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Article
FLUKA Simulations of Kβ/KαIntensity Ratios of Copper in
Ag–Cu Alloys
Aneta Maria Gójska * , Karol Kozioł , Adam Wasilewski , Ewelina Agnieszka Mi´sta-Jakubowska ,
Piotr Mazerewicz and Jakub Szymanowski
Citation: Gójska, A.M.; Kozioł, K.;
Wasilewski, A.; Mi´sta-Jakubowska,
E.A.; Mazerewicz, P.; Szymanowski, J.
FLUKA Simulations of Kβ/Kα
Intensity Ratios of Copper in Ag–Cu
Alloys. Materials 2021,14, 4462.
https://doi.org/10.3390/ma14164462
Academic Editor: Savo Rikanovi´c
Received: 24 May 2021
Accepted: 7 August 2021
Published: 9 August 2021
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Attribution (CC BY) license (https://
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4.0/).
National Centre for Nuclear Research, ul. A. Soltana, 7 05-480 Otwock, Poland; karol.koziol@ncbj.gov.pl (K.K.);
adam.wasilewski@pwr.edu.pl (A.W.); ewelina.mista@ncbj.gov.pl (E.A.M.-J.); piotr.mazerewicz@ncbj.gov.pl (P.M.);
jakub.szymanowski@ncbj.gov.pl (J.S.)
*Correspondence: aneta.gojska@ncbj.gov.pl
Abstract:
The numerical simulations of Cu
Kα
and Cu
Kβ
fluorescence lines induced by Rh X-ray
tube and by monoenergetic radiation have been presented. The copper
Kβ/Kα
intensity ratios for
pure elements as well as for Ag–Cu alloys have been modeled. The results obtained by use of the
FLUKA code, based on the Monte-Carlo approach, have been compared to available experimental
and theoretical values. A visible relationship was found between the simulated
Kβ/Kα
intensity
ratios and the copper content of the Ag–Cu alloy: as the Cu content increases, the
Kβ/Kα
coefficient
decreases. The results can play role in elemental material analysis, especially in archaeometry.
Keywords: FLUKA simulation; Kβ/Kαintensity ratios; Ag-Cu alloy
1. Introduction
The
Kβ/Kα
intensity ratios have been intensively studied since 1969. Daoudi et al. [
1
]
reports that after more than thousand measurements, 127 theoretical and experimental
publications have been created in the last half century. The experimental measurements
of X-ray spectra are crucial in examining theoretical models [
2
–
4
]. The values of
Kβ/Kα
intensity ratios are used for estimation the vacancy transfer probability (e.g., transfer
of hole from
K
shell to
L
shell [
5
–
7
])) and
Kα
and
Kβ
X-ray production cross-sections
[8–11]
. In experiments the excitation mediums were mainly radioisotopes:
241Am
[
10
,
12
–
28
],
109Cd
[
29
–
31
],
137Cs
[
32
,
33
],
57Co
[
34
], and
238Pu
[
35
]. X-ray tubes were also used for
Kβ/Kαintensity ratios studies [36–39] as well as proton beam [40,41].
There are a couple of aspects linked to the study on
Kβ/Kα
intensity ratios. At first,
experiments in which chemical compounds were tested showed that the
Kβ/Kα
X-ray
intensity ratios are sensitive to the chemical environment for 3
d
elements [
13
,
20
,
42
,
43
]. The
results were explained by the change in screening of 3
p
electrons by 3
d
valence electrons.
The studies on 3
d
metal alloys and compounds have shown dependence of the
Kβ/Kα
intensity ratios on alloy composition or chemical state through changes in electron binding
and electron configuration of the valence states [
16
]. Since Cu 3
d
states do not overlap
energetically with the Ag 4
d
band, energy mismatch between Ag 4
d
and Cu 3
d
states is
the main contributor to the sharpness and degeneracy of the Cu 3
d
states. Despite the lack
of overlap of silver and copper wave functions in the Ag–Cu alloy, the charge density is
transferred between Ag and Cu [
44
]. Raj et al. [
17
] report that in alloys the 3
d
electron
transfer/delocalization is the main factor causing change in the
Kβ/Kα
intensity ratios.
Thus the changes in
Kβ/Kα
intensity ratios of alloy’s element indicate changes in the
valence electronic configurations or charge transfer effect caused by presence of second
elements [
25
]. The measurement of
Kβ/Kα
intensity ratios can be a sensitive probe of 3
d
charge transfer [45].
