Microchim Acta (2007)

DOI 10.1007/s00604-007-0821-0

Printed in the Netherlands

Original Paper

Estimation of uncertainty in electron probe microanalysis: irondetermination in manuscripts, a case study

Kristina Virro1, Enn Mellikov2, Olga Volobujeva2, Vaino Sammelselg3, Jelena Asari3,

Lilli Paama1, Jaana Jurgens1, Ivo Leito1

1 Institute of Chemical Physics, University of Tartu, Tartu, Estonia2 Department of Materials Science, Tallinn Technical University, Tallinn, Estonia3 Institute of Physical Chemistry, University of Tartu, Tartu, Estonia

Received 16 November 2006; Accepted 31 May 2007; Published online 6 August 2007

# Springer-Verlag 2007

Abstract. A case study of ISOGUM-based estimation

of measurement uncertainty of quantitative surface

elemental analysis is presented. The analytical task

was the measurement of iron content in the ink writing

on the surface of an 18th century manuscript by elec-

tron probe microanalysis using a scanning electron

microscope equipped with an energy-dispersive X-

ray spectrometer (SEM=EDS). The problems that arise

in uncertainty estimation of quantitative surface anal-

ysis are outlined (defining the measurand, preparing

calibration standards, non-uniformity of the surface,

etc) and ways of overcoming them are suggested.

The average iron content on the ink-covered surface

of the sample manuscript was found to be 0.12�0.04mgmm�2 (at confidence level 95.5%). Paper sheets

with ink lines of known iron contents were used as

calibration standards. The main source of uncertainty

was the variability of the parallel measurement results

from different locations of the sample surface (mostly

due to the variations in the surface structure of paper),

which contributed 78% of the total uncertainty. It is

concluded that EPMA using a SEM=EDS is suitable

for at least semi-quantitative determination of iron in

the writing of ink-written manuscripts. Several specific

issues in uncertainty analysis are pointed out that need

further investigation.

Keywords: Measurement uncertainty; scanning electron micros-

copy (SEM); electron probe microanalysis (EPMA); SEM=EDS;iron-gall ink; manuscript

Significant progress in uncertainty estimation of

chemical measurement results has been made in the

recent years. In addition to conceptual works quite

some practical case studies and examples are available.

These are included as examples in guidance docu-

ments, such as the ISO GUM [1], the EURACHEM=CITAC uncertainty estimation guide [2] or the

Nordtest handbook [3], in journal articles [4–9] and

also on some measurement uncertainty related web-

sites [10].

All these, however, relate only to the conventional

‘‘macroscopic’’ chemical analysis. Besides this, local

and=or surface-analysis techniques are nowadays usedmore and more frequently. These measurements as

any other have uncertainty. Due to the problems with

defining the measurand, the (often high) non-unifor-

mity of the surface and other problems, estimating

Electronic Supplementary Material. The full uncertainty calcu-

lation file in MS Excel 2002 format is available as electronic

supplementary material (ESM). This material is available online

at http:==www.ut.ee=katsekoda=GUM_examples=EDS_ESM.xls.

Correspondence: Ivo Leito, Institute of Chemical Physics,

University of Tartu, Jakobi 2, 51014 Tartu, Estonia, e-mail: ivo.

leito@ut.ee

uncertainty in surface analysis is not an easy task.

Hence, not surprisingly, although there is no shortage

of papers devoted to surface analysis of a great variety

of objects, including quantitative analysis, the uncer-

tainty of such measurements, especially for materials

outside the alloy, mineral and semiconductor domain,

has received only limited attention. The sources of

uncertainty in such analysis have been studied, but

far less is known about the evaluation of their magni-

tude [11–15]. Works where the different uncertainty

sources are quantified and combined according to a

measurement model to yield the combined uncertainty

of the result are to the best of our knowledge com-

pletely absent from the literature. Such approach would

be valuable, as it would enable to analyze the contribu-

tions of different uncertainty sources and to point out

ways to improve the measurement procedure.

Among the objects that are analyzed using the sur-

face analysis methods, old manuscripts have an im-

portant place. Manuscripts and drawings made with

the iron-gall ink (active components are tannic acid

and iron(II)sulfate) form a major part of the European

cultural heritage. Iron-gall ink was the most common

ink used before the 19th century [16]. Unfortunately,

all such manuscripts are threatened by the so-called

ink corrosion – deterioration of paper by ink compo-

nents like acids and ions of metals (e.g. Fe, Cu) [16,

17]. The paper looses its mechanical strength and the

unique artifacts are damaged forever. This has become

a serious problem in the field of cultural heritage pres-

ervation. The problem has been recognized and large

collaborative efforts have been undertaken to find ways

of stopping manuscript degradation. See the InkCor

project homepage [17] and the Ink Corrosion Website

[16] for a good overview of the state of the art in this

field. The mechanisms of the processes are still under

investigation [16, 17]. Various parameters of the ink

and the paper have their influence on the corrosion

and on the possible means of retarding and stopping

it. An important factor among them is the iron content

in the ink writing on the paper surface. Exact cor-

relations have not been published, because there is

not enough iron content data in manuscripts available.

