Stern and Gerlach Experiment

Principle:

When a magnetic diople is passed through an inhomogeneous magnetic field it experienced both torque and translatory force.

Theory:

Consider an atomic magnet of length 'l' pole strength 'p' is placed inside inhomogeneous magnetic field having field gradient \(\frac{dB}{dX}>0 by making an angle 0 with the direction of magnetic field as shown in fig.

The net magnetic force experience by magnetic dipole inside non-uniform field is given by,

$$ F= F_2-F_1= p( B+ \frac{dB}{dx}\cdot lcos\theta)- pB$$

$$=\frac{dB}{Dx} plcos\theta$$

$$=\frac{dB}{dx}\cdot(Hcos\theta)\dotsm(1)$$

Where, pl= magnetic dipoe moment

let,

m= mass of atomic magnet.

L= length of inhomogeneous magnetic field.

V= Velocity of atomic magnet as it enters magnetic field.

Acceleration of atomic magnet is given by

$$a=\frac Fm= \frac{dB}{dx} \frac{\mu cos\theta}{m}\dotsm(2)$$

Deflection of atomic magnet due to magnetic force.

$$or, d=\frac 12 at^2$$

$$or, d=\frac 12\biggl(\frac{dB}{dx}\biggl)\frac{\mu cos\theta}{m}\biggl(\frac Lr\biggr)^2\dotsm(3)$$

Experimental arrangement:

Atomic beam of silver (Ag) atom evaporated in an electric even is passed through two slit \(s_1\) and \(s_2|0 and then throuhg inhomogeneous magnetic field as shown in figure(1).

In the absence of magnetic field gradient a single peak of intensity of silver atoms is observed on photographic plate in the presence of magnetic field gradient two peak of intensity is observed as in fig(3)

Explanation:

The last electron of silver atom is in S-orbital (5s). For an electron S-orbital

$$ l= 0$$

$$\mu_1= l\mu_\beta=0$$

If magnetic moment of Ag- atom due to orbital motion of electron then the value of d must be zero.

\(A\; single\; peak,\frac{dB}{dx}\ne0\)

It couldn't explain two peaks on the intensity distribution.

For an electron in s-orbital

\(\mu\)= \(\cos\theta\)= component of \(\mu\) along \( \overrightarrow{B}\)

$$=\frac{2e\hbar}{2m}\cdot ms\;\;\;\;( ms= cos\theta)$$

$$=\pm\frac 12\times 2\biggl(\frac{e\hbar)}{2m}\biggr)$$

$$or, \mu cos\theta= I\mu B\dotsm(4)$$

The two peaks on the intensity distribution is due to spin magnetic moment of electron in s-orbital. The projection of spin magnetic moment has two value that is why equation (3) gives two value of displace with equation (4).

$$ i.e\;\; d=\pm\frac 12(\frac{dB}{dX} )(\frac{\mu B}{m})(\frac Lr)^2\dotsm(5)$$

Importance:

1. It gives the existance of spin magnetic moment of atom

2. It also gives the space quantisation ( two values of \mu) due to spinnin gof an electron.

Spectral Notation:

The state of an atom is denoted by capital letters S,P,D,F,G etc for orbital quantum number L= 0,1,2,3,4,... etc. respectively, value or total angular momentum quantum number J is written as, subscript right side of symbol indicated by value of L. The multiplicity of state i.e 2s+1 is written as, superscript of the symbol on left side.

\(\therefore\)Spectral Notation =\(^{2s+1}[Symbol for L]_{Value\; of\; J}\)

Sodium atom at ground state: \(1s^22s^22p^6s^1\)

The valence electron is on 3s orbital

For electron in 3s-orbital

For electron in 3s-orbital: l=0, =1\2, j= 1\(\pm\) S = 0\(\pm \)1\2= 1\2

Multiplicity = 2s+1= 2 X 1\2+1=2

Spectral notation for electron in 2s state is \(^2S_{\frac 12}\) read as Doublet S-half.

All the optical properties of atom is determined by electron in valence shell.

For sodium atom, L=0, S=1\2, J=1\2

Multiplicity= 2 x 1\2 + = 2

Spectral notation for sodium atom \(^2S_{1/L}\)

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.