Atomic spectra of one and two-electron systems

Transcription

1 Atomic spectra of one and two-electron systems Key Words Term symbol, Selection rule, Fine structure, Atomic spectra, Sodium D-line, Hund s rules, Russell-Saunders coupling, j-j coupling, Spin-orbit coupling, Good quantum number Object It aims to obtain atomic spectra of various elements and relate them to basic concepts learned before in a physical chemistry course. Introduction Interaction of a molecule with light can be described as the response of nuclei and electron charges of the molecule to an oscillating electric field (classical light = electromagnetic wave). Such response entails induced dipole moment oscillating with the frequency of the external field that is light. Different molecular processes generating such an oscillating dipole moment have characteristic frequencies, because nuclei and electrons move in a fundamentally different time unit. Atomic spectra result from transitions between different electronic states of an atom via emission or absorption of photons. They are powerful evidences of quantization of energy of atoms. The discrete values of wavelengths that atoms emit explain about it. The atomic spectra can be used to explain the electron configuration of atoms. In this experiment, we will obtain the wavelengths of spectral lines of various atoms (He, Na, Hg) by a diffraction spectrometer. Background Information Theory 1. Electronic transition Nuclei and electrons in a molecule move in a fundamentally different time unit. - Electron: attosecond (10-18 s) It takes 160 attoseconds for an electron to orbit a hydrogen atom. Therefore corresponding frequency of the orbit motion is given by 1 ν = s = s 1 which falls into the visible and ultraviolet region. - Nuclei: femtosecond (10-15 s) 1

2 Molecular vibration and rotation are the result of the motion of nuclei. In general vibrational motion is faster than the rotation, since in the vibration mode nuclei just oscillate around their equilibrium positions with slight deviation. The corresponding frequency of such oscillation is around s that falls into the infrared region. Characteristic frequency of a rotating molecule strongly depends on the size of the molecule and thus has a broad range from s for large molecules to s for small molecules, which fall into the microwave and far-infrared regions, respectively. - Bohr frequency Such a characteristic frequency for a specific molecular motion can be measured through detecting absorption and emission of photons for molecules, which is called the spectroscopy. Observed frequency in the spectroscopy, namely the Bohr frequency is given by the energy difference between initial and final states as follows. ν = ΔE h or more often in the unit of wavenumber (cm -1 ) v = ΔE hc By analyzing various characteristic frequencies in spectra, one can obtain structural information of molecules. In addition, it is possible to study chemical dynamics such as reaction mechanisms by measuring the spectrum as a function of time. 2. Term symbols of multi-electron atoms and good quantum number For a hydrogen atom, the angular momentum operators L 2 and Ŝ 2 commute with its Ĥ. Therefore, they share the same eigenfunctions so that each orbital can be specified by the quantum number n, l, s. So these are called the good quantum numbers. For atoms with N electrons, however, the individual angular momentums l i do not commute with the Ĥ due to the electron-electron interaction terms. But rather the vectorial sum of the individual angular momentums does commute with the Hamiltonian. Consequently, the atoms share their energy eigenfunctions with the total angular momentum and total spin, which means that each state of the atoms can now be specified by a set of L, S, and J as shown below. 2S+1 L J L is the quantum number indicating the total orbital angular momentum of an electronic state. S is the quantum number representing the total spin angular momentum of the electronic state. J is the quantum number representing the total angular momentum of the electronic state. The quantity 2S+1 is called the multiplicity of the state. 2

3 The value of L is obtained by coupling the individual orbital angular momentum of two electrons. This is done using Clebsch-Gordan series: L= l 1 + l 2, l 1 + l 2-1,, l 1 l 2 If more than two electrons are present in an atom, the orbital angular momentum of each additional electron is coupled to the previous value of L, using the same formula. The Clebsch-Gordan series terminates at l 1 l 2 because the total orbital angular momentum L, like each individual orbital momentum l, cannot be less than zero. A closed shell has zero orbital angular momentum S is the total spin quantum number. 2S+1 is the spin multiplicity: the maximum number of different possible states of J for a given (L, S) combination is given by (S = s 1 + s 2, s 1 + s 2-1,, s 1 s 2 ). If more than two electrons are present in an atom, the spin angular momentum of each additional electron is coupled to the previous value of S, using the same formula. For a single electron, recall that s = Spin-orbit coupling In atoms or molecules, electrons have spin motions as well as orbit motions. Since the electrons are charged objects, their orbit motions produce a finite magnetic field which influences on their own spins or vise versa. We may need to consider the so-called spin-orbit coupling term, when we calculate correct energy of electrons. - Spin-orbit coupling in hydrogenic atoms Because an electron has spin angular momentum, and because a moving charge generates a magnetic field, an electron has a magnetic moment that arises from its spin. Also, the same electron has an orbital angular momentum and possesses a magnetic moment that arises from its orbital angular momentum. The interaction of the spin magnetic moment and the magnetic field arising from the orbital angular momentum is known as spin-orbit coupling. The extent of the spin-orbit coupling depends on the relative orientation of the spin and orbital magnetic moments, and therefore on the relative orientation of the spin and orbital angular momenta. The total angular momentum of an electron is described by the quantum numbers l and s. j = l + s When the two angular momenta are in the same direction, j = l + 1, and when they are in the 2 opposite direction, j = l - 1. The different values of j that arise for a given value of l label 2 levels of a term. 3