Therefore development of quantitative analysis meth-

ods are very important.

The broader goal of our research in this direction is

to evaluate the electron probe microanalysis approach

based on SEM=EDS for elemental quantification in

investigation of historic objects. The narrower aim of

this paper is to tackle the problem of estimating un-

certainty of quantitative surface analysis on the exam-

ple of measuring iron content on the paper surface

(surface concentration) of historic manuscripts. In this

context the term surface is to be understood in a broad

sense: it means not only the monomolecular surface

layer of the paper but a significantly deeper layer,

since a large part of the ink infiltrates deeper into the

paper at the time of writing. The exact definition of

the term surface concentration is given below.

Several rather difficult problems are associated with

this task: quantification in the actual surface analysis,

non-uniformity of the surface (e.g. in its porosity) and

the elemental content on the surface area, preparation

of calibration objects that are sufficiently similar to

the samples of unknown concentration.

We present here an uncertainty estimation procedure

according to the ISO GUM [1] and the EURACHEM=CITAC guide [2] applicable to SEM=EDS quantifica-

tion of iron in paper surface of historic manuscripts.

We explore and discuss the influence of various ex-

perimental parameters on the uncertainty of the mea-

surement result. In particular, our attention is devoted

to the difficult issues of measurand definition and

sample non-uniformity. Several specific problems in

uncertainty analysis are outlined, that need further

investigation.

Different spectroscopic techniques are used to ana-

lyze old written manuscripts or drawings to identify

inks [15, 18–20], gain information about the deterio-

ration of paper [21–24], investigate the distribution

and amounts of components in inks [11–14, 22, 25–

30] and the influence of inhibitors to iron or evaluate

conservation techniques or strategies [11, 29–35]. The

most important of them in the viewpoint of non-de-

structiveness are X-ray microprobe [29], microscopic

X-ray fluorescence analysis [15, 22, 27], Particle (pro-

ton) Induced X-ray Emission (PIXE) spectroscopy

[11–13, 18, 20, 31, 36], X-ray absorption near edge

spectroscopy [27, 28] and SEM=EDS [14, 26, 33]

(some of these methods, depending of the sample type,

are more or less destructive). Although PIXE is gen-

erally considered the most powerful of them, fully

meeting the requirements of analyzing objects of his-

torical value, it has poorer lateral resolution and is less

accessible than SEM=EDS. If quantitative results are

desired, then if the investigated object is not a homo-

geneous material, the quantification with SEM=EDS

has several difficulties (see below) but the method has

the important advantages of being sensitive, fast and it

is possible to measure the analyte content on a highly

K. Virro et al.

localized area. All in all the SEM=EDS method is cur-

rently perhaps the most widely used method for local-

ized elemental analysis of various surfaces and bulk

materials.

Experimental

Description of the analysis procedure

The intensity of the X-rays corresponding to iron atoms (i.e. the

EDS signal intensity of the Fe K� line at 6.404 keV) generated in the

paper surface by the electron probe was used as the analytical signal

to determine the iron content of the sample surface. Quantification

was carried out using the calibration graph method. The EDS signal

intensity of standards (paper sheets with ink lines) of known surface

concentration of iron was used for calibration.

Preparation of the calibration standards

It is very important to carry out calibration with standards that re-

semble the actual analysis objects as closely as possible. For our

analysis of manuscripts we prepared calibration standards in the

form of paper sheets with ink lines of known iron surface concen-

tration. The widths of the ink lines were adjusted to be as similar to

the widths of the lines on the sample manuscript as possible.

Pure Whatman cotton (without additives) paper (www.whatman.

com) with gelatin sizing was used. Gelatin was a common surface

sizing agent until 19th century. The other widespread method – the

alum-rosin sizing method was invented only in the early 19th century

(and was in use until the 1970s) [37].

Test sheets for simulation of old writing were prepared by plot-

ting 40 longitudinal lines (about 265mm long and 0.54mm wide)

with three different model-ink solutions on the paper. The fountain

pen used for plotting the lines was weighed before and after the

plotting in order to find out how much ink was consumed per area

unit of a line.

