Zeeman effect( Quantum mechanical treatment), Critical potential

Quantum mechanical explanation of Normal Zeemman effect:

Debye explainnormal Zeeman effect only cosidering orbital motion of electron (i.e spinning motion of electro is not taken into account).

The orbital angular momentum of ann electron is

\(L=l\hbar\dotsm(1)\)

Where, l=orbital angular number

\(\hbar=\frac{h}{2\pi}\)

The orbital magnetic moment of electron in terms of L is

\(\overrightarrow\mu = \frac{-e\overrightarrow L }{2m}\dotsm(2)\)

where, \(\overrightarrow\mu_L \) and \(\overrightarrow L\) are antiparallel.

The magnetic interaction energy of an electron inside uniform magnetic field \(\overrightarrow B\) is given by

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

\(\;\;\;=-\biggl(\frac{-e\overrightarrow L}{2m}\cdot\overrightarrow B\biggr)\)

\(\;\; =\frac{e}{2m}LBcos\theta, \) Where, \(\theta\)= angle between \(\overrightarrow B \; and\overrightarrow L\)

\(\;\;= \frac{e\hbar}{2m}(lcos\theta).B[L= l\hbar]\)

\(\Delta E= \frac{e\hbar}{2m}\cdot mlB\) where, \(ml=lcos\theta\)= projectrion of orbital quantum number alnog \(\overrightarrow B\)

\(\Delta E=\frac{eh}{4\phi m\cdot mlB}\dotsm(3)\)

Let \(E\circ\) be energy of electron in the absence of magnetic field and \(E_B\) in the presence of magnetic field.

\(\therefore\;\;\; E_B= E_\circ+ \Delta E\)

\(\therefore\;\;\; E_m= E_\circ+\frac{eh}{4\pi m }\cdot ml. b\dotsm(3)\)

Let \(E_0'\) and \(E_B'\) be energy of electron in p orbital (i.e l=1) and \(E_B''\) and \(E_\circ''\) be energy of electron in d-orbital (i.e l=2) in the absence of \(\overrightarrow B\) and in presence of\(\overrightarrow B\) respectively.

For, l=1

\(E_B'=E_\circ'+\frac{eh}{4\pi m}\cdot ml'B\dotsm(5)\)

where, ml= -1,0,1

For l=2

\(EB''=E_\circ''+\frac{eh}{4\pi m}\cdot ml''.B\dotsm(6)\)

where ml=-2,-1,0,1,2

Each level splits into (2l+1) levels inside magnetic field.

The energy emitted during transition of electron from l= 2 to l=1 inside magnetic field is given by,

\(EB''-E_B'=E_\circ''-E_\circ'+\frac{3h}{4\pi m}(ml''-ml')B\)

\(h\nu'=h\nu+\frac{eh}{4\pi m}.\Delta m. B\)

\( \nu'=\nu+\frac{eB}{4\pi m}\cdot \Delta ml\dotsm(7)\)

Where, \(\nu'\)= Frequency of radiation inside magnetic field

\(\nu\)= frequency of radiation in the absence of magnetic field.

According to selection rule of ml only those transition are possible in which \(\Delta ml= 0,+1,-1..\)

When\( \Delta ml= +1\)

\(\nu'=\nu+\frac{eB}{4\pi m}\dotsm 8(a)\)

\(\Delta ml=0\)

\(\nu'=\nu\dotsm 8(b)\)

\(\Delta ml =-1\)\

\(\nu'=\nu-\frac{eB}{4\pi m}\dotsm 8(c)\)

Equation 8(a), 8(b), 8(c) can explain the three frequency component inside magnetic field. (i.e normal Zeeman effect)

Critical potential:

Minimum amount of energy required in electron volt(ev) to jump electron from ground sate to any one of higher energy state is known as critical potential. There are two type of critical potential:

I. Excitation potential

II. Ionisation potential

I. Excitation potential:

The minimum amount of energy in ev required to bring an atom into excited state is called exicitation potential. For eg, for H-atom

\(1^st\) exicitation potential= \(E_2-E_1\)

\(\;\;\;= \biggl(\frac{-13.6ev}{2^2}\biggr)-\biggl(\frac{-13.6}{1}\biggr)ev\)

\(\;\;\;= -13.6 ev\biggl(\frac 14-1\biggr)\)

\(\;\;\;=\frac 34\times 13.6ev\)

\(\;\;\;=10.2ev\)

\(2^nd\) excitation potential=\( E_3-E_1\)

\(\;\;\;=13.6(1-\frac 19)\)

\(\;\;\;=\frac89\times 13.6ev\)

II. Ionization potential:

Minimum amount of energy in ev required to completely ionized an atom or to jump an electron from ground sate to infinity is known as ionization potential.

Eg:- For H-atom

Ionization potential = \(E_\infty- E_1\)

Ionization potential = \(E_\infty-E_1\)

\(\;\;=\frac{13.6ev}{\infty}-\biggl(\frac{-13.6ev}{12}\biggr)\)

\(\;\;\;=13.6ev\)

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.