Ch 10.4 Enrichments
Detailed derivation of the Marshall-Hall (1966) method for quantitative x-ray
microanalysis of biological thin sections.
10.4er.1 The Characteristic Signal
We will consider the derivation of the Marshall - Hall (1966) method from first
principles. More information on the physics of characteristic and continuum xray generation will be found in Chapter 6.
We can predict the number of characteristic X-rays generated into a full sphere
(4π steradians) from the following relation:

CA

N

ICh =
O A  QA FANedz
A

10.4er-1
where N0 is Avogadro's number,  is the density of the target in the analyzed
volume, CA is the weight fraction of the analyte in the volume and AA is the
atomic weight of element "A", QA is the ionization cross section for the shell of
interest and has dimensions of area; A, the fluorescent yield, is the probability
that an X-ray will be emitted due to the ionization of a given shell; F A is the
probability of emission of the X-ray line of interest relative to all the lines that can
be emitted because of ionization of the same shell; Ne is the number of electrons
that have irradiated the target during the measurement time; and dz is the target
thickness in the same units of length as used in Q and .
The quantity in parentheses in equation 10.4er-1 is the number of atoms of analyte
per unit volume of target and is obtained by the following rationale. By definition
the weight fraction of an element "A" is the mass of "A" divided by the total
mass. The mass of "A" per cm3 is the weight fraction of "A" times the density  in
the analytical volume given in g/cm3. We emphasize that density is a measured
quantity and refers to the mass of all the atoms per unit of volume. To convert
mass to number of atoms we use Avogadro's number, N0= 6.02•1023: the number
of atoms in a mole of an element. Therefore,the number of atoms of element "A"
per cm3 is (CA•)•(N0/AA) where AA is the gram atomic weight of a mole of
element "A". The total number of atoms in a volume of one cm3 ,N, is then the
sum over all the elements "i" in the volume:
N = NO

i
 C 
 i
A
 i
10.4er-2
The basic functional form of the characteristic cross section is due to Bethe
(1930):
-20
Q = 6.51•10
ns bs
2
UEC
ln  cs U
10.4er-3
where the constant is the product πe4 (in keV2- cm2; e is the charge of an
electron), ns is the number of electrons that populate the sth shell or subshell of
interest (nk=2), Ec is the energy required to remove an electron from a given shell
or subshell (the critical excitation energy), U is the overvoltage ratio E0/Ec (where
E0 is the energy of the impinging beam electron) and bs and cs are constants for
the s'th shell or subshell. The "area" in cm2 of Q is essentially the size of the K,L
or M shell "target" that a beam electron must "hit" to produce an ionization of that
shell. In general this area is about 100 times smaller than the area of the entire
atom.
9.6.2 The Continuum Signal
The interaction of a large number of beam electrons with a thin foil of a given
element produces an emitted continuum spectrum having a distribution
approximately shaped as 1/E, where E is the photon energy. The magnitude
(height) of this distribution is proportional to the number of atoms comprising the
foil. Other factors, such as beam current and measurement time will scale the
magnitude linearly. The beam energy, E0 , and observation angle, Ø, have a
second order effect on the shape of the distribution and will not be discussed here.
The observation angle is the angle between the x-ray detector and the direction of
travel of the electron beam and is not to be confused with the x-ray take-off angle,
usually designated by . We must make this distinction for thin targets because
the only effect of tilting a specimen away from the nomal is to thicken the
specimen while changing the observation angle will cause the detector to intercept
a different part of the non-uniform (anisotropic) distribution of the continuum
distribution.
We can predict the number of continuum X-rays generated into a unit steradian
(1/4π of a full sphere) from the following relation:
ICO =
 NQ N


i

i i
e
10.4er-4
where Ni is the number of atoms of element i and Qi is the continuum cross
section for that element and has dimensions of area and Ne is the number of
electrons that have irradiated the Ni target atoms during the measurement time. Qi
is usually assumed to be differential in photon energy and observation angle, Ø.
Consequently, it is not necessary to include dE and dØ (solid angle) terms in
equation 10.4er-4. We note that continuum cross sections usually are defined in
units of one steradian while the characteristic cross sections are usually specified
as radiating into the full sphere of 4π steradians. The reasons for this are beyond
the scope of this book and the reader is refered to Fiori, et al (1986) for a detailed
treatment and further references. Using the fact that the total number of atoms in
one cm3 is:
NO

i
 C 
 i
A
 i
10.4er-5
Equation 10.4er-5 can be rewritten for a thin film as:
 Q 
 Ci i  N dz
e
Ai
i 

10.4er-6
where dz is the target thickness in the same units of length as used in Qi and .
ICO = NO