Preparation of the ink solutions

In order to achieve ink writing with known iron concentration

(expressed in mgmm�2), the iron concentration of the ink must be

known. Iron gall inks were made from aqueous solutions of iron(II)

sulfate, gallnuts (source of tannin) and gum arabic [32]. Preparation

of iron-gall ink with known iron concentration is not straightfor-

ward. The main reason is that the ink is not completely homogenous

and tends to form a precipitate. The generally used ink recipe, de-

scribed by Neevel [32] was taken as the basis for standard solutions.

Precise gravimetric preparation of the ink solution was complicated

because of precipitation. Hence, we prepared three ink solutions:

12, 18 and 24g of Mohr’s salt (Reakhim, the former USSR, no

website available) was dissolved in 200mL of aqueous solution of

gum arabic (Aldrich, www.sigmaaldrich.com) and then powdered

tannin (Reakhim, the former USSR, no website available) was

added. The tannin and gum arabic content was the same in all

three models of inks: 0.05 and 0.03gmL�1, respectively. The me-

dium concentration of the three inks with the Fe to tannin molar

ratio of 5.5 has been selected according to J. Neevel as a represen-

tative ratio in old inks [32]. These solutions were left standing and

were filtered twice through a plug of cotton-wool. After such treat-

ment the ink solutions were reasonably stable. The concentrations of

iron in the filtered ink solutions were measured by the flame AAS

method (using the standard conditions for iron determination at

wavelength 248.3 nm).

Calculation of the surface concentrations of iron

on the calibration samples

With the term ‘‘surface concentration’’ we denote the mass (in mg)of iron per unit of surface area (mm2) irrespective of how deep iron

has diffused into the paper. Under our experimental conditions the

X-rays are emitted throughout the thickness of the paper (standards

and sample).

The surface concentrations of iron in the calibration standards

were calculated using the following data: the lengths and widths of

the lines, iron concentration and the density of the ink solution, and

the mass difference of the fountain pen before and after drawing the

lines.

The original manuscript

Three small samples from different places located diagonally

over the investigated manuscript (18th century, from the Estonian

National Archive) were analyzed. The spots were chosen with letters

that visually looked similar to the model ink lines.

Analysis procedure

A SEM model Leo Supra 35 equipped with R€oontec EDS Xflash

3001 detector was used. Accelerating voltage of 15.00 kV was cho-

sen for the probe electrons. The beam cross-section was 0.5 mm,

the beam current was 1–2 � 10�8 A. From each of the three stan-

dard sheets several cutouts were subjected to the analysis. Select-

ed samples of approximately 0.5 cm2 were mounted on copper

supports using double-sided adhesive tape. The standards and the

manuscript samples were coated with a thin platinum layer of

2 nm thickness.

The standards and samples were examined using SEM magnifi-

cation around 300–400 times (Fig. 1). For analysis both wide X-ray

spectra and intensities of the Fe K� lines on the manuscript sample

(Fig. 2) and on standards (Fig. 3) were measured on the rectangular

excitation areas on the ink lines (Fig. 1). More details are given in

Fig. 1. Scanning electronmicrograph of the manuscript paper. The

black rectangle on the ink line (the slightly lighter horizontal line)

marks the excitation area

Uncertainty in electron probe microanalysis

the Electronic Supplementary Material (ESM). The averaged rate of

Fe K�-radiation counts measured with EDS over the rectangle areas

was used as the analytical signal. The iron content of the clean orig-

inal and standard paper was negligible so that no correction was

necessary.

The three most important matrix interferences (ZAF-corrections)

in our case are the following: generation of characteristic X-ray

radiation (involving Z-(atomic number) correction), absorption by

the matrix (involving A-(absorption) correction) and secondary fluo-

rescence (involving F-correction). All three have low effect on the

results of our analysis: differences in the X-ray generation condi-

tions in standards and in the original paper are relatively small as all

the matrixes are rather similar; absorption by the matrix is low be-

cause the matrix consists predominantly of light elements and the Fe

K� analytical line is relatively energetic; fluorescence by the matrix

is also negligible because there are no heavier elements than iron in

the matrix. Nevertheless it is reasonable to expect that there is at

least some mismatch between the sample matrix and the matrices of

the standards and this will be taken into account below.

Derivation of the uncertainty estimation procedure

Specification of the measurand and defining

the mathematical model

The measurand is Csurf_orig – the average total iron

content (Fe2þ and Fe3þ) in the surface layer (ex-

Fig. 2. EDS spectrum of the manu-

script paper

Fig. 3. EDS spectrum of a standard

K. Virro et al.

pressed in mg per paper mm2 of paper surface) of the

writing of the original manuscript:

Csurf orig ¼mFe orig

A; ð1Þ

where mFe_orig is the mass of iron in the written region

of paper and A is the surface area of the written re-

gion. We have defined our measurement as a relative

measurement: we used calibration with standards hav-

ing similar matrix. The electron beam acceleration

voltage was 15.00 kV in all our experiments and it is

an intrinsic part of the measurand definition. The pen-

etration depth (depth of diffusion into the paper) of

ink is not evaluated in this work, as the detected X-

rays are emitted throughout the paper thickness. This

is evidenced by the appearance of the weak copper

lines originating from the top layer of copper sup-

ports on which we firstly mounted the paper samples.