Another advantage of
Kβ/Kα
X-ray intensity ratio studies is that this parameter can
be used for determination of depth profile distributions of the elements in thick targets [
40
].
Materials 2021,14, 4462. https://doi.org/10.3390/ma14164462 https://www.mdpi.com/journal/materials
Materials 2021,14, 4462 2 of 11
This technique can be used in archaeometry to determine the silver enrichment taking place
in antique silver–copper coins [
39
,
46
–
48
]. Since many decorative objects are composed
of silvered copper, gilded copper, or silver, the use of
Kβ/Kα
X-ray intensity ratio from
different chemical element can allow estimating the thickness of the surface layer [49].
The knowledge on
Kβ/Kα
X-ray intensity ratio can also be a tool to find elements
which
K
or
L
lines overlap the lines of main element. So the
Kβ/Kα
X-ray intensity ratio
can be used to find the intensities of the unresolved lines of neighbor elements [50].
Research on
Kβ/Kα
intensity ratios of complex materials motivated us the use of
advanced Monte-Carlo tools, by means of the FLUKA code [
51
,
52
], to simulate complex
X-ray spectra. The Monte-Carlo simulation method was introduced in 1949 and since then
it has been successively used in many areas of physics, such as atomic physics, high energy
physics, medical physics, as well as in material engineering, construction of accelerator
structures, and in other fields of science, including mathematics, biology, economics, and
archaeometry. Although the FLUKA code does not include in-depth quantum-mechanic
features at atomic level such as the charge transfer between 3
d
electrons, in our paper the
analytical challenge is based on a comprehensive and accurate description of the spectrum
features such as the shape of the primary radiation spectrum, i.e. the intensity and the
shape of the X-ray tube anode lines, and the intensity, the centroid, and the shape of all
emission lines of the tested material. Each of the individual elements of the spectrum
provides relevant analytical information. The simulation allows the determination of
the
Kβ/Kα
intensity ratios without the need to transform the radiation intensity of the
characteristic sample obtained in the detector. Self-absorption and detector performance
corrections, which are usually necessary in conventional quantitative analysis based on
main peak analysis, are therefore eliminated, which means that the FLUKA code just
simulates an experimental output, not a detector input. The reliability of Monte-Carlo
tools, in addition to the subjective modeling of the composition and structure of the sample,
depends on the analytical model adopted, the description of the radiation source, and the
settings of equipment specifications, operational parameters, and experimental geometry.
Since
Kβ/Kα
intensity ratios of Cu have been extensively explored the simulation of this
element is an appropriate test point.
In this work, X-ray spectra of silver–copper alloys were modeled. The copper
Kβ/Kα
intensity ratios were calculated for pure Cu as well as for Ag–Cu alloys. Two kinds of
the FLUKA simulations have been performed. The first kind includes primary electron
beam and radiation of X-ray tube equipped with a Rh anode operating at 40.8 kV. The
second kind includes a monoenergetic 59.9 keV photon beam. The fluorescence spectra
of silver–copper alloys are an output for both kind of simulations. The results obtained
are critically evaluated by comparison with available experimental and theoretical values
for pure elements. It is worth underlining that the obtained results are not exactly the
same kind as the experimental results. It is because the experimental results are based on
the X-ray photons counted by the detector and then the values are corrected by detector
efficiency and air and sample absorption coefficients. In contrast, the FLUKA simulation
results are based on the X-ray photons emitted directly from the sample.
2. Experiment Simulations
Monte-Carlo simulations were performed using FLUKA 2011 code version 2c.8 in-
stalled on a computer cluster at ´
Swierk Computing Center [
53
]. FLUKA code uses the
Evaluated Photon Data Library (EPDL97) [
54
]. The EPDL library consists of tabulations of
photon interaction data including photoionization, photoexcitation, coherent and incoher-
ent scattering, and pair and triplet production cross sections.
The experimental setup reproduced in the calculations, consisting of an X-ray tube
model with a 1 mm thick Rh anode, a 1 mm thick Be window, and two irradiated sample
groups with a diameter of 1 cm and a thickness of 2 mm and 1
µ
m, is shown in Figure 1.
