The vector atom model basically deals with the total angular momentum of an atom, which is results of the combination of orbital and spin angular momenta. The two fundamental features of the vector atom model are (i) space quantization of orbits and (ii) spinning electron hypothesis.
The angular momentum is a vector quantity, hence its direction must be specified to describe it completely. To specify the orientation or direction of an orbit, a reference is required. The direction of the magnetic field applied to the atom is chosen as the reference line.
The rotating electron about the nucleus forms a current loop which has a magnetic moment µ = IA, where I is the current in the loop and A is the area vector. The energy of loop-field system is given by U = -µ. B = – |µ| |B| cosθ, where θ is the angle between the magnetic moment θ and magnetic field B. As such classically any energy value between -µB to +µB is possible for the loop.
An electron orbiting around the nucleus in an atom possess angular momentum L which interacts with external applied magnetic field B. According to quantum theory, there are fixed directions of magnetic moment 1.1. of magnetic dipole (formed by closed loop motion of electron about nucleus) with respect to the magnetic field B. The magnetic moment µ and angular momentum L can be related as
This relationship is true from both classical and quantum mechanical point of view. Since discrete directions of µ are allowed, as such the direction of L will also be quantized in space. Here, the quantization refers the projection of L along the Z-direction (L2) which can have discrete values only. The orbital magnetic quantum number ml gives the direction of L and tells the possible components of L in the Z-direction (the field direction). The phenomenon of quantization of L in the direction of magnetic field B is commonly known as space quantization.
The direction of B is along the Z axis. As such the component of L along Z-direction is given as LZ = ml, h, the ml can table the values from –l to +l including zero.
The above discussed space quantization of orbits and possible components of L can be understood with the following example.
Let us calculate the allowed projections of L for l = 2. The L can be visualized as a vector lying on the surface of a cone (see Figure 10.2).
Figure 10.2 (a)
Figure 10.2 (b)
Figure 10.2 (c)
The possible values of ml for l = 2 can be -2, -1, 0, 1, 2 hence LZ = m1h = -2h – h, 0, h, 2h. Figure 10.2(b) and 10.2(c) show space quantization of L for l=2.
Goudsmit and Uhlenbeck in 1925 proposed that electron ‘spins’ about an axis through its centre of mass and further it has both angular momentum and a magnetic moment. The spin of electron is analogous to the planetary motion about the Sun in our solar system. Quantum mechanical treatment has demonstrated that the spin of electron should be quantized. As such a new quantum number 5 has been introduced. The spin angular momentum S is related with spin quantum number s as
The spin can be either clockwise or anticlockwise as such s can have two values i.e., ± ½
Component of S along the direction of magnetic field is governed by quantum, number ms as
where ms is the spin magnetic quantum number. The ms can take (2s + 1) values.
We have seen that the orbital and spin motions of an electron are quantized in magnitude as well as in direction. These motions are represented by quantized vectors, as such the atomic model is termed as vector atom model.
The Quanturrr’Numbers and Spectroscopic Notations
The Bohr-Sommerfeld atomic theory uses four quantum numbers which are (i) Principal quantum number (n) (ii) Orbital quantum number (1) (iii) Spin quantum number (s) and (iv). Total angular momentum quantum number (j). In addition to these, three other quantum numbers have been introduced in vector atom model which are (i) Magnetic orbital quantum number (ml), (ii) Magnetic spin quantum number (ms) and (iii) Magnetic total an momentum quantum number (mj). The general description and possible values of these quantum numbers is given below.
i. Principal Quantum Number (n): This quantum number represents the serial number of the atomic shells starting from the innermost. The ‘n’ can have only positive integral values excluding zero.
n = 1,2,3,4
ii. Orbital Quantum Number (I): The Orbital quantum number ‘l’ can have the values from a to (n -1), where ‘n’ is the principal quantum number.
For example, if n = 3, then 1 = 0, 1,2. I
Conventionally we call
1 = 0 s electron
1 = 1 p electron
1 = 2 d electron
1 = 3 f electron
iii. Spin Quantum Number (s): Spin quantum number s has only one value
iv. Total Angular Momentum Quantum Number (j): The total angular momentum quantum number j is the sum of orbital angular momentum ‘l’ and spin quantum number’s’. As such j can have l ± s or l ± ½ values.
v. Magnetic Orbital Quantum Number ‘ml‘: As discussed previously in article 10.3.1, the magnetic orbital quantum number ml can have 2l + 1 values.
For example, I = 2, ml can have – 2, – 1, 0, 1, 2 i.e., total 5 values.
vi. Magnetic Spin Quantum Number ‘ms‘: The magnetic spin quantum number take (2s + 1) values. We know that s = ½ and hence ms, can take (2 × ½ + 1) = 2 in total. The values of ms‘ therefore are – s and + s or – ½ and + ½.
vii. Magnetic Total Angular Momentum Quantum Number mj: The total momentum J of an atom is a vector quantity and is the vector sum of orbital momentum L and spin angular momentum S i.e., J = L + S. We define quantum number mj which is known as the magnetic total angular mom quantum number and specify the orientation of J in space with respect to Z-axis.
According to quantum mechanics
The possible values of mj are 2j + 1 i.e., mj can take values from – j to + j including zero in integral steps.
Now Jz = Lz ± Sz
which implies that mj = ml ± ms. Now ml is an integer and ms is ± ½ as such mj will have the half integral values only. The mj, ml and ms can have the maximum value j, l and s respectively.