Atomic fluorescence 1. Introduction Transitions in multi-electron atoms Energy levels of the single-electron hydrogen atom are well-described by EE nn = RR nn2, where RR = 13.6 eeee is the Rydberg constant. Radiative transitions between initial and final levels nn ii > nn ff have energies ωω iiii = RR 1 nn ff 2 1 nn ii 2. Exercise: check that the first lines of the Balmer series (nn ff = 2) are HH αα = 6563 AA and HH ββ = 4861 AA (note: 124 eeee corresponds to λλ = 1 nnnn). States of a multi-electron atom can be obtained in a first approximation by adding electrons one-by-one in energy levels of the hydrogen atom following the Pauli principle (no two electrons in one state). This gives, for instance, the configuration 1ss 2 2ss 2 2pp 3 for the ground state of the ZZ = 7 element (nitrogen). This approach has two main problems: it neglects electron -- electron interactions and it does not consider the permutation symmetry of indistinguishable particles in the multi-electron quantum state. It is often illustrated with electron orbits or orbitals and is not recommended because it must be modified with Slater determinants to add the required permutation symmetry. One is also more interested in the energy levels rather than in the states. There is no simple result for EE nn and ωω iiii in the case of multi-electron atoms, similar to that for hydrogen. Several models (Hartree, Thomas-Fermi, Hartree Fock, Ritz variational) can be applied to calculate the energy levels in these cases. In practice, the permutation symmetry also implies that we should assign quantum numbers to multi-electron states (the entire atom), rather than to its individual electrons. Then, we have one atomic SS, LL, JJ instead of the many electronic ss, ll, jj. Selection rules on the atom quantum numbers apply equally well in multi-electron atoms in the dipole approximation: the parity change Δππ = ±1, ΔSS =, ΔLL = ±1 and ΔJJ =, ±1, even though in these cases the spatial representations of the initial and final states are not known (in contrast, for the hydrogen atom we could check the selection rules directly by doing the integrals in the transition matrix element). The intensity of a transition line can be described with a transition probability inversely proportional to the radiative lifetime AA kkkk 1 ττ, or with the oscillator strength ff iiii = 2mmωω iiii xx iiii 2. These must be supplemented in practice by the statistical weights gg ii,kk = 2JJ ii,kk + 1 of the two transition levels, accounting for the number of states with different mm JJ and the same SS, LL, JJ. The spontaneous emission of this experiment is different from the stimulated emission and a complete derivation requires a quantized light field. An expression for it can be obtained with the Einstein rate equations for blackbody radiation modified to our case (narrow lines) or with a Page 1
classical analogy. The result for the hydrogen 2pp 1ss transition is 1 ττ ssss = ee2 ωω 3 3ππħcc 3 xx 12 2. Typical values for transitions in the visible are 1 nnnn (exercise: check this), but the strong ωω 3 dependence reduces the emission rate for small energy difference. 2. Setup Fig.1 Neutral helium ( He I ) and neutral mercury ( Hg I ) plasma emission spectra The atomic fluorescence spectrum can be measured either in emission or absorption, depending on whether the atom emits or absorbs light on transition. In practice, transition lines were first observed in absorption in the spectrum of the Sun. This experiment is in emission, with radiative transitions from higher to lower states. The light source is a spectrum tube filled with He (2 electrons, when neutral) or Hg (8 electrons, when neutral) vapor. The higher states that give the radiative glow are excited with an electrical discharge in the tube [Fig. 2(a)]. A narrow central capillary between the electrodes lights up when power is turned on, with atoms ionized in a plasma and a ma-range current flowing between electrodes kept at a kv-range potential difference. Note: keep the tube on during measurements only to increase its lifetime. Note: keep fingers away from electrodes when in use. Note: an arc discharge also can occur when a current is passing through a gas. This is related to emission of electrons from the electrodes and is strongest near the electrodes. In contrast, the light in our case is related to electrons emitted from the gas atoms and it is strongest away from the electrodes, inside the narrow capillary tube, where the electron current density is larger. The large-area electrodes are chosen on purpose in these spectrum tubes, to suppress the irregular arc discharge. (b) Page 2
Observe lines for He and Hg with the grating glasses [the two gratings have 1 lines/mm (purple one) and 5 lines/mm (yellow one)] and compare to the plasma emission lines in Fig. 1. Note how the lines are clustered in the 1 st and 2 nd diffraction orders. We can easily see the Hg yellow doublet, separated by 2 AA, with the 1 lines/mm grating. The resolution of this grating is at least Δλλ = 2 = Δkk 2kk Δkk 2 λλ 576 kk 576 2, ccmm 1 576 7 ccmm 1. Note: spectroscopy is done in momentum kk space. Because of this, the resolution is written in momentum units ccmm 1, rather than in distance units AA or nnnn [865 ccmm 1 = 1 eeee] Consulting the NIST Atomic Database (next section) reveals more details about the transitions. The strongest mercury lines are (no red lines gives a bluish light overall): 436 nm (blue) line: 6s7s, 3 SS1 6ss6pp, 3 PP1 546 nm (green) line: 6s7s, 3 SS1 6ss6pp, 3 PP2 577 nm (yellow) line: (6ss6dd, 3 DD2 6ss6pp, 1 PP1 ) 579 nm (yellow) line: (6ss6dd, 1 DD2 6ss6pp, 1 PP1 ) An "oo" superscript means that the level has odd parity, no superscript means even parity Note: 1 DD is a term, 1 DD2 is a level, 1 DD2,mm JJ =2 is a state The two strongest helium lines are: 587 nm (yellow) (several transitions betweeen the terms 1ss3dd, DD 1ss2pp, PP 668 nm (red) (1ss3dd, DD2 1ss2pp, PP1 ) 3 ) 3 PP 1 3 PP 2 3 II 2 3DD 2 1DD 2 1SS 3SS 1 1 PP 1 3 PP 2 3 PP 1 3 PP Page 3 Fig. 2: Grotrian diagrams illustrate the ΔLL = ±1 selection rule very well. An example with the strongest (AA > 1 1 7 ss 1 ) transitions in the visible range for neutral mercury