9.6.3 Derivation of the Hall Procedure in Terms of X-ray Cross
Sections
The density of the target within the analytical volume of the electron probe is
generally not known unless the material being irradiated is a standard reference
material. We also note that the thickness of a target at the site of impact of the
electron beam is rarely a known quantity. However, it is obvious that a change in
either local density or thickness causes a proportionate change in the number of
atoms with which a beam electron is likely to interact in passing through a thin
target. Characteristic and continuum generation are equally affected. If mass loss
is occuring during the measurement period then we must assume that atoms are
being lost proportionally. To treat more easily these changes, density and
thickness are frequently combined into a single quantity, "mass thickness",
denoted by the notation (dz), that has the somewhat confusing dimensions of
g/cm2 (i.e., g/cm3•cm=g/cm2). We must stress that the concept of mass thickness
is useful only if the beam energy does not change significantly in traversing the
target thickness, dz. A typical biological or polymer target should be less than
several hundred nano meters in thickness for the above assumption to be accurate.
With this caveat we form the ratio of equations 10.4er-1 and 10.4er-6.
ICh
ICO
=

CA
 NO  Q  F Ne dz
A A A
AA



NO
i
 Q 
 Ci i  Ne dz
Ai


10.4er-7
We note that (dz), the mass thickness, Avogadro's number and the number of
electrons incident on the film during the measurement period all cancel in the
ratio. To further simplify the formula we take advantage of the fact that, for the
analysis of a given element, the fluorescent yield and associated relative transition
probability are constants that can be gathered into one grand constant, k. Thus we
have:
ICh
ICO
CA
= k

i
QA
AA
 Q 
 Ci i 
Ai


10.4er-8
Where QA is the characteristic cross section for a particular element "A" and Qi is
the continuum cross section for the ith element; the subscript i must span all the
elements present in the electron beam-target interaction volume. Equation 10.4er8 is the most general formula for the method proposed by Hall.
We now put into the numerator of equation 10.4er-8 the characteristic cross
section used by Marshall and Hall (the basic Bethe equation, Ch 3) and in the
denominator their simplified choice of a continuum cross section (for a derivation
see Fiori, et al 1986). We collect all numerical constants into k to obtain:
CA
ICh
=k
ICO

A
 A
U
2 
EC
ln CS U
 C Z2i  
EO
i  i Ai  ln 1.166 Ji 

 EO E  
10.4er-9
We can further simplify this formula by taking advantage of the fact that the
electron beam energy, E0 , is held constant during the analysis and the energy
band dE of continuum measurement, centered at the energy E, is usually not
changed during the analysis. Consequently, dE, E, E0 (and hence U) can be
absorbed into k resulting in the simpler form:
CA
ICh
= k
ICO
AA
  Z2
i   Ci Aii 

EO 

ln 1.166 J  
i 

10.4er-10
where Ji is a function of atomic number such as given in equation 3.10.
This is the usual representation of the Marshall-Hall equation which is a more
rigorous expression of the continuum correction concept proposed by Hall et al.
earlier (1966) and often referred to as the "Hall" method, or correction.
The equation can be used in the following manner: A measured characteristic to
continuum ratio from a well known material is set equal to the right side of
equation 10.4er-10. Since the target being irradiated is a reference standard we
will know the atomic numbers Zi, atomic weights Ai and weight fractions Ci.
Since all other terms except the constant k are known, k can be calculated for the
given set of experimental conditions. Next, we hold these conditions constant and
measure the characteristic and continuum intensities from the specimen. To
calculate a weight fraction of analyte, CA, from the specimen we must know the
weight fraction of each and every one of the elements that comprise that part of
the specimen irradiated by the electron beam. Crucial to the Hall procedure is the
following assumption: The analytes of interest are minor or trace constituents
such that the quantity ∑(Ci•Zi2/Ai) for the biological or polymeric specimen to be
analyzed is dominated by the matrix. In this case the unknown contribution of the
analyte, CA, to the sum may be neglected. Furthermore, the value of the sum is
known or can be estimated from other information about the detail being analyzed
(the actual, localized, material being irradiated by the electrons, and not some
bulk property). Some representative values of ∑(Ci•Zi2/Ai) are 3.67(water),
3.01(Nylon), 3.08(Polycarbonate) and 3.28(Protein with S). Typically the range is
between 2.8 to 3.8 for most biological and many polymeric materials. This basic
theme can be extended and several analytes may be analyzed simultaneously and
at higher concentrations. The details of this extention are beyond the scope of this
book but a full description and derivation can be found in Kitazawa, et al. (1983).
Most modern commercial computer x-ray analyzer systems have the full Hall
procedure included in their suite of analysis tools.
The Hall procedure works well for "thin" specimens in the energy regime of the
SEM and electron probe microanalyzer and the analytical electron microscope.
The method will not work for specimens where the average atomic number is
expected to vary significantly from one analysis point to another, or to that of the
standard. Consequently, most applications in materials science can not be
accommodated by the procedure.
References
Bethe, H. (1930). Ann. Phys. (Leipzig). 5, 325.
Fiori, C. E., Swyt, C. R., and Ellis, J. R. (1986). In Principles of Analytical Electron
Microscopy, Eds. D.C. Joy, A.D. Romig and J.I. Goldstein, Plenum Press, New York,
Chapter 13.
Hall, T.A. et al. (1966). In The Electron Microprobe, Eds. McKinley, T.D.
Heinrich, K.F.J. and Wittry, D., John Wiley and Sons, New York, 805.
Marshall, D. J. and Hall, T. A. (1966). In X-Ray Optics and Microanalysis, Castaing,
Deschamps, and Philibert, eds, Hermann, Paris, 374.