Zeeman Effect in Mercury The goal in this experiment is to measure the shift of atomic energy levels due to an external magnetic field. This effect is known as the Zeeman effect and was first observed by the Dutch physicist Peter Zeeman in 1896. Measurements of energy shifts are performed for the green line and both yellow lines in the mercury spectrum, and the qualitatively different results for the different lines are explained in terms of the properties of the initial and final states for each line. The results for the energy shifts are compared to theoretical predictions. 1 Background Many texts offer a detailed discussion of the Zeeman effect [1, 2, 4]. Here, we offer here only a brief overview of the theory, and encourage you to consult these texts or one of your own choosing for a more comprehensive treatment. Electrons in atoms occupy states with well-defined energies. When an electron transitions from a higher energy state to a lower energy state, a photon with energy equal to this difference is emitted by the atom. By analyzing the properties of the emitted photons, namely wavelength and type of polarization, one can learn about the states of the electron before and after the transition. For some atoms, application of a magnetic field will change the properties of the emitted light (such as its wavelength or polarization), which is in turn caused by changes in the initial and final states of the electrons in the atoms. Indeed, in the early development of quantum mechanics studying the effects of external perturbations (applied magnetic and electric fields, for example) on these transitions provided major clues leading to a detailed understanding of atomic structure. Atomic transitions were first studied in the optical (humanly visible) portion of the electromagnetic spectrum. Transitions were observed using an instrument called a spectrograph which separates the incoming light by wavelength. Transitions at different wavelengths appear as distinct vertical lines in the spectrograph view-field, hence the term “line” to denote a transition at a particular wave- length. With the apparatus used in this experiment (a Fabry-Perot interferometer), the different “lines” are actually observed as different sets of concentric circles of different radii. 1.1 Magnetic moments Magnetic moments are a key part of understanding the Zeeman effect, so let’s briefly review their basic properties. When an electrical current I circulates in a loop of wire with area A (= πa2), there is an associated magnetic moment ~µ of magnitude µ = IA , (1) and direction given by the right-hand rule, as shown in Fig. 1. If this loop is placed in an external magnetic field B~ , the loop will experience a torque ~τ ~τ = ~µ B~ . (2) × Work is done by the magnetic field on the loop when the orientation of the loop changes relative to that of the magnetic field, and the potential energy associated with the orientation of the loop is Emagnetic = ~µ B~ . (3) − · 1 Figure 1: A current loop and its associated magnetic moment. 1.2 Angular momentum and magnetic moments in the atom It is simplest to first consider an atom with only one electron outside the closed inner shells. An example of such “hydrogen-like” atoms is sodium. The optical spectrum of such an atom is due to transitions by the outer electron, and such an electron is called “optically active”. Later we will consider the case for more than one electron outside the closed shells, an example of such an atom being mercury, the subject of investigation in this experiment. 1.2.1 Orbital angular momentum and magnetic moment Each state that an electron can occupy in an atom is characterized not only by a well defined energy, but also by a specific angular momentum. There are two contributions to the angular momentum, first from the orbital motion of the electron about the nucleus, and second from the intrinsic angular momentum of the electron. The first is known as orbital angular momentum, and the second as spin angular momentum. Let us consider each one separately, and then see how they combine to form the total angular momentum. The orbital angular momentum of the electron can be thought of as due to circulation of the electron about the nucleus, and as such can be regarded as a current circulating about a loop. Just as in the macroscopic case discussed above, this circulation gives rise to a magnetic moment. The ~ relationship between the orbital angular momentum l and the associated magnetic moment ~µl is e ~µ = ~l , (4) l − 2m µ e ¶ where e is the