THE UNIVERSITY OF QUEENSLAND DEPARTMENT OF PHYSICS PHYS2041 ATOMIC SPECTROSCOPY

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1 THE UNIVERSITY OF QUEENSLAND DEPARTMENT OF PHYSICS PHYS2041 ATOMIC SPECTROSCOPY Warning: The mercury spectral lamps emit UV radiation. Do not stare into the lamp. Avoid exposure where possible. Introduction This laboratory involves the study of hydrogen-like atomic systems. The study will be performed by direct observation of the optical transitions in Na (sodium). The experimental equipment consists of a sodium spectral lamp and a prism spectrometer. A mercury spectral lamp is provided for calibration of the spectrometer. Theory When an element (or salt of an element) is heated in a flame, or when an element in the form of a low pressure gas is subjected to an electric discharge, the atoms are excited and subsequently emit light when they de-excite. Upon analysis with a spectrometer, it is found that the radiation emitted, or the emission spectrum, contains discrete wavelengths characteristic of the element. What is observed is a series of bright lines of pure colour on a dark background - each line corresponding to one of the characteristic wavelengths. This is the line spectrum. Any successful theory of atomic structure must be able to explain the observed spectral lines for the elements. It was the spectrum of the hydrogen atom which was first explained on a fundamental basis in the development of atomic physics. Balmer in 1885 found that the frequency of the lines in the visible region of the spectrum of the H atom were given by the equation ν = Rc (1) 2 n 2 where c is the speed of light, R is the Rydberg constant, n is an integer (n > 2), and v = c/λ, where λ is the wavelength of the emitted light. Following relationship (1), it was shown that a similar expression describing all spectral lines of H existed. This expression is 1

2 ν = Rc 1 m 1 (2) 2 n 2 where m is a positive integer and n > m. The line series for m = 1,2,3 are known as the Lyman (ultraviolet), Balmer (visible) and Paschen (infrared) series respectively. No fundamental explanation of these equations existed until the atomic model of Niels Bohr in However, earlier models of atomic structure existed. The first one was based on Thompson's discovery of the electron. Thompson, in 1898, proposed a "plum pudding", model of the electron. This was later found to be incorrect following Rutherford's α-particle scattering experiment, which supported the existence of an atomic nucleus and led to Rutherford's "planetary" atomic model. In this model, electrons orbit the nucleus. However, classical electromagnetism requires such an electron to continuously radiate energy and thereby spiral into the nucleus. This fundamental difficulty together with the failure of this model to explain the existence of characteristic spectra, provided the stimulus for Bohr which led to the development of his theory of the atom. The basis of Bohr's theory can be expressed as four postulates. These are: 1. An electron in an atom moves in a circular orbit about the nucleus under the influence of the Coulomb attraction between the electron and the nucleus, and obeys the laws of classical mechanics. 2. Only certain orbits are allowed. It is only possible for an electron to move in an orbit for which its orbital angular momentum L is an integral multiple of Planck's constant h, divided by 2π. That is L = mvr = nh = nh (3) 2π where n is a positive integer, h = h /2π, m is electron mass, r is the radius of the electron orbit, and v is the speed. Equation.(3) shows that angular momentum is quantized. 3. An electron moving in such an orbit does not radiate electromagnetic energy despite the fact that it is constantly accelerating. 4. Radiation is emitted if an electron initially moving in an orbit of total energy E i discontinuously changes its motion so that it moves in an orbit of total energy E f, E f < E i. The frequency of the emitted radiation from a single electron is ν = E i E f h (4) 2

3 Similarly, if E i < E f, radiation of frequency v can be absorbed, causing an electron in an orbit of energy E i to jump to an orbit of energy E f. Postulate 1 is based on the idea of an atomic nucleus. Postulates 2, 3 and 4 introduce new ideas, conflicting with classical physics. Exercise 1 1 : Using these postulates derive the formula for energies of the allowed orbits in a hydrogen atom. Show also that equation (2) can be derived from the Bohr model. Discuss how you would modify the Bohr model in order to be able to use it for other one-electron systems, where one valence electron can be pictured as moving independently in a constant potential due to the nuclear charge and all the other electrons averaged together. The theory as presented thus far has assumed that the electron moves about a fixed nucleus. This cannot be correct unless the nucleus is infinitely massive. A refinement to the theory is necessary to account for the fact that the nucleus and electron will rotate about the centre of mass of the two body system. We introduce the reduced mass µ given by µ = mm m + M (5) The reduced mass correction introduces a small numerical change in the values for the allowed energies. The mass of an electron m is kg. The mass of a proton or neutron is kg. Using equation (5) we can see that the reduced mass for hydrogen is kg. As the number of protons and neutrons in the nucleus increases, the reduced mass for a one-electron atom will approach the mass of an electron - the reduced mass correction becoming negligible. The Bohr theory successfully describes the line spectra of the hydrogen atom. However, it does not explain the spectra of other one electron atoms. Some predicted lines are observed to consist of two or more closely spaced lines. The Bohr model also fails to explain the variation in intensity of different spectral lines. However, a complete description of atomic structure is provided by quantum mechanics. With the advent of quantum mechanics, we gave up thinking of electrons as moving in well defined orbits. It was also found that in some experiments, particles were behaving as waves. In quantum mechanics, electrons are thought of as waves and the behaviour of an electron is described by a wave function which can depend on position x and time t. The wave function, unlike a conventional wave, has no real physical meaning. However, Ψ * Ψ, where Ψ * is the complex conjugate of the function Ψ, gives the probability of where the 1 Include answers to all exercises in your report. 3

