Page 54 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

36 MEMS MATERIALS AND THEIR PREPARATION

where n is a positive nonzero integer, m c is the free-electron mass, e is its electronic
charge, h is the Planck's constant divided by 2n, and EQ is the permittivity of free
2
space . These are the only allowed energy values for the electron. These energy levels
are characterised by the integer n, which is called the principal quantum number. These
levels, or the orbits corresponding to them, are called shells; shells corresponding to
n = 1, 2, 3,... , are labeled the K, L, M,... , shells, respectively, where the K shell is
the one nearest to the nucleus. Some shells have subshells determined by the angular
momentum of the electron in its orbit. We observe that these energy values are negative,
meaning that the electron is restricted to move close to the nucleus. Such an elec-
tron is generally referred to as a bound electron, that is, the electron is bound to the
nucleus. If this electron acquires energy from an external source such that its energy E n
becomes equal to or greater than zero, then the electron frees itself from the nucleus
and is then called a free electron. As we shall see later, these free electrons are moving
charges and are therefore important in defining the properties of electronically conducting
materials.
The possible values of the orbital angular momentum for the electron in an atom are
1/2
given by [l(l + l)] h, where the quantum number / takes the values / = 0, 1, 2, 3,... ,
(n — 1); that is, / < (n — 1). Here again, just as the energy of the electron in a one-
electron atom is quantised, so is its orbital angular momentum. The orbital quantum
number / governs the angular momentum of the electron. In an external magnetic field,
we also find that the possible (allowed) values of the z-component of the orbital angular
momentum are denoted by m, where the quantum number m is called the magnetic
quantum number. It is called as such because in the absence of a magnetic field, all
the states defined by mh have the same energy values, that is, the states are degen-
erate. However, the application of a magnetic field lifts the degeneracy and each of these
states would have a different energy. The value of m can only be an integer between
2
-/ and +/, that is, m is -/, (-/ + 1), (-/ + ),..., 0, 1, 2 (/ - 2), (/ - 1), / or
l < m < +l.
To sum up, while in a specific shell n, the electron can have any one of a number of
allowed orbital angular momenta characterised by the quantum number / = 0, 1, ,... ,
2
(n — 1). These different / values correspond to different subshells, or orbitals, each
1/2
of angular momentum value of [l(l + l)] h. Hence, although an electron occupies a
certain shell n of certain energy E n, its orbital angular momentum can assume different
values. Within the same orbital, the projection of the angular momentum along the z-axis
can take certain values as described earlier. The quantum number for the z-component
of the angular momentum is m, where —l < m < +l. Therefore, at this point it can
be said that the state of the electron in a hydrogen atom is characterised by n, /,
and m.
So far, we have implied that the electronic distribution in an isolated atom can be
characterised by only three quantum numbers n, I, and m. However, in addition to orbital
angular momentum, it was found that each individual electron also has an intrinsic spin
angular momentum. The value of this spin angular momentum is characterised by the
quantum number 5 that assumes only one value — ^. The magnitude of this spin angular
1/2
momentum is [S(S + l)] h = * 3 h / 2 and is called the total spin. Actually, the spin of
an electron in a magnetic quantum number has two states identified by the spin quantum

2
The values of the fundamental constants are given in Appendix D.