- Email: [email protected]
MottCalc: A new tool for calculating Mott scattering differential cross sections for analytical purposes
Nuclear Inst. and Methods in Physics Research B 461 (2019) 44–49
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
MottCalc: A new tool for calculating Mott scattering differential cross sections for analytical purposes
T
M. Kokkorisa, , A. Lagoyannisb, F. Maragkosa ⁎
a b
Department of Physics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece Tandem Accelerator Laboratory, Institute of Nuclear and Particle Physics, N.C.S.R. “Demokritos”, Aghia Paraskevi, 15310 Athens, Greece
ARTICLE INFO
ABSTRACT
Keywords: Mott Scattering ERDA Differential cross sections MottCalc Software tool
MottCalc, the result of the present work, is a new software tool, tuned to support the implementation of several IBA techniques, such as ERDA. The code is able to calculate both angular and energy distributions in the case of pure Mott elastic scattering and create the appropriate R33 files, suitable for direct implementation in all analytical codes. Many isotopes are available for the user to choose from, while the screening effect is taken into account by implementing the Andersen model. The tool is available in two different versions, namely as Excel spreadsheet (which provides detailed information about all the calculations in both the laboratory and center of mass reference frames) and as a stand-alone application (which can produce R33 files for energy or angular distributions for analytical purposes). Both versions are already available to the scientific community for downloading and testing via the web page of the Nuclear Physics group at NTUA (http://nuclearphysics.ntua.gr/ downloads.php).
1. Introduction ERDA (Elastic Recoil Detection Analysis), especially due to the recent development of ToF (Time of Flight) capabilities, has emerged as one of the most prominent IBA (Ion Beam Analysis) techniques for the accurate quantitative determination of elemental depth profile concentrations in near surface layers of various matrices. In certain cases, especially when a relatively light ion beam is used (e.g., 12C, 16O or 28 Si), the occurrence of Mott scattering in certain ERDA implementations, e.g., traditional (with an absorber foil in front of the detector), transmission, or even ToF, requires a careful treatment in the subsequent spectral analysis, since the corresponding differential cross sections may present very strong deviations from the Rutherford ones. These deviations may in certain cases, depending on the detection angle, enhance the probability for the detection of certain light elements (up to a factor of 2), as has been shown in the past [1]. Mott scattering occurs only during the interaction of identical nuclei (beam–target) and, according to the indistinguishability principle, the elastic scattering in such case can only be calculated quantum mechanically, leading to an interference term contributing to the total differential cross section for elastic scattering. This term strongly depends on the J of the ground state of the interacting nuclei and the phenomenon is described by a well-known analytical formula in literature, e.g., [2]. ⁎
Although Mott scattering plays only a minor role in several practical ToF-ERDA implementations using a standard 35Cl, or 197Au beam, as it will be shown in the following sections, it is important to note that all the widely used analytical codes (e.g., SIMNRA, DF, POTKU), do not presently (current distributed versions) provide suitable theoretical differential cross-section calculations, although the presence of Mott scattering has been thoroughly and carefully documented [3]. Moreover, to the authors′ best knowledge, the only publicly available Mott scattering detailed calculator for heavy ions is the single-value one included in LISE+ (developed at MSU), which, however, does not currently include the screening effect and cannot generate either energy distributions (excitation functions) or the standard R33 files for energy and angular distributions required for IBA applications [4]. In order to address this problem, a new software support tool has been developed, named MottCalc, which is able to calculate both angular and energy distributions in the case of pure Mott elastic scattering (i.e., excluding nuclear perturbations) and create the appropriate R33 files, suitable for direct implementation in all analytical codes. 314 stable (along with a few radioactive) isotopes are currently available as both target and incident nuclei for the user to choose from. This list can be arbitrarily expanded to include more radioactive isotopes if necessary. The screening effect is taken into account by implementing the Andersen model (as in [3]), although the future inclusion of other models is also possible. The software tool is available in two different
Corresponding author. E-mail address: [email protected] (M. Kokkoris).
