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Anomalous scattering factor using proton induced X-ray emission technique
Radiation Physics and Chemistry 107 (2015) 103–108
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Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Anomalous scattering factor using proton induced X-ray emission technique P. Latha a,n, P. Magudapathy b, K.K. Abdullah c, K.G.M. Nair b, B.R.S. Babu d, K.M. Varier e a
Department of Physics, Providence Women's College, Calicut, Kerala 673 009, India Materials Science Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, India c Department of Physics, Farook College, Calicut, Kerala 673 631, India d Department of Physics, Sultan Quaboos University, Oman e Department of Physics, Kerala University, Kariavattom Campus, Thiruvananthapuram , Kerala 695 581, India b
H I G H L I G H T S
Anomalous scattering factors determined from the attenuation data. PIXE technique is used for getting the attenuation data. Our results are in close agreement with the available theoretical values. PIXE technique is a reliable tool for determining anomalous scattering factors.
art ic l e i nf o
a b s t r a c t
Article history: Received 15 June 2014 Accepted 27 September 2014 Available online 22 October 2014
Atomic scattering factor is in general a complex number represented by the sum of normal scattering factor (f0) and anomalous scattering factors [real (f ′) and imaginary (f ″) ]. Anomalous scattering factors in Ag, In, Cd and Sn were determined experimentally from attenuation data measured using PIXE and compared with theoretical values. The data cover the energy region from 10 to 30 keV and atomic number Z from 47 to 50 keV. Our results found to be in close agreement with theoretical values. & Elsevier Ltd. All rights reserved.
Keywords: Mass attenuation coefficient Total cross section Photoelectric cross section PIXE Dispersion correction Anomalous scattering factor
1. Introduction In the X-ray energy range, the primary interactions of photons with atoms are photoabsorption, coherent (elastic) scattering and Compton (inelastic) scattering. Nuclear scattering and absorption including pair production and Delbrück scattering from the nuclear field; and nuclear resonant processes (such as nuclear Thomson scattering) are relevant at energies above around 1 MeV. The normal atomic scattering factor f0 describes the strength of X-rays scattered by the electrons in an atom just like free oscillators. However, in actual situations the scattering electrons are bound in atomic orbitals and they act instead as a set of damped oscillators with resonant frequencies matched to the absorption frequencies of the electron shells. The total atomic scattering factor f is then a complex number, and is n
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http://dx.doi.org/10.1016/j.radphyschem.2014.09.016 0969-806X/& Elsevier Ltd. All rights reserved.
represented by the sum of the normal scattering factor (f0) and anomalous scattering factor [real (f ′) and imaginary (f ″) also called the dispersion correction]. Knowledge of complex X-ray scattering factors is very important in various applications such as crystallography, medical diagnosis, radiation safety, and X-ray absorption fine structure studies (XAFS). A detailed calculation of real and imaginary parts of anomalous scattering factor using the dispersion relation is carried out in this work. The absorption cross sections needed for this study are measured using Proton Induced X-ray Emission Technique (Appaji Gowda and Umesh, 2006). In the present work, we have used protons of 2 MeV energy to excite characteristic X-rays from a set of targets kept inside a scattering chamber (PIXE chamber). This technique has two distinct advantages as compared to photon induced x-ray emission. The X-ray flux available from the PIXE technique is much more than the flux available from radioactive sources. Also, the background effects from the PIXE technique are much less as compared to other methods.
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2. Theoretical description of anomalous scattering factor
Table 1 Relativistic corrections (high energy limit, f ′(∞) = ▵ ).
Theory of anomalous scattering and dispersion of X-rays is treated in detail by James (1948). In this theory the atoms interacting with photons are treated like electric dipole oscillators having certain definite natural frequencies which correspond to the absorption frequencies of the atoms. The electric field of incident photon modifies the frequency and the amplitude of the oscillators by a dispersive term and an absorptive term. The dispersive term depends on the proximity of the impressed frequency to the natural resonant frequency of the system and the absorptive term depends on the damping factor. At very low photon energy, the photon has no sufficient energy to excite any of the available electronic transitions. The elastic scattering cross-section (or the probability that the photon is scattered) may be adequately described by the normal atomic scattering coefficient f0 only, with no phase delay (imaginary component f ″ is zero). When the incident photon has enough energy, some of them are scattered normally, while some of them are either absorbed and then re-emitted at lower energy (fluorescence) or absorbed and then re-emitted at the same energy (strong coupling to absorption edge energy). The scattered photon gains an imaginary component to its phase ( f ″ becomes non-zero); i.e. it is retarded compared to a normally scattered photon. Thus, in anomalous X-ray scattering from an atom the scattering factor becomes a complex quantity and is written as
f = f0 + f ′ + if ″
(1)
where first term f0 is the normal scattering form factor, f ′ is the real part of anomalous scattering factor representing the dispersion effect in scattering near the resonant level and f ″ is the imaginary part of anomalous scattering factor representing the absorptive part of the elastic scattering near a resonant state of the bound scattering electron. f ′ and f ″ are together known as dispersion corrections. From the optical theorem
f″ =
mc ϵ 0E e2 ?
σtot
(2)
In terms of classical electron radius r0, we obtain the expression as
f″ =
E σtot 2hcr0
(3)
2
▵KP S-matrix correction
▵CM Multiple correction
▵CL Dipole corrections
Silver Cadmium Indium Tin
0.264 0.277 0.291 0.305
0.285 0.300 0.315 0.331
0.471 0.496 0.522 0.548
correction to the high energy limit f ′(∞) of forward scattering. Kissel and Pratt (1990) used numerical S-matrix approach to determine f ′(∞) to a high degree of accuracy and the values have been tabulated for all neutral atoms. The attenuation coefficient data were determined experimentally for the elements Ag, Cd, In and Sn using the PIXE method, for evaluating the dispersion correction in these elements. Details of the calculation and the results obtained therefrom are given in the following section.