4 The energies of the levels with quantum numbers l, s, and j are given by the equation using the first order perturbation theory to consider it as a correction term to the energy of hydrogen-like atoms: E n (1) = Z 4 j(j+1) l(l+1) s(s 1) 2(137) 2 n 3 2(l+1)(l+ 1 2 ) E h (1) The strength of the spin-orbit coupling increases sharply with atomic number (as Z 4 ). So there is less spin-orbit coupling in He than in Hg. Observed transitions in the electronic spectrum of atoms arising from levels of different values of j are known as fine structure of the spectrum. The total angular momentum quantum number J gives the relative orientations of the spin and orbital angular momenta for several electrons. The strength of the spin-orbit coupling affects the value of J. If the spin-orbit coupling is weak (as in atoms of low atomic number, Z<30), the total angular momentum is determined by the Russell-Saunders coupling scheme. This scheme is based on the assumption that if the spin-orbit coupling is weak, then it is effective only when all the orbital momenta operate cooperatively. This allows the total angular momentum quantum number J to be determined from a Clebsch-Gordan series of L and S: J = L + S, L + S 1,, L S If the spin-orbit coupling is large (as in heavy atoms, Z>40), then the total angular momentum quantum number J is determined from the j-j coupling scheme. In this case, the individual total angular momentum (j) of each electron is determined, and these j values for each electron are consecutively coupled together using the Clebsch-Gordan series: J = j 1 + j 2, j 1 + j 2-1, j 1 + j 2 2,, j 1 j 2 For multi-electron atoms, the energies of the levels arising from spin-orbit coupling are given by Eq Hund s rules States with different L and S have different energies because the magnitude of the electron-electron repulsion differs between the states. States with the same L and S, but different J, differ in energy because of spin-orbit interactions. The relative energy ordering of possible states is given by a set of generalizations known as Hund s Rules: i. The term level with maximum multiplicity and hence maximum spin quantum number is the lowest in energy due to the Pauli exclusion principle. States with lower multiplicities follow in order of decreasing multiplicity. ii. For states with equal multiplicities, the state with the maximum value of L is lowest in energy. 4

5 iii. States that have equal L and S values differ in energy because of spin-orbit coupling. If there is a single subshell (given n and l values) that is less than half filled, the state with minimum J is lowest in energy. If the subshell is more than half filled, the state with maximum J is lowest in energy. Any state of the atom and any spectral transition can be specified by using term symbols. 5. Selection rules However, there are a set of selection rules governing which transitions are allowed or forbidden: These selection rules arise from the conservation of angular momentum during a transition and from the fact that an absorbed or emitted photon has a spin of 1. These selection rules are: 1) ΔL = 0, ±1 and ΔS = 0 (for Russell-Saunder coupling: Z<30) 2) J = 0, ±1 (for j-j coupling: Z>30) 6. Sodium D-Line (Fine structure) The sodium D-line is responsible for the familiar orange glow of many street lights. The origin of the glow is emission of photons in the visible region of the electromagnetic spectrum from excited sodium atoms. The excited atoms emit light and return to their ground electronic states. The sodium D-line gets its name because there are really two closely-spaced emissions possible, or a doublet, as shown in Figure 1a. These transitions occur at wavelengths of 5890 and 5896 Å. Figure 1. Transitions involved in the sodium D-line. Term symbols for transitions involved in the sodium D-line The doublet observed in the sodium D-line transition involves the outer electron in the sodium atom which undergoes a transition from an excited 1s 2 2s 2 2p 6 3p 1 configuration to the ground state 1s 2 2s 2 2p 6 3s 1 configuration. To see why this electronic transition corresponds to a doublet, the atomic term symbols for the different electronic configurations must be determined. In both cases, only the outer open shell needs to be considered. 5