magnitude of the electron charge, and me is the electron mass; the minus sign is due to the negative charge of the electron. Exercise (Optional) Derive Eq. (4) from Eq. (1) under the assumption that the current I is from a single circulating electron. This relationship is conventionally stated as e¯h ~l ~l ~µ = gl , = glµB , (5) l − 2m ¯h − ¯h · e ¸ Ã ! Ã ! where gl = 1 for a single optically active electron and e¯h/2me µB is known as the Bohr mag- ≡ neton. The dimensionless quantity gl is known as the “g factor” and is a measure of the ratio of the magnitude of the magnetic moment in units of µB to the magnitude of the orbital angular 2 momentum in units ofh ¯; in other words, gl is defined by Eq. (5). Since gl = 1 for a single optically active electron, having such a factor in the equation for µl may not seem to make much sense. But as we shall discover, g can take on values quite different from unity when we consider all the contributions to the angular momentum. Now let’s take a “quantum leap” over much important physical theory and simply state some results of quantum mechanics: 1. The variable l is known as the orbital angular momentum quantum number and can take on the values 0, 1, 2,... The magnitude of the orbital angular momentum is l(l + 1)¯h. 2. The projection of the orbital angular momentum along a particular axis isp quantized in units ofh ¯. When the atom is placed in an external magnetic field, as when observing the Zeeman effect, the axis of quantization is parallel to the field. By convention the magnetic field is taken to point in the +z direction. 3. The quantity ml is the quantum number specifying the projection of the orbital angular momentum along the z axis and can take on the values l,l 1,l 2,..., l, for a total of − − − 2l + 1 values. It follows that the possible values of the z component of the orbital angular momentum, lz, are just ml¯h. Figure 2 shows the situation for l = 2 , with the 5 possible values of ml (all magnitudes are in units ofh ¯). As an aside we note that quantum mechanics allows for only one component of the angular momentum to be specified exactly—the other two components have indefinite value and average to zero over time [2, p.258]. In our scheme the z component has a definite value, so the x and y components can be anywhere on the cone denoted by the dashed line. Since the magnetic moment and orbital angular momentum are proportional, the magnetic moment will also have quantized projections along the z axis. The question arises: what is the potential energy associated with a particular projection of the magnetic moment along the z axis? Let E(B~ ) be the energy resulting from the interaction of the atom with the external field B~ . When we turn on B~ (in the +z direction) the change in energy is ~ ~ E(B) E(0) = Emagnetic = ~µ B = µlz B , (6) − − l · − where µlz is the component of ~µl in the z direction. From Eq. (5) in terms of the z component alone, we obtain lz µlz = glµB = glµBml , (7) − ¯h − µ ¶ where again the negative sign comes from the charge of the electron. Thus, the energy change from the magnetic field is Emagnetic = ( glµBml)B = glmlµBB . (8) − − 1.2.2 Spin angular momentum and magnetic moment The treatment of the electron spin angular momentum, ~s, and spin magnetic moment ~µs is similar to the above treatment of orbital angular momentum and its magnetic moment. The quantity s is known as the spin quantum number and can take on only the numerical value of 1/2 when just one electron is involved. The magnitude of the spin angular momentum is s(s + 1)¯h. p 3 Figure 2: Angular momentum l = 2 where ~l = 2(2 + 1)¯h. The z-projections of ~l are in units of | | ¯h. p The projection of the spin angular momentum along a particular axis, again taken to be the z axis, is quantized in units ofh ¯, with possible values +¯h/2 and ¯h/2. Here, ms is the quantum number − specifying this projection and takes on the values +1/2 and 1/2. See Fig. 3. When the atom is − placed in an external magnetic field, as when observing the Zeeman effect, the axis of quantization is again parallel to the field. Figure 3: The spin angular momentum vector ~s for a single electron. The relation between the spin angular momentum ~s and the associated magnetic moment ~µs is similar to the relations for the orbital quantities: ~s ~µ = gsµB , (9) s − ¯h µ ¶ 4 and likewise, the energy shift for a spin placed in an external magnetic field B~ is Emagnetic = gsmsµBB . (10) Is gs = 1, just like gl? No. It turns out that gs 2, or more precisely, gs = 2.00232 ..