Introduction

In 1896, it was observed by the Zeeman (the Dutch physicist Pieter Zeeman) that, when an atom is placed in an external magnetic field, and then it excited, the spectral lines emits in the deexcitation process are split into several components. These several components were determined by their frequencies.

SPC’s for Zeeman Effect

The Supporting Physical Concepts (SPCs) for the Zeeman Effect are given below to understand the topic;

  1. Basic atomic structure and nomenclature
  2. Atomic dipole moment
  3. The transition of an electron in different states and selection rule
  4. Interaction of atomic dipole moment with the external magnetic field
  5. A contribution of this interaction in the total energy of the atom
  6. The emitted radiation interpretation in terms of frequencies

These are basic concepts which are required to explain and understand the several spectral lines in the magnetic field from the excited atom.

Types of Zeeman Effect

The Zeeman effect is of two types one is called normal Zeeman effect while the second one is called Anomalous Zeeman effect.

Normal Zeeman effect

In general, we observe single line emission from the excited state to the ground state transition of electrons in the absence of an external magnetic field. But when this atom is placed in the magnetic field, this was observed;

  1. two lines when views parallel to the magnetic field; these two lines are known as sigma lines
  2. three lines when views perpendicular to the magnetic field; the central line is called pi line while the two extreme lines are called sigma lines

Normal Zeeman effect we can observe only when the external magnetic field is strong (in Tesla) as compared to the internal magnetic field of the atom.

Anomalous Zeeman effect

If the applied external magnetic field is weak than the atom’s magnetic field then spectral line split into several components.

Classical Theory of Zeeman effect

Normal Zeeman effect explained in a simple way by Lorentz by using the Bohr’s atomic model. The complete description of this theory you can watch through this video discussed in Hindi and English. The classical theory of Zeeman Effect Part-3.

 

Quantum Theory of Zeeman Effect

In this section, we will first consider about to the atomic dipole moment and then its interaction with the magnetic field. The first question to you, have you an idea about to the magnetic dipole moment and its physical picture if yes, great if not then first you understand this concept because it will not help you not only for this topic but you will understand many more other topics too.

As you know that the Zeeman effect is very important in applications such as nuclear magnetic resonance spectroscopy, electron spin resonance spectroscopy, magnetic resonance imaging (MRI) and Mössbauer spectroscopy. So it is must for the understanding of any such phenomenon.

Watch also: Link of the video for ATOMIC DIPOLE MOMENT

So the magnetic moment of the atom in a uniform magnetic field

\dpi{120} \fn_cm \large \vec{\mu_{l}}=-\frac{e}{2m}\vec{L}                                ———–(1)

Here, mu l is the atomic dipole moment due to the orbital motion and vector L is the orbital angular momentum of the electron. The ‘e’ is the charge and m is the mass of the electron. the negative sign indicates the direction of the atomic dipole moment and orbital angular momentum, which is opposite to each other.

When an atom is placed in the magnetic field B, the torque acting on it defined by;

\dpi{120} \fn_cm \large \vec{\tau}=\vec\mu_{l}\times\vec{B}\\ \tau=\mu_{l}B Sin\theta                                ———-(2)

So it means the magnetic field works on the atomic dipole to rotate it. As you know this work done by the field on the system will store in terms of potential energy. That we have to find by rotation of this dipole from pi by 2 to the angle theta. This can be written as

\dpi{120} \fn_cm \large U=\int_{\pi/2}^{\theta}\tau d\theta\\ = \mu_{l}B\int_{\pi/2}^{\theta}\sin\theta d\theta=-\mu_{l}B\left [ \cos\theta \right ]_{\pi/2}^{\theta}\\ =-\mu_{l}B \cos\theta=-\vec{\mu_{l}}.\vec{B}

Again, as you know in the theoretical derivations we always say that a magnetic field is applied in the Z-direction, then the potential energy can be written as,

\dpi{120} \fn_cm \large U=-\vec{\mu_{l}}.\vec{B}\\ \vec{\mu_{l}}=-\frac{e}{2m}\vec{L}\\ U=-[-\frac{e}{2m}\vec{L}].\vec{B}\\ U=\frac{eB}{2m}.L_{z}\\ U=\frac{eB}{2m}m_{l}\hbar \Rightarrow \because L_{z}=m_{l}\hbar\\ U=\frac{e\hbar}{2m}m_{l}B\\ U=\mu_{B}m_{l}B

So the potential energy stored into the atom is U described above and it is additional. This potential energy will increase the total energy of the atom and hence a change will be observed in the ground state and excited state by this factor.

Watch the video here  Quantum theory of ZEEMAN EFFECT Part-4

How we are introducing this energy and how it plays the role when electron transition takes place from excited state to the lower energy state. Just observe it by this relation introducing two cases one without the magnetic field and the second with a magnetic field.

In absence of the magnetic field when an electron jumps from high energy state to the lower energy state, the frequency of the emitted spectral line is given by the relation,

\dpi{120} \fn_cm \large \nu_{0}=\frac{E_{2}-E_{1}}{h}\\

If the atom is placed in a magnetic field then the total energy of the atom is

\dpi{120} \fn_cm \large E^{'}=E+m_{l}\mu_{B}B\\ so\\ E_{f}=E_{2}+m_{l}^{f}\mu_{B}B\\ E_{i}=E_{1}+m_{l}^{i}\mu_{B}B\\

Now the transition of an electron from high energy state to the lower energy state,

\dpi{120} \fn_cm \large \nu=\frac{E_{f}-E_{i}}{h}=\frac{E_{2}-E_{1}}{h}+\frac{m_{l}^{f}-m_{l}^{i}}{h}\mu_{B}B\\ \nu=\nu_{0}+\frac{\Delta m_{l}}{h} \mu_{B}B=\nu_{0}+\Delta m_{l} \frac{e \hbar}{2 m h}B\\ \nu=\nu_{0}+\Delta m_{l}\frac{eB}{4 \pi m}\\ \because \frac{e\hbar}{2mh}=\frac{e}{4\pi m} \\

 

Zeeman effect splitting

According to the selection rule, you can see the transition between different states in the absence and presence of an external magnetic field in the above picture. Now we have to interpret the splitting of spectral lines in terms of the frequency.

There are three cases,

(i) The frequency of line corresponding to

\dpi{120} \fn_cm \large \Delta m_{l}=0 \\ \nu=\nu_{0}

 

(ii) The frequency of line corresponding to

\dpi{120} \fn_cm \large \Delta m_{l}=+1 \\ \nu=\nu_{0}+\frac{eB}{4\pi m}\\

(iii) The frequency of line corresponding to,

\dpi{120} \fn_cm \large \Delta m_{l}=-1 \\ \nu=\nu_{0}-\frac{eB}{4\pi m}\\

Zeeman effect frequency difference

This is the quantum theory of Normal Zeeman effect.