3. Analysis For evaluating the dispersion corrections, it is essential to determine the total mass attenuation coefficients of the elements of interest over a wide range of energies around the corresponding K-edges. Using these values, the total attenuation cross section s was determined. From the measured values of total attenuation cross section, the total photoelectric cross sections were estimated by subtracting the sum of the coherent and incoherent scattering contributions. The scattering contributions required for this were extracted from the XCOM (Berger et al., 1990) data base. 3.1. Calculation of imaginary part of the anomalous scattering factor f″ The photo effect cross sections obtained as mentioned above were used to calculate f ″ using Eq. (3). These values are tabulated in Tables 2–5. Table 2 Photo effect cross-sections (sph), imaginary part (f ″) and real part (f ′) of the anomalous scattering factor for Ag. Energy
sph
(keV)
(barns/atom)
10.50 11.51 12.62 12.94 13.38 14.76 15.73 16.57 17.43 17.67 18.62 19.61 21.10 22.08 23.08 23.82 24.11 24.94 25.16 25.51 26.10 27.28 28.49
17,068.027 1016.23 12,113.447 721.53 11,482.46 7 683.59 11,013.89 7 658.85 8945.25 7 536.84 7971.52 7 476.49 6010.05 7 357.70 5635.107 335.40 4762.85 7 283.50 4879.427 290.81 4295.08 7 256.10 3575.747 213.24 2903.72 7172.91 2429.227 144.68 2109.89 7 125.76 2100.52 7 125.72 1800.377 107.38 1687.627 101.13 1498.787 89.46 1244.00 774.76 8933.097 542.19 8428.03 7 515.94 7161.69 7 441.15
f″
Real part, f ′
2
Here, r0 = e /4π ϵ0mc is the classical electron radius. The total cross section stot is given by σtot = τ + σBBT − σBPP , where τ, σBBT , and σBPP are the photo effect, photo excitation, and bound pair production cross-sections, respectively. For energies sufficiently away from absorption edges of a particular element, σBBT , σBPP are expected to be insignificant for Z > 10, below the pair production threshold (Wang, 1986; Wang and Pratt, 1983). In the energy region of current interest, if we neglect the spin flip process, the f ′ and f ″ are connected by the modified Kramers–Kronig transform (Zhou et al., 1992; Henke et al., 1982)
fR ′(E) = f ′(∞) −
Element
2 P π
∫0
∞
E′f ″(E′) dE′ E2 − E′ 2
(4)
where P is the Cauchy principal value of the dispersion integral. The second term on the R.H.S. of the above equation represents the non-relativistic values fNR ′(E). The factor f ′(∞) is called the high energy limit or relativistic correction. The f ′(∞) values of Cromer and Liberman (1970) [CL], Creagh and McAuley (1992) [CM] and Kissel and Pratt (1990) [KP] for elements Ag, Cd, In and Sn which have been used in the present work are given in Table 1. Cromer and Liberman (1970) and Henke et al. (1982) considered electric dipole approximation in their estimate of the relativistic
2.56 7 0.15 1.99 7 0.12 2.077 0.12 2.047 0.12 1.71 70.10 1.687 0.10 1.357 0.08 1.34 7 0.08 1.197 0.07 1.23 7 0.07 1.147 0.07 1.007 0.06 0.88 7 0.05 0.777 0.05 0.707 0.04 0.727 0.04 0.62 7 0.04 0.60 7 0.04 0.54 7 0.03 3.36 7 0.20 3.337 0.20 3.29 7 0.20 2.92 7 0.18
(KP)
(CM)
(CL)
0.03 0.35 0.55 0.46 0.50 0.65 0.74 0.84 0.95 0.98 1.01 1.12 1.33 1.53 1.83 2.08 2.22 2.98 3.40 4.89 2.91 1.74 1.22
0.05 0.37 0.57 0.48 0.52 0.67 0.76 0.86 0.97 1.00 1.04 1.14 1.36 1.55 1.85 2.10 2.25 3.00 3.42 4.91 2.93 1.76 1.24
0.23 0.56 0.76 0.66 0.71 0.86 0.94 1.05 1.16 1.19 1.22 1.32 1.54 1.74 2.04 2.29 2.43 3.19 3.60 5.10 3.12 1.94 1.42
P. Latha et al. / Radiation Physics and Chemistry 107 (2015) 103–108
Table 3 Photo effect cross-sections (sph), imaginary part (f ″) and real part (f ′) of the anomalous scattering factor for Cd. Energy
sph
(keV)
(barns/atom)
7.06 8.04 8.12 8.91 9.34 9.67 10.50 11.51 12.62 13.38 14.76 15.73 16.57 17.67 18.62 19.61 21.10 22.08 23.08 23.82 24.11 24.94 25.16 26.10 26.71 27.28 28.49
58,118.077 5018.40 40,704.777 2422.02 40,664.80 72443.87 30,609.78 71825.65 28,733.09 71711.34 25,470.21 71517.58 21,946.85 71303.44 15,555.55 7924.11 13,525.69 7802.60 10,266.22 7615.50 8745.37 7 521.36 7294.85 7432.65 6293.777 373.29 5417.777 321.70 4150.93 7246.64 3896.707 231.58 3001.38 7 178.13 2565.39 7152.25 2180.247 129.50 1949.46 7116.39 2045.25 7121.57 1920.05 7114.68 1691.18 7100.60 1607.36 796.34 1557.007 94.35 9497.69 7583.26 7470.677 460.41
f″
5.86 70.51 4.687 0.28 4.727 0.28 3.90 70.23 3.84 70.23 3.52 70.21 3.29 70.20 2.56 70.15 2.447 0.14 1.96 70.12 1.85 70.11 1.647 0.10 1.357 0.09 1.377 0.08 1.117 0.07 1.09 70.06 0.917 0.05 0.81 70.05 0.72 7 0.04 0.667 0.04 0.707 0.04 0.687 0.04 0.617 0.04 0.60 70.04 4.08 70.25 3.707 0.23 3.047 0.19
Real part, f ′
Energy
sph
(keV)
(barns/atom)
10.50 11.51 12.62 13.38 14.76 15.73 16.57 17.43 17.67 18.62 19.61 21.10 22.08 23.08 23.82 24.11 24.94 25.16 26.10 27.28 27.94 28.49 29.78 30.40
18,430.23 7 1284.99 16,096.30 7 1123.33 13,411.20 7 818.19 11,093.75 7 803.45 9021.14 7 563.16 7402.56 7436.41 6510.687 383.64 5574.59 7 328.26 5439.32 7322.04 4762.05 7282.01 4018.77 7 237.50 3200.62 7188.56 2733.19 7 160.98 2435.80 7143.64 2280.75 7135.62 2037.74 7 120.27 1948.39 7115.81 1817.07 7 107.34 1890.62 7112.97 1656.34 799.62 1542.007 95.78 8779.56 7528.06 7520.007 467.09 7131.007 442.92
f″
2.777 0.19 2.65 70.18 2.42 70.15 2.127 0.15 1.90 70.12 1.667 0.10 1.54 70.09 1.39 70.08 1.377 0.08 1.277 0.08 1.137 0.07 0.977 0.06 0.86 70.05 0.80 70.05 0.78 70.05 0.707 0.04 0.69 70.04 0.65 70.04 0.717 0.04 0.65 70.04 3.75 70.23 3.58 70.22 3.20 70.20 3.107 0.19
Table 5 Photo effect cross-sections (sph), imaginary part (f ″) and real part (f ′) of the anomalous scattering factor for Sn. Energy
sph
f″
