Magnetic Moment
Orbital Magnetic Dipole Moment and Zeeman Effect
Currents in a wire loop create a magnetic dipole field perpendicular to the plane of the loop. Surely, the same is true for a charged particle in a circular orbit.
Fig. 1 electron on Bohr orbit
Figure 1 shows Bohr orbit and we can consider electron mobbing with velocity (v) in a circular Bohr orbit or radius r, Produces a current,
(1)
where T is the orbital period of the electron. In this case, current loop produces a magnetic field, with a moment
(2)
where i is the current and A is the area of the loop. In the Figure 1, the orbital angular momentum expresses as below.
(3)
Therefore, combine eq.1 and eq.2, we can obtain,
(4)
Divide each term by equation3, then
(5)
In the above equation, μl/L would be a combination of constants. We customarily wrote this ratio as
(6)
The ratio of the magnetic moment and of the angular momentum is the gyromagnetic ratio, gl, which is always g=e/2m for a rotating charged particle. The fixed gyromagnetic ratio tells one that rotation is always accompanied with angular momentum and magnetic moment at the same ratio. In the quantum mechanics, the angular momentum is quantized. The quanta of angular momentum (at least of the z-component) are integer multiples of ħ. So, the combination that always appears in quantum physics is the product of the gyromagnetic ratio and ħ, called the Bohr magneton.
(7)
This could be figured out from dimensional considerations (up to the multiplier 1/2) if we try to find quantities of the dimension of a magnetic diplole moment constructed from constants characterizing the electron e and m, and universal constants, ħ, c.
The presence of a magnetic moment for nonzero angular momentum has important consequences on the behavior of atoms in magnetic field. First of all a dipole has a nonzero energy in magnetic field. The magnetic energy of a dipole is
(8)
It can be shown from electromagnetic theory that the energy of a magnetic dipole in an external magnetic field B is,
(9)
where ml is the magnetic quantum number and wL is the Larmor Frequency. In the equation 9, ωL would be,
(10)
Equation 9 shows that the magnetic energy of an atomic electron depends on the magnetic quantum number ml and hence, is quantized. The total energy of this electron is the magnetic energy U plus the energy it had in the absence of an applied field (E0). So we have,
(11)
For atomic hydrogen, E0 depends only on the principal quantum number n. We can see that in presence of the magnetic field the energy levels are no longer degenerate. For example, for the first excited state n=2, l=1, ml can take three values -1, 0, and 1. Thus the energies are
(12)
Therefore, as shown in the figure 2, the single line will split into three equally spaced.
Figure 2. The spectrum with and without magnetic field
We call the phenomenon splitting of energy levels of magnetic field as the Zeeman Effect. Sometimes, Zeeman Effect can be used to measure the magnetic field in stars. The Zeeman Effect is the reason why ml is called the magnetic quantum number. In fact, after the invention of quantum mechanics it became soon evident that the observed splitting of lines, though similar to the ones predicted, frequently differed from the prediction considerably. This is the anomalous Zeeman effect which is due to the existence of the electron spin
Larmor Precession
The Larmor Frequency in the equation 10 has an additional significance. A dipole in a magnetic field is acted upon by the torque,
(13)
According to the laws of dynamics the angular momentum will change as
(14)
Since μ is parallel to L the dL is perpendicular to L and B. In other words the magnitude of L will be unchanged but its direction will be keep changing describing a0rotating motion around the direction of B. This is called precession. The frequency of this precession is exactly the Larmor Frequency as
(15)
Now |L|sinθ=|L_prep | just the rotating, perpendicular (to the B axis) component of L. Since Lz is unchanged we can write,
(16)
Now in a rotation motion, the |dL|sinθ=dϕ|L_prep |, so we can obtain,
(17)
The frequency of procession in equation 17 is equal to the Larmor Frequency in equation 10.
Electron Spin
Discrepancy with the observed and predicted Zeeman Effect leads to the discovery of electron spin by Pauli. It is not tied to any rotational motion but rather it is an intrinsic property. This angular momentum is called the spin. According to this, electrons not only possess magnetic moment due to orbital motion but also possess magnetic moment due to spin. The spin magnetic moment for a charged particle (of charge q) is,
(18)
g is called is the Lande g-factor. For electrons it can be shown from relativistic quantum mechanics that the value of g is slightly greater than 2. So the total magnetic moment is the sum of the orbital and spin moments:
(19)
For electrons, it is
(20)
The existence of a spin moment of the electron was first demonostrated in a classic experiment by O. Stern and W. Gerlach. In their experiment, a beam of neutral silver atoms was passed through a non-uniform magnetic field (created between the poles of a larger magnet - see figure below). The beam was detected by being deposited on a glass collector plate. A non-uniform magnetic field exerts a force on any magnetic moment, so that each atom is deflected in the gap by an amount governed by the orientation of its moment with respect to the direction of inhomogeneity (the z axis). Thus, the atomic beam should split into a number of discrete components, one for each distinct moment orientation present in the beam. In the Stern-Gerlach experiment, it was observed that the silver atomic beam was clearly split into two components. On the other hand the silver atoms in their ground state have zero angular momentum (l=0) as outermost electron is in the s-state. The splitting of the beam in two cannot be explained from the space quantization where one would predict the beam to split in 2l+1 = 1 component. To explain this, we have to introduce the concept of spin and the spin quantum number ms.
References
http://hyperphysics.phy-astr.gsu.edu/Hbase/magnetic/larmor.html
http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/orbmag.html
en.wikipedia.org/wiki/Magnetic_moment
en.wikipedia.org/wiki/Larmor_precession
Robert Eisberg, Robert Resnick, "Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles", John Wiley & Sons, New York
Raymond A. Serway, Clement J. Moses, and Curt A. Moyer, "Modern Physics", Saunders College Publishing
