Shirley Background
Shirley Background
Raw Spectrum – O (1s) Signal
Example: Raw spectrum of O (1s) signal from a silicon wafer – no background
O (1s) Signal with Iterated Shirley Background
Iterated Shirley Background was added. The start and finish endpoints are located at 524 eV and 540 eV. The next spectrum is a Vertical Expansion to show the endpoints.
Vertically Expanded View of O (1s) Signal with Shirley Background
The O (1s) spectrum was vertically expanded to verify that the Shirley Background endpoints were added to flat regions of the spectrum.
Shirley Background compared to Linear Background
Raw spectrum of Ni (2p3) from pure Nickel metal
Overlay of Shirley Background and Linear Background – having identical endpoints – for Ni (2p3) of pure Nickel metal
Overlay of 3 different Shirley Backgrounds – for Ti (2p) spectrum of fresh exposed bulk of a single crystal of TiO2
Which is the best choice? It some sense, it depends on what sort of information you want.
Best choice depends on whether or not you are “peak-fitting” or measuring “atom% values”.
Two slightly different Shirley Backgrounds on a C (1s) spectrum. (Ignore the linear backgrounds)
The 2 Shirley backgrounds have different endpoints at the High BE end. This small difference produces a small 0.1/25.57 variance in the atom%, a small effect in this example.
From J. Vegh, Surface Science 563, (20040 183-190
From Practical Surface Analysis: Appendix 3, written by Peter Sherwood.
From J. Vegh, Surface Science 563, (20040 183-190
3. The Shirley cross-section function
In order to derive the Shirley-equivalent cross section function, we have some physical conditions the sought function needs to meet, a mathematical condition that defines it and some experiences that help to determine the parameters of the function.
3.1. Physical conditions
A cross-section function describes the probability of losing a certain amount of energy, so it must be non-negative and finite all over its range of interpretation. The electrons cannot gain energy due to inelastic collisions, so at negative loss
energies its value must be exactly zero. One can
expect that very large energy losses have insignificant
probability, i.e. the value of the function
must disappear as the lost energy approaches
infinity, in the same way as the Tougaard-type and
the experimental cross-sections do. Conversely, at
low loss energies the value of the trial cross-section
function must be much higher than that of the
Tougaard type functions, in order to explain the
greatly different behavior of the two background
correction procedures near to photopeaks. The
integral of the cross-section function over the
interval [0,1] means the total probability of
loosing energy via inelastic collision, so it must
converge. The integral of both types of cross-section
must be comparable. Exact equivalence
cannot be expected, because in the practical evaluation
the cross-sections are integrated over a
finite rather than an infinite energy range. The
background correction procedure uses the integral
over a finite energy range as a measure of the
inelastic background and so the integrals of these
two total cross-sections over a limited energy
J. Vegh / Surface Science 563 (2004) 183–190 185
range need to be nearly equal, rather than the
integral over the energy interval [0,1].
3.2. Mathematical form
We have two independent derivations of the
Shirley-type inelastic background. Once the
empirical procedure results in Eq. (1) and through
summing up the contributions due to inelastically
scattered electrons we get Eq. (6). Because both of
these expressions represent the same physical object,
they must be equal and the sought KðEÞ
function must satisfy the condition:
k
Z þ1
E
PðE0ÞdE0 ¼
X1
n¼1
PðEÞ KðEÞn: ð7Þ
This equation provides a possibility for finding
the ‘‘Shirley-equivalent inelastic energy loss crosssection’’
function KðEÞ. Omitting the arbitrary
constant k and Fourier transforming Eq. (7), the
right side transforms into a geometrical series that
can be immediately summed up. After transforming
the result back one gets the final result that the
form of the kernel is
KðxÞ ¼ dðxÞ
1 þ ix
1 þ x2
: ð8Þ
We are obviously interested in the real part of
the function and because of the arbitrary constant
k in Eq. (1), some physical assumptions are also
needed to derive the exact function form. For
comparison with the Tougaard’s two-parameter
‘‘universal’’ cross-section function
KTðTÞ ¼
BT T
ðCT þ T 2Þ2 ; ð9Þ
the ‘‘Shirley-equivalent’’ cross-section function
derived by the author is sought in form
KSðTÞ ¼
BS
CS þ T 2
; ð10Þ
where T is the lost energy. Here and below, the
coefficients B and C are used in connection with
both the Shirley-equivalent scattering function and
the Tougaard scattering function. In order to
avoid confusion, the coefficients are subscripted
with the first letter of the corresponding method.
Note that because of the dissimilarity of the
function forms, BT and BS have different units.
It has been shown [9,12] that the integral of the
cross-section function has a physical meaning: it
gives the intensity of the first loss spectrum. In case
of the Tougaard ‘‘universal’’ cross-section function
its integral A is expressed as
A ¼
BT
2 CT
: ð11Þ
Replacing BT and CT with the well-known values
[7] or the ones found in the elaborate experimental
work by Seah [12] (or the ones adjusted to
the spectrum in question), one can calculate the
value of A. As it has been pointed out above, this
number shall be equal to the integral of the sought
function. This latter can be expressed as
A ¼
BS ffiCffiffiffiffiffi S
p
p
2
; ð12Þ
i.e. one can derive only the ratio of the coefficients
from the area of the function.
Fortunately, there exists some further analogy
between the shape of the Tougaard function and
that of the present one. The BS coefficient is the
‘‘scattering intensity’’ and CS describes the ‘‘distribution
width’’ of the function. The Tougaard
function drops to half of its reached maximum
height at ca. 25–30 eV higher after reaching its
maximum. If one requires a similar feature for the
present trial function, one finds that CS should be
about 600–900 eV2 and correspondingly BS about
15–20 eV. These assumptions only serve as a rough
guide in guessing the magnitude, the correct values
of the coefficients need to be derived from experimental
spectra, a work in progress.