https://doi.org/10.1016/j.nimb.2019.09.022 Received 24 July 2019; Received in revised form 3 September 2019; Accepted 16 September 2019 Available online 20 September 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 461 (2019) 44–49
M. Kokkoris, et al.
versions, namely as a MS Excel spreadsheet with several worksheets (which provide detailed information about the effect of the screening factor, along with the differential cross sections in both the laboratory and center of mass reference frames, as well as comparisons with the classical Rutherford values if the interference term is ignored) and as stand-alone application (which can produce standard R33 files for energy or angular distributions, to be directly implemented in codes used for analytical applications). Both versions of MottCalc have already been made available to the scientific community for downloading, testing and reporting via the web page of the Nuclear Physics group at the National Technical University of Athens – NTUA (http:// nuclearphysics.ntua.gr/downloads.php) and they will be included in the upcoming version of SIMNRA (v. 7.03). MottCalc is an open–source code. The current application has been developed in C++, while the graphical interface is based on the Qt libraries. The MS Excel spreadsheet, on the other hand, is based on Visual Basic. Both versions are distributed freely under the revisited BSD license, including the embedded source code. It is the authors′ sincere hope that future releases of the popular analytical codes for IBA techniques will also eventually incorporate the MottCalc algorithm for both educational and practical purposes, and thus solve the existing problem in the elastic scattering of identical nuclei in the most elegant possible way.
interference pattern for bosons or fermions, since the possible change of sign is included in the (-1)2j factor. J corresponds to the total angular momentum (usually reported as ‘spin’ or ‘total spin’ in literature) of the ground state of the identical interacting nuclei, (2 J + 1) in the denominator is the weighing factor for the spin multiplicity and ηs is the Sommerfeld parameter, described by the following formula: s
d d
)|2
(1)
where f corresponds to the scattering amplitude and the ± sign reflects the symmetric or antisymmetric nature of the wave function, that is, whether the interacting nuclei are bosons (integer spin) or fermions (semi-integer spin) respectively. Eq. (1) can also be rewritten analytically as follows:
d d
= |f (
d d
)|2
+ |f (
)|2
+ 2Re f ( ) f (
= |f ( )|2 + |f (
)|2
2Re f ( ) f (
= , cm
e 2 Z2 4 o 4E
2
·
1 sin4
+ 2
1 cos4
+ 2
cm lab
=
µc 2 2Elab
(3)
(1 + 2 cos + 2)3/2 |1 + cos |
(1 +
(4)
)
1 V1 2 2 E 2 2
1+
V1 E
+
V1 2Esin
() 2
(5)
where E and θ are the initial beam energy and the detection angle respectively in the CM system and V1 is the increase in the kinetic energy given in keV. This increase is equal to
) (for bosons)
2 V1 (keV ) = 0.04873·Zprojectile ·
) (for fermions)
2( 1)2jcos
Z2
o
fAndersen =
2/3 2·Zprojectile
(6)
It should be noted here, that the screening factor, which for typical small accelerator energies (of the order of 0.1–1 MeV/nucleon) affects the differential cross-section values by no more than 1–3%, is nonetheless the result of a phenomenological model and therefore the obtained results should be treated with prudence. It is also important to mention that in order to ensure the validity of the reported formulas, the implemented beam energies should be well below the corresponding interaction barrier. In the MS Excel version of MottCalc a simplified Coulomb barrier is mentioned for reasons of completeness, based on the following equation:
In the classical limit, when the particles are distinguishable (i.e., not identical), the interference term mentioned above would vanish and the differential cross section for scattering would simply become equal to the sum of the individual probabilities for each particle. When Coulomb (potential) scattering is the only occurring process, there is a way to rewrite the above equations for the scattering of identical particles in a compact way [2] in the CM system, taking also into account that in the case of spin multiplicity (spin 0 ) the cross section must be an incoherent sum of all the various possible total spin states, as follows:
d d
4
=
