Biomedical Engineering Reference
In-Depth Information
field lines up only in certain discrete directions with respect to the external field.
The magnetic quantum number gives the component of the orbital angular mo-
mentum in the direction of the external field. Accordingly,
m
can have any inte-
gral value between
+
l
and
-
l
;
viz.,
m
= 0, ±1, ±2,
...
, ±
l
.With
l
= 1
, for example,
m
= -1, 0, 1
.
Although the Bohr-Sommerfeld theory explained a number of features of atomic
spectra, problems still persisted. Unexplained was the fact that the alkali-metal
spectra (e.g., Na) show a doublet structure even though these atoms have only a sin-
gle valence electron in their outer shell (as we show in the next section). In addition,
spectral lines do not split into a normal pattern in a weak magnetic field (anom-
alous Zeeman effect). These problems were cleared up when Pauli introduced a
fourth quantum number of “two-valuedness,” having no classical analogue. Then,
in 1925, Uhlenbeck and Goudsmit proposed that the electron has an intrinsic an-
gular momentum
2
due to rotation about its own axis; thus the physical signifi-
cance of Pauli's fourth quantum number was evident. The electron's intrinsic spin
endows it with magnetic properties. The spin quantum number,
s
, has two values,
s
1
2
. In an external magnetic field, the electron aligns itself either with “spin
up” or “spin down” with respect to the field direction.
The Pauli exclusion principle states that no two electrons in an atom can occupy
a state with the same set of four quantum numbers
n
,
l
,
m
,and
s
. The principle can
also be expressed equivalently, but more generally, by saying that no two electrons
in a system can have the same complete set of quantum numbers. Beyond atomic
physics, the Pauli exclusion principle applies to all types of identical particles of
half-integral spin (called fermions and having intrinsic angular momentum
2
,
3
2
=±
, etc.). Such particles include positrons, protons, neutrons, muons, and others.
Integral-spin particles (called bosons) do not obey the exclusion principle. These
include photons, alpha particles, pions, and others.
We next apply the Pauli principle as a basis for understanding the periodic sys-
tem of the elements.
2.7
Atomic Theory of the Periodic System
The K shell, with
n
1, can contain at most two electrons, since
l
0,
m
0,and
=
=
=
1
2
are the only possible values of the other three quantum numbers. The two
electrons in the K shell differ only in their spin directions. The element with atomic
number
Z
s
=±
2 is the noble gas helium. Like the other noble-gas atoms it has a
completed outer shell and is chemically inert. The electron configurations of H and
He are designated, respectively, as 1s
1
and 1s
2
. The symbols in the configurations
give the principal quantum number, a letter designating the azimuthal quantum
number (s denotes
l
=
3)
, and a superscript
giving the total number of electrons in the states with the given values of
n
and
l
.
The electron configurations of each element are shown in the periodic table in the
=
0
; p denotes
l
=
1;
d,
l
=
2;
and f,
l
=