(KP)
(CM)
(CL)
(keV)
(barns/atom)
0.11 0.03 0.02 0.08 0.10 0.10 0.07 0.17 0.24 0.32 0.53 0.57 0.72 0.72 0.83 0.98 1.15 1.33 1.55 1.79 1.88 2.14 2.26 3.15 5.24 2.75 1.71
0.13 0.05 0.04 0.10 0.13 0.12 0.09 0.20 0.26 0.34 0.56 0.59 0.74 0.74 0.85 1.00 1.18 1.35 1.57 1.81 1.90 2.17 2.28 3.17 5.26 2.78 1.74
0.33 0.25 0.24 0.30 0.32 0.32 0.29 0.39 0.46 0.54 0.75 0.79 0.94 0.94 1.05 1.20 1.37 1.54 1.77 2.01 2.10 2.36 2.48 3.37 5.46 2.97 1.93
6.40 7.06 8.04 8.12 8.91 9.34 9.67 10.50 11.51 12.62 13.38 14.76 15.73 16.57 17.43 17.67 18.62 19.61 21.10 22.08 23.08 23.82 24.11 24.94 26.10 27.28 28.49 29.19 30.97 31.82
81,222.56 7 5327.24 76,751.36 7 6735.16 50,470.527 3013.94 48,906.52 7 2937.24 36,224.07 72167.12 33,295.257 1988.73 28,352.827 1693.59 24,380.057 1453.69 17,169.53 7 1024.14 14,461.80 7 861.92 11,008.127 661.54 8945.09 7 535.15 8034.53 7 478.71 7112.83 7 423.81 6171.26 7 367.73 5991.067 357.45 5059.58 7 302.01 4374.107 261.13 3233.07 7192.76 2640.88 7 157.46 2505.02 7 149.47 2363.52 7 141.74 2021.92 7 120.73 2256.447 135.37 1832.917 110.34 1613.727 97.65 1533.957 93.36 1491.007 91.07 7422.007 453.34 6958.007 425.00
Table 4 Photo effect cross-sections (sph), imaginary part (f ″) and real part (f ′) of the anomalous scattering factor for In. Real part, f ′ (KP)
(CM)
(CL)
0.45 0.26 0.30 0.36 0.49 0.55 0.62 0.70 0.73 0.80 0.89 1.07 1.22 1.40 1.51 1.57 1.81 1.88 2.23 3.07 5.06 2.96 1.71 1.44
0.42 0.28 0.32 0.39 0.51 0.58 0.64 0.73 0.76 0.83 0.92 1.10 1.25 1.42 1.53 1.60 1.83 1.90 2.26 3.09 5.09 2.98 1.73 1.46
0.21 0.49 0.53 0.59 0.72 0.78 0.85 0.94 0.96 1.03 1.13 1.31 1.45 1.63 1.74 1.80 2.04 2.11 2.46 3.30 5.30 3.19 1.94 1.67
105
7.43 7 0.49 7.747 0.68 5.80 7 0.35 5.6770.34 4.617 0.28 4.45 7 0.27 3.92 7 0.23 3.667 0.22 2.83 7 0.17 2.617 0.16 2.117 0.13 1.89 7 0.11 1.81 7 0.11 1.69 7 0.10 1.54 7 0.09 1.517 0.09 1.3570.08 1.23 7 0.07 0.98 7 0.06 0.83 7 0.05 0.83 7 0.05 0.80 7 0.05 0.7070.04 0.80 7 0.05 0.687 0.04 0.63 7 0.04 0.62 7 0.04 3.577 0.22 3.29 7 0.20 3.177 0.19
Real part, f ′ (KP)
(CM)
(CL)
0.60 0.55 0.44 0.45 0.33 0.36 0.32 0.22 0.09 0.02 0.14 0.46 0.55 0.56 0.60 0.62 0.68 0.75 0.91 1.10 1.28 1.35 1.42 1.60 1.81 2.23 3.08 5.01 1.87 1.41
0.63 0.58 0.41 0.43 0.30 0.33 0.29 0.20 0.07 0.05 0.16 0.49 0.58 0.58 0.63 0.64 0.70 0.78 0.94 1.12 1.31 1.38 1.45 1.63 1.83 2.25 3.11 5.03 1.90 1.44
0.85 0.79 0.20 0.21 0.09 0.12 0.08 0.02 0.15 0.26 0.38 0.71 0.79 0.80 0.85 0.86 0.92 0.99 1.15 1.34 1.52 1.59 1.67 1.84 2.05 2.47 3.32 5.25 2.12 1.66
the upper limit was 1500 keV. To evaluate the integral numerically, the energy region used for integration was divided into a large number of small intervals. Within each interval (Ei, Ei + 1), the energy dependence of fi″ was determined by a linear function
fi″ = ai + bi E
(5)
In this interval, the dispersion integral assumes the form
Ii, i + 1(Es) =
2 P π
∫E
Ei + 1 i
E′f ″(E′) dE′ Es2 − E′ 2
(6)
Mathematically the above equation is equivalent to
Ii, i + 1(Es)
=
⎡ E 2 − Ei2+ 1 2 ⎢ ai ln s 2 π ⎢⎣ 2 Es − Ei2 ⎛ (E + Es)(Ei − Es) E +bi ⎜⎜Ei + 1 − Ei − s ln i + 1 2 (Ei + 1 − Es)(Ei + Es) ⎝
⎞⎤ ⎟⎟⎥ ⎠⎥⎦
(7)
Here Es is the energy of interest at which the value of the dispersion integral is to be determined. The details of the calculation can be found elsewhere (Appaji Gowda and Umesh, 2006). Using the coefficients ai and bi the dispersion integral in Eq. (6) was calculated as explained (Appaji Gowda and Umesh, 2006) in the following section.
3.2. Calculation of real part of the anomalous scattering factor
4. Evaluation of the dispersion integral
The dispersion correction f ′(E) was determined by the numerical evaluation of the dispersion integral in Eq. (4). For this purpose, the lower limit of integration was chosen to be 1 keV and
Photelectric cross sections extracted from the attenuation data determined using PIXE were used for evaluating the integral in Eq. (6). Below the lower energy limit Emin of the present
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P. Latha et al. / Radiation Physics and Chemistry 107 (2015) 103–108
measurements, down to 1 keV, and above the upper limit Emax up to 1500 keV, XCOM (Hubbell et al., 1975; Hubbell and Overbo, 1979) values were used for the evaluation of the integral. The required integrals in Eq. (4) have been evaluated using a semi-analytical, semi-numerical approach. At each energy Ei, the integral to be evaluated is
I (Ei) =
∫E
E2
Eσph(E) dE
1
Ei2
where ϕ(E) = steps:
2
−E
=
∫E
E0
ϕ(E) dE
(8)
1
Eσph(E)/(Ei2
2
− E ). The integration is done in three
(a) from E1 to Ei − ϵ (b) from Ei −ϵ to Ei + ϵ (c) from Ei +ϵ to E2
Fig. 2. PIXE chamber and the Si(Li) detector.
Here ϵ is an arbitrarily chosen small energy interval (typically taken as Ei/1000). Now,
I (Ei) =
∫E
Ei −ϵ
1
ϕ(E) dE +
∫E
Ei +ϵ i −ϵ
ϕ(E) dE +
∫E
E2 i +ϵ
ϕ(E) dE
(9)
i.e.
I (Ei) = I1(Ei) + I2(Ei) + I3(Ei).
(10)
For evaluating I1(Ei) and I3(Ei), the relevant energy interval is subdivided further into a large number of equal sub-intervals (on a logarithmic scale). Within each sub interval the integration is carried out using Simpson's rule, assuming a linear variation of the photoelectric cross section
ln(σph(E)) = a + b ln(E).
(11)
In the energy region below the lower energy limit and above the upper energy limit of the present measurements, the XCOM values of the photoelectric cross sections are directly used for the integration. For evaluating I2, Eq. (7) has been used. Based on the above procedure, the dispersion integral and consequently the real part of the anomalous scattering factor were evaluated numerically using a fortran program.