where γ = 1 for the elastic scattering (Q = 0) of identical particles. Since Mott scattering is essentially a forward scattering process, the detection angle in the CM system has a range of 0° < θ < 180°, while in the laboratory one this range translates to 0° < θlab < 90°. Moreover, as a result of the reaction kinematics, Mott scattering is symmetrical around 90° in the CM system and around 45° in the laboratory one. An interesting key point is that in order to render the theoretical calculations directly comparable to the experimental results, the screening by the electron shells of both interacting nuclei must also be taken into account. This can be achieved via a multiplication of the differential cross section as obtained in the CM system with a correction factor, f, derived according to the Andersen model, as shown in [3], for both the projectile and the recoil nucleus (due to the indistinguishability principle), given by the following expression:
Mott scattering is a pure quantum mechanical effect originating from the indistinguishability principle, meaning that in the case of the elastic scattering of two identical particles the detection system cannot distinguish between the beam ion and the recoil nucleus originating from the target. Thus, in the center of mass (CM) system, for every particle emitted towards Ω ≈ (θ,φ) there will exist an identical one emitted in the opposite direction, namely (π − θ, φ + π). Therefore, in the case of φ-invariance, the differential cross section can be expressed as follows:
= |f |2 = |f ( ) ± f (
Z 2e 2
where α is the fine-structure constant, Elab the energy of the beam in the laboratory system and μ the reduced mass of the interacting nuclei. Eq. (2) is the famous Mott formula for elastic Coulomb scattering of identical nuclei in the CM system and has profound consequences in the obtained differential cross section values, leading to strong oscillatory patterns, due to the complex trigonometric dependence of the interference term. This equation can be transformed to the lab reference system in the classical way (i.e., excluding relativistic corrections), taking into account that:
2. Theoretical background
d d
=
2 s lntan 2
UCoulomb =
(2J + 1) sin2 2 cos2 2
1.438 Z2 0.5 + 1.36·2·A1/3
(7)
However, more accurate models for the calculation of such an interaction barrier exist in literature (e.g., the Bass model [5] or the proximity one – and the user may try http://nrv.jinr.ru/nrv/webnrv/ qcalc/ for the corresponding calculations). Naturally, the existence of any such barrier is simply indicative and should also be treated with
(2) where the first two terms correspond to the classical case for Rutherford elastic scattering of the beam ion and the recoil nucleus from the target respectively. The third term expresses the combined 45
Nuclear Inst. and Methods in Physics Research B 461 (2019) 44–49
M. Kokkoris, et al.
a
b
c
d
Fig. 1. a–d. (a) Angular distribution of 12C on 12C ions at 6 MeV in the laboratory system (b) angular distribution of energy distribution of 12C on 12C ions at θlab = 30° (d) energy distribution of 12C on 12C ions at θCM = 60o.
prudence (for example strong deviations from Mott scattering have been reported in the past at relatively low energies for carbon ions, attributed to the formation of quasi-molecular states [6]). In any case, at high beam energies, Mott scattering in combination with the nuclear one would occur, yielding diffraction patterns in the differential cross section values and leading to EFS (Elastic Forward Spectroscopy) and not to the classical energy regime suitable for ERDA. Moreover, as far as ultra-light ions are concerned (e.g., p-p, d-d or α-α scattering), when strong deviations from pure Coulomb scattering are expected, experimental differential cross-section datasets should be implemented when available, as e.g., those reported in IBANDL (Ion Beam Analysis Nuclear Data Library, https://www-nds.iaea.org/exfor/ibandl.htm) or EXFOR (https://www-nds.iaea.org/exfor/).
12
C on
12
C ions at 6 MeV in the CM system (c)
MottCalc were tested against the LISE + calculator, yielding practically identical results, with the exception of course of the screening factor contribution. In the case of odd-even or even-odd nuclei and more particularly for J = 1/2 ones, the oscillatory pattern is definitely less pronounced, yielding smaller differences from the classical approach, as shown in Fig. 2a–d for the corresponding case of 6.5 MeV (0.5 MeV/nucleon) 13C ions impinging on 13C nuclei, where the angular and energy distributions in the CM and lab systems are depicted. Following an oscillatory behavior, a minimum value occurs, centered around π/2 in the CM system and π/4 in the laboratory one, as expected for fermions. For higher spin fermions, namely for ones with J = 3/2, the oscillatory behavior persists in both the angular and energy distributions, but the obtained differences are even less pronounced and do not exceed 25% in the case of the widely used 35Cl beam at 35 MeV (1 MeV/ nucleon) and 33% for 197Au ions at 197 MeV (1 MeV/nucleon). Moreover, the effect of the ion mass and energy in the periodicity of the oscillatory behavior is evident, via the influence of the Sommerfeld parameter in the interference term. The resulting angular and energy distributions for the 35Cl and 197Au ions are depicted in Fig. 3a–d for the lab reference system. In the case of very high spin fermionic beams, such as 127I (J = 5/2), 45 Sc (J = 7/2) and 113In (J = 9/2), the results, obtained at energies corresponding to 1 MeV/nucleon and shown in Fig. 4a–f for the laboratory system, demonstrate that the oscillatory patterns in both the angular and energy distributions are severely dumped, and for the highest spin value they practically converge towards the classical limit, reducing the impact of the interference term, as expected by the presence of the (2J + 1) weighing factor in its denominator, shown in Eq. (2). At the other end, the case of high spin bosonic beams presents some interesting features, although their implementation for practical applications is severely limited, due to the fact that there exist only four
3. Results, discussion and comparisons The consequences of Mott scattering are more profound in the case of spinless bosons. This is crucial, because some of the low-Z, even-even stable nuclei, commonly implemented as beam ions in small accelerator labs, have J = 0 in their ground state. A typical example is shown in Fig. 1a–d for 6 MeV (0.5 MeV/nucleon) 12C ions impinging on 12C nuclei, where the angular and energy distributions in the CM and laboratory systems are depicted. For all the energy distributions the detection angle in the laboratory system was typically set to 30° (and 60° in the CM one accordingly). The differences from the classical case (denoted in the graphs as ‘Rutherford’, i.e., ignoring the interference term) are evident and the obtained values can be up to a factor of 2 higher than the classical ones. Following a strong oscillatory behavior, a maximum value occurs, centered around π/2 in the CM system and π/4 in the laboratory one, as expected for bosons. At the other extreme, very sharp minima can also be observed (with the cross section attaining values very close to zero). Random values obtained with the use of 46
Nuclear Inst. and Methods in Physics Research B 461 (2019) 44–49
M. Kokkoris, et al.
Fig. 2. a–d. (a) Angular distribution of 13C on 13C ions at 6.5 MeV in the laboratory system (b) angular distribution of 13C on 13C ions at 6 MeV in the CM system (c) energy distribution of 13C on 13C ions at θlab = 30° (d) energy distribution of 13C on 13C ions at θCM = 60o.
a
b
c
d
Fig. 3. a–d. (a) Angular distribution of 35Cl on 35Cl ions at 35 MeV in the laboratory system (b) angular distribution of 197Au on 197Au ions at 197 MeV in the laboratory system (c) energy distribution of 35Cl on 35Cl ions at θlab = 30° (d) energy distribution of 197Au on 197Au ions at θlab = 30o. 47
Nuclear Inst. and Methods in Physics Research B 461 (2019) 44–49
M. Kokkoris, et al.
Fig. 4. a–f. (a) Angular distribution of 127I on 127I ions at 127 MeV in the laboratory system (b) energy distribution of 127I on 127I ions at θlab = 30° (c) angular distribution of 45Sc on 45Sc ions at 45 MeV in the laboratory system (d) energy distribution of 45Sc on 45Sc ions at θlab = 30° (e) angular distribution of 113In on 113In ions at 113 MeV in the laboratory system (f) energy distribution of 113In on 113In ions at θlab = 30o.
stable odd-odd nuclei in nature yielding integer values for J greater than zero (namely 2H, 6Li, 10B and 14N). Angular distributions in the laboratory system have been generated with the use of MottCalc, at energies corresponding to 0.5 MeV/nucleon, more specifically for 6Li (J = 1), 10B (J = 3), as well as, for the exotic radioactive nucleus 134Cs (J = 4), commonly used as radioactive tracer and are shown in Fig. 5a–c. The effect of the ion mass in the periodicity of the oscillatory behavior is again evident, while the differences from the classical case are increasingly dumped at higher J values. The main difference from the fermionic case, however, is the occurrence of the interesting phenomenon of ‘transverse isotropy’ at typical accelerator energies suitable for analytical applications [7], meaning the relative invariance (small variation) of the obtained differential cross section values and the practical convergence towards the classical limit over a large angular range, after a critical value of the Sommerfeld parameter has been 3J + 2 . This typically extends between surpassed, namely for s ~30 and 60° in the laboratory system, as depicted in Fig. 5a–c.
