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I'm reading a standard modern physics history book ("Inward Bound" by A. Pais), and I realized I don't really understand Zeeman splitting well. In the section I'm reading, there's a short discussion of the $D_1$ and $D_2$ spectral lines in sodium, which correspond to the $2P_{1/2}\rightarrow 2S_{1/2}$ and $2P_{3/2}\rightarrow 2S_{1/2}$ transitions in sodium respectively. It goes on to say that,in the presence of an external magnetic field, the $D_1$ line splits into $2\times 2=4$ components and that the $D_2$ line splits into $4\times 2-2=6$ components. I would like a detailed explanation for these splitting rules. If you can point to a helpful piece of standard literature, please do so.

First off, in modern notation the $D_1$ and $D_2$ lines correspond to the $3P_{1/2}\rightarrow 3S_{1/2}$ and $3P_{3/2}\rightarrow 3S_{1/2}$ respectively. Two's were used because those are the numbers that would be showing up in the Balmer formula (for hydrogen).

Second, whenever I try to explain those numbers I think about spin-orbit interaction and normal Zeeman splitting, but I can't seem to come up with a coherent (and physical) explanation using just these two phenomena (which I know should be all I need).

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In addition to the answer by hsinghal it is worth point out some historical notational quirks.
An expression such as $^2P_{3/2}$ is called a Term Symbol. The superscript is the multiplicity of the electron spins, i.e. 2$S$+1 for total spin S. The capital letter, P in this example is the total orbital angular momentum and has letters and values of S=0, P=1, D=2, etc (as for atomic orbitals) and the subscript is the total spin and orbital angular momentum given by J. Thus the term symbol is $^{2S+1}L_J$

It should also be noted that when calculating values such as $L= |l_1−l_2| $ to $|l_1+l_2|$ the values are taken in unit steps, i.e. each values differs by 1 from the next until no more can fit with the formula. Sometimes there is only one value.

The origin of the Zeeman effect is in the magnetic field produced by a 'spinning' charge. In the absence of an external field the sub-levels of J (2$J$+1 of them) have the same energy. In an external magnetic field the magnetic field caused by spin interacts with the external field and levels split. (The hamiltonian is $H=-\mu . B$, where $\mu$ is the magnetic dipole of the atom and is directly proportional to J, and B is the external magnetic field strength). You can use Hund's Rules to decide which state is highest or lowest. See chapter 2 in 'Molecules and Radiation' by J. Steinfeld. The figure shows an example of a $p^2$ configuration. Russel-saunders


Nowadays the Zeeman effect is tremendously important, not in atomic spectroscopy but in nuclear magnetic resonance (NMR). This is widely used in Chemistry to determine the structure of molecules and even of proteins when in solution. NMR is the basis of MRI imaging widely used in hospitals. In this case it is the nuclear Zeeman effect, usually using spin 1/2 protons, but also very many other nuclei with integer and half integer spin can be used. See 'Spin Dynamics' M. Levitt.

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As you know that zeeman splitting is due to the phenomena known as spatial quantization. i.e. if there is a fixed or preferred direction in the space (i.e. symmetry of the space is broken by the electric or magnetic field) then the atom can not assume arbitrary orientation. This orientation depends on the angular momentum of the atomic/spectroscopic state. For example the $^2S$ spectroscopic levels have $0$ orbital angular momentum and $\frac{1}{2}$ spin angular momentum hence it can assume only two states (corresponding to magnetic quantum number $m=-\frac{1}{2}$ and $+\frac{1}{2}$.

There can be two types of coupling i.e. LS coupling and jj coupling. These coupling arises due multi-electronic interaction and tells us how the spin and orbital angular momentum of different electrons are interacting with each other. In LS coupling the orbital angular momentum of the different electrons interact with each other to give the final angular momenta and the spin angular momentum will interact with each other to give final spin momenta. Then the final angular momentum is obtained from these final orbital and spin momenta. In the case of two electrons with momenta $(l_1,s_1)$ and $(l_2,s_2)$ (of course $s_i$ are just $\frac{1}{2})$

L= $|l_1-l_2|$ to $|l_1+l_2|$

S=$|s_1-s_2|$ to$|s_1+s_2|$

Final momentum

J=$|L-S|$ to $|L+S|$

similarly in jj coupling first the $L_i$ and $s_i$ combine to make $j_i$ and then these $j_i$ combine to make final $J$.

in a magnetic field an atom can attain $m$ spatial states where

$m=-J$ to $+J$.

Hence you will see splitting in the spectral lines, this splitting increases with the increase in magnetic field.

It may be noted here that if the magnetic field is very strong the LS or jj coupling might break and you will see change in splitting. This is known as Paschen-Back effect.

I would like to add here that the terms written in your question $^2P_{3/2}$ are known as spectroscopic terms and calculated on the basis of LS coupling.

For entry level knowledge please consult the book "Introduction to Atomic Spectra" by H.E. White. You may find some portions of this book little difficult. The zeeman effect is described in chapter X. To understand that I suggest you to read earlier chapters.

The understanding of the atomic spectra is a little laborious job and take some effort.

EDIT: To explain sodium D lines

The sodium D lines are from transitions $3^2P_{3/2}\rightarrow3^2S_{1/2}$ ($D_2$) and $3^2P_{1/2}\rightarrow3^2S_{1/2}$ ($D_1$)

Now under the action of magnetic field the upper level of $D_2$ line is split into 4 levels corresponding to m$=\frac{3}{2},\frac{1}{2},-\frac{1}{2},-\frac{3}{2}$ and lower level is split into two levels corresponding to m$=\frac{1}{2},-\frac{1}{2}$. The selection rule require $\Delta$m=$\pm1,0$ and hence you will see transition from m=$\frac{3}{2}\rightarrow\frac{1}{2},\frac{1}{2}\rightarrow\frac{1}{2},\frac{1}{2}\rightarrow-\frac{1}{2},-\frac{1}{2}\rightarrow\frac{1}{2},-\frac{1}{2}\rightarrow-\frac{1}{2}$ and $-\frac{3}{2}\rightarrow-\frac{1}{2}$, simmilarly you can calculate 4 components of $D_1$ line.

I hope this will help

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