Afterwards we used aluminum supports and as a result

the copper lines are missing from the spectra. The

emitted X-rays are affected by the porosity and inho-

mogeneity of the studied material.

The surface concentration of the ink on the sample

from the manuscript’s sample Csurf_orig (mgmm�2)

was found using the calibration graph method.

Calibration line was constructed with the following

equation:

IstdðiÞ ¼ I0 þ b � Csurf stdðiÞ; ð2Þ

where Istd(i) denotes the average rate of signal counts

observed with the i-th standard and Csurf_std(i) denotes

the average surface concentration (mgmm�2) of iron

in the written region of the i-th standard. I0 and b are

the intercept and the slope of the calibration line,

respectively.

The iron content on the sample surface was found

as follows:

Csurf orig ¼Iorig � I0

b� fmatrix eff ; ð3Þ

where Iorig is the averaged value of count rates from

different locations of ink lines of the original manu-

script and fmatrix_eff is the factor taking into account

the slight mismatch between the matrices of the sam-

ples and the calibration standards. It is known that the

EDS method is linear with respect to the surface con-

centration if the matrix composition of the samples is

similar [38]. In the narrow range of concentrations

that we used for the calibration graph we can assume

that the matrix similarity is sufficient.

The slope and intercept of the calibration line are

found according to the linear regression equations as

follows:

b ¼Pn

i¼1 Csurf stdðiÞ � IstdðiÞ � n � Csurf std � IstdPni¼1 C

2surf stdðiÞ � n � C2

surf std

ð4Þ

I0 ¼Istd

Pni¼1C

2surf stdðiÞ �Csurf std

Pni¼1Csurf stdðiÞIstdðiÞPn

i¼1 C2surf stdðiÞ � n � C2

surf std

;

ð5Þ

where Csurf_std(i) and Istd(i) are the same as above and n

is the number of points on the calibration line. Surface

concentrations of the model samples Csurf_std(i) were

found as follows:

Csurf stdðiÞ ¼mstdðiÞ � cðiÞ AASlðiÞ � wðiÞ � �stdðiÞ

� 1000; ð6Þ

where mstd(i) (mg) is the mass of the i-th ink standard

solution used for plotting the lines on the paper (found as

the difference between themasses of fountain pen before

and after plotting the 40 lines), c(i)AAS (mgmL�1) is the

concentration of the i-th standard ink solution deter-

mined by AAS, l(i) (mm) is the total length of the 40

lines of the i-th model sample, w(i) (mm) is the average

width on the lines of the i-th model sample and �std(i)(mgmL�1) is the density of the i-th standard solution,

and the multiplier 1000 (mgmg�1) converts the units.

Equations (3) to (6) taken together form our math-

ematical model.

Identifying uncertainty sources

Uncertainty of weighing includes the following

sources: 1) repeatability uncertainty of weighing; 2) un-

certainty caused by the drift of balance; 3) uncertainty

caused by rounding of the reading of the balance. Air

buoyancy is not taken into account here (densities of

the sample and the standard are similar and the result-

ing uncertainty is small) and the weights read from the

balance are considered to be the masses.

Uncertainty of ink solution concentrations mea-

surement by AAS. Detailed description of uncertainty

estimation of the AAS results will be published else-

where. For the purpose of this work we consider the

uncertainty of the AAS results as a single uncertainty

source.

Uncertainty of the surface area of the lines. The

uncertainty of length and width of the lines contribute

to the uncertainty of the surface areas of the lines.

Uncertainty in electron probe microanalysis

The contribution of the length uncertainty is small

and is determined by the accuracy of the length mea-

surement. The uncertainty of the width of the lines

is caused mainly by the uneven width of the lines

themselves and to a lesser extent by the measurement

uncertainty of measuring the line widths (using a mea-

suring microscope).

Uncertainty arising from the SEM=EDS measure-

ments. The following uncertainty sources affect the

signal intensity:

1) The variability of the count rates is caused by a)

the non-uniformity of the paper surface; b) the non-

uniform distribution of the ink particles on the sur-

face; c) the non-uniform distribution of the Pt coating

on the surface; d) the fluctuations of the electron beam

intensity, high voltage and time interval measurement;

e) the fundamentally random nature of emission of X-

ray quanta.