6 - Na Atom Ground State 3s 1 Term symbol Since the ground state of sodium only has one outer electron, the total angular momentum quantum number L and total spin angular momentum quantum number S are identical to the orbital and spin angular momentum quantum numbers of the outer electron. Thus, S = s 1 = 1 2 and L = l 1 = 0 The multiplicity 2S+1 is therefore 2 (a doublet) and the state corresponds to a 2 S state. Then, all that is needed is to determine the total angular momentum quantum number J. The total angular momentum quantum number J ranges from L S to L + S. For 2 S state, L=0 and S = 1 leads to J= 1. Therefore, the only possible term symbol for the sodium 3s1 2 2 ground state is 3 S 1/2. - Sodium Atom Excited State 3p 1 Term symbol The 3p 1 excited state of sodium only has one outer electron, so the total orbital angular momentum quantum number L and total spin angular momentum quantum number S are identical to the orbital and spin angular momentum quantum numbers of the outer electron. Thus, S = s 1 = 1 2 and L = l 1 = 1 The multiplicity 2S+1 is therefore 2 (a doublet) and the state corresponds to a 2 P state. Then, all that is needed is to determine the total angular momentum quantum number J. The total angular momentum quantum number J ranges from L S to L + S. For 2 P state, L=1 and S = 1 leads to two possible values of J, J= 1 and J= 3. Therefore, there are two possible term symbols for the sodium 3p 1 excited state: 2 P 1/2 and 2 P 3/2. Spin-orbit coupling leads to energy splitting between these two terms. The term symbols determined for the ground and excited states of sodium can be used to label the transition responsible for the sodium D-line emission, as shown in Figure Another example of energy splitting of atomic terms Consider an example of an atomic electron configuration 1s 1 2p 1. There are 12 ways of choosing the individual quantum numbers for the two electrons in this configuration. In the absence of electron-electron repulsions, all these states are degenerate. The possible term symbols for the 1s 1 2p 1 configuration are 1 P and 3 P (not including the J value). Hund s first rule states that terms with higher multiplicity will be lower in energy. Thus, including electron-electron repulsion, 3 P will be lower in energy than 1 P. For the 1 P, the only possible value of J is 1; thus, the only term symbol for this state is 1 P 1. For the 3 P term, the possible 6

7 values of J are 0, 1, and 2; this leads to term symbols 3 P 0, 3 P 1, and 3 P 2. The total degeneracy of the 1 P and 3 P terms is 3, 1, 3, and 5, respectively, for a total of 12. The 3 P 0, 3 P 1, and 3 P 2 states are split in energy by a very small amount. This splitting is due to the coupling of spin angular momentum(s) with total orbital angular momentum (L). The spin-orbit coupling splits levels within the same term (that is, the same values of L and S) that has different values of J. Finally, if the atom is placed in a magnetic field, the levels with the same values of L, S, and J, but with different values of M J are split. All of the energy splitting for the 1s 1 2p 1 electron configuration are summarized in Figure 2. As this figure, in two electron systems energy states are split with various reasons. Figure 2. Energy splitting of atomic terms in the 1s 1 2p 1 configuration Equipment 1. Diffraction spectrometer 7

8 Figure 3. Brief scheme of diffraction spectrometer A diffraction grating is an array of identical, equidistant, parallel lines on a surface. Gratings are used to produce optical spectra from a single source, parallel beam of light. There are two types of grating: reflection & transmission Figure 4. The transmission grating For transmission grating, if light of a wavelength λ falls on to a grating of constant d it is diffracted. Intensity maxima are produced if the angle of diffraction θ satisfies the following conditions: nλ =dsinθ (n = 0, 1, 2, ) Typically, diffraction grating is described in term of the number of lines (slits) per unit length N. From this value the slit separation can be found as d = 1/N. 2. Reading a Vernier scale Practice reading the angle from a precise protractor scale on the rim of the black table. Use the Vernier scale with the little magnifying glass to read the angle to the nearest arc minute. (1 arcmin = 1' = 1/ 60 degree.) The following is an example: 8

9 In this example, the zero line of the Vernier scale (the upper scale) is between 40.5 and 41, so the angle is somewhere between 40 30' and 41. The Vernier scale tells exactly where in between. Look along the Vernier for the line that exactly lines up with the line below it. In this case, it's the 17' line. So the angle is 40 47', which we get by adding 17' to 40 30'. Before using this angle in equation (2), we must convert it to decimal degrees: 40 + (47/60) degrees = Pre-Laboratory Questions 1. What is the Bohr frequency condition? 2. What is the selection rule and Hund s rule? 3. How is the atom s term symbol constructed? 4. What is the sodium D-line? 5. What is the meaning of the good quantum number? 6. How can one lift degenerate energy levels in general? 7. How does the spin-orbit coupling result in energy level splitting of atoms? Materials Apparatus Spectrometer/goniom. W. vernier 1 Diffraction grating, 600 lines/mm 1 Spectral lamp He, pico 9 base 1 Spectral lamp Na, pico 9 base 1 Spectral lamp Hg, pico 9 base 1 Power supply for spectral lamps 1 Lamp holder 1 Tripod base PASS 1 Safety and Hazards Lamps can be burst so lamps should be connected with a lamp holder and concealed by a safety guard before supplying power on them. Lamps can get damaged if you handle them strongly. Even though lamps may be jammed to a socket, please treat them carefully. Lamps 9

