Anomalous Zeeman effect

Anomalous Zeman effect:

when an atomic source is placed inside weak magnetic field a spectral line splits into more than 3-components lines known as anomalous Zeeman effect.

It can be explain by consider orbital as well as spinning motion of electron in an atom.

The total angular momentum of electron in an atom is give by,

\(\overrightarrow J=\overrightarrow L+\overrightarrow S\dotsm(1)\)

Where, \(\overrightarrow L\)= Orbital angular momentum

\(\overrightarrow S\)= spinning angular momentum

The orbital magnetic moment \(\mu_l=\frac{eh}{4\pi m}\cdot l\dotsm(2)\)

and the spin magnetic moment \(\mu_s=\frac{eh}{4\pi m}.(2s)\dotsm(3)\)

Orbital magnetic moment is opposite to orbital angular momentum and spin. Magnetic moment is opposite to spin angular momentum, but total magnetic moment \(\mu\) is not opposite 'j' . Resolving total magnetic moment '\(\mu\)' into two components one along 'j' ,'\(\mu_j\)' and another perpendicular to j, \(\mu_1\). the perpendicular component over 1 complete cycle average to zero.

Now,

\(\mu_1\)= Component of \mu_k along j + component of \(\mu_s\) along j

\(\;\;\;= \frac{eh}{4\pi m}\cdot l cos(l,j)+\frac{eh}{4\pi m}cos (s,j)\dotsm(4)\)

\(\;\;\;=\frac{eh}{4\pi m}\biggl(\frac{l^2+j-sL}{2-j}\biggr)+\frac{eh}{4\pi m}\biggl(\frac{s^2+j^2-l^2}{2sj}\biggr)\)

\(\;\;\;=\frac{eh}{4\pi m}\bigg[\frac{l^2+j^2-s^2}{2j}+\frac{s^2+j^2-l^2}{j}\biggr]\)

\(\;\;\;=\frac{eh}{4\pi m}\biggl[\frac{l^2+j^2-s^2+2s^2+2j^2-2l^2}{2s}\biggr]\)

\(\;\;\;=\frac{ehj}{4\pi m}\biggl[\frac{2j^2}{2j^2}+\frac{j^2+s^2-l^2}{2j^2}\biggr]\)

\(\;\;\;=\frac{eh}{4\pi m}\cdot j\biggl[1+\frac{l^2+s^2-l^2}{2j^2}\biggr]\)

\(\;\;\;=\frac{eh}{4\pi m}\cdot j\cdot g\)

\(\;\;\;\overrightarrow mu_j= \frac{-eh}{4\pi m}\cdot \overrightarrow j\cdot g\dotsm(6)\)

where,

\(g=1+\frac{j^2+s^2-l^2}{2j^2}\) is called Lande's splitting factor.

From vector atom model:

\(j^2=j(j+1)\hbar^2\)

\(s^2+s(s+1)\hbar^2\)

\(l^2=l(l+1)\hbar^2\)

\(\therefore\;\;\; g=1+\frac{j(j+1)+s(s+l)-l(l+1)}{2j(j+1)}\dotsm(7)\)

Now, magnetic interaction energy of electron inside weak magnetic field is given by,

\(\Delta E=-\overrightarrow \mu_j\cdot\overrightarrow B\)

\(\;\;\;=\frac{eh}{4\pi m}(\overrightarrow J\cdot\overrightarrow B)\cdot g\)

\(\;\;\;=\frac{eh}{4\pi m}B(jcos\theta)m.g\)

\(\;\;\;\Delta E= \frac{eh}{4\pi m}\cdot m_jBg\dotsm(8)\)

where \(m_j\)= jcos\(\theta\)=projection of j along B

For each value of j there are 2j_1 values of \(m_j\) which can explain the anomalous zeeman effect.

\(^2P_\frac 12\rightarrow^2S_\frac12\rightarrow D_i-line\)

-1/2 +1/2 -1/2+1/2

Sodium \(D_1\) line is due to transition of electron from \(^2P_\frac12\rightarrow ^2S_\frac12\)

For \(^2P_\frac12\), j=1/2, l=1, 2s+1=2\(\rightarrow\)s=1/2

There are 2j+1 possible values of \(m_j\)

\(2\times\frac12+1\) possible values of \(m_j\)

=2-possible values of \(m_j\)

\(m_j=\frac{-1}{2},\frac{1}{2}\)

\(g=1+\frac{\frac12\biggl(\frac12+1\biggr)+\frac12\biggl(\frac12+1\biggr)-1(1+1)}{2\times\frac12\biggl(\frac12+1\biggr)}\)

\(\;\;\;=1+\frac{\frac34+\frac34-2}{\frac32}\)

\(\;\;\;=\frac23\)

\(\;\;\;gm_j=\frac23\times\frac{-1}{2},\frac 23\times\frac 12\)

\(\;\;\;gm_j=\frac {-1}{3}, \frac{+1}{3}\)

Inside weak magnetic field \(^2P_\frac12\) level splits into 2-levels having \(gm_j=\frac{-1}{3},\frac{+1}{3}\)

\(^2S_\frac12, j=\frac12, l=0, 2s+1=2\Rightarrow S=\frac12\)

There are 2j+1 possible values of \(m_j\)

=\(2\times\frac12+1\)possible values of \(m_j\)

=2 possible value of \(m_j\)

\(m_j=-\frac12, \frac12\)

g=1+\(g=1+\frac{\frac12\biggl(\frac12+1\biggr)+\frac12\biggl(\frac12+1\biggr)-0}{2\times\frac12(\frac12+1)}\)

\(g= 2\)

\(gm_j=-2\times\frac 12, 2\times \frac 12\)

\(gm_j=-1,+1\)

\(^2S_\frac12\) level splits into 2-levels.

From above consideration it is found that \(D_s\) line inside weak magnetic field splits into 4-component lines which explains anomalous 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.