4. Conclusions In the present work a new software tool has been presented, named MottCalc, which has been developed for the calculation of Mott scattering of heavy ions (elastic Coulomb scattering of identical nuclei), particularly useful for educational purposes and in the case of specific ERDA applications. The results obtained with the use of MottCalc (which include the screening effect and can be recorded in R33 format) have been tested against the simpler online LISE + calculator. Angular and energy differential cross-section distributions generated by the code for various fermionic and bosonic beams revealed the impact of the ion mass, energy and ground state properties in the obtained results. Both versions of the code, namely as Excel spreadsheet (written in Visual Basic) and as stand-alone application (written in C++, along with Qt libraries for the graphical interface) can be easily accessed and downloaded from the web site of NTUA (http://nuclearphysics.ntua.gr/ downloads.php), following the corresponding hyperlinks. MottCalc will 48
Nuclear Inst. and Methods in Physics Research B 461 (2019) 44–49
M. Kokkoris, et al.
Fig. 5. a–c. (a) Angular distribution of 6Li on 6Li ions at 3 MeV in the laboratory system (b) angular distribution of 10B on 10B ions at 5 MeV in the laboratory system (c) angular distribution of 134Cs on 134Cs ions at 67 MeV in the laboratory system.
also be included in SIMNRA v. 7.03 in the near future. The current version of MottCalc has been tuned and tested for 32 bit and 64 bit MS Windows 10, but the source code is also readily available, should a user wishes to adapt the program to other operating systems (e.g., Linux, or MacOS).
References [1] I. Bogdanović-Radović, M. Jakšić, F. Schiettekatte, J. Anal. At. Spectrom. 24 (2009) 194–198. [2] C.J. Joachain, Quantum Collision Theory, North Holland Publishing Company, 1975, pp. 148–160. [3] M. Mayer, Nucl. Instrum. Meth. B332 (2014) 176–180. [4] O.B. Tarasov, D. Bazin, Nucl. Instrum. Meth. B376 (2016) 185–187. [5] R. Bass, Nucl. Phys. A 231 (1974) 45–63. [6] D.A. Bromley, J.A. Kuehner, E. Almqvist, 123, Phys. Rev. 3 (1974) 878–893. [7] M.S. Hussein, L.F. Canto, R. Donangelo, W. Mittig, Few Body Syst. 57 (2016) 195–203.
Declaration of Competing Interest The authors declared that there is no conflict of interest.
49
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
MottCalc: A new tool for calculating Mott scattering differential cross sections for analytical purposes
T
M. Kokkorisa, , A. Lagoyannisb, F. Maragkosa ⁎
a b
Department of Physics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece Tandem Accelerator Laboratory, Institute of Nuclear and Particle Physics, N.C.S.R. “Demokritos”, Aghia Paraskevi, 15310 Athens, Greece
ARTICLE INFO
ABSTRACT
Keywords: Mott Scattering ERDA Differential cross sections MottCalc Software tool
MottCalc, the result of the present work, is a new software tool, tuned to support the implementation of several IBA techniques, such as ERDA. The code is able to calculate both angular and energy distributions in the case of pure Mott elastic scattering and create the appropriate R33 files, suitable for direct implementation in all analytical codes. Many isotopes are available for the user to choose from, while the screening effect is taken into account by implementing the Andersen model. The tool is available in two different versions, namely as Excel spreadsheet (which provides detailed information about all the calculations in both the laboratory and center of mass reference frames) and as a stand-alone application (which can produce R33 files for energy or angular distributions for analytical purposes). Both versions are already available to the scientific community for downloading and testing via the web page of the Nuclear Physics group at NTUA (http://nuclearphysics.ntua.gr/ downloads.php).
