9.2 Xray series
The Xrays emitted in transitions between innershell energy levels will have energies corresponding
to the difference in binding energy of the electrons in the two shells concerned. The binding energy
of an electron in a given shell of quantum number,
n
, may be expressed using a hydrogenic model:
E
n
=
R
(
Z

σ
n
)
2
n
2
(211)
Where
R
is Rydberg’s constant.
σ
n
is a screening factor that accounts for the effect of the other
electrons.
For the Kshell (
n
= 1) there are 2 electrons. As the ejected electron moves outwards the remaining
electron provides a spherically symmetric shell around the nucleus of
Z
protons and reduces the
effective nuclear charge to (
Z

1).
The other electrons in higher shells also make a contribution
to the screening. The total screening factor,
σ
k
is then approximately 2. It is difficult to calculate
screening factors, although good estimates can be made using atomic structure calculations. Usually
we reply on experimental (empirical) values for
σ
. (The screening factors also depend on the angular
momentum of the states involved.)
Transitions from higher shells to a vacancy in the Kshell give rise to a series of lines.
The
wavenumber (
ν
= 1
/λ
) of these lines will be given by the differences in the binding energies:
Kseries:
ν
K
=
R
(
Z

σ
K
)
2
1
2

(
Z

σ
i
)
2
n
2
i
(212)
Where
n
i
= 2
,
3
,
4 etc.
In general:
ν
X
=
R
(
Z

σ
i
)
2
n
2
i

(
Z

σ
j
)
2
n
2
j
(213)
With
n
i
,
n
j
integers and
n
i
< n
j
.
Kseries
Lseries
Mseries
n=4
n=3
n=2
n=1
K
L
M
N
abg
abg
abg
Figure 37: Origin of Xray series from vacancies created in inner shells.
The longest wavelength series member is labelled
α
, with successive lines denoted
β, γ
etc.
57
Atomic Physics, P. Ewart
9 XRays: transitions involving inner shell electrons
9.3 Fine structure of Xray spectra
A single vacancy in an otherwise full shell has the properties of a single electron in an otherwise empty
shell. The Xray energy levels therefore resemble those of hydrogen or alkali atoms. The energy levels
are split into
terms
and the terms are split by spinorbit interactions giving “fine structure”. The
energy splitting due to fine structure can be written
Δ
E
fs
=
5
.
8
Z
4
n
3
l
(
l
+ 1)
(214)
The levels are labelled by quantum numbers (
n, l, s, j
). The “
Z
4
” factor results in very large “fine
structure” splitting for heavy elements (large
Z
) eg. 10
7
cm

1 in Uranium!
This structure was relatively easy to measure and for a long time such measurements gave the
most accurate values of
α
, the fine structure constant. The Xray lines have a multiplet structure
governed by selection rules:
Δ
l
=
±
1
,
Δ
j
= 0
,
±
1
(215)
e.g. the K
α
line becomes a doublet K
α
1
, K
α
2
.
9.4 Xray absorption
Absorption spectra in the visible or UVrange of the spectrum consist of a series of discrete lines
whose wavelengths coverage to a series limit; the ionization limit. The strength of the lines decreases
also towards the ionization limit. For shorter wavelengths than this limit the absorption is spectrally
continuous and continues to decrease in strength as the absorbed wavelengths get shorter.
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 Physics, Radiation, Atomic physics, central field, central field approximation, P. Ewart