The joint contribution of these uncertainty sources

can be estimated by making multiple measurements

from different surface regions and averaging the re-

sults. It is important to mention that these uncertainty

sources originate from the objects under study and are

not caused by the SEM-EDS system.

2) Different wetting properties of the inks used

in the originals and in the calibration samples. This

effect causes different levels of ink diffusion into the

depth of the paper material and thus different iron

concentration depth profiles. In our case this uncer-

tainty source was minimized by a) the fact that the

emission depth of the X-rays (determined by the pen-

etration depth of the electrons) is deeper than the

thickness of the papers (as evidenced by the presence

of copper lines from the in the spectrum originating

from the copper support) and b) careful selection of

the paper and the width of the lines on the calibration

sheets (both of these were matched to the original as

closely as possible).

3) Matrix mismatch between the standards and the

samples. Matrix mismatch is caused by different com-

positions and structure of the paper samples and the

inks used for preparation of the standards and the

actual samples. This effect causes different generation

rate and attenuation of the X-rays emitted from the

paper (see above). The paper for the preparation of the

calibration standards was made of pure cotton fibers,

without any additives found in modern papers (see the

Description of the Analysis Procedure). In addition,

the paper was sized with gelatin to mimic as closely

as possible the original paper. The same approach has

been used in other works dedicated to instrumental

analysis of historic manuscripts [25, 33–35].

4) Overlap between the EDS peaks of iron and pos-

sible interfering elements. The only element that can

significantly interfere under our conditions is manga-

nese. However, in our case the possibility of overlap

between the peaks of iron (K� 6.404 keV) and man-

ganese (K� 6.49 keV) in the EDS spectrum negligible,

because even the strongest manganese line (K�) at

5.899 keV was not detected in the investigated sam-

ples and standards (see the spectra in Figs. 2 and 3).

Additionally to this, literature data indicate that man-

ganese is not common to be found in paper or in ink,

or if found, then its concentration is very low. Budnar

et al. [12] and Remazeilles et al. [13] have detected

manganese content in paper=ink to be by 100–1000

times lower than that of iron. In our original paper

or samples, no traces of manganese were detected at all

under our experimental conditions. Therefore we can

leave this uncertainty source out of consideration.

Application example, quantifying the uncertainty

components, calculating the combined uncertainty

Uncertainty of ink mass measurement

Sartorius ME235S balance was used for weighing the

ink for preparation of the standards. The data on the

repeatability of the balance was determined experi-

mentally in the laboratory in the following manner:

The fountain pen was weighed for 10 times before

and 10 times after plotting with each standard ink

solutions. From these data the uncertainty of the dif-

ferences of the masses was calculated according to the

EURACHEM=CITAC Guide [2]. The u(mstd(i), rep)

was found 0.0138mg. The drift of the balance was

estimated from long-term experience from our labo-

ratory: uðmstdðiÞ; driftÞ ¼ 0:01=ffiffiffi3

p¼ 0:0058mg. The

digital display of the balance has five decimal places,

hence the uncertainty caused by rounding of the dig-

ital reading is: uðmstdðiÞ; roundÞ ¼ 0:000005=ffiffiffi3

p¼

0:0029mg.

The equation of uncertainty of the mass of ink stan-

dards was found:

uðmstdðiÞÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuðmstdðiÞ; repÞ2 þ uðmstdðiÞ; driftÞ2

þ uðmstdðiÞ; roundÞ2

vuut ð7Þ

The values of uncertainties of the ink masses used for

preparation of the standards are given in Table 1.

K. Virro et al.

Uncertainty of the density and iron content

of the ‘‘Self-Prepared Ink’’

The densities of the model ink solutions at tem-

perature 22.6� 0.2 �C and their standard uncertainties

expressed in milligrams per milliliter were the follow-

ing: �1¼ 1033.1, u(�1)¼ 0.5, �2¼ 1045.0, u(�2)¼ 0.6

and �3¼ 1060.4, u(�3)¼ 0.7. Pure water (density at

the same temperature was 9976.6mgmL�1) was used

as the reference.

The iron contents of the three calibration ink so-

lutions measured by AAS were as follows: 7.33,

11.12, 14.57mgmL�1. The standard uncertainties

of these concentrations are 0.14, 0.20, 0.26mgmL�1,

respectively . The AAS analysis method and the un-

certainty estimates were validated by analyzing the

11.12mgmL�1 solution also with a different method

(ICP-AES) at a different laboratory. The concen-

tration was found as 11.00� 0.44mgmL�1, k¼ 2,

norm., which is well in line with our result.