1. Introduction ERDA (Elastic Recoil Detection Analysis), especially due to the recent development of ToF (Time of Flight) capabilities, has emerged as one of the most prominent IBA (Ion Beam Analysis) techniques for the accurate quantitative determination of elemental depth profile concentrations in near surface layers of various matrices. In certain cases, especially when a relatively light ion beam is used (e.g., 12C, 16O or 28 Si), the occurrence of Mott scattering in certain ERDA implementations, e.g., traditional (with an absorber foil in front of the detector), transmission, or even ToF, requires a careful treatment in the subsequent spectral analysis, since the corresponding differential cross sections may present very strong deviations from the Rutherford ones. These deviations may in certain cases, depending on the detection angle, enhance the probability for the detection of certain light elements (up to a factor of 2), as has been shown in the past [1]. Mott scattering occurs only during the interaction of identical nuclei (beam–target) and, according to the indistinguishability principle, the elastic scattering in such case can only be calculated quantum mechanically, leading to an interference term contributing to the total differential cross section for elastic scattering. This term strongly depends on the J of the ground state of the interacting nuclei and the phenomenon is described by a well-known analytical formula in literature, e.g., [2]. ⁎
Although Mott scattering plays only a minor role in several practical ToF-ERDA implementations using a standard 35Cl, or 197Au beam, as it will be shown in the following sections, it is important to note that all the widely used analytical codes (e.g., SIMNRA, DF, POTKU), do not presently (current distributed versions) provide suitable theoretical differential cross-section calculations, although the presence of Mott scattering has been thoroughly and carefully documented [3]. Moreover, to the authors′ best knowledge, the only publicly available Mott scattering detailed calculator for heavy ions is the single-value one included in LISE+ (developed at MSU), which, however, does not currently include the screening effect and cannot generate either energy distributions (excitation functions) or the standard R33 files for energy and angular distributions required for IBA applications [4]. In order to address this problem, a new software support tool has been developed, named MottCalc, which is able to calculate both angular and energy distributions in the case of pure Mott elastic scattering (i.e., excluding nuclear perturbations) and create the appropriate R33 files, suitable for direct implementation in all analytical codes. 314 stable (along with a few radioactive) isotopes are currently available as both target and incident nuclei for the user to choose from. This list can be arbitrarily expanded to include more radioactive isotopes if necessary. The screening effect is taken into account by implementing the Andersen model (as in [3]), although the future inclusion of other models is also possible. The software tool is available in two different
Corresponding author. E-mail address: [email protected] (M. Kokkoris).
https://doi.org/10.1016/j.nimb.2019.09.022 Received 24 July 2019; Received in revised form 3 September 2019; Accepted 16 September 2019 Available online 20 September 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 461 (2019) 44–49
M. Kokkoris, et al.
versions, namely as a MS Excel spreadsheet with several worksheets (which provide detailed information about the effect of the screening factor, along with the differential cross sections in both the laboratory and center of mass reference frames, as well as comparisons with the classical Rutherford values if the interference term is ignored) and as stand-alone application (which can produce standard R33 files for energy or angular distributions, to be directly implemented in codes used for analytical applications). Both versions of MottCalc have already been made available to the scientific community for downloading, testing and reporting via the web page of the Nuclear Physics group at the National Technical University of Athens – NTUA (http:// nuclearphysics.ntua.gr/downloads.php) and they will be included in the upcoming version of SIMNRA (v. 7.03). MottCalc is an open–source code. The current application has been developed in C++, while the graphical interface is based on the Qt libraries. The MS Excel spreadsheet, on the other hand, is based on Visual Basic. Both versions are distributed freely under the revisited BSD license, including the embedded source code. It is the authors′ sincere hope that future releases of the popular analytical codes for IBA techniques will also eventually incorporate the MottCalc algorithm for both educational and practical purposes, and thus solve the existing problem in the elastic scattering of identical nuclei in the most elegant possible way.
