GRID AND PRISM SPECTROMETERS

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1 FYSA230/2 GRID AND PRISM SPECTROMETERS 1. Introduction Electromagnetic radiation (e.g. visible light) experiences reflection, refraction, interference and diffraction phenomena when entering and passing an (optically) active medium like a grid and prism. The phenomena can be explained partly with classical wave-mechanics but are basically quantum mechanical or better quantum electrodynamical in character. Grids and prisms belong to such media. Visible light (spectra) emitted by atoms of gas-discharge tubes are well suited to study properties of atomic excitations and functioning principles of optically active devices, simultaneously. 2. Devices 2.1 Spectrometer A spectrometer is an (optically) active medium (device) sensitive to the energy, polarisation and intensity of electromagnetic or particle radiation (beam) selecting it e.g. according energy into energy spectra. When the energy is 1 kev, an electromagnetic radiation is customary to classify with wavelength (E = h = hc/). The grid and prism spectrometers used here are designed to handle electromagnetic radiation in visible regime. A schematic principle is presented in figure 1 and in more detail in appendix 2. prism red light source slit collimator grid telescope violet Figure 1. Grid and prism spectrometers. Grid deflects and prism refracts the light into different angles according to the wavelength creating energy spectra that can be observed by looking at the transmitted light with a telescope in various angles. Knowing the geometry (number of slits per

2 - 2 - distance) of the grid or the refraction index of the prism the angle information can be translated to wavelength. In practise the energy or wavelength response of a spectrometer is calibrated using known spectra of standard sources and the resulting calibration function = f(θ) is used to analyse spectra of radiation studied. Various optical elements like slits, collimators and focusing devices are needed in experiments with some accuracy desired. A slit (S) narrows the beam from the light source. The collimator (C) parallels it before the grid or prism. The refracted rays with different wavelengths are focused with the objective of the telescope on the position of a hair grate located between the focus point and the ocular. The ocular magnifies the picture formed by the hair grate and the radiation to be observed by naked eye. Read appendix 2! 2.2. Diffraction grid Monochromatic light passing a narrow slit experiences diffraction i.e. deflects forming a diffraction pattern characteristic to the wavelength and the width b of the slit. The intensity minima of the diffraction pattern are located at angles sin(θ) = n/b, (n = 0, 1, 2, ). With several slits the diffracted light from different slits interfere and an interference pattern modulated by the diffraction one appears. The location (angle) of intensity minima and maxima depend now also on the density (location) of the diffraction slits. A regular, 2-dimensional diffraction grid with parallel, rectangular slits (width b, distance a, s.c. grid constant) forms principal intensity maxima at angles defined by n sin( ) (1) a If the light is non monochromatic, each wavelength yields its own diffraction pattern. Usually the main components of a light source are well separated in energy and a clean diffraction pattern of a grid for each wavelength will be observed. The grid and prism can be characterised with their ability to resolve different wavelengths, the dispersion D, which derived from eq. (1) is d n D (2) d acos( ) Irregular grids with 2- or 3-dimensional geometry's yield very complicated diffractioninterference patterns and are used as versatile, optically active devices Prism Light passing through a prism will be reflected and refracted by the entrance and exit surfaces. Monochromatic light (beam) will be refracted by an angle from its original direction. A minimal refraction ( = min ) follows, when the light passes the prism symmetrically (see fig. 2). Now it holds sin(½( min A)) n, (3) sin(½a)

3 - 3 - where A is the angle of the deflecting edge and n = c/c n is the refraction coefficient of the material of the prism (c and c n are the speed of light in air (or vacuum) and in the prism, correspondingly). Since the refraction coefficient n depends of the wavelength, each wavelength has a characteristic min. A 1 2 n min 1 A Figure 2. Minimal deflection. Non-monochromatic light will thus disperse in prism according to the wavelengths. The spectrum created differs in structure from the corresponding grid spectrum. The dispersion of a prism can be derived using eq. (3) and relation n = c/c n d min 2sin(½A) 2B D cos(½( 3, (4) d min A)) where B is a constant. 3. Radiation emitted by an excited atom Possible excitation of an atom are quantum mechanical states of (many) electron configurations. The lowest in energy (E 1 ) is the (stable) ground state and other (meta stable) configurations have higher energies E i (i = 2, 3,...), decaying to lower lying states either directly or via transition states (figure 3). The energy shift occurs emitting electromagnetic radiation, a photon (e.g. visible light) corresponding to the energy difference E i E j (i > j; i = 2, 3,...). The frequency of the photon is thus h Ei E 2 (5) where = c/ and h the Planck s constant. Each atom has a discrete, characteristic energy spectrum.

4 - 4 - E 4 transition E 3 excited E 2 E 1 ground Figure 3. States and transitions of an atom. Here we study electromagnetic radiation of excited atoms in a gas-discharge tube in the regime of visible light. The discrete spectra predicted by the theory presented before are analysed with the aid of grid and prism spectrometers. J. Balmer observed on 19th century that the wavelengths of visible spectral lines of a hydrogen atom obey the rule where RZ 2 '2 n n R = Rydberg s constant = 1, m -1 Z = 1 = atomic number of hydrogen n = 2 = the quantum number of the final state n' = the quantum number of the initial state = 3 for the hydrogen - line = 4 for the hydrogen - line = 5 for the hydrogen - line. (6) Eq. (6) is valid for all atoms with a single electron (like H, He +, Li 2+ ) and it can be derived from Bohr s model of the atom Lymanlines Balmerlines Figure 4. Partial decay scheme of a hydrogen atom.