5. Experimental setup The entire experiment is performed in a PIXE chamber attached to the beam line of a Tandetron accelerator at the Indira Gandhi Atomic Research Centre, Kalpakkam, Tamil Nadu, India. Schematic diagram of the chamber and a photograph of actual PIXE setup are shown in Figs. 1 and 2 respectively. The PIXE chamber is made of steel and has a diameter of 35 cm and a height
of 38 cm and is installed in one of the beam lines of the Tandetron accelerator. Two graphite collimators of 3 mm diameter are used to collimate the proton beam. The chamber is provided with various ports like Proton beam port (P), Detector port (D), Beam view port (B) and Vacuum pump port (V) as shown in Fig. 1. At the centre of the chamber there is a target ladder for holding six targets at a time. The position of the ladder can be adjusted from outside without breaking the vacuum and the position of the target as well as the absorber can be fixed. There is an absorber ladder positioned in front of the detector port which can hold five absorber foils at a time. A charge integrator is coupled directly to the target holder to measure the proton current. The vacuum inside the chamber was maintained through the port (V). The absorbers in the form of thin foils of Ag, Cd, In, Sn of size 1.5 cm 1.5 cm are mounted on the movable ladder kept in front of the detector port. Protons of energy 2 MeV were allowed to fall on suitable target foil mounted on the target ladder. The X-rays emitted by this foil at 45° to the incident proton beam were then allowed to fall on the Si(Li) detector, with and without the absorber. A mylar foil of sufficient thickness is placed between the detector and the target to filter out the back scattered protons. The spectra were recorded with a MCA for a pre-set total charge as measured by the current integrator connected to the target ladder. Typical X-ray spectra for the direct beam and the beam transmitted through the absorbers are shown in Fig. 3. Counting times are roughly around 30 min for each spectrum, keeping the total proton charge at a constant value of 10 μC, as measured by the current integrator. From the net counts under the characteristic peaks, the mass attenuation coefficient was calculated using the
22.16keV
12000
without absorber with In absorber
10000
Counts
8000
6000
4000
24.94keV 2000
0 21
22
23
24
25
26
Energy(keV) Fig. 1. Schematic diagram of the experimental set up for X-ray attenuation measurements using the PIXE method.
Fig. 3. Typical X-ray spectra for the direct beam and the beam transmitted through the indium absorber.
The values of photoelectric cross sections (sph) obtained as outlined in Section 3 were used to extract the imaginary part f ″ of anomalous scattering factor as per Eq. (3). The f ′ values were obtained by evaluating the dispersion integral as per the procedure mentioned above. f ′ values were thus calculated corresponding to f ′(∞) values of Cromer and Liberman (1970)[CL], Creagh and McAuley (1992) [CM] and Kissel and Pratt (1990) [KP]. The f ′ and f ″ values around K-edge are tabulated for Ag, Cd, In and Sn in Tables 2–5. The f ′ values are plotted in Figs. 4–7. The theoretical data of Chantler et al. (2005) and Cullen et al. (1997) is also shown for comparison of the present results. It is seen from the figures that the present values of f ′ follow the trend as suggested by these theories.
7. Conclusions
Real part of anomolous scattering factor
Real and imaginary parts of anomalous scattering factors were determined from the attenuation data determined using PIXE technique. The expected error in the measurement of cross section includes the error due to counting statistics and the error in the measurement of mass thickness of the samples. The overall error is always less than 7%. The calculated scattering factors are in agreement with the theoretical values reported by Chantler et al. (2005) and Cullen et al. (1997). It is observed that the f ′ values corresponding to f ′(∞) values of Creagh and McAuley (1992) [CM] and Kissel and Pratt (1990) [KP] show better agreement with theoretical values than those corresponding to Cromer and Liberman (1970) [CL]. The KP corrections are based on the S-matrix calculations. The CM values are based on use of multipole terms in the self-consistent field term, rather than limiting to the dipole term only as has been done in the CL estimates. This is probably the reason for the better agreement of the experimental results with the theoretical values using the KP and CM corrections. No earlier studies have been reported for the measurement of anomalous scattering factors using PIXE technique. However, with the development of tunable synchrotron radiation sources, some measurements have been carried out using synchrotron radiation (Chantler et al., 2001). Our results clearly establish that
Silver K-edge = 25.51keV
2 0 -2
Cadmium K-edge = 26.71keV
0 -2 Chantler with KP correction with CM correction with CL correction Cullen
-4 -6
-4 -6 5
10
15
20 25 Energy(keV)
10
15
20 25 Energy(keV)
30
35
Fig. 5. Plot of real part of the anomalous scattering factor as a function of photon energy around the K-edge of cadmium.
2
Indium K-edge = 27.94 keV
0 -2 Chantler with KP correction with CM correction with CL correction Cullen
-4 -6 5
10
15
20 25 Energy(keV)
30
35
Fig. 6. Plot of real part of the anomalous scattering factor as a function of photon energy around the K-edge of indium.
2
Tin K-edge = 29.19 keV
0 -2 Theoretical with KP correction with CM correction with CL correction Cullen
-4 -6 5
Chantler with KP correction with CM correction with CL correction Cullen
107
2
5
Real part of anomolous scattering factor
6. Result and discussion
Real part of anomolous scattering factor
known absorber mass thickness. Measurements were repeated for all possible combinations of exciter foil and absorbers.
Real part of anomolous scattering factor
P. Latha et al. / Radiation Physics and Chemistry 107 (2015) 103–108
10
15
20 25 Energy(keV)
30
35
Fig. 7. Plot of real part of the anomalous scattering factor as a function of photon energy around the K-edge of tin.
30
35
Fig. 4. Plot of real part of the anomalous scattering factor as a function of photon energy around the K-edge of silver.
in situations where mono energetic X-rays are not available from radioactive sources, it is possible to use PIXE technique as a reliable tool to generate the required X-ray energies in the region of interest using suitable targets.
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P. Latha et al. / Radiation Physics and Chemistry 107 (2015) 103–108
Acknowledgments We are thankful to the technical staff at the Materials Science Group, IGCAR, Kalpakkam, for their help during the PIXE studies.
References Appaji Gowda, S.B., Umesh, T.K., 2006. Nucl. Instr. Methods Phys. Res. B 243, 2. Berger, M.J., Hubbell, J.H., Seltzer, S.M., Coursey, J.S., Zucker, D.S., 1990. XCOM: Photon Cross Section Database (Version 1.2), National Institute of Standards and Technology, Gaithersburg, MD, 〈http://physics.nist.gov.xcom〉. Chantler, C.T., Tran, C.Q., Barnea, Z., Paterson, D., Cookson, D.J., Balaic, D.X., 2001. Phys. Rev. A 64, 062506. Chantler, C.T., Olsen, K., Dragoset, R.A., Chang, J., Kishore, A.R., Kotochigova, S.A., Zucker, D.S., 2005. X-Ray Form Factor, Attenuation and Scattering Tables (version 2.1). National Institute of Standards and Technology, Gaithersburg, MD. Available: 〈http://physics.nist.gov/ffast〉.