Uncertainty of the lengths and widths

of the standard ink lines

Uncertainty of the length of the lines was calculated

through the following equation:

uðllineÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuðlline; calÞ2 þ uðlline; repÞ2

q; ð8Þ

where the repeatability of uncertainty u(lline, rep) of the

line lengthwas estimated as the standard deviation of the

mean of 20 readings (of the same length) and the cali-

bration uncertainty was estimated as �0.5mm (rectan-

gular distribution): uðlline; calÞ ¼ 0:5=ffiffiffi3

p¼ 0:29mm.

The total lengths of standard (1), standard (2) and

standard (3) were 10538.7, 10582.4 and 10451.1 re-

spectively. The standard uncertainty of the three lengths

was the same: 11.6mm. Each of the total length of the

standard consisted of 40 short lines.

The widths (w(i)) were determined making mea-

surements in 22 different locations picked on the sheet

diagonal (every location was on a different ink line)

using a measuring microscope. The uncertainty aris-

ing from the measurement itself is negligible com-

pared to the uncertainty due to the uneven width of

the lines. The widths of the lines on the standards with

their uncertainties are presented in Table 1.

Uncertainty due to the SEM=EDS

The electron beam (the probe current) of the used

SEM=EDS is of high stability – better than 0.2%

per hour for analytical applications. Hence, the instru-

ment instability contribution to the uncertainty of Csurf

was considered of no importance. All the other direct

uncertainty sources of the SEM=EDS measurement as

well as uncertainty due to the uneven iron distribution

Table 1. The uncertainty budgets of the surface concentrations of iron on the calibration standards

Csurf_std(1) Csurf_std(2) Csurf_std(3)

Values (mgmm�2) 0.121 0.151 0.185

uc(Csurf_std(i)) (mgmm�2) 0.005 0.006 0.007

Relative uc (%) 3.7 3.7 3.7

Components of uc Csurf_std(1) Csurf_std(2) Csurf_std(3)

Value u % Value u % Value u %

mstd(i) (mg) 97.62 0.02 �0 81.17 0.02 �0 76.34 0.02 �0

C(i)AAS (mgmL�1) 7.33 0.14 24.8 11.12 0.20 24.0 14.57 0.26 23.4

�std(i) (mgmL�1) 1033.1 0.5 �0 1045.0 0.6 �0 1060.4 0.7 �0

l(i) (mm) 10540.5 11.6 0.1 10582.4 11.6 0.1 10451.1 11.6 0.1

w(i) (mm) 0.54 0.018 75.1 0.54 0.018 75.9 0.54 0.018 76.5

Table 2. The uncertainty budget of the surface concentration of

iron in the original manuscript

Results Value uc

Set I Set II Set I Set II

Csurf_orig_I (mgmm�2) 0.124 0.121 0.010 0.020

Components of uc(xI in text)

Value u

Csurf_std(1) (mgmm�2) 0.121 0.005

Csurf_std(2) (mgmm�2) 0.151 0.006

Csurf_std(3) (mgmm�2) 0.185 0.007

Sstd(1)_I 31.1 22.7 1.3 1.2

Sstd(2)_I 44.1 31.3 1.8 1.7

Sstd(3)_I 48.7 35.0 2.0 1.8

Soriginal_I 32.7 23.2 2.2 3.8

fmatrix_eff 1.000 1.000 0.032 0.032

Uncertainty in electron probe microanalysis

and surface non-uniformity will be accounted for by

the repeatability of the measurements. Parallel mea-

surements on different locations of the lines were per-

formed with each of the three standards and with the

original manuscript (see the ESM for the calcula-

tions). The mean values and the standard uncertainties

of the Sorig and Sstd(i) values are given in Table 2.

The uncertainty sources 2 and 3 (see above) are taken

into account by the matrix mismatch factor fmatrix_eff

(Eq. (3)). This factor is defined in such a way that its

value is equal to unity in order not to change the value

of Csurf_orig. Its uncertainty is defined as the estimated

relative standard uncertainty due to the matrix mis-

match. To get a rough estimate of this contribution a

model experiment was carried out. ZAF correction

factors for ’’pure paper’’ (c(C)¼ 98.35m%, c(Fe)¼0.88m% and c(S)¼ 0.77m%) and a paper with ad-

ditives (c(C)¼ 81.65m%, c(Fe)¼ 2.99m%, c(S)¼0.95m%, c(Si)¼ 8.54m%, c(Al)¼ 0.36m%, c(Na)¼0.17m%, c(K)¼ 4.50m%, c(Cl)¼ 0.12m% and

c(Ca)¼ 0.72m%) were calculated using EPMA cor-

rection algorithm Stratagem [39] switched into bulk

mode. This algorithm also permits to take the Pt coat-

ing into account. The difference between the calculat-

ed ZAF corrections was small: 3.2%. This value was

used as the estimate of standard uncertainty due to the

matrix mismatch (fmatrix_eff).

