends on the kind of atoms involved in the emission process and to some extent the excitation mechanism. Spectrum of Atomic Hydrogen In our first spectroscopy experiment we will focus on historically important case of the hydrogen spectrum obtained from a hydrogen-filled discharge tube. Throughout the development of quantum mechanics, atomic spectra have played a central role. The problem of the origin of atomic spectra was one of the original motives for the beginning of quantum mechanics. For many years the analysis of spectra has been the most important analytical tool in the investigation of atomic and molecular structure. The central idea in the relation of atomic structure to spectra is the existence of discrete energy levels in atoms. An atom in a state with energy E1 can make transition to a state with a lower energy E2 by emitting a photon whose energy is E1- E2. The energy of the photon is in turn related to its frequency and wavelength by the familiar Planck's relation

E1 − E 2 = hf =

hc

λ

,

(1)

where c is the speed of light and h is the Planck's constant. Conversely, an atom can be raised from lower-energy state to a higher-energy state by absorbing a photon whose energy equals the difference of energy of the two states. The hydrogen is the simplest of all atoms, consisting of a single electron and a single proton. Thus, it is not surprising that the spectrum of hydrogen has a corresponding simplicity. Elementary quantum mechanics shows (for more details consult a standard textbook on modern physics), that the energy levels of a single-electron atom are given by a simple formula

฀฀Z 2e 4 m ฀฀฀฀ 1 ฀฀ , En = −฀฀ 2 ฀฀ ฀฀฀฀ ฀฀8ε 2 h 2 ฀฀ ฀฀฀฀n ฀฀ ฀฀ 0

(2)

where e is the electron charge, εo is the permittivity of free space, n is a positive integer called the principal quantum number, and Z stands for number of protons in the nucleus (1 in case of H). The lowest energy state or ground state is the state with n=1. The energy E=0 corresponds to a state in which the electron has been completely separated from the proton (corresponding to n → ∞ ).

7

If the atomic nucleus were infinitely massive, the m in Eq.(2) would be simply the electron mass. For the finite nuclear mass the right result is obtained by replacing the electron mass by the reduced mass of the system, defined by mM , (3) µ= m+ M where M stands for the actual mass of the nucleus. This correction has the effect of decreasing the calculated energy for all levels by 0.05%, as compared to the values they would have with M = ∞ . It is also responsible for the difference between wavelengths of different isotopes (e.g. the ordinary hydrogen and the heavy hydrogen, called deuterium). The lines observed in the hydrogen spectrum correspond to all possible transitions between energy levels. Their wavelengths are given by the energy - wavelength relation:

1

λ

=

E1 − E 2 µ ( Ze 2 ) 2 1 1 = ( 2 − 2 ), 2 3 hc 8ch ε 0 n 2 n1

(4)

where n1 and n2 denote the quantum numbers of the initial and final states, respectively. The combination of constants in front of the last parenthesis is often called the Rydberg constant and abbreviated R. When an ordinary hydrogen is considered and Z=1, transitions from higher initial states to the final state n2=2 yield wavelengths in the visible region of spectrum allowing for their naked eye observation. The corresponding series of lines is called the Balmer series after Johann Balmer, who discovered Eq.4 empirically in 1885, long before its relation to the structure of the hydrogen atom was understood. Table 1 shows the agreement between observed wavelengths and those computed from Balmer's formula (Eq. 4). It also shows the usual spectroscopic notation used to designate these lines.

Line

Quantum #'s: n1, n2

λ observed (nm)

λ computed (nm)

H H H H H H

3,2 4,2 5,2 6,2 7,2 8,2

656.279 486.133 434.047 410.174 397.007 388.906

656.280 486.133 434.048 410.175 397.008 388.906

α

β

γ

δ

ε

ζ

8

Experimental Aim: construct a prism spectrometer and calibrate it using a He discharge light.

How to align your spectroscope: The ocular of the telescope is adjusted for a sharp vision of the crosshairs. The viewing telescope is adjusted for sharp imaging of parallel rays by focusing it on some distant object, e.g. a tree behind the window. The light source is placed in front of the slit. The slit of the collimator is viewed directly (with no prism on the platform) with the telescope. A sharp image ascertains that the slit is in focus of the collimator lens and a parallel beam is produced. The slit is moved closer and further, to determine the optimum spot. The crosshairs are aligned along with the slit image. The position of the telescope is recorded since it represents the zero deviation position. The prism is placed in position on the platform. The refracting edge should be parallel to the axis of rotation. When spectrum is observed, prism orientation should be adjusted for minimum deviation and fixed (you can use a special holder provided by your instructor). -

After that, the spectrometer is ready to be used for qualitative spectra observation only. You can use a helium discharge tube as a source of light to obtain a sharp and clear spectrum. You can -Consult instructor-provided information on helium spectrum (pictures, tables) and identify the lines which you can see.

What relationship (i.e. between which measurable quantities) do we need to establish if we wan to use it to determine the wavelength of an unknown spectral feature (line)? Take the appropriate measurements and make an appropriate graph.

9

Note, that as long as we now use the same prism at the same orientation, the relation between the beam deviation and λ, given by the dispersion curve, remains valid even if a different light source is put into use. Accordingly, to find wavelengths produced by an unknown source we only need to determine the telescope position at which lines are observed and read the corresponding λ from the graph. If there is time and an appropriate Hydrogen source is available we shall use this approach to determine the wavelengths in spectrum of hydrogen, observe the Balmer lines, and compare our results with the accepted values. How can you find out if the wavelengths follow Eq. (4)? (Hint: Do an appropriate graph). Determine the Rydberg constant from your graph and compare it with the accepted value.

Thinking about the lab results 1. There is a difference between the spectrum obtained with a grating and one obtained with a prism. Which color is deviated most in each case? Why? 2. Why are the higher-order spectra more accurate than the first-order spectrum in determining wavelengths of light? What determines the number of orders of spectra that can be observed? 3. What would be the effect on the results of the experiment if a grating with fewer lines per cm had been used?

10...