4 electron is. This wave function Ψ must satisfy a type of wave equation. If we assume that electrons move in a general spherically symmetric, time-independent central field, V, the wave function Ψ must satisfy the time independent Schrödinger equation h 2µ 2 + V Ψ = EΨ (6) where E is the total energy, µ is the reduced mass, and 2 is the Laplace operator. The Coulomb potential is a central field potential. If we solve equation (6) for a pure Coulomb field, the energy is given by the old Bohr formula, and dependent only on the principal quantum number n. If we solve the Schrödinger equation for a general central field, it is found that the energy is given by an expression which is dependent on both n and l, where l is called the orbital angular momentum quantum number. l describes the magnitude of an electron's orbital angular momentum. It can take on the values l = 0,1,2,..., (n-1) where n is the principal quantum number and can have the values n = 1,2,3,... It is customary to specify electron angular momentum states according to the convention l = 0, 1, 2, 3, s, p, d, f, Exercise 2: find out the reason for the choice of letters and correlate with your observations. In general, electrons having the same n values but different l values have slightly different energies. However, for the pure Coulomb potential, states with the same n value have the same energy. These levels are then called degenerate. Electrons having the same principal quantum number n are said to occupy the same atomic shell. Electrons in a shell that share a certain l value are said to occupy the same subshell. A full quantum mechanical treatment of atomic structure would reveal two more quantum numbers, m l and m s in addition to n and l. m l is the quantum number for the spatial orientation of the orbital angular momentum vector and assumes the 2l + 1 values over the range m l = -l, - l +1,, + l. 4

5 m s is the intrinsic spin quantum number of the electron and assumes the two values m s = ± 1 2. According to the Pauli principle, two electrons cannot occupy the same state defined by n, l, m l and m s. Therefore a maximum of 2(2l+1) electrons can exist with the same value of n and l, i.e. the same energy. Each subshell is identified by its principal quantum number n, followed by the letter corresponding to its orbital angular momentum quantum number l. A superscript indicates the number of electrons in that subshell. According to this, we are left with a spherically symmetric closed atomic core consisting of completely filled subshells and one or more electrons in an outer shell. In an alkali element, or one electron atom, there is 1 outer or valence electron. For example, the electronic configuration of Na is written 1s 2 2s 2 2p 6 3s 1 which means that the 1s and 2s subshells contain two electrons each, the 2p subshell contains six electrons and the 3s subshell contains one electron. In an optical excitation of an alkali atom, only the outer electron is excited. The excited states can be described completely by the single optically active electron. As the closed shells are spherically symmetric, we can, with good approximation, assume that the outer electron is moving in a central field, produced by the nucleus and the other electrons. The presence of these shells however means that the valence electron is no longer moving in a strictly Coulomb field since we are not dealing with point charges. The energy levels with the same n and different l values consequently have different energies. Therefore, the Na spectrum, as well as the spectrum for any other one electron atom will be different from the Hydrogen spectrum. However, empirically it has been found that it is possible to modify the expression describing energy levels of a non Hydrogen-like one-electron atom. In order to establish what modification to this expression is needed we will study the spectrum of Na in detail. Experimental Considerations The equipment available for this laboratory consists of a prism spectrometer and spectral lamps (Hg, Na low and high pressure lamps). Observe one of the lamps through the spectrometer and see if you can find sharp spectral lines. You may need to adjust the entrance slit size and the focus. Exercise 3: Draw a diagram showing the paths taken by light of different wavelengths through the spectrometer. 5

6 Devise a method that will allow you to calibrate the spectrometer (the readings on the dial are not necessarily accurate). You may assume that some of the spectral lines of mercury are well known, and these are given in table 1. In table 2, you will find some of the spectral lines of sodium and a guide to which lines you will be able to identify and measure. Be aware that other lines may exist in the lamp so you may need to use your analysis to correctly identify them. After all the lines have been recorded, construct an energy level diagram (called a Grotrian diagram) and indicate which transitions you have been able to measure. You may use the fact that the ground state (3s) has an energy of ev. Then construct a plot of const/ E versus principal quantum number. Determine the constant and comment on its physical significance. By studying this plot predict how the Bohr Model s energy formula should be modified to be applicable to the spectrum for Na (Hint: there should be some dependence on l). Table 1. Mercury Emission Lines Wavelength (nm) Colour violet violet blue blue/green green (v. strong) yellow yellow With reference to the Grotrian diagram that you constructed, does it give any information about what rules should be applied for the observed transitions? Comment on these results and investigate whether there are any ways of accommodating these results within the Bohr model. Comment on why the spectral lines that you measured and observed have different intensities. After you have completed the study of the sodium spectrum using the low pressure spectral lamp, study the spectrum of Na using a high pressure lamp for the first 20 minutes after the lamp is switched on. Comment on your observations, especially the behaviour of the yellow spectral line. 6

7 Table 2. Na transitions Colour Transitions Wavelength [nm] IR 4s 3p 1139 IR 3d 3p 819 red 5s 3p yellow 3p 3s green 4d 3p green 6s 3p green 5d 3p blue-green 7s 3p blue 6d 3p violet 8s 3p violet 7d 3p violet 8d 3p violet 9s 3p UV 4p 3s UV 5p 3s UV 6p 3s very weak lines 7

Line spectrum (contd.) Bohr s Planetary Atom

Line spectrum (contd.) Bohr s Planetary Atom Line spectrum (contd.) Hydrogen shows lines in the visible region of the spectrum (red, blue-green, blue and violet). The wavelengths of these lines can be calculated by an equation proposed by J. J. Balmer: