Quantum number associated with vector atomic model.

Quantum number associated with vector atomic model:

Magnetic spin quantum numbers \(m_s\):

It is used to indicate the projection of spin angular momentum along the direction of magnetic field. For each value of s, there are two possible values of \(m_s\).

$$i.e\;\; m_s=\pm\frac 12$$

Where,

$$m_s= +\frac 12= spin up (\uparrow) (\overrightarrow S and \overrightarrow B \;and \;parallel)$$

$$= -\frac 12 = spin down (\downarrow) (\overrightarrow S \; and\; \overrightarrow B\; are \;anti-parallel)$$

7. Magnetic total angular momentum quantum number (\(m_j\)):

It is used to specify the projection of total angular momentum along the direction of magnetic field. It is denoted by ( \(m_j\)) . For a given value of j there are 2j+1 values of \(m_j\) from –j to +j excluding 'zero' increasing each step by +1.

For example,

When j= 3/2 there are 2j+1 value of \(m_j\)

\(m_j\)= -3/2, -1/2, +1/2, 3/2 [ 2 x 3/2 + 1= 4 values]

Diagrammatic representation: radius j= 3/2

Example 2: j=7/2 there are 2j+1= 2 . 7/2 +1 = 8 values of \(m_j\)

\(m_j\)= -7/2 , -5/2, -3/2, -1/2, +3/2, +7/2

Radius j= 7/2 (semi circle)

Coupling scheme ( Addition of angular momentum)

An atom consists of large number of electron. The angular momentum of an atom can be specified by two ways.

1. L-S coupling ( Russel – Sounders coupling)
2. f-j coupling

L-S coupling:

The angular momentum in L-S coupling is obtained by sum of total orbital angular momentum L-vector and total spin angular momentum of all elements in an atom i.e

$$\overrightarrow J=\overrightarrow L+ \overrightarrow S$$

Where, $$\overrightarrow L= sum\;of\; overrightarrow L_i= \sum_{i=1}\overrightarrow L_i= orbital\; angular\; momentum\; of \; i^th \; electron.$$

$$\overrightarrow S= Sum\; of \;\overrightarrow S= \sum_{i=1} \overrightarrow S_i$$

= spinning angular momentum of ith electron.

The total orbital quantum number is always an integers.

The total spin quantum number of an atom is an integer for eve number of electrons and half-integral multiple of odd integer for odd no. of electrons.

For two electron atom:

The possible value of s:

For two electron and three electrons

2. J-J Coupling Sheme:

In this type of coupling sheme the total angular momentum of atom is obtained by the vector sum of total angular momentum of ndividual electrons.

Mathematically: $$\overrightarrow J= \overrightarrow J_1+ \overrightarrow J_2 + \overrightarrow J_3 ­+ …$$

$$= \sum_{i=0}^n \overrightarrow J_i$$

Where \(\overrightarrow J_i\)= Vector sum of \(\overrightarrow s_i\) and \(\overrightarrow L_i\)

\(S_i\)= orbital angular momentum of ith electrone.

\(J_i\)= Total angular momentum of \(i^th\) electron.

For two electron system:

Orbital magnetic moment (\(\overrightarrow \mu_L\))

Associated with the orbital motion of an electron in an atom an electron has orbital magnetic moment.

Consider an electron of mass 'm', charge 'e' is revolving in an elliptical orbit is shown in figure. The orbital magnetic moment is defined as product of current and the area of loop enclosed by the current. i.e \(\mu_l= I.A\)

$$or, \mu_l= \frac eT A\dotsm(1)$$

Where, T= time period of revolution

A= area of ellipse.

We have, areal velocity of electron is \(\frac{dA}{dt}= \frac 12 r^2 \frac{d\phi}{dt}\dotsm(2)\)

The angular momentum of an electron in central force field is constant

i.e $$\;\; L= mr^2\frac{d\phi}{dt}= constant $$

$$or, \;\; r^2\frac {d\phi}{dt}= \frac Lm\dotsm(3)\;\; (\frac 1m \; is \; constant)$$

From (2) and (3)

$$ or, \;\; \frac{dA}{dt}= \frac{1}{2m}dt$$

$$or,\;\; A=\int_0^T\frac{L}{2m}dt$$

$$or,\;\; A= \frac{L}{2m}\int_0^T dt$$

$$or,\;\; A= \frac{LT}{2m}\dotsm(4)$$

Using equation (4) in (1)

$$\mu_L= \frac eT \frac{LT}{2m}$$

$$or,\;\; \mu_L= \frac{eL}{2m}\dotsm(5)$$

From vector atom model

L= orbital angular momentum = \(l\hbar\dotsm(6)\)

l=orbital quantum nmber

From (5) and (6)

Orbital magnetic moment of electron is directly proportional to orbital quantum number.

When, l=1

$$\mu_1=\frac{e\hbar}{2m}$$

Putting the value of constant we ge,

\(\mu= 9.274\times10^{-24} J/T\)

This minimum value of orbital magnetic moment is known as Bohr's Magneton.

Bohrs Magnetion \(\mu_B=\frac{e\hbar}{2m}=\frac{eh}{2\pi m}= 9.24\times 10^{-24} J/T \)

Gyromagnetic ratio:

The ratio of magnitude of orbital magnetic moment to orbital angular moment is known as gyromagnetic ratio It is constant quantity.

$$\therefore\;\; Gyromagnetic \; ratio= \frac {\mu_l}{\mu}= \frac {e}{2m}= \frac{1.602\times 10^{-19}}{2\times 9.1\times 10^{-31}}= 8.8\times 10^9 $$

By knowing value of orbital magnetic moment we can calculate magnetic interaction.

Spin magnetic moment:

A spinning electron behaves as a tiny magnet and has spin magnetic moment.

Analogous with orbital magnetic moment (\(\overrightarrow {\mu_1}= \frac{e}{2m}\overrightarrow L\))

The spin magnetic moment is given by,

\(\therefore\;\; \mu\overrightarrow S= -2(\frac{e}{2m}\cdot\overrightarrow S\)

The factor '2' is due to experimental observation

Taking magnitude only,

$$\therefore \;\; \mu_s= 2(\frac {e}{2m}s$$

$$=2(\frac{e}{2m}\cdot s\hbar$$

{s= s\(\hbar\)= spinning angular momentum}

$$=\frac{e\hbar}{2m}$$

$$\mu_s=\mu_B$$

Hence, \(\mu_s=\mu_B=9.27\times 10^{-24} J/T\)

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