Ch. 6 Enhancements
E6.1
Quantum Numbers and Characteristic X-ray Production
Characteristic x-ray emission consists of very narrow lines of specific
energies which appear in definite groups for each element. Groups of lines with similar
energies are designated the K-series, L-series, and M-series. The energies of these
characteristic x-ray emission lines are a consequence of electron transitions between
specific energy levels in the atom; energy levels defined by quantum numbers.
Quantum numbers. The Pauli exclusion principle tells us that only one electron in
an atom can have the same set of quantum numbers. Thus, to understand atomic energy
levels, we must learn how each electron in its energy level is described by a specific set of
quantum numbers: (n, l , s, j, mj).
(1) n, the principal quantum number, denotes a shell in which the electrons have
nearly the same energy: n = 1 corresponds to the shell designated K in x-ray terminology,
n = 2 (L shell), n = 3 (M shell), n = 4 (N shell), etc. Changes in n have the largest effect
on the energies of the atomic energy levels.
(2) l , the orbital or azimuthal quantum number, characterizes the orbital angular
momentum of an electron in a shell. This quantum number is restricted to the values of l
= 0 or l= n–1.
(3) s, the spin quantum number, describes that part of the total angular
momentum due to the electron spinning on its own axis. It is restricted to values for
positive and negative electron spin: ±1/2.
(4) j , the total angular momentum quantum number, describes the magnetic
coupling between the spin and orbital angular momenta. This quantum number takes on
the positive values of j = l ± s. Thus, for the L-levels, n = 2, l = 0 or 1, and j takes on
values of 0, 1/2, and 3/2 giving rise to the three subshells of LI, LII, and LIII. The number
of electrons occupying each subshell is given by 2j+1. So the levels LI, LII, and LIII can
have up to 2, 2, and 4 electrons, respectively.
(5) mj, the magnetic quantum number, represents the different orientations taken
by the angular momentum vector under the influence of an applied magnetic field. This
quantum number does not play a large part in x-ray microanalysis, and will not be
discussed further.
The Pauli exclusion principle restricts the number of electrons in an energy level
to two, one with spin up and the other with spin down. Thus, each electron has a unique
set of quantum numbers which describes it.
Electron transitions. Electromagnetic radiation is emitted when an electron moves
from one energy level to another to fill an electron vacancy arising, say, from electron
ionization by the SEM electron beam. The specific energy of a characteristic x-ray
photon is the energy difference between two inner electron levels that are normally filled
with electrons. Table E6.1 lists the shells and subshells, their quantum numbers, the
modern chemistry notation, and the maximum electron occupation of each level. For
characteristic x-ray emission, the following selection rules must be satisfied for allowed
transitions: n ≥ 0, l = 1, j = 0 or 1, but j = 0 is forbidden if j = 0 initially. This
means that not all transitions between energy levels result in characteristic x-rays.
Consider the example of the copper K-series. Electron transitions to fill a vacant K state
from the L energy levels in the Cu atom can illustrate this point:
Electron moves from LIII  K
(l = 1, j = 0)
K1 x-ray of EK1 = EK – ELIII
Electron moves from LII  K
(l = 1, j = 1)
Electron moves from LI  K
(j = 0, j = 0 initially)
K2 x-ray of EK2 = EK – ELII
No x-ray emission
There are other lines that do not follow the above selection rules, so called “non-diagram”
and “satellite” lines. Non-diagram lines arise from very low probability transitions where
l > 1. Satellite lines are weak lines that occur when an atom is doubly ionized, perhaps
missing both a K and an L electron. These lines are slightly higher in energy because the
electron screening of the nucleus is less and the binding energy of the electron is greater.
Thus, there are several weak satellite lines, that can barely be detected with a WDS
system, at energies slightly higher than the K1 line.
Critical Ionization Energy. Inner-shell ionization occurs when an electron is
removed from a shell and ejected from the atom. Because the energy of each energy level
(shell or subshell) is sharply defined, the minimum energy necessary to remove an
electron from a specific is specific and characteristic, the critical ionization energy Ec
(also known as EK , ELIII, etc.). Each shell and subshell of an atom requires a different
critical ionization energy; for example, Ec for copper K, LI, LII, and LIII are respectively
8979, 1096, 951, and 931 eV. Thus, the energy of the K1 line of Cu is
EK 1  EK  ELIII  8979 931  8048 eV
Cu
EK 2  EK  EL II  8979 951  8028 eV .
Cu
Because there are twice as many electrons occupying the LIII level compared to the LII
level (Table E6.1), the x-ray intensity for K1 is twice that of K2. However, with only 20
eV separating these lines, a WDS system is usually required to resolve the separate
members of the K1 /K2 doublet.
Table E6.1
Quantum Numbers for Electrons in Atomic Electron Shells
Quantum Numbers for Energy Levels
Maximum
electron
population
X-ray
notation
K
Modern
notation
1s
n
l
j=l+s
(2j + 1)
1
0
1/2
2
LI
LII
LIII
2s
2p1/2
2p3/2
2
2
2
0
1
1
1/2
1/2
3/2
2
2
4
MI
MII
MIII
MIV
MIV
3s
3p1/2
3p3/2
3d3/2
3d5/2
3
3
3
3
3
0
1
1
2
2
1/2
1/2
3/2
3/2
5/2
2
2
4
4
6
NI
NII
NIII
NIV
NV
NVI
NVII
4s
4p1/2
3/2
4p
4d3/2
4d5/2
4f5/2
4f7/2
4
4
4
4
4
4
4
0
1
1
2
2
3
3
1/2
1/2
3/2
3/2
5/2
5/2
7/2
2
2
4
4
6
6
8