Additionally, for samples with thickness of 1
µ
m the monoenergetic
241Am
(59.9 keV) have
been used. As one can see from Figure 1, in the case of 1
µ
m sample a part of radiation is
Materials 2021,14, 4462 3 of 11
going through the sample and another part is reflected back off the sample. The first part is
called the forward output flux and the second part is called the backward output flux (this
part is usually incoming to the detector). In the case of the 2 mm sample there is no forward
output flux, because all radiation going trough the sample is absorbed or re-emitted in
a backward direction. Calculations for each alloy were made by dividing them into 500
parallel processes. The 1 keV photon and electron transport energy cut-off was set to best
reproduce photon and electron behaviour for the used beam energy range. The Rh-X-ray
was induced by a 2
·
10
11
monoenergetic electron beam (40.8 keV) with flat distribution,
Φ
= 1 cm. The Rh anode X-ray spectrum filtered by a 1 mm Be layer is presented in Figure
2. The calculated K X-ray spectra of Ag–Cu alloy registered on a flat, irradiated sample
surface are presented in Figure 3.
Figure 1.
Experimental setup and photon fluence reproduced in the calculations for the sample thickness of 2 cm (
left
) and
1 µm (right).
Rh Kα1
Rh Kα2
Rh Kβ1,3
Rh Kβ2
Bremsstrahlung
0 5 10 15 20 25 30 35 40
0.0
2.0
4.0
6.0
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
Figure 2. Rh anode X-ray spectrum.
Materials 2021,14, 4462 4 of 11
5 10 15 20 25 30
0
0.5
1
1.5
2
Ag Kα1
Ag Kα2
Ag Kβ1,3
Ag Kβ2
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
Pure Ag
5 10 15 20 25 30
0
0.5
1
1.5
2
Ag Kα1
Ag Kα2
Ag Kβ1,3
Ag Kβ2
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
90% Ag 10% Cu
5 10 15 20 25 30
0
0.5
1
1.5
Ag Kα1
Ag Kα2
Ag Kβ1,3
Ag Kβ2
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
80% Ag 20% Cu
5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
Ag Kα1
Ag Kα2
Ag Kβ1,3
Ag Kβ2
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
65% Ag 35% Cu
5 10 15 20 25 30
0
1
2
3
4
Ag Kα1
Ag Kα2Ag Kβ1,3
Ag Kβ2
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
50% Ag 50% Cu
5 10 15 20 25 30
0
2
4
6
Ag Kα1,2
Ag Kβ1,3
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
30% Ag 70% Cu
5 10 15 20 25 30
0
2
4
6
Ag Kα1,2
Ag Kβ1,3
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
25% Ag 75% Cu
5 10 15 20 25 30
0
2
4
6
8
10
Ag Kα1,2
Ag Kβ1,3
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
10% Ag 90% Cu
5 10 15 20 25 30
0
2
4
6
8
10
Ag Kα1,2
Ag Kβ1,3
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
5% Ag 95% Cu
5 10 15 20 25 30
0
2
4
6
8
10
12
Cu Kα1,2
Cu Kβ1,3
Energy (keV)
Photon flux (N−1
e0cm−2GeV−1)
Pure Cu
Figure 3. X-ray spectra in studied Ag–Cu alloys.
Materials 2021,14, 4462 5 of 11
3. Results and Discussion
The simulated Cu
Kα
and Cu
Kβ
intensities as well as the different Ag–Cu alloys are
presented in Tables 1,2, and 3. In our work we have considered the following K-x-ray
transitions:
Kα1,2
(
K
-
L2,3
),
Kβ1,3
(
K
-
M2,3
), and
Kβ2
(
K
-
N2,3
). The dependence of
Kβ/Kα
intensity ratio on copper concentration in Ag–Cu alloy is presented in Figures 4–6. Three
cases are studied: backward output flux, forward output flux, and weighted average
output flux. The error bars arise from statistical uncertainties. In real experiment the errors
are attributed to uncertainties from various parameters used in the determination of the
Kβ/Kα
intensity ratio, including errors caused by the evaluation of peak area, detector
efficiency, self-absorption factors, target thickness, and counting statistic. Table 4presents
the available theoretical and experimental values for pure copper while in Table 5values
for Cu alloys obtained from available literature. The literature data for pure Cu and Cu–Ag
alloy are also presented collectively in Figures 4–6.
