Vector atom model

Vector atom model:

Assumption:

1. The concept of space or spatial quantization.

2. Spinning electron hypothesis.

1. The concept of spatial quantization:

According to Bohr's theory and Sommerfield atomic model the magnitude ( shape and size ) of orbit is quantized ( discrete). In odd atomic mode orientation of electronic orbit in space is not taken into account. To specify the orientation of electronic orbit of fixed reference axis or lines is selected along the direction of externally applied magnetic field with respect to the reference axis the orientiation of an electronic orbit is also quantized. The projection of an electronic orbit along the direction of magnetic field is also quantized. This quantization if orbit in shape is known as space quantisation ( spatial quantization).

2. Spinning electron hypothesis:

According to " Uhlenbeck and Goudsmit" spinning electron hypothesis of an electron has two types of motion

(1) Orbital motion around nucleus indicated by orbital angular momentum vector \(\overrightarrow L\)

(2) Spinninng motion about an axis passing through it's own centre indicated by spin angular momentum vector \(\overrightarrow S\). They proposed new quantum number to explain the spinning motion of an electorn.

Ie Spin quantum number (s)= \(\frac 12\)

In this atomic model vector quantitiy such as orbital magnetic moment (\(\overrightarrow\mu_L)\), spin magnetic moment (\(\overrightarrow\mu_s)\) orbital angular momentum (\(\overrightarrow S\)) are defined hence this model is known as vector atom model. This atom model used to explanin ' Zeeman effect ' , ' Experimental evidences of Stern-Gerlach experiment' etc.

Quntum number associated with 'Vector atom model'

1. Principle quantum number(n):

The principal quantum number is used to specify the magnitude of an electron orbit (Shape and size) in an atom value of princial quantum no. is n= 1,2,...

when n=1 ( K- shell )

n=2 ( L- shell )

n= 3 ( M- shell ) so on.

2. Orbital quantum number (L):

Orbital quantum number is used to specify the orbital angular momentum in an atomic model. The value of orbital quantum numer is L= 0,1,2,3,...,(n-l) { i.e all possible values of \(n_\phi-1\) }

According ot wave mechanics:

Angular momentum of electron in a given value of 'L' is ,

L= l\(\hbar\) , where \(\hbar\)=\(\frac{h}{2\pi}\)

when, L=0 (S- orbital )

L= 1( P- orbital )

L= 2 ( d- orbital )

L= 3 ( f- orbital )

According to quantum mechanics:

$$L= \sqrt{L(L+1)\hbar}$$

For eg, For a electron in p-orbital

L=1

$$\therefore L= \sqrt{1(L+1)\hbar}$$

$$=\sqrt{2\hbar}$$

3. Spin quantum number (s):

Spin quantum number s=1/2 is used to indicate the spinning motion of electron in an atom.

The spin angular momentum, \(P_s= s\hbar\)

According to wave mechanics, spin angular mmomentum is given by

Spin angular momentum= \(P_s\)=\(\sqrt{ s(s_1)\hbar}\)

$$=\sqrt{\frac 12 (\frac 12+1)\hbar}$$

$$=\sqrt{\frac 34 \hbar}$$

4. Total angular momentum quantum number (J):

To indicate the total angular momentum of an electron in an atom we use total angular momentum quantum number (j). Total angular momentum is vector sum of orbital angular momentum and spin angular momentum.

i.e $$\overrightarrow j= \overrightarrow l+ \overrightarrow s$$

If we take magnitude only two, \(P_j= \sqrt{j(j+1) \hbar}\)

For an electron in p- orbital

L=1, S=1/2

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

$$=\overrightarrow L+\overrightarrow S\;\; (\overrightarrow L\; and \;\overrightarrow S \; are\; parallel)$$

$$=\overrightarrow L=\overrightarrow S\;\;(\overrightarrow L \; and \;\overrightarrow S\; are \; antiparallel)$$

Possible value of j= 1+1/2, 1-1/2

$$j= \frac 32, \frac 12$$

For an electron in s-orbital

L=0, s=1/2

$$=\overrightarrow L\pm\overrightarrow S= 0\pm\frac 12= 0+\frac 12= \frac 12$$

(j can not be negative)

5. Magnetic orbital quantum number (ml):

To specify the projection of orbital angular momentum of an electron in an atom along the direction of magnetic field \(m_l\) is used.

There are (2l+1) possible value of ml from -L to +L increasing each step by +1

$$\therefore\;\; M_L=L,-(L-1), -(L-2),...,-1,0,-1,...,(L-2), (L-1), L$$

The projection of orbital angular momentum is given by,

$$Lcos\theta= projection\;\overrightarrow L\; along\;\overrightarrow B$$

When L=2, there are 2L+1= 2\times2+1=5 possible value of \(m_L\)

\(M_L\)= -2,-1,0,1,2

In diagramatic notation: ( radius L=2 , semi circle)

There are 5 orientation of angular momentum vector for , L=2. This phenomena of discrete orientation of \(\overrightarrow L\) vector in space with respect to \(\overrightarrow B\) is known as space quantization. The possible angular orientation are given by, \(\cos\theta=\frac {mL}{L}\therefore\) a has (2L+1) values in space.

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.