Finding the combined uncertainty

The combined standard uncertainty is calculated ac-

cording to the ISO GUM approach [1], that is, the

calculations were performed according to Eq. (9).

ucðCsurf origÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXi

�@Csurf orig

@xiuðxiÞ

�2

vuut ; ð9Þ

where xi denotes the values of seven components of

uncertainty in Table 2. The calculations were carried

out with the MS Excel package using the Kragten

Method [40] for approximating the partial derivatives.

The electron probe microanalysis with SEM=EDSwas carried out twice on different days (Sets I and II)

using the same standards and original manuscripts

but different cutouts. The uncertainty budgets of iron

concentrations on the surface of the standards are pre-

sented in Table 1. The results of the two separate anal-

yses are presented in Table 2. The average surface

concentration of iron in the original manuscript togeth-

er with its combined uncertainty and the uncertainty

budget containing the uncertainty contributions of the

input quantities are presented in Table 3.

Results and discussion

Contributions of the uncertainty sources

The obtained iron contents in the ink writing of the

manuscript were 0.124 and 0.121 mgmm�2 with stan-

dard uncertainties 0.010 and 0.020mgmm�2, respec-

tively. The large difference between the uncertainties

of the two sets is caused by the different number of

parallel measurements on the original (9 and 3, re-

spectively, see the ESM). The average iron content

on the ink-covered surface of the sample manuscript

found as the weighed average from these measure-

ment results was 0.123 mgmm�2 with the standard

uncertainty of 0.016 mgmm�2 (see the ESM for the

calculations). The uncertainty budget is heavily dom-

inated by the variability of the count rates, which in

turn is caused by inhomogeneity of the ink lines. Thus

the number of effective degrees of freedom of this

result can be estimated as completely determined by

the low number of parallel measurements made from

different locations of the sample leading to df¼ 10

(see the ESM). The coverage factor at 95.45% confi-

dence level (corresponding to the k¼ 2 level for the

Normal Distribution) is therefore 2.28 leading to the

expanded uncertainty 0.037mgmm�2 at 95.45% con-

fidence level (relative expanded uncertainty 30%).

The main source of uncertainty with our sample

type and equipment was the variability of the parallel

Table 3. The final result

Csurf_orig (mgmm�2) uc(Csurf_orig) (mgmm�2) Degrees of freedom k(95.45%) U(95.45%) (mgmm�2)

0.123 0.016 10 2.28 0.036

The uncertainty contributions to the result (%)

u(Csurf_std(1)) u(Csurf_std(2)) u(Csurf_std(3)) u(Sstd(1)) u(Sstd(2)) u(Sstd(3)) u(Soriginal) u(Matrix effect)

6.18 0.45 0.11 10.51 0.05 0.02 77.84 4.84

K. Virro et al.

measurement results from different locations of the

manuscript surface contributing 54 and 84% of the

overall uncertainty in the first and second analy-

sis set, respectively, and 78% to the uncertainty of

the average result. This contribution is followed by

the signal variability of the first calibration standard

(11%) and the iron content of the first calibration

standard (6%). The prevailing influence of the sam-

ple heterogeneity fully validates the use of only 3

points for calibration in our case. Even in the impos-

sible case that calibration graph would contribute no

uncertainty to the result (meaning that there is infi-

nite number of points on the graph and there are no

systematic effects affecting the points – both condi-

tions unrealizable), the standard uncertainty of the

result would decrease only marginally: from 0.016

to 0.015 mgmm�2. The high contribution of the fist

calibration standard is not unexpected: the average

rate of counts from the sample is very similar to that

of the first standard.

The dominance of the signal variability (both in

the case of samples and standards) in the uncertainty

budget does not mean that the sample heterogeneity

is the only uncertainty source worth considering. With

a different sample the heterogeneity can be far lower

and the calibration graph points can have large rela-

tive contribution to the uncertainty. The uncertainty

contribution due to the variability of the signal can

be reduced by increasing the number of measurements

both from the calibration standards and from the sam-

ple. Both the purpose of the measurement and the eco-

nomic effect have to be carefully considered.

The uneven widths of the ink lines play the key role

in determining standards preparation as the second

most important source of uncertainty. The uncertain-

ty contribution arising from the uneven width of the

ink lines of the standards could in principle be re-

duced by drawing wider lines using the so-called red-

dish pen. The absolute uncertainty of the line widths

would remain more or less the same but the relative

uncertainty would drop several times. Because the

uncertainty contribution of the width of the line is

dependent on the relative uncertainty, the combined

uncertainty of the result would seemingly decrease.