Creagh, D.C., McAuley, W.J., 1992. In: Wilson, A.J.C. (Ed.), International Tables for Crystallography, vol. C, Kluwer Academic Publishers, Dordrecht, p. 206. Cromer, D.T., Liberman, D., 1970. J. Chem. Phys. 53, 1898. Dermott E. Cullen, John H. Hubbell, Lynn Kissel, 1997. Evaluated Photon Data Library, '97 Version 〈http://www-nds.iaea.org/epdl97〉. Henke, B.L., Lee, P., Tanaka, T.J., Shimambukuro, R.L., Fujikawa, B.K., 1982. Atom. Data Nucl. Data Tables 27, 1. Hubbell, J.H., Overbo, I.J., 1979. J. Phys. Chem. Ref. Data 8, 69. Hubbell, J.H., Veigele, W.m.J., Briggs, E.A., Brown, R.T., Cromer, D.T., Howerton, R.J., 1975. J. Phys. Chem. Ref. Data 4, 417. James, R.W., 1948. The Crystalline State, vol. II, Bell & Hyman, Ltd. Republished as The Optical Principles of the Diffraction of X-rays Ox Bow Press, Woodbridge, CT, 1962. Kissel, L., Pratt, R.H., 1990. Acta Cryst. A 46, 170. Wang, M.S., 1986. Phys. Rev. A 34, 636. Wang, M.S., Pratt, R.H., 1983. Phys. Rev. A 28, 3115. Zhou, B., Kissel, L., Pratt, R.H., 1992. Nucl. Instr. Methods Phys. Res. B 66, 307.
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Anomalous scattering factor using proton induced X-ray emission technique P. Latha a,n, P. Magudapathy b, K.K. Abdullah c, K.G.M. Nair b, B.R.S. Babu d, K.M. Varier e a
Department of Physics, Providence Women's College, Calicut, Kerala 673 009, India Materials Science Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, India c Department of Physics, Farook College, Calicut, Kerala 673 631, India d Department of Physics, Sultan Quaboos University, Oman e Department of Physics, Kerala University, Kariavattom Campus, Thiruvananthapuram , Kerala 695 581, India b
H I G H L I G H T S
Anomalous scattering factors determined from the attenuation data. PIXE technique is used for getting the attenuation data. Our results are in close agreement with the available theoretical values. PIXE technique is a reliable tool for determining anomalous scattering factors.
art ic l e i nf o
a b s t r a c t
Article history: Received 15 June 2014 Accepted 27 September 2014 Available online 22 October 2014
Atomic scattering factor is in general a complex number represented by the sum of normal scattering factor (f0) and anomalous scattering factors [real (f ′) and imaginary (f ″) ]. Anomalous scattering factors in Ag, In, Cd and Sn were determined experimentally from attenuation data measured using PIXE and compared with theoretical values. The data cover the energy region from 10 to 30 keV and atomic number Z from 47 to 50 keV. Our results found to be in close agreement with theoretical values. & Elsevier Ltd. All rights reserved.
Keywords: Mass attenuation coefficient Total cross section Photoelectric cross section PIXE Dispersion correction Anomalous scattering factor
1. Introduction In the X-ray energy range, the primary interactions of photons with atoms are photoabsorption, coherent (elastic) scattering and Compton (inelastic) scattering. Nuclear scattering and absorption including pair production and Delbrück scattering from the nuclear field; and nuclear resonant processes (such as nuclear Thomson scattering) are relevant at energies above around 1 MeV. The normal atomic scattering factor f0 describes the strength of X-rays scattered by the electrons in an atom just like free oscillators. However, in actual situations the scattering electrons are bound in atomic orbitals and they act instead as a set of damped oscillators with resonant frequencies matched to the absorption frequencies of the electron shells. The total atomic scattering factor f is then a complex number, and is n
Corresponding author. Tel.: þ 91 9847154956. E-mail address: [email protected] (P. Latha).
http://dx.doi.org/10.1016/j.radphyschem.2014.09.016 0969-806X/& Elsevier Ltd. All rights reserved.
represented by the sum of the normal scattering factor (f0) and anomalous scattering factor [real (f ′) and imaginary (f ″) also called the dispersion correction]. Knowledge of complex X-ray scattering factors is very important in various applications such as crystallography, medical diagnosis, radiation safety, and X-ray absorption fine structure studies (XAFS). A detailed calculation of real and imaginary parts of anomalous scattering factor using the dispersion relation is carried out in this work. The absorption cross sections needed for this study are measured using Proton Induced X-ray Emission Technique (Appaji Gowda and Umesh, 2006). In the present work, we have used protons of 2 MeV energy to excite characteristic X-rays from a set of targets kept inside a scattering chamber (PIXE chamber). This technique has two distinct advantages as compared to photon induced x-ray emission. The X-ray flux available from the PIXE technique is much more than the flux available from radioactive sources. Also, the background effects from the PIXE technique are much less as compared to other methods.
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P. Latha et al. / Radiation Physics and Chemistry 107 (2015) 103–108
2. Theoretical description of anomalous scattering factor
Table 1 Relativistic corrections (high energy limit, f ′(∞) = ▵ ).
Theory of anomalous scattering and dispersion of X-rays is treated in detail by James (1948). In this theory the atoms interacting with photons are treated like electric dipole oscillators having certain definite natural frequencies which correspond to the absorption frequencies of the atoms. The electric field of incident photon modifies the frequency and the amplitude of the oscillators by a dispersive term and an absorptive term. The dispersive term depends on the proximity of the impressed frequency to the natural resonant frequency of the system and the absorptive term depends on the damping factor. At very low photon energy, the photon has no sufficient energy to excite any of the available electronic transitions. The elastic scattering cross-section (or the probability that the photon is scattered) may be adequately described by the normal atomic scattering coefficient f0 only, with no phase delay (imaginary component f ″ is zero). When the incident photon has enough energy, some of them are scattered normally, while some of them are either absorbed and then re-emitted at lower energy (fluorescence) or absorbed and then re-emitted at the same energy (strong coupling to absorption edge energy). The scattered photon gains an imaginary component to its phase ( f ″ becomes non-zero); i.e. it is retarded compared to a normally scattered photon. Thus, in anomalous X-ray scattering from an atom the scattering factor becomes a complex quantity and is written as
f = f0 + f ′ + if ″
(1)
where first term f0 is the normal scattering form factor, f ′ is the real part of anomalous scattering factor representing the dispersion effect in scattering near the resonant level and f ″ is the imaginary part of anomalous scattering factor representing the absorptive part of the elastic scattering near a resonant state of the bound scattering electron. f ′ and f ″ are together known as dispersion corrections. From the optical theorem
f″ =
mc ϵ 0E e2 ?
σtot
(2)
In terms of classical electron radius r0, we obtain the expression as
f″ =
E σtot 2hcr0
(3)
2
▵KP S-matrix correction
▵CM Multiple correction
▵CL Dipole corrections
Silver Cadmium Indium Tin
0.264 0.277 0.291 0.305
0.285 0.300 0.315 0.331
0.471 0.496 0.522 0.548
correction to the high energy limit f ′(∞) of forward scattering. Kissel and Pratt (1990) used numerical S-matrix approach to determine f ′(∞) to a high degree of accuracy and the values have been tabulated for all neutral atoms. The attenuation coefficient data were determined experimentally for the elements Ag, Cd, In and Sn using the PIXE method, for evaluating the dispersion correction in these elements. Details of the calculation and the results obtained therefrom are given in the following section.