Some general conclusions can be drawn based on the presented data : (i) In the case
of backward and average output fluxes, there is very small difference between results
calculated for 1
µ
m sample for both Rh X-ray tube and monoenergetic 60 keV radiations,
but here is distinct difference between these results and results calculated for the 2 mm
sample. (ii) There is no forward output flux in the case of the 2 mm thick sample, because
all radiation is absorbed in this direction while moving through the sample. For 1
µ
m
samples the difference between results for Rh X-ray tube and monoenergetic radiations is
bigger than in the case of backward output flux. (iii) As can be seen from Figure 4, a major
part of experimental results is placed in between the FLUKA results calculated for very
thin (1
µ
m) and very thick (2 mm) samples. The present results can also partially explain
the differences between various experimental results for pure copper as a result of different
thickness of samples used in experiments. (iv) The Cu
Kβ/Kα
intensity ratio is sensitive
to alloy composition. As the Cu content increases, the
Kβ/Kα
coefficients decrease. The
alloying effect is in order of a few percent and this size of effect is consistent with the size
of the alloying effect reported by Dhal et al. [28].
Table 1.
Simulated
Kβ/Kα
intensity ratio for 1
µ
m thick samples induced by Rh X-ray tube radiation,
calculated for backward and forward direction and weighted average of them.
Cu (%) Kβ/Kα
Average Backward only Forward only
10 0.1285(12) 0.1283(12) 0.1373(81)
20 0.1292(8) 0.1289(8) 0.1367(41)
30 0.1286(7) 0.1282(7) 0.1358(28)
40 0.1285(6) 0.1281(6) 0.1335(21)
50 0.1282(5) 0.1277(5) 0.1336(17)
60 0.1278(5) 0.1272(5) 0.1328(14)
70 0.1275(4) 0.1269(4) 0.1320(13)
80 0.1272(4) 0.1265(4) 0.1318(11)
90 0.1266(4) 0.1259(4) 0.1308(10)
100 0.1269(4) 0.1263(5) 0.1299(10)
Table 2.
Simulated
Kβ/Kα
intensity ratio for 1
µ
m thick samples induced by monoenergetic radiation,
calculated for backward and forward direction and weighted average of them.
Cu (%) Kβ/Kα
Average Backward only Forward only
10 0.1299(2) 0.1300(3) 0.1298(3)
50 0.1277(1) 0.1275(1) 0.1280(1)
90 0.1258(1) 0.1258(1) 0.1258(1)
100 0.1253(1) 0.1254(1) 0.1253(1)
Materials 2021,14, 4462 6 of 11
Table 3.
Simulated
Kβ/Kα
intensity ratio for 2 mm thick samples induced by Rh X-ray tube radiation.
Only backward direction is calculated.
Cu (%) Kβ/Kα
10 0.1474(19)
20 0.1498(13)
35 0.1470(9)
50 0.1462(7)
70 0.1447(6)
75 0.1433(6)
90 0.1406(5)
95 0.1402(4)
100 0.1392(4)
0 20 40 60 80 100
0.120
0.130
0.140
0.150
Cu content in the Cu-Ag alloy (%)
I(Kβ)/I(Kα)
Average
FLUKA, Rh X-ray tube, 1 µm sample
FLUKA, Rh X-ray tube, 2 mm sample
FLUKA, X-ray monoenergetic 60 keV, 1 µm sample
Experiment (X-ray induced)
Experiment (γinduced and other)
Other theory
Figure 4. Kβ/Kα
intensity ratio arising from the primary X-ray simulated with FLUKA, calculated
for average output flux in 1 µm and 2 cm samples.
0 20 40 60 80 100
0.120
0.130
0.140
0.150
Cu content in the Cu-Ag alloy (%)
I(Kβ)/I(Kα)
Backward photons only
FLUKA, Rh X-ray tube, 1 µm sample
FLUKA, Rh X-ray tube, 2 mm sample
FLUKA, X-ray monoenergetic 60 keV, 1 µm sample
Experiment (X-ray induced)
Experiment (γinduced and other)
Other theory
Figure 5. Kβ/Kα
intensity ratio arising from the primary X-ray simulated with FLUKA, calculated
for backward output flux in 1 µm and 2 cm samples.
Materials 2021,14, 4462 7 of 11
0 20 40 60 80 100
0.120
0.130
0.140
0.150
Cu content in the Cu-Ag alloy (%)
I(Kβ)/I(Kα)
Forward photons only
FLUKA, Rh X-ray tube, 1 µm sample
FLUKA, X-ray monoenergetic 60 keV, 1 µm sample
Experiment (X-ray induced)
Experiment (γinduced and other)
Other theory
Figure 6. Kβ/Kα
intensity ratio arising from the primary X-ray simulated with FLUKA, calculated
for forward output flux in 1 µm sample.
