Electron spin, Pauli's exclusion principle

Pauli's exclusion principle:

Statement:

No two electrons in an atom can be in the same quantum state. OR

Two electron in an atom cannot have same set of quantum numer ( n,l,\(m_l,m_s\))

Explanation:

The state of an electro in an atom is represented by a set of four quantum number n,l,\(m_l,m_s\).

When n=1 (k- shell)

The possible value of l=0 (s- orbital)

For l=0

The possible values of \(m_l\) are (2l+1)

= 2 . 0 +1

= 1- value.

\(m_l\)=0

For each value of \(m_l\), \(m_s\) has 2-values \(m_s=\pm\frac{1}{2}\)

The two possible set of quantum number are (1,0,0,+\(\frac 12\)) & (1,0,0,\(\frac{1}{2}\))

It is represented by \(1s^2\).

2.When n=2

\(\therefore\) l=0, 1 ( s & p orbital )

For l=0, \(m_l\)=0 & \(m_s\)= \(\pm\frac12\)

The two different sets of quantum number are

(2,0,0 , \(\pm\frac12\))= \(2s^2\)

For l=1 (p-orbital)

The possible value of \(m_l\) are \(m_l\)= -1,0,+1( 3 - values ).

The six different sets of quantum number are (2,1,-1,+1/2) , (2,1,0,\(\pm\frac12\)), (2,1,+1,\(\pm\frac12\))=2\(p^6\)

3. When n=3 ( M-shell)

\(\therefore\) l= 0,1,2

For l=2 ( d-orbital)

There are 2l+1=2 \(\times\) 2= 5 possible value of \(m_l\)

\(m_l= -2,-1, 0, +1, +2\)

Te 10 different sets of quantum numbers are ( 3, 2, -2, \(\pm\frac12\)), (3, 2, -1, \(\pm\frac12\)), (3, 2, 0, \(\pm\frac12\)), ( 3, 2, 0, +1, \(\pm\frac12\)) & ( 3, 2, +2, \(\pm\frac 12\)).

Total number of electrons in a shell.

For each value of 'n' there are n-values of 'l' from 0 to n-1.

i.e l= o,1, 2, 3, .... , n-1

For each value of l

There are (2) values of \(m_l\), there are two values of \(m_s\) (\(pm\frac12\))

\(\therefore The total number of states for a given value of 'n'

=\(2^{n-1} \sum_{l=0} (2l+1\))

= 2[ 1 + 3 + 5 + 7 + ... + 2(n-1)+1]

=2[ 1+3+5+7+.... + 2(2n-1)] Contain 'n' terms

\= 2{ n/2} 2\times 1 + (n-1) 2} [ \(s_n=\frac{n}{2} 2a+(n-1) d]\)

=n[ 2+ 2n - 2]

= \(2n^2\)

\(\therefore\) number of electrons in a shell = 2n^2

For K-shell:

n=1,

Total number of electrons =\( 2\times 1^2= 2\)

\(\Rightarrow 1s^2\)

For l shell:

n=2 , total number of electron = \( 2\times 2^2= 8\Rightarrow 2s^22p^6\)

For m- shell

n=3, total number of electron = \(2\times 3^2= 12 \Rightarrow 3s^23p^63d^10\)

For n- shell

n=4, total number of electron =\( 2\times 4^2\)= 32

\(\Rightarrow 4s^24p^64d^{10}4f^{14}\) and so on

Electron Spin:

In 1891, the Irish physicist, George Stoney, believed that electricity should have a fundamental unit. He called this unit the electron. The electron was discovered by J.J. Thomson in 1897. The electron was the first subatomic particle ever found. It was also the first fundamental particle discovered. As is commonly known, the hydrogen atom is the smallest atom that exists in nature. The mass ratio between an electron and a hydrogen atom is approximately 1:1836. From this numerical comparison, it is evident that the mass of an electron is much smaller than that of a hydrogen atom. Besides its electric charge and mass, an electron has one more key property, which is called “electron spin.?Just as the earth rotates around its axis, the electron is also constantly “spinning.? The concept of electron spin was discovered by S.A. Goudsmit and George Uhlenbeck in 1925. The electron has three basic properties: electric charge, mass, and spin. As far as we can understand, electron spin is the electron’s intrinsic angular momentum. As far as we can tell, the electron is still regarded as a point-like a particle, with no internal structure and no physical size. How can a point particle, without any physical size, spin and have intrinsic angular momentum? The spinning of the point particle is meaningless. What matters is where the intrinsic angular momentum originates from inside the electron. There are many people who believe that an electron’s mass may have an electromagnetic origin. Is there a possibility then that the electron spin also has an electromagnetic origin? The book “What Is the Electron Spin? tries to answer this question. This book is based on the assumption that the electron spin has an electromagnetic origin. That is, the electron’s intrinsic angular momentum results from an electromagnetic field. The electron is a unit with a single electrical charge and has an intrinsic electric field. Because the electron has an intrinsic angular momentum, we know that the electron must have an intrinsic magnetic field. Similar to a magnetic dipole field, the electron is nature’s smallest magnet. As is commonly known, the electromagnetic field can have energy and momentum as well as angular momentum. “What Is the Electron Spin??makes a basic assumption that the electron itself has an electromagnetic origin. It then extends that theory to all of the electron’s properties, such as mass and spin, claiming they have an electromagnetic origin as well. For example, an electron’s self-energy comes from its electromagnetic field energy, and the electron spin is the angular momentum of the electron’s electromagnetic field. The first point this book attempts to make is that the electron is no longer regarded as a point particle. Rather, it purports that there is a continuum spherical distribution of both electric and magnetic charges inside the electron.

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.