3. Analysis For evaluating the dispersion corrections, it is essential to determine the total mass attenuation coefficients of the elements of interest over a wide range of energies around the corresponding K-edges. Using these values, the total attenuation cross section s was determined. From the measured values of total attenuation cross section, the total photoelectric cross sections were estimated by subtracting the sum of the coherent and incoherent scattering contributions. The scattering contributions required for this were extracted from the XCOM (Berger et al., 1990) data base. 3.1. Calculation of imaginary part of the anomalous scattering factor f″ The photo effect cross sections obtained as mentioned above were used to calculate f ″ using Eq. (3). These values are tabulated in Tables 2–5. Table 2 Photo effect cross-sections (sph), imaginary part (f ″) and real part (f ′) of the anomalous scattering factor for Ag. Energy
sph
(keV)
(barns/atom)
10.50 11.51 12.62 12.94 13.38 14.76 15.73 16.57 17.43 17.67 18.62 19.61 21.10 22.08 23.08 23.82 24.11 24.94 25.16 25.51 26.10 27.28 28.49
17,068.027 1016.23 12,113.447 721.53 11,482.46 7 683.59 11,013.89 7 658.85 8945.25 7 536.84 7971.52 7 476.49 6010.05 7 357.70 5635.107 335.40 4762.85 7 283.50 4879.427 290.81 4295.08 7 256.10 3575.747 213.24 2903.72 7172.91 2429.227 144.68 2109.89 7 125.76 2100.52 7 125.72 1800.377 107.38 1687.627 101.13 1498.787 89.46 1244.00 774.76 8933.097 542.19 8428.03 7 515.94 7161.69 7 441.15
f″
Real part, f ′
2
Here, r0 = e /4π ϵ0mc is the classical electron radius. The total cross section stot is given by σtot = τ + σBBT − σBPP , where τ, σBBT , and σBPP are the photo effect, photo excitation, and bound pair production cross-sections, respectively. For energies sufficiently away from absorption edges of a particular element, σBBT , σBPP are expected to be insignificant for Z > 10, below the pair production threshold (Wang, 1986; Wang and Pratt, 1983). In the energy region of current interest, if we neglect the spin flip process, the f ′ and f ″ are connected by the modified Kramers–Kronig transform (Zhou et al., 1992; Henke et al., 1982)
fR ′(E) = f ′(∞) −
Element
2 P π
∫0
∞
E′f ″(E′) dE′ E2 − E′ 2
(4)
where P is the Cauchy principal value of the dispersion integral. The second term on the R.H.S. of the above equation represents the non-relativistic values fNR ′(E). The factor f ′(∞) is called the high energy limit or relativistic correction. The f ′(∞) values of Cromer and Liberman (1970) [CL], Creagh and McAuley (1992) [CM] and Kissel and Pratt (1990) [KP] for elements Ag, Cd, In and Sn which have been used in the present work are given in Table 1. Cromer and Liberman (1970) and Henke et al. (1982) considered electric dipole approximation in their estimate of the relativistic
2.56 7 0.15 1.99 7 0.12 2.077 0.12 2.047 0.12 1.71 70.10 1.687 0.10 1.357 0.08 1.34 7 0.08 1.197 0.07 1.23 7 0.07 1.147 0.07 1.007 0.06 0.88 7 0.05 0.777 0.05 0.707 0.04 0.727 0.04 0.62 7 0.04 0.60 7 0.04 0.54 7 0.03 3.36 7 0.20 3.337 0.20 3.29 7 0.20 2.92 7 0.18
(KP)
(CM)
(CL)
0.03 0.35 0.55 0.46 0.50 0.65 0.74 0.84 0.95 0.98 1.01 1.12 1.33 1.53 1.83 2.08 2.22 2.98 3.40 4.89 2.91 1.74 1.22
0.05 0.37 0.57 0.48 0.52 0.67 0.76 0.86 0.97 1.00 1.04 1.14 1.36 1.55 1.85 2.10 2.25 3.00 3.42 4.91 2.93 1.76 1.24
0.23 0.56 0.76 0.66 0.71 0.86 0.94 1.05 1.16 1.19 1.22 1.32 1.54 1.74 2.04 2.29 2.43 3.19 3.60 5.10 3.12 1.94 1.42
P. Latha et al. / Radiation Physics and Chemistry 107 (2015) 103–108
Table 3 Photo effect cross-sections (sph), imaginary part (f ″) and real part (f ′) of the anomalous scattering factor for Cd. Energy
sph
(keV)
(barns/atom)
7.06 8.04 8.12 8.91 9.34 9.67 10.50 11.51 12.62 13.38 14.76 15.73 16.57 17.67 18.62 19.61 21.10 22.08 23.08 23.82 24.11 24.94 25.16 26.10 26.71 27.28 28.49
58,118.077 5018.40 40,704.777 2422.02 40,664.80 72443.87 30,609.78 71825.65 28,733.09 71711.34 25,470.21 71517.58 21,946.85 71303.44 15,555.55 7924.11 13,525.69 7802.60 10,266.22 7615.50 8745.37 7 521.36 7294.85 7432.65 6293.777 373.29 5417.777 321.70 4150.93 7246.64 3896.707 231.58 3001.38 7 178.13 2565.39 7152.25 2180.247 129.50 1949.46 7116.39 2045.25 7121.57 1920.05 7114.68 1691.18 7100.60 1607.36 796.34 1557.007 94.35 9497.69 7583.26 7470.677 460.41
f″
5.86 70.51 4.687 0.28 4.727 0.28 3.90 70.23 3.84 70.23 3.52 70.21 3.29 70.20 2.56 70.15 2.447 0.14 1.96 70.12 1.85 70.11 1.647 0.10 1.357 0.09 1.377 0.08 1.117 0.07 1.09 70.06 0.917 0.05 0.81 70.05 0.72 7 0.04 0.667 0.04 0.707 0.04 0.687 0.04 0.617 0.04 0.60 70.04 4.08 70.25 3.707 0.23 3.047 0.19
Real part, f ′
Energy
sph
(keV)
(barns/atom)
10.50 11.51 12.62 13.38 14.76 15.73 16.57 17.43 17.67 18.62 19.61 21.10 22.08 23.08 23.82 24.11 24.94 25.16 26.10 27.28 27.94 28.49 29.78 30.40
18,430.23 7 1284.99 16,096.30 7 1123.33 13,411.20 7 818.19 11,093.75 7 803.45 9021.14 7 563.16 7402.56 7436.41 6510.687 383.64 5574.59 7 328.26 5439.32 7322.04 4762.05 7282.01 4018.77 7 237.50 3200.62 7188.56 2733.19 7 160.98 2435.80 7143.64 2280.75 7135.62 2037.74 7 120.27 1948.39 7115.81 1817.07 7 107.34 1890.62 7112.97 1656.34 799.62 1542.007 95.78 8779.56 7528.06 7520.007 467.09 7131.007 442.92
