The Continuum Normalisation Method for Quantification of X-ray Spectra in Biological
Microanalysis. -- “ The Hall theory revisited”.
1. Generalised Bremsstrahlung production cross-sections and analysis using standards
Theoretical development using bremsstrahlung cross-sections.
The assumptions made or conditions required in all earlier developments were that measurements
on standards are made at the same electron energy as used for the specimen and that both the
specimen and standard are sufficiently thin that energy loss within the sample is negligible.
The concept of replacing Kramers (1923) simple model for bremsstrahlung production as used by
Hall (1971) in the CN method, appears in Nicholson and Chapman (1983), Fiori et al (1989). In
Kramers (1923) the photon intensity is described by a function proportional to the square of the
atomic number of the sample and inversely proportional to the electron energy and photon energy
and is considered isotropic with respect to emission angle. Fiori et al 1989, considered some further
aspects and noted that the most important parameter is the relative mean atomic numbers of the
sample Zsp and standard Zstd. They examined the effect that various models may have on the
accuracy of quantification using the CN method when Zsp and Zstd differ. The above analyses were
restricted to the approximate Hall equation (see eqn 18, below).
Here we incorporate a generalised cross-section into the full equation which is suitable for any type
of sample and then give a useful approximation suitable for samples which have low concentrations
of heavy elements (i.e. Z>11) in an organic matrix. The ‘thin sample’ condition above is still
required so that the bremsstrahlung production cross-sections can be considered constants.
The analytical development of the final equation in Hall 1971 is not particularly simple, so the
present development will follow it as closely as possible and to facilitate direct comparison some of
the original equation numbers used in Hall (1971) are given in square brackets. The original theory
is written in terms of the local weight fraction Cx, of the analysed element x, which is defined by
the ratio of the local mass per unit area of the analysed element and the total local mass per unit
area.
As an electron travels a distance ds in the specimen it generates an average number of characteristic
X-rays  x ds , given by
x ds   cx N x ds
(1)[70]
where the characteristic X-ray production cross-section per atom  cx   x S x ix , with x the
fluorescence yield, ix the ionisation cross-section per atom and Sx the partition function. Nx is the
number of atoms of type x per unit volume. In Hall (1971) the number of bremsstrahlung quanta
bdkds in the photon energy range from k to (k+dk) is expressed using the Kramers (1923) theory for
bremsstrahlung production so that:b dkds 
 dk
 N Z2 ds
t0 k r r r
[71]
1
where  is a constant, t0 the energy of the incident electrons, Z the atomic number and r is an index
running over all elements in the sample. If the experimental conditions are chosen appropriately, as
discussed below, this formulation will only lead to small errors. Here we prefer to describe the
bremsstrahlung production in terms of a general bremsstrahlung production cross-section.
2
A more accurate formulation of the bremsstrahlung intensity may then be used once the derivation of
the continuum normalisation method is complete.
 b dkds
  br N r ds
(2)
r
where br is the bremsstrahlung production cross-section for r type atoms integrated over the photon
 dk 2
Z
energy range dk. Thus for Kramers theory  br 
t0 k r
Dividing equation 1 by equation 2 for the specimen by the same ratio for the standard, we define Rx
Rx 
(character istic count/brem sstrahlung count) sp
(character istic count/brem sstrahlung count) std

Px
Px
W sp
W std


 Nx



  N r  br 
 r
 sp



 Nx



  N r br 
 r
 std
(3)[72]
We assume we know what the specimen contains in the way of `interesting' elements and matrix,
just not how much of each. Here the subscripts sp and std refer to the specimen and standard
respectively. The unknowns here are Nx and Nr in the specimen.
In the Hall (1971), version of this equation (numbered [72] in that work), the cross-sections appear
as terms in Z2. It is here the implicit assumption of constant electron energy is made. In this
formulation that condition may be relaxed so that ‘old’ standard data measured at a different
electron energy may be used. Since cross-sections are used throughout, this is true whether they be
calculated by Kramer’s theory (above) or another formulation.
We define a constant Gx, called the standard factor

Gx   Nx

N 
r
r
br


 std
We may put equation 3 into terms of weight-fractions using the identity C x  N x A x
(4)
N A
r
r
,
r
where Ax is the atomic weight of element x. Using this identity, substituting Gx, and re-arranging
equation 3 becomes:
C x  R x A x G x  C r  br A r
(5)[73]
r
Hence for any two elements a and b
Ca R a A a G a

Cb R b A b G b
(6)
since the sum terms cancel.
3
C
If we sum equation 6 over all r, since
 1 , we can also say that C x  C x
r
r
C
r
and thus for
r
any element x
Cx 
R x A xG x
 R r A rGr
(7)
r
Separating the denominator into matrix elements m, for which we record NO characteristic lines and
u the unknown (interesting) elements which include x
Cx 
R x A xG x
 R m A mGm   R u A uGu
m
(8)[76]
u
The problem here is to evaluate the
R
m
A m G m , since all the other terms on the right hand side are
m
known or can be measured. Re-arranging equation 5
R xA xG x 
Cx
 C r  br A r
(9)
r
Equation 9 is valid for any element x including those in group m. Multiplying both sides of
equation 9 by  bx A x and sum both sides over m
R
m
m
G m  bm
C 

C 
m
bm
Am
r
br
Ar
m
(10)
r
In particular, note that if we extend the sum over all elements we see
R G
r
r
 br = 1 since the
r
numerator and the denominator of the right hand side will be equal, from which it can be deduced
that:
R
m
G m bm  1   R u G u  bu
m
Summing equation 9 over m
R
(11)
u
m
A mG m 
m
C
m
m
C 
r
(12)
Ar
br
r
Divide equation 12 by equation 10 to eliminate
C 
r
br
A r and re-arrange
r
R
m
m
A mG m 
C
C
m
m
m
 bm A m
R
m
G m  bm
m
m
4

C
C
m
m
m
 bm A m


 1   R u G u  bu 


u
(13)
m
The object of the next section is to eliminate the ratio term in equation 13 since this has terms in Cm,
the concentrations of the various matrix elements in the sample which we don't know. We replace
them with terms involving the fractions of matrix elements m, in the matrix C/m, for which we can
choose a sufficiently accurate approximation. Hence C /m is defined as:
C /m 
Cm
 Cm
(14)[78]
m
Multiply both sides of 14 by  bm A m and sum over m therefore:
C
/
m
 bm A m 
C
m
m
m
 bm A m
(15)
 Cm
m
Substituting the above into equation 13 it becomes:
1   Ru Gu bu
u
Rm Am Gm 

C
m
 m/  bm Am
[77]
m
Substituting this into equation 8
R xA xG x
Cx 
1   R u G u  bu
u
 R A G
 C /m bm A m u u u u
16[79]
m
Defining    C  bm A m
/
m
and substituting this into Equation 16
m
Cx 
R x Ax G x 
1   Ru Gu bu  
u
R
u
Au Gu
(17)
u
The only X-ray lines which must be observed apart from that of x, are the non-matrix elements
which may contribute significantly to the bremsstrahlung. In the case of biological thin sections, this
is not usually necessary. If all of the non-matrix elements including x are present in low
concentration, e.g. less than a few percent by weight at the time of analysis, their contribution to the
bremsstrahlung will be insignificant. From this follows that the terms Ru in equation 17 which are
the ratios of the peak to bremsstrahlung ratio of the elements u in the sample to those in the standard,
then these terms will be very small and the equation may be simplified to
5
Cx  R x A xG x
(18)
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