Note: there must be ionized He-II and Hg-II in the spectrum tubes when they are on, since the atoms are ionized, with free electrons flowing between electrodes. However, one can verify with the NIST database that the intensity of their lines is much smaller than of He-I and Hg-I. Marker 2SS+1 LLLJJJ Electrical discharge Spontaneous Emission glow Spectrum tube θθ θθθ 2SS+1 LLJJ (a) (b) Photodiode Fig. 3: (a) A transition between different levels. (b) Top view of the experimental geometry. 3. Experiments Measuring transition line wavelengths In this part, you will measure the wavelength of one emission line Use the black clamp as a marker to read the angles as you rotate the table (an accuracy of 1 is achievable). Look from the top to find the grating angle θθ Choose a spectrum tube and one intense line Rotate the table until the photodiode gives a peak intensity Calculate the angle θθ (note that this angle is defined in Fig. 3(b) with respect to the normal to the grating) The wavelength can be calculated from λλ = dd(sin θθ + sin θθ ) [exercise: derive this equation]. The result should be within 1 2% of the listed value Choose a different grating angle θθ, obtain the new θθ and confirm that the result for the wavelength λλ is the same Optional: measure other lines from a different spectrum tube or with a different grating 4. The NIST Atomic Spectra Database Open Explorer (a Java program you will need later does not currently work with Firefox), go to www.physics.nist.gov, select Lines, enter the element (He I is neutral helium, He II is singly-ionized helium, etc.) and the wavelength interval at the top Page 4
You can generate PDF files of the plasma emission spectra (Fig. 1) [different electron temperatures TT ee will give slightly different relative intensities, but the same lines and overall spectrum] and transition Grotrian diagrams (Fig. 2). Grant permission to NIST Java applications, if a warning appears when generating diagrams. If there are too many lines, you can select only those with a minimum value for AA. The vertical scale is the energy in ccmm 1 (divide by 865 to obtain the energy in ev), and can be narrowed with the Zoom feature. Clicking Retrieve Data gives a table with transition lines energy, intensity, upper and lower level configurations and quantum numbers. Tables with the transitions in the visible for He-I and Hg-I can be applied to illustrate exchange energy and fine structure. 4.1. Exchange energy in helium The exchange energy is the energy difference between levels with the same LL, JJ and different SS. In helium, examples are 1ss2pp, 3 PP1 (2.96 ev) and 1ss2pp, 1 PP1 (21.22 ev). The triplet state, with the larger spin (SS = 1) is lower in energy than the singlet state (SS = ), or spins on the same site orient parallel to each other in the ground state (also called the Hund s 2 nd rule). This is because the spatial part is anti-symmetric, to minimize the electrostatic interaction VV eeee between the two electrons, so the spin part must be symmetric. 4.2. Fine structure in mercury The fine structure is the energy difference between levels with the same LL, SS and different JJ. It is easier to see in Hg because of its magnitude is ZZ 2. For instance, the 3 DD term is split into 2SS + 1 = 3 levels 3 DDJJ=1,2,3. In practice, since the 1 DD2 and 3 DD1 levels happen to be very close, the yellow doublet in Hg you observed, separated by 2 AA, is a good indication of the fine structure splitting. Exercise: obtain the 3 DDJJ=1,2,3 energies and check the Lande interval rule ΔEE JJ,JJ 1 = AAAA (one of the few simple general results for the lines of multi-electron atoms). 4.3. Intercombination transitions in mercury The combination 2SS + 1 that appears to the upper left in a term notation is called the multiplicity, because it indicates into how many levels a term is split. Selecting the term multiplicity feature will give the singlet level Grotrian diagram SS = if "" is selected, the triplet levels if "1" is selected, and so on. This separation of diagrams is usually advantageous because the selection rule ΔSS = implies that transitions will not occur between levels on diagrams with different SS. Page 5
This ΔSS = selection rule does not always hold. For instance, the strong Hg-I transition at 579 nm (one of the yellow doublet lines) has ΔLL = 1, ΔJJ = 1 and ΔSS = 1. This spin-flip (does not obey the ΔSS = selection rule) transition is an electric dipole intercombination transition that can occur when the LLLL coupling in not entirely justified and labelling states with LL and SS quantum numbers does not work as well as in lighter elements. Note: other spin-flip transitions with ΔSS = ±1 are due to the magnetic dipole interactions and, in contrast to the electric dipole transitions, satisfy the ΔLL = selection rule [exercise: show this]. 5. Conclusion The detailed spectrum of an element is its fingerprint and spectroscopic measurements are often applied to determine the composition of a sample. For instance, X-ray resonance fluorescence, inelastic Raman and resonance fluorescence light scattering have evolved into powerful tools in the experimental studies of materials. Page 6
Name Phys-62 Quantum Mechanics Laboratory Atomic fluorescence lab report Dates of measurements: Page 7