Table 4. Kβ/Kαintensity ratio for copper taken from the literature.
Kβ/KαReference Excitation Source
Experiment:
0.1382(16) [12]241Am
0.1370(110) [13]241Am
0.1330(33) [14]241Am
0.1212(90) [10]241Am
0.1211(19) [15]241Am
0.1360(6) [17]241Am
0.1340(130) [18]241Am
0.1343(12) [19]241Am
0.1374(113) [20]241Am
0.1390(130) [21]241Am
0.1220(100) [22]241Am
0.1360(60) [13]241Am
0.1359(30) [24]241Am
0.1314(87) [25]241Am
0.1289(86) [26]241Am
0.1296(66) [27]241Am
0.1360(10) [28]241Am
0.1394(70) [55]241Am
0.1360(60) [35]238Pu
0.1366(330) [56]109Cd
0.1370 [30]109Cd
0.1390(56) [31]109Cd
0.1240(30) [33]137Cs
0.1240(90) [32]137Cs
0.1339 [34]57Co
0.1360(20) [38]K-capture
0.1372(10) [41] 1 MeV protons
0.1358(17) [36] 50 kV W X-ray tube
0.1383(55) [37] 35 mA W X-ray tube
0.1370(20) [38] 30 kV Mo X-ray tube
0.123(7) [57] 10 keV synchrotron radiation
Theory:
0.1379 [2]
0.1377 [3]
0.1340, 0.1350, 0.1366, 0.1377 [4] *
* different approaches
Materials 2021,14, 4462 8 of 11
Table 5. Kβ/Kα
intensity ratio and relative
Kβ/Kα
ratio (compared to
Kβ/Kα
ratio of pure Cu) for
copper alloys taken from literature.
Cu Alloy Kβ/KαRelative Kβ/KαReference
Cu29Ag71 0.1391(7) - [28]
Cu94Sn60.1351(6) - [28]
Cu48.4Sn51.6 0.1419(72) 1.0949 [27]
Cu14Sn86 0.1429(73) 1.1026 [27]
Cu6.1Sn93.9 0.1381(70) 1.0656 [27]
Co25Cu74Ag10.1388(92) 1.0563 [25]
Co31Cu68Ag10.1444(96) 1.0989 [25]
Co36Cu63.6Ag0.4 0.1371(91) 1.0434 [25]
Co10.7Cu89.1Ag0.2 0.1341(89) 1.0205 [25]
CuAl 0.1335(6) - [16]
4. Conclusions
The Cu
Kβ/Kα
intensity ratios for pure copper and for a sequence of nine Ag–Cu
alloys (from 10% to 100% Cu) have been simulated with the FLUKA code. The results can
play role in elemental material analysis, especially in archaeometry. Silver and copper are
used in jewelry and minting from antique times [
39
,
58
–
60
]. Thus it is in the interest of
archaeologists to explore the ancient technologies of silver jewelry production. Copper
was often added to silver to make sterling silver, increasing its strength. The concen-
tration of more than 2.6% Cu indicates a deliberate addition by ancient manufacturers.
The spectroscopic techniques like ED-XRF, SEM-EDX, or PIXE are commonly used in
compositional research. The elemental content is determined by using intensity of peaks
recorded in energetic spectra. However, these techniques can be used for surface and
subsurface analysis. The
Kβ/Kα
X-ray intensity ratio analyses can be applied for elemental
composition analysis as well as for determination of depth profile distributions of the
elements in studied artifacts. The thickness of coating in double layers artifacts and silver
surface enrichment of silver–copper alloys can be also determined. Moreover, since the
Ag–Cu alloying system has many other applications, among others it is often used in
nanotechnology [
61
,
62
] and it is estimated as the best material for improving oxidation
resistance with only a slight reduction in electrical conductivity [
63
], knowledge about
alloying effects may play important role in those areas.
Author Contributions:
A.M.G.: conceptualization, methodology, writing—original draft. K.K.:
investigation, formal analysis, writing—review and editing, visualization. A.W.: investigation,
formal analysis, writing—review and editing, visualization. E.A.M.-J.: investigation, writing—review
and editing. P.M.: investigation, writing—review and editing. J.S.: investigation, writing—review
and editing. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
The data presented in this study are available on request from the
corresponding author.
Acknowledgments: The authors are grateful to Jacek Ratajczyk for providing language help.
Conflicts of Interest: The authors declare no conflict of interest.
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