5 Measurements 4.1. Focusing the spectrometer The spectrometer used is described in details in appendix 2. Fig. 5 shows a schematic diagram of the optics used. A B C D E F G Figure 5. Optical elements of the spectrometer: Slit A, objective of the collimator B, objective of the telescope C, hair grate D, ocular lenses of the telescope E and G, half-transparent mirror F which illuminates the hair grate. The distance between the ocular lenses E and G can be varied so that the image and the hair grate are seen sharp, simultaneously. Focusing the spectrometer is done as follows: If the lines with different colours and the hair grate are seen sharp using the grid, focusing is not necessary. Otherwise, set the lens G so that the hair grate is seen sharp. Then the entire set D, E, F, G is moved so that an object in distance is seen sharp and does not move with the position of the eye (elimination of s.c. parallax). Now the image of the hair grade and the object coincide. The position of the hair grate can be adjusted with the focusing ring M (see figure A1). The collimator B is focused by moving the slit A so that the slit is seen sharp independent of the position of the eye. The slit should be set vertical, exactly, and as narrow as possible depending of the intensity of the light source Grid Place the grid (about 900 slits per mm) on the turn table perpendicular to the optical axis. Optimise the height of the table. Use the helium lamp (He) and measure positions of the 6 brightest lines of the first order and the 3 corresponding of the second order on the both sides of the straight incoming light (angles 1 and 2 ). The deviation is =1/2( 1-2 ). Calculate the grid constant with eq. (1). By using the mercury-cadmium lamp (Hg/Cd), measure deflection angles of the 9 brightest lines of the first order. For hydrogen (H 2 ) observation of 3 brightest lines is sufficient. If you are not able to discern all the lines on both sides of the straight incoming light, record the angles from the the straight incoming light and the line on the other side. The deviation can then be calculated with help of these.

6 Prism With the Hg/Cd lamp optimise the set-up for the prism according to fig. 1 so that the spectrum is clearly visible. Search the minimal deflection angle. This can be done as follows: monitor the position of a selected spectral line and rotate the prism, until you find an angle, where the lines start to move in reverse direction when rotating the prism to same direction. In the limits of the accuracy of the spectrometer it is sufficient to use only one line (e.g. the green one) to define the minimal angle for the entire spectrum. For the other lines move now the position of the telescope, only, to measure the angles (θ 1 ). The minimal deflection for each line is now the difference between the angles measured and the angle (θ 0 ) for straight light when the prism is removed. The wavelengths of the lines are taken from the measurements with the grid. 5. Analysis of results Grid: By using your results for the He lamp and the wavelengths in appendix 1, determine the grid constant a. For H and Hg/Cd calculate the wavelengths using eq. (1) and the measured grid constant a. Verify and compare your values for Hg/Cd to those given in appendix 1 and in the literature to find the lines of mercury. Furthermore draw the curve () using measured results for the Hg/Cd lamp. Prism: Draw the minimal deflection min as a function of the wavelength. Furthermore, calculate the reflection coefficient n in eq. (3) (A = 60) and plot n as a function of. For several materials the so-called Cauchy s formula is valid B C n A, (7) 2 4 where A, B and C are constant parameters characteristic to the material, but independent of the wavelength. Define these parameters and study how well Cauchy s formula is valid. Both grid and prism: To determine the dispersions D = dθ/dλ and D = dδ min /dλ, draw a few tangents of curves θ(λ) and δ min (λ) with fit intervals and calculate their slopes. Plot in the same figure d/d and d min /d as a function of. Balmer s formula Calculate with eq. (6) the wavelengths of -, - and - lines of hydrogen. By comparing the calculated values with your experimental ones (obtained with the grid) define the colours of -, - and - lines.

7 - 7 - Answer to the following questions: What are most striking differences of the spectra obtained with the grid and the prism? Based on the curves D(), what can you tell about properties of the grid and the prism? What is dispersion? Why you don t see the Lyman-lines? Derive eq. (3) using fig. 2 and the Snell s law. Recall that a degree ( ) is not a SI unit. Appendix 1 Wavelengths given in literature (nm): He: Hg: red yellow yellow yellow green (strong) green green (weak) blue blue green violet blue

8 - 8 - Appendix 2 The Spencer-spectrometer In principle the spectrometer is simple (fig. L1), but has many components worth to study and try out their functions carefully before the measurements. Figure A1. Spencer-spectrometer. Slit A, objective of the collimator B, objective of the telescope C, prism D, hair grate E, fixing screw of the table I, fine-tuning screw of the table J, fine-tuning screw of the telescope K, ring of ocular L, focusing ring of telescope M, cover of the Nonie-ring N, height-setting screws of the table O, height-setting screws of the collimator and the telescope P, fixingscrew of the telescope Q, fixing-screws of the telescope R, fixing-screw of the collimator S. The telescope and the optical table rotate around common axis and they can be fixed with screws K and J. The height of the table is set by screw I. Do not over tighten them. When screws K and J are fixed, the position of the table and telescope can be tuned with fine tuning screws. To determine the angles precisely either the table (usually) or the telescope has to be kept fixed. The minimal deflection angle is best to find out with the fine tuning the table. The spectrometer is equipped with two 30' Nonie-scales. It is sufficient to use one of them, only. Learn to read the Nonie-scale correctly!

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