3.3. Performance
Using the function form and the estimated
coefficients, one can re-formulate the Shirley
background correction method (see Eq. (1)) again
in complete analogy with the Tougaard method
(see Eq. (3)) as
SSðEÞ ¼
Z þ1
E
KSðE0 EÞ ðjðE0Þ SS;0ðE0ÞÞdE0;
ð13Þ
186 J. Vegh / Surface Science 563 (2004) 183–190
where KSðEÞ is the cross-section function given by
Eq. (10). In the followings Eq. (13) is called the
‘‘cross-section based’’ and Eq. (1) the ‘‘classic’’
Shirley background correction method. One can
test the features of the trial cross-section function
through comparing them directly or comparing
the resulting backgrounds.
In Fig. 1 the trial cross-section function is shown
together with Tougaard’s ‘‘universal’’ cross-section.
The present cross-section has its maximum at
zero loss energy and gives much more emphasize to
the very low energy inelastic energy losses as it
could be expected on the basis of theoretical
assumptions or experimental results. The line
‘‘ShirleyCS’’ is calculated with parameters
CS ¼ 2500, BS ¼ 27:8, which assumes exact agreement
of the integral with that of the Tougaard’s
function. When using the trial cross-section function
for evaluating a measured Ag 3d spectrum (see
Fig. 2) the coefficients shall be adjusted to another
values (BS ¼ 36, CS ¼ 900), which results in using
the cross-section marked by ‘‘Shirley’’ in Fig. 1.
In Fig. 2 the backgrounds fitted to a measured
Ag 3d spectrum by the different Shirley methods are
compared. The classic Shirley method with no
iteration [1] does not provide a satisfactory background,
because the difference in the background
levels is too big here. After iterating that initial
background as described in [3], the method produces
a perfect Shirley background. The cross-section
based Shirley background (generated in the
same way as one produces a Tougaard background)
is nearly identical with the classic background, if the
parameter BS is adjusted properly. Note that the
parameters of the backgrounds shown in Fig. 2 are
slightly changed from their ‘‘best’’ values in order to
make the individual curves visible.
The present results strongly contradict to the
former statements on the shape of this cross-section
function like in [13]: ‘‘Shirley’s method is traditionally
used but does not represent the accurate
energy loss spectrum responsible for the peak
background to be subtracted: it is assumed that the
probability for one electron of energy E to lose an
energy T does not depend on E or on T ’’. The presented
cross-section function proves that the
Shirley phenomenon can be explained assuming that
the inelastic scattering process is governed by an
inelastic cross-section function. The present paper
does not state that the real inelastic energy loss
cross-section can be described with the derived
function, rather it explains that such a cross-section
function is implicitly assumed when one uses the
Shirley background correction procedure.
3.4. Peak tails
As has been pointed out earlier [5,8,9], attaching
the contributions due to inelastically scattered
electrons in form of ‘‘tails’’ to the individual
photopeaks results in some advantages. One can
Fig. 1. Comparing the present 1=ð1 þ x2Þ-type cross-sections
(the one resulting the same inelastic loss than the Tougard’s
‘‘universal’’ function and the one with parameters resulting in a
reasonable fit to the background on the left side of the Ag 3d
peak shown in Fig. 2) to the Tougaard’s ‘‘universal’’ function
[7].
Fig. 2. Different Shirley backgrounds to an Ag 3d spectrum:
the non-iterated Shirley method [1], the iterated Sherwood
method [3] and the present method.
J. Vegh / Surface Science 563 (2004) 183–190 187
derive cross-section based photopeak tail for the
present Shirley cross-section in the same way as it
was done for the Tougaard cross-section [9]. As
Eq. (6) shows, the total contribution due to scattered
electrons is given by a sum, i.e. in principle it
should be calculated trough convoluting the primary
distribution with the inelastic energy loss
cross-section function a large number of times and
summing up the contributions.
In the linear convolution algorithms, the convolved
array does not change during the convolution
and the returned result contains the array
convolved exactly once. In this special case the
features of the inelastic scattering process allow to
make modifications [9] on the algorithm due to
which the returned result will contain the sum of
all contributions up to infinite order, i.e. Eq. (6)
can be calculated in one single linear-convolution
like step as outlined on the flow chart in Fig. 3.
The key points here are the shaded rectangles.
Since the electrons can not gain energy during
inelastic collisions (i.e. the cross-section function is
exactly zero for negative loss energies), if one calculates
the contribution due to the inelastic collision
of electrons with energy Ei, it surely will not affect
contributions at energies higher than Ei. Because of
this, if one starts the calculation at the highest
available energy and proceeds towards the lower
energy points, the already calculated contributions
will not change any more. It is known from the
physics of the scattering that the probability of
being scattered does not depend on the previous
energy losses, i.e. scattered electrons with their new
energy are to be accounted immediately when calculating
the contributions at energies Ej, j < i.
Increasing the amount of the electrons at electron
energy Ej with the amount of scattered electrons
from higher energies accounts for this physical effect
and so after repeating the steps until the lowest
energy reached, the total contribution due to all
inelastic collisions will be produced. The mathematical
correctness of this one-step procedure was
proved by Graat et al. [14] using Fourier-transform.
In Fig. 4 different peak tails due to inelastic
collisions (calculated using the present method and
the ones in [5,9]) are added to a peak. As shown, in
case of the Shirley-type backgrounds the tail is
constant in the low energy side of the spectrum, at
reasonable distance from the peak. As expected, in
the vicinity of the photopeak the shape of the tail
depends strongly on the parameters of the crosssection
function.