f″
2.777 0.19 2.65 70.18 2.42 70.15 2.127 0.15 1.90 70.12 1.667 0.10 1.54 70.09 1.39 70.08 1.377 0.08 1.277 0.08 1.137 0.07 0.977 0.06 0.86 70.05 0.80 70.05 0.78 70.05 0.707 0.04 0.69 70.04 0.65 70.04 0.717 0.04 0.65 70.04 3.75 70.23 3.58 70.22 3.20 70.20 3.107 0.19
Table 5 Photo effect cross-sections (sph), imaginary part (f ″) and real part (f ′) of the anomalous scattering factor for Sn. Energy
sph
f″
(KP)
(CM)
(CL)
(keV)
(barns/atom)
0.11 0.03 0.02 0.08 0.10 0.10 0.07 0.17 0.24 0.32 0.53 0.57 0.72 0.72 0.83 0.98 1.15 1.33 1.55 1.79 1.88 2.14 2.26 3.15 5.24 2.75 1.71
0.13 0.05 0.04 0.10 0.13 0.12 0.09 0.20 0.26 0.34 0.56 0.59 0.74 0.74 0.85 1.00 1.18 1.35 1.57 1.81 1.90 2.17 2.28 3.17 5.26 2.78 1.74
0.33 0.25 0.24 0.30 0.32 0.32 0.29 0.39 0.46 0.54 0.75 0.79 0.94 0.94 1.05 1.20 1.37 1.54 1.77 2.01 2.10 2.36 2.48 3.37 5.46 2.97 1.93
6.40 7.06 8.04 8.12 8.91 9.34 9.67 10.50 11.51 12.62 13.38 14.76 15.73 16.57 17.43 17.67 18.62 19.61 21.10 22.08 23.08 23.82 24.11 24.94 26.10 27.28 28.49 29.19 30.97 31.82
81,222.56 7 5327.24 76,751.36 7 6735.16 50,470.527 3013.94 48,906.52 7 2937.24 36,224.07 72167.12 33,295.257 1988.73 28,352.827 1693.59 24,380.057 1453.69 17,169.53 7 1024.14 14,461.80 7 861.92 11,008.127 661.54 8945.09 7 535.15 8034.53 7 478.71 7112.83 7 423.81 6171.26 7 367.73 5991.067 357.45 5059.58 7 302.01 4374.107 261.13 3233.07 7192.76 2640.88 7 157.46 2505.02 7 149.47 2363.52 7 141.74 2021.92 7 120.73 2256.447 135.37 1832.917 110.34 1613.727 97.65 1533.957 93.36 1491.007 91.07 7422.007 453.34 6958.007 425.00
Table 4 Photo effect cross-sections (sph), imaginary part (f ″) and real part (f ′) of the anomalous scattering factor for In. Real part, f ′ (KP)
(CM)
(CL)
0.45 0.26 0.30 0.36 0.49 0.55 0.62 0.70 0.73 0.80 0.89 1.07 1.22 1.40 1.51 1.57 1.81 1.88 2.23 3.07 5.06 2.96 1.71 1.44
0.42 0.28 0.32 0.39 0.51 0.58 0.64 0.73 0.76 0.83 0.92 1.10 1.25 1.42 1.53 1.60 1.83 1.90 2.26 3.09 5.09 2.98 1.73 1.46
0.21 0.49 0.53 0.59 0.72 0.78 0.85 0.94 0.96 1.03 1.13 1.31 1.45 1.63 1.74 1.80 2.04 2.11 2.46 3.30 5.30 3.19 1.94 1.67
105
7.43 7 0.49 7.747 0.68 5.80 7 0.35 5.6770.34 4.617 0.28 4.45 7 0.27 3.92 7 0.23 3.667 0.22 2.83 7 0.17 2.617 0.16 2.117 0.13 1.89 7 0.11 1.81 7 0.11 1.69 7 0.10 1.54 7 0.09 1.517 0.09 1.3570.08 1.23 7 0.07 0.98 7 0.06 0.83 7 0.05 0.83 7 0.05 0.80 7 0.05 0.7070.04 0.80 7 0.05 0.687 0.04 0.63 7 0.04 0.62 7 0.04 3.577 0.22 3.29 7 0.20 3.177 0.19
Real part, f ′ (KP)
(CM)
(CL)
0.60 0.55 0.44 0.45 0.33 0.36 0.32 0.22 0.09 0.02 0.14 0.46 0.55 0.56 0.60 0.62 0.68 0.75 0.91 1.10 1.28 1.35 1.42 1.60 1.81 2.23 3.08 5.01 1.87 1.41
0.63 0.58 0.41 0.43 0.30 0.33 0.29 0.20 0.07 0.05 0.16 0.49 0.58 0.58 0.63 0.64 0.70 0.78 0.94 1.12 1.31 1.38 1.45 1.63 1.83 2.25 3.11 5.03 1.90 1.44
0.85 0.79 0.20 0.21 0.09 0.12 0.08 0.02 0.15 0.26 0.38 0.71 0.79 0.80 0.85 0.86 0.92 0.99 1.15 1.34 1.52 1.59 1.67 1.84 2.05 2.47 3.32 5.25 2.12 1.66
the upper limit was 1500 keV. To evaluate the integral numerically, the energy region used for integration was divided into a large number of small intervals. Within each interval (Ei, Ei + 1), the energy dependence of fi″ was determined by a linear function
fi″ = ai + bi E
(5)
In this interval, the dispersion integral assumes the form
Ii, i + 1(Es) =
2 P π
∫E
Ei + 1 i
E′f ″(E′) dE′ Es2 − E′ 2
(6)
Mathematically the above equation is equivalent to
Ii, i + 1(Es)
=
⎡ E 2 − Ei2+ 1 2 ⎢ ai ln s 2 π ⎢⎣ 2 Es − Ei2 ⎛ (E + Es)(Ei − Es) E +bi ⎜⎜Ei + 1 − Ei − s ln i + 1 2 (Ei + 1 − Es)(Ei + Es) ⎝
⎞⎤ ⎟⎟⎥ ⎠⎥⎦
(7)
Here Es is the energy of interest at which the value of the dispersion integral is to be determined. The details of the calculation can be found elsewhere (Appaji Gowda and Umesh, 2006). Using the coefficients ai and bi the dispersion integral in Eq. (6) was calculated as explained (Appaji Gowda and Umesh, 2006) in the following section.
3.2. Calculation of real part of the anomalous scattering factor
4. Evaluation of the dispersion integral
The dispersion correction f ′(E) was determined by the numerical evaluation of the dispersion integral in Eq. (4). For this purpose, the lower limit of integration was chosen to be 1 keV and
Photelectric cross sections extracted from the attenuation data determined using PIXE were used for evaluating the integral in Eq. (6). Below the lower energy limit Emin of the present
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P. Latha et al. / Radiation Physics and Chemistry 107 (2015) 103–108
measurements, down to 1 keV, and above the upper limit Emax up to 1500 keV, XCOM (Hubbell et al., 1975; Hubbell and Overbo, 1979) values were used for the evaluation of the integral. The required integrals in Eq. (4) have been evaluated using a semi-analytical, semi-numerical approach. At each energy Ei, the integral to be evaluated is
I (Ei) =
∫E
E2
Eσph(E) dE
1
Ei2
where ϕ(E) = steps:
2
−E
=
∫E
E0
ϕ(E) dE
(8)
1
Eσph(E)/(Ei2
2
− E ). The integration is done in three
(a) from E1 to Ei − ϵ (b) from Ei −ϵ to Ei + ϵ (c) from Ei +ϵ to E2
Fig. 2. PIXE chamber and the Si(Li) detector.
