Zeeman effect

Zeeman effect:

The phenomen of splitting of spectral lines when an atomic source is placed inside magnetic field is known as zeeman effect. There are two types of effect depending upon strength of magnetic field and angular momentum of electron.

1. Normal zeeman effect

2. Anomalous zeeman effect.

Normal Zeeman effect:

The phenomena of splitting of spectral lines into 3-component lines inside strong magnetic field is known as normal Zeeman effect. Depending upon the direction of observation with respect to the direction of applied magnetic field it can be further divided into two categories.

i. Longitudinal normal Zeeman effect.

ii. Transverse normal Zeeman effect.

i. Longitudinal normal Zeeman effect:

When viewed along the direction of applied magnetic field a single line is found to split into two lines. This is known as Longitudinal normal Zeeman effect.

ii. Transverse normal Zeeman effect:

When viewed perpendicular to the direction of applied magnetic field a single line is found to split into 3-component lines. This is known as transeverse normal zeeman effect.

Normal zeeman effect can be explained by classical physcis

Experimental set up for Zeeman effect:

The experimental setup for normal Zeeman effect is shown in above figure. It consists of two cylindrical pole pices of electromagnet NS drilled along it's axis is connected to external current source. For the longitudinal Zeeman effect spectrograph \(S_2\) is used along the axis of magnet. For the observation of transverse Zeeman effect spectrograph \(S_1\) is used perpendicular to direction of magnetic field. With \(s_2\) two circularly polarised component are observed as in fig (ii). With \(s_1\) three-component are observed as in fig.

Classical explanation of normal Zeeman effect:

According to classical theory of radiation the frequency of electromagnetic radiation is equal to the frequency of oscilating electron in an atom.

Electron revolving around nucleus experience centripital force given by,

\(F=\frac {mv^2}{r}=\frac{mr^2\omega^2}{r}= mr\omega^2\dotsm(1)\)

where,

m= mass of electron

r= radius of electronic orbit

\(\omega\)= angular frequency

If we applied magnetic field in a direction perpendicular to plane of orbit then electron experience extra force evB. This force speed up or slow down electron.

\(F=(-e)(-\hat j\times\hat K)\)

\(\overrightarrow F=e\hat i\)

Let(\(\omega+\delta\omega\)) be new angular frequency of electron inside magnetic field.

\(\therefore\;\; net\;force= m_r(\omega+\delta\omega)^2\)

For clockwise motion:

\(or\;\; F-evB=mr(\omega+\delta\omega)^2\)

\(or\;\;mr\omega^2-evB=mr\omega L+2mr\omega\cdot\delta\omega+mr(\delta\omega)^2\)

Since, \(\delta\omega\) is to small we can neglect (\(\delta\omega)^2\)

\(or\;\; -er\omega B= 2mr\omega\cdot\delta\omega\)

\(or\;\;\delta\omega=\frac{-eB}{2m}\dotsm(3)\)

For anticlockwise motion.

\(or\;\; F+evB= mr(\omega+\delta\omega)^2\)

\(or\;\;mr\omega^2+evB=mr(\omega+\delta\omega)^2\)

\(or\;\;\delta\omega=\frac{eB}{2m}\dotsm(r)\)

Combining (3) & (4)

\(\delta\omega=\pm\frac{eB}{2m}\dotsm(5)\)

Equation (5) gives change in angular frequency of electron due to magnetic field.

sicne, \(\omega=2\pi\gamma, \;\;\gamma=\)freuency

\(or\;\;\delta\omega=2\pi\delta\gamma\dotsm(6)\)

from (5) and (6)

\(\delta\gamma=\pm\frac{eB}{4\pi m}\dotsm(7)\)

The equation (7) gives shift in frequency i.e Zeeman shift inside magnetic field.

Again,

\(\gamma=\frac {c}{\lambda}\) [ c= speed of light in vaccum ]

\(or\;\;\frac{\delta\gamma}{\delta\lambda}=\frac{-c}{\lambda^2}\)

\(or\;\;\delta\gamma=\frac{-c}{\lambda^2}\cdot\delta\lambda\dotsm(8)\)

From equaiton (7) and (8)

\(\frac{-c}{\lambda^2}\delta\lambda=\pm\frac{eB}{4\pi m}\)

\(\delta\lambda=\pm\frac{eB}{4\pi mc}\times\lambda^2\dotsm(9)\)

Which is required expression for Zeeman shift in wavelength.

The wavelength of 3-componenet lines are

\(\lambda_1+\lambda_\circ-d\lambda\)

\(=\;\;\lambda_\circ-\frac{eB}{4\pi m c}\cdot\lambda_{\circ}^2\)

\(\lambda= \lambda_\circ\) [middle or modified component]

\(\lambda_3= \lambda_\circ+d\lambda=\lambda_\circ+\frac{eB}{r\pi mc}\cdot\lambda_{\circ}^2\)

Important:

1. It can be use to calculate value (e/m) by measuring by d\(\lambda\)

i.e \(d\lambda=\frac{eB}{4\pi mc}\cdot\lambda^2\)

\(\Rightarrow\frac em=\frac{d\lambda\times 4\pi c}{\lambda^2B}\)

2. It is used to calculate the magnetic moment of electron

i.e \(\mu S= \frac{e\hbar}{2m}=\frac em = \frac {h}{4\pi}\)

3. To find the number of component lines in normal Zeeman effect.

Reference:

Reviews of Modern Physics. Lancaster, P.A.: Published for the American Physical Society by the American Institute of Physics, 1952. Print.

Wehr, M. Russell, and James A. Richards. Physics of the Atom. Reading, MA: Addison-Wesley Pub., 1984. Print.

Young, Hugh D., and Roger A. Freedman. University Physics. Boston, MA: Pearson Custom, 2008. Print.

Adhikari, P.B. A Textbook of Physics. 2070 ed. Vol. II. Kathmandu: Sukunda Publication, 2070. Print.