10 can be hot after using them so you should wait a while to cool them and wear cotton work gloves when pulling out lamps. Experimental Procedure Figure 4. Experimental set up of the diffraction spectrometer 10

11 Figure 5. Experimental set up of the diffraction spectrometer Figure 6. Experimental set up of the diffraction spectrometer Setting up a lamp 1. You will first use a helium lamp for calibration. Turn off the power supply for spectral lamps and plug in it. Insert a He lamp to the lamp holder. Conceal the lamp holder with a safety guard. Connect the cord of the lamp holder to the power supply tightly. Check whether the lamp works well by turning on the power supply. If the lamp doesn t work well, repeat turning off and turning on the power supply or pulling out and inserting the lamp in. If the lamp still doesn t work you can change it to a different one. 11

12 Aligning a spectrometer 1. Set up of the diffraction spectrometer is shown in Figure 4, 5, and 6. Fix the collimator and the telescope by screws. An eyepiece can be assembled with the telescope. The collimator and the telescope should be set up so that the slit part of the collimator and the eyepiece part of the telescope are heading outside. Small holes on the collimator and the telescope are for fixing them by screws so you should put them beside screws and tighten the screws. Make the collimator and the telescope horizontally by turning vertical screws. Align the telescope and the collimator in one line. 2. Locate the He lamp on the lamp holder close to the slit part of the collimator. You should put it close to the slit so that the light that doesn t come from the lamp passes the slit less. 3. Turn on the lamp and look the light that passes through the slit of collimator by the eyepiece of the telescope. By adjusting the slopes of the collimator and the telescope, the location of the telescope, the location and the height of the lamp, the width of the slit so that the image of the light passing through the slit seen by the telescope is clear and nice. 4. Rotate the observing part of the telescope so that one line of the cross hair of the telescope overlaps with the observed light. 5. Insert the diffraction grating to the diffraction grating support so that the letters on the grating are upside down or right side up. Put the diffraction grating between the collimator and the telescope. 6. Confirm whether the spectrum of the He lamp is observed by moving the telescope left or right. If the spectrum lines are not clear, adjust the slopes of the collimator and the telescope, the location of the telescope, the location and the height of the lamp, and the width of the slit. 7. Move the telescope so that the telescope and the collimator are aligned in one line again. The goniometer for this experiment has a ring-shaped scale and an arc-shaped scale. Match 0 of the arc-shaped scale with 0 of the ring-shaped scale. You can match them precisely by a rotatable magnifying glass. Measuring spectrum 1. For a He lamp, move a telescope left or light so that each spectrum lines of He overlaps with one line of a cross hair of the telescope. Measure the angles for each spectrum lines by a Vernier scale of goniometer and write down the colors and the angles. 12

13 2. For a Na lamp, repeat the procedure of measuring the angles of its spectrum lines. (Confirm whether the sodium D-line is observed.) 3. For a Hg lamp, repeat the procedure of measuring the angles of its spectrum lines. Data Evaluation 1. For helium color Red Yellow Green Greenish blue Bluish green Blue Angle 2. For sodium color angle 3. For mercury color angle 13

14 Post-Laboratory Data Evaluation 1. Red Yellow Green Greenish blue Bluish green Blue Table 1. Wavelength of the He spectrum 667.8nm 587.6nm 501.6nm 492.2nm 471.3nm 447.1nm Draw a calibration curve of the diffraction spectrometer (sinθ versus λ) for the first order (n=1) and measured angles θ using Table 1. Determine the constant of the grating d using the formula nλ =2dsinθ. 2. Determine the wavelength of the spectrum of Na. color Wavelength (nm) Transition Ex) 3 1 D 2 1 P 3. Determine the fine structure splitting of Na (Sodium D-line) 4. Determine the wavelength of the spectrum of Hg. color Wavelength (nm) Transition Ex) 6 1 D P 1 5. Discuss about the similarity and difference between He and Hg spectra. 14

15 References 1. Fine structure and one-electron spectrum experimental guide(phywe) 2. Two-electron spectra with the prism spectrometer experimental guide(phywe) 3. Illinois state university lecture note Atomic term symbols and energy splitting 4. Fayetteville state university lecture note Spin-orbit coupling 5. KAIST physical chemistry 1 lecture notes Chapter 8 Multielectron atoms, Chapter 13 Molecular Spectrocopy 6. G. Herzberg, Atomic Spectra and Atomic Structure (Dover Publ.) 7. D.R. Bates, Quantum Theory II (Academic Press Inc.) 15