Page 55 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

OVERVIEW 37

Table 3.1 Quantum numbers, their allowed values, and the parameters they
quantise
Quantum number Quantised parameter Allowed values
n Total energy 1, 2, 3, 4, etc.
l Orbital angular momentum 0, 1,2,3, ...,(n- 1)
m Orbital angular momentum -l,-l + 1, -l + 2,...,-!,
component along the axis 0, , . . . l - 2 , l - 1,l
l
of quantisation
i
m s Spin angular momentum + o r - I
component along the axis
of quantisation

number m s, which takes two values +| and — |. Therefore, four quantum numbers n, l,
m, and m s are required for a complete description of the electronic states in an isolated
one-electron atom. The four quantum numbers are listed in Table 3.1 along with the
parameters that they represent and the limitations on the values of each of these quantum
numbers.
When an atom containing more than one electron is treated quantum mechanically, a
useful first-approximation is that the electrons do not exert forces on one another. Thus,
in this approximation, the states occupied by these electrons are still characterised by the
quantum numbers in Table 3.1. However, the arrangement of these electrons in the atom
satisfies Pauli's exclusion principle. This principle states that in a multielectron system no
two electrons can have identical sets of quantum numbers. Stated differently, the Pauli's
exclusion principle requires that no two electrons may have the same spatial distribution
and spin orientation and that no more than one electron may have the same wave function
when spin is included. Therefore, four quantum numbers n, I, m, and m s are required to
describe the exact state of an electron in an atom.
The lowest-energy, or ground-state, electron configuration for any atom can be
explained using the results summarised in Table 3.1 and the Pauli's exclusion principle.
The number of combinations of m and m s for a given subshell or orbital (n, /) gives the
maximum number of electrons in that subshell. For each value of /, there are (21 + 1)
values of m, and for each value of / and m, there are two values of m s (±1/2). Therefore,
the maximum number of electrons that can be placed in a given subshell, in accordance
with Pauli's exclusion principle, is 2(21 + 1). As stated earlier, the shells associated with
n = 1, 2, 3, 4, 5, and 6 are labeled as the K, L, M, N, O, and P shells, respectively.
The orbitals associated with / — 0, 1, 2, and 3 are labeled as the s, p, d, and f orbitals,
respectively. The quantum number / specifies the shape of the envelope in which the
electron is likely to be found. Figure 3.1 shows the calculated (from quantum theory)
envelopes for s, p, and d electrons. In the case of s electrons, the envelope is spherical;
for p electrons, it is dumbbell-shaped; and for d electrons, it is clover-shaped in four cases
and dumbbell-shaped in one.
Using the results on the atomic structure outlined earlier, we can now proceed to set up
the periodic table of elements. The first few elements of the periodic table are shown in
Table 3.2. The first element is hydrogen with only one electron in the lowest energy state
defined by n = 1, l = 0, and m = 0. This configuration of the hydrogen atom is designated
1
I
as s . The number 1 to the left stands for the shell n = 1, s specifies the orbital l = 0,