The Zeeman Effect in Atomic Mercury Taryl Kirk 2001 Introduction A state with a well defined quantum number breaks up into several sub states when the atom is in a magnetic field The final energies are slightly more or slightly less than the energy of the state in the absence of the magnetic field This phenomena leads to a splitting of individual lines into separate lines when atoms radiate in a magnetic field with the spacing of the lines dependent on the magnitude of the magnetic field The splitting up of these spectral lines of atoms within the magnetic field is called the Zeeman Effect Abstract The neutral mercury Hg atom in its ground state has 80 electrons in the configuration 1s22s22p63s23p63d104s24p64d104f145s25p65d106s2 where 2S 1L J notation is used in which the n 1 2 3 and 5 electronic energy levels are completely filled The optical emission spectrum of Hg results from transitions of the two valence electrons between various excited two electron configurations The Hg spectrum therefore has many features in common with the two electron helium system In a helium like system the total angular momentum J of the atom i s determined solely by the total angular momentum to the two valence electrons since the orbital and intrinsic spin angular momenta of the electrons in the closed shell inert core are coupled to zero In the Russell Sauders or LS coupling scheme the l orbital angular momentum quantum numbers l1 and l2 of the two valence electrons are coupled to form a resultant angular momentum quantum number L and similarly the intrinsic spin angular momentum quantum numbers S1 and S2 are coupled to resultant intrinsic spin angular momentum quantum number S When the conditions for the LS coupling approximation are satisfied the operators L and S commute with Hamiltonian operator H for the atomic system and the allowed energy levels may be labeled directly in terms to the angular momentum quantum numbers L and S but not the individual quantum numbers l1 l2 S1 and S2 The total angular momentum operator J also commutes with H and therefore the total angular momentum J L S may also be sued to label atomic energy levels The angular momentum addition theorem restricts the possible values of an angular momentum quantum number L resulting from the sum of two individual angular momenta l1 and l2 A similar restriction governs the sum of S1 and S2 to form S and the sum of L and S to form J These angular momentum restrictions may be used to predict the quantum numbers of the low lying excited states of the neutral Hg system If one considers only single electron excitations of the 6s2 ground state the lowest configurations should be 6s6p 6s6d 6s 7s 6s 7p and 6s7d For a two electron system S1 1 2 and S2 1 2 So that the total intrinsic angular momentum quantum number S of the atom is limited to the values 0 and 1 corresponding to what are called singlet and triplet terms respectively The electric dipole selection rules allow transitions that involve only the following changes 1 2 3 4 DS 0 DL 1 0 DJ 0 1 but not J 0 J 0 DMJ 0 1 but not MJ 0 MJ 0 when D J 0 Figure 1 Geometry of the Zeeman effect On the left the total dipole moment m precesses around the total angular momentum J On the right J precesses much more slowly about the magnetic field The total magnetic dipole moment of the electron is 1 2 b h L 2S 1 where b is the Bohr Magneton Because of the difference in the orbital and spin gyromagnetic ratios of the electron the total magnetic dipole moment is not in general parallel to J L S 2 So as L and S precess about J the total dipole moment also precesses about J Assuming the external field to be in the z direction this field causes J to precess about the z axis If the external field is much weaker that 1 Tesla 10 000 Gauss then the precession of J around the z axis will take place much more slowly that the precession of around J The Hamiltonian of the Zeeman effect is DHz B BB 3 where B is the projection of the dipole moment onto the direction of the field the zaxis Because of the difference in the precession rates it is reasonable to evaluate b by first evaluating the projection of onto J called J and then evaluating the projection of this onto B thus giving some average projection of onto B First the projection of onto J is J J J b h L 2S L S J 4 Then B J J B J B J Jz J b h L 2S L S Jz J2 5 Evaluating the dot product using again that J2 L2 S2 2L S this becomes B b h 3J2 S2 L2 Jz 2J2 6 So when first order perturbation theory is applied the energy shift is DEz bB g mJ 7 g 1 j j 1 s s 1 l l 1 2j j 1 8 where is called the Land g factor for the particular state being considered Note that if S 0 then j 1 so g 1 and if l 0 j s so g 2 The Land g factor thus gives some effective gyromagnetic ratio for the electron when the total dipole moment is partially from the orbital angular momentum and partially from the spin From equation 8 it can be seen that the energy shift caused by the Zeeman effect is linear in B and mj so for a set of states with particular values of n l and j the individual states with different mj will be equally spaced in energy separate by bBg However the spacing will in general be different for a set of states with different n l and j due to the difference in the Land g factor Figure 2 Diagram of the set up for experiment Procedure 1 Be sure to remove the Hg Pen Ray lamp from between the electromagnet pole pieces before turning on the magnet power supply Note This is to assure that the Hg lamp is not crushed by the pole pieces as they are drawn together when the magnet current is turned on 2 Turn the electromagnet power supply and advance the magnet current to 20 amps With the current at 20 amps determine if the Hg lamp can be placed between the pole tips If the lamp cannot be placed between the pole pieces you will have to remove the Hg lamp turn the magnet current down to zero and rotate the pole piece adjustment so that the pole pieces are farther apart With the Hg lamp removed turn the magnet back up to 20 amps See if the Hg lamp can now be put between the pole pieces 3 Measure the magnetic field at the center of the region between the pole tips using the Hall effect guassmeter Note The Hall effect guassmeter should be oriented perpendicular to the …