Here ϵ is an arbitrarily chosen small energy interval (typically taken as Ei/1000). Now,
I (Ei) =
∫E
Ei −ϵ
1
ϕ(E) dE +
∫E
Ei +ϵ i −ϵ
ϕ(E) dE +
∫E
E2 i +ϵ
ϕ(E) dE
(9)
i.e.
I (Ei) = I1(Ei) + I2(Ei) + I3(Ei).
(10)
For evaluating I1(Ei) and I3(Ei), the relevant energy interval is subdivided further into a large number of equal sub-intervals (on a logarithmic scale). Within each sub interval the integration is carried out using Simpson's rule, assuming a linear variation of the photoelectric cross section
ln(σph(E)) = a + b ln(E).
(11)
In the energy region below the lower energy limit and above the upper energy limit of the present measurements, the XCOM values of the photoelectric cross sections are directly used for the integration. For evaluating I2, Eq. (7) has been used. Based on the above procedure, the dispersion integral and consequently the real part of the anomalous scattering factor were evaluated numerically using a fortran program.
5. Experimental setup The entire experiment is performed in a PIXE chamber attached to the beam line of a Tandetron accelerator at the Indira Gandhi Atomic Research Centre, Kalpakkam, Tamil Nadu, India. Schematic diagram of the chamber and a photograph of actual PIXE setup are shown in Figs. 1 and 2 respectively. The PIXE chamber is made of steel and has a diameter of 35 cm and a height
of 38 cm and is installed in one of the beam lines of the Tandetron accelerator. Two graphite collimators of 3 mm diameter are used to collimate the proton beam. The chamber is provided with various ports like Proton beam port (P), Detector port (D), Beam view port (B) and Vacuum pump port (V) as shown in Fig. 1. At the centre of the chamber there is a target ladder for holding six targets at a time. The position of the ladder can be adjusted from outside without breaking the vacuum and the position of the target as well as the absorber can be fixed. There is an absorber ladder positioned in front of the detector port which can hold five absorber foils at a time. A charge integrator is coupled directly to the target holder to measure the proton current. The vacuum inside the chamber was maintained through the port (V). The absorbers in the form of thin foils of Ag, Cd, In, Sn of size 1.5 cm 1.5 cm are mounted on the movable ladder kept in front of the detector port. Protons of energy 2 MeV were allowed to fall on suitable target foil mounted on the target ladder. The X-rays emitted by this foil at 45° to the incident proton beam were then allowed to fall on the Si(Li) detector, with and without the absorber. A mylar foil of sufficient thickness is placed between the detector and the target to filter out the back scattered protons. The spectra were recorded with a MCA for a pre-set total charge as measured by the current integrator connected to the target ladder. Typical X-ray spectra for the direct beam and the beam transmitted through the absorbers are shown in Fig. 3. Counting times are roughly around 30 min for each spectrum, keeping the total proton charge at a constant value of 10 μC, as measured by the current integrator. From the net counts under the characteristic peaks, the mass attenuation coefficient was calculated using the
22.16keV
12000
without absorber with In absorber
10000
Counts
8000
6000
4000
24.94keV 2000
0 21
22
23
24
25
26
Energy(keV) Fig. 1. Schematic diagram of the experimental set up for X-ray attenuation measurements using the PIXE method.
Fig. 3. Typical X-ray spectra for the direct beam and the beam transmitted through the indium absorber.
The values of photoelectric cross sections (sph) obtained as outlined in Section 3 were used to extract the imaginary part f ″ of anomalous scattering factor as per Eq. (3). The f ′ values were obtained by evaluating the dispersion integral as per the procedure mentioned above. f ′ values were thus calculated corresponding to f ′(∞) values of Cromer and Liberman (1970)[CL], Creagh and McAuley (1992) [CM] and Kissel and Pratt (1990) [KP]. The f ′ and f ″ values around K-edge are tabulated for Ag, Cd, In and Sn in Tables 2–5. The f ′ values are plotted in Figs. 4–7. The theoretical data of Chantler et al. (2005) and Cullen et al. (1997) is also shown for comparison of the present results. It is seen from the figures that the present values of f ′ follow the trend as suggested by these theories.
7. Conclusions
Real part of anomolous scattering factor
Real and imaginary parts of anomalous scattering factors were determined from the attenuation data determined using PIXE technique. The expected error in the measurement of cross section includes the error due to counting statistics and the error in the measurement of mass thickness of the samples. The overall error is always less than 7%. The calculated scattering factors are in agreement with the theoretical values reported by Chantler et al. (2005) and Cullen et al. (1997). It is observed that the f ′ values corresponding to f ′(∞) values of Creagh and McAuley (1992) [CM] and Kissel and Pratt (1990) [KP] show better agreement with theoretical values than those corresponding to Cromer and Liberman (1970) [CL]. The KP corrections are based on the S-matrix calculations. The CM values are based on use of multipole terms in the self-consistent field term, rather than limiting to the dipole term only as has been done in the CL estimates. This is probably the reason for the better agreement of the experimental results with the theoretical values using the KP and CM corrections. No earlier studies have been reported for the measurement of anomalous scattering factors using PIXE technique. However, with the development of tunable synchrotron radiation sources, some measurements have been carried out using synchrotron radiation (Chantler et al., 2001). Our results clearly establish that
Silver K-edge = 25.51keV
2 0 -2
Cadmium K-edge = 26.71keV
0 -2 Chantler with KP correction with CM correction with CL correction Cullen
-4 -6
-4 -6 5
10
15
20 25 Energy(keV)
10
15
20 25 Energy(keV)
30
35
Fig. 5. Plot of real part of the anomalous scattering factor as a function of photon energy around the K-edge of cadmium.
2
Indium K-edge = 27.94 keV
0 -2 Chantler with KP correction with CM correction with CL correction Cullen
-4 -6 5
10
15
20 25 Energy(keV)
30
35
Fig. 6. Plot of real part of the anomalous scattering factor as a function of photon energy around the K-edge of indium.
2
Tin K-edge = 29.19 keV
0 -2 Theoretical with KP correction with CM correction with CL correction Cullen
-4 -6 5
Chantler with KP correction with CM correction with CL correction Cullen
107
2
5
Real part of anomolous scattering factor
6. Result and discussion
Real part of anomolous scattering factor
known absorber mass thickness. Measurements were repeated for all possible combinations of exciter foil and absorbers.
Real part of anomolous scattering factor
P. Latha et al. / Radiation Physics and Chemistry 107 (2015) 103–108
10
15
20 25 Energy(keV)
30
35
Fig. 7. Plot of real part of the anomalous scattering factor as a function of photon energy around the K-edge of tin.
30
35
Fig. 4. Plot of real part of the anomalous scattering factor as a function of photon energy around the K-edge of silver.
in situations where mono energetic X-rays are not available from radioactive sources, it is possible to use PIXE technique as a reliable tool to generate the required X-ray energies in the region of interest using suitable targets.
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Acknowledgments We are thankful to the technical staff at the Materials Science Group, IGCAR, Kalpakkam, for their help during the PIXE studies.
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