However, this would destroy the similarity between

the sample and standards (see the Preparation of the

Calibration Standards Section): the diffusion of the

ink in the broad lines would not be the same any more

and a hard-to-quantify additional uncertainty source

would be introduced.

Limit of detection

All our samples, both standards and the original, had

non-detectable content of iron on the unwritten area.

Iron is a widespread element in the environment

and the paper of any manuscript contains iron traces

originating from contamination e.g. by paper verso

sides. Based on the baseline noise level of the EDS

spectra and the calibration line parameters a crude

estimate of the detection limit of iron was obtained

as 0.02mgmm�2 (signal to noise ratio 3) under our

conditions. The signal=noise ratio for the lowest cali-

bration standard was around 15. Thus, SEM=EDS is a

suitable method to determine, at least semi-quantita-

tively, the surface concentration of iron in inks on the

paper.

Comparison of our results with those of other authors

This type of analysis is not done by laboratories on a

regular basis. Thus it would be very desirable to com-

pare both the element content levels and the uncer-

tainties with those obtained by other authors in similar

measurements. Studies on quantification of iron in old

ink-written manuscripts have been published before

[11, 13, 14, 25, 33]. Budnar et al. [11, 12, 36] and

Remazeilles et al. [13] have done quantification of the

elemental components of ink with PIXE. With de-

structive GFAAS method quantitative analysis of iron

and copper in old manuscripts has been carried out

by Wagner et al. [25]. The same group has quantified

elements in iron-gall ink with energy dispersive X-ray

fluorescence analysis [14]. Semiquantitative analy-

sis of inorganic components has been carried out by

Odlyha et al. [33].

It is nevertheless difficult to compare our results to

the literature results. Our results are presented as iron

mass per area of written paper (unwritten area not

included). The literature results are mostly presented

in the form of mass of iron per mass of paper (unwrit-

ten area included). Analyzing our manuscript sample

by AAS would have been a possible approach for

comparing with an independent method, but that would

destroy the sample and also it would be extremely dif-

ficult to find the area of the ink lines as they were

irregular and the whole manuscript was patchy.

In some of the above-cited papers the issue of un-

certainty has been addressed but a full ISO GUM-

based uncertainty analysis has not been carried out.

In Ref. [13] the authors have given the estimates of

Uncertainty in electron probe microanalysis

the variations between the results obtained from sev-

eral spectra (measured with PIXE) from the same

manuscript, but at different locations. Each spectrum

was processed the same way and the authors looked at

the variabilities of the results for the most typical ele-

ments. For iron the variability was found as �16%

(presented as full spread of seven measurement re-

sults). In Ref. [11] the estimation of the variation

has also been presented as the full spread of �10%

(the number of measurements was not given). These

uncertainties are lower than our uncertainty estimate.

In Ref. [36] the accuracy of the analysis method

(PIXE) was under discussion. The systematic error

(5–20%) and precision (5%) were evaluated for the

analysis of Fe and Cu. The relative standard un-

certainty extracted from these data is in the range of

7–21%. This compares quite well with our relative

standard uncertainty estimate of 13%.

Conclusions

SEM=EDS was used to determine the surface concen-

tration of iron in the ink writing of an old manuscript.

For quantification a calibration graph method utilizing

AAS analyses of 3 standard ink solutions with dif-

ferent iron content was used. To estimate roughly

the surface concentration of iron in the ink writing on

the manuscript we recorded spectra on two days from

altogether six different locations. The mean iron surface

concentration on the written area was 0.123mgmm�2

with the expanded uncertainty 0.037 mgmm�2 at

95.45% confidence level, which we consider accept-

able for such a complicated analysis task.

Electron probe microanalysis with SEM=EDS is

a suitable method for evaluating the iron content

in old manuscripts. The method is universal (enables

to quantify other elements in a similar manner), quite

sensitive (down to few atomic percent detection lim-

its) and fast (it takes around 5min to record one

spectrum and around 1 h to prepare a set of 10–20

samples). Although non-destructive methods, like

PIXE, m-XRF, etc exist and are in use for the analysis

of old paper, the necessary equipment may not be

available for everyday analysis, and getting high pre-

cision quantitative results from ink-on-paper samples

is not a routine task also with those methods.

Acknowledgments. We are deeply indebted to Mr. Jaan Lehtaru, Dr.

Viljar Pihl and Dr. Koit Herodes for valuable suggestions. This work

was supported by the Estonian Science Foundation grants 5475 and

6651 and by the Basis financing grant 06902 of University of Tartu.

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Uncertainty in electron probe microanalysis