interference pattern for bosons or fermions, since the possible change of sign is included in the (-1)2j factor. J corresponds to the total angular momentum (usually reported as ‘spin’ or ‘total spin’ in literature) of the ground state of the identical interacting nuclei, (2 J + 1) in the denominator is the weighing factor for the spin multiplicity and ηs is the Sommerfeld parameter, described by the following formula: s
d d
)|2
(1)
where f corresponds to the scattering amplitude and the ± sign reflects the symmetric or antisymmetric nature of the wave function, that is, whether the interacting nuclei are bosons (integer spin) or fermions (semi-integer spin) respectively. Eq. (1) can also be rewritten analytically as follows:
d d
= |f (
d d
)|2
+ |f (
)|2
+ 2Re f ( ) f (
= |f ( )|2 + |f (
)|2
2Re f ( ) f (
= , cm
e 2 Z2 4 o 4E
2
·
1 sin4
+ 2
1 cos4
+ 2
cm lab
=
µc 2 2Elab
(3)
(1 + 2 cos + 2)3/2 |1 + cos |
(1 +
(4)
)
1 V1 2 2 E 2 2
1+
V1 E
+
V1 2Esin
() 2
(5)
where E and θ are the initial beam energy and the detection angle respectively in the CM system and V1 is the increase in the kinetic energy given in keV. This increase is equal to
) (for bosons)
2 V1 (keV ) = 0.04873·Zprojectile ·
) (for fermions)
2( 1)2jcos
Z2
o
fAndersen =
2/3 2·Zprojectile
(6)
It should be noted here, that the screening factor, which for typical small accelerator energies (of the order of 0.1–1 MeV/nucleon) affects the differential cross-section values by no more than 1–3%, is nonetheless the result of a phenomenological model and therefore the obtained results should be treated with prudence. It is also important to mention that in order to ensure the validity of the reported formulas, the implemented beam energies should be well below the corresponding interaction barrier. In the MS Excel version of MottCalc a simplified Coulomb barrier is mentioned for reasons of completeness, based on the following equation:
In the classical limit, when the particles are distinguishable (i.e., not identical), the interference term mentioned above would vanish and the differential cross section for scattering would simply become equal to the sum of the individual probabilities for each particle. When Coulomb (potential) scattering is the only occurring process, there is a way to rewrite the above equations for the scattering of identical particles in a compact way [2] in the CM system, taking also into account that in the case of spin multiplicity (spin 0 ) the cross section must be an incoherent sum of all the various possible total spin states, as follows:
d d
4
=
where γ = 1 for the elastic scattering (Q = 0) of identical particles. Since Mott scattering is essentially a forward scattering process, the detection angle in the CM system has a range of 0° < θ < 180°, while in the laboratory one this range translates to 0° < θlab < 90°. Moreover, as a result of the reaction kinematics, Mott scattering is symmetrical around 90° in the CM system and around 45° in the laboratory one. An interesting key point is that in order to render the theoretical calculations directly comparable to the experimental results, the screening by the electron shells of both interacting nuclei must also be taken into account. This can be achieved via a multiplication of the differential cross section as obtained in the CM system with a correction factor, f, derived according to the Andersen model, as shown in [3], for both the projectile and the recoil nucleus (due to the indistinguishability principle), given by the following expression:
Mott scattering is a pure quantum mechanical effect originating from the indistinguishability principle, meaning that in the case of the elastic scattering of two identical particles the detection system cannot distinguish between the beam ion and the recoil nucleus originating from the target. Thus, in the center of mass (CM) system, for every particle emitted towards Ω ≈ (θ,φ) there will exist an identical one emitted in the opposite direction, namely (π − θ, φ + π). Therefore, in the case of φ-invariance, the differential cross section can be expressed as follows:
= |f |2 = |f ( ) ± f (
Z 2e 2
where α is the fine-structure constant, Elab the energy of the beam in the laboratory system and μ the reduced mass of the interacting nuclei. Eq. (2) is the famous Mott formula for elastic Coulomb scattering of identical nuclei in the CM system and has profound consequences in the obtained differential cross section values, leading to strong oscillatory patterns, due to the complex trigonometric dependence of the interference term. This equation can be transformed to the lab reference system in the classical way (i.e., excluding relativistic corrections), taking into account that:
2. Theoretical background
d d
=
2 s lntan 2
UCoulomb =
(2J + 1) sin2 2 cos2 2
1.438 Z2 0.5 + 1.36·2·A1/3
(7)
However, more accurate models for the calculation of such an interaction barrier exist in literature (e.g., the Bass model [5] or the proximity one – and the user may try http://nrv.jinr.ru/nrv/webnrv/ qcalc/ for the corresponding calculations). Naturally, the existence of any such barrier is simply indicative and should also be treated with
(2) where the first two terms correspond to the classical case for Rutherford elastic scattering of the beam ion and the recoil nucleus from the target respectively. The third term expresses the combined 45
Nuclear Inst. and Methods in Physics Research B 461 (2019) 44–49
M. Kokkoris, et al.
a
b
c
d
Fig. 1. a–d. (a) Angular distribution of 12C on 12C ions at 6 MeV in the laboratory system (b) angular distribution of energy distribution of 12C on 12C ions at θlab = 30° (d) energy distribution of 12C on 12C ions at θCM = 60o.
prudence (for example strong deviations from Mott scattering have been reported in the past at relatively low energies for carbon ions, attributed to the formation of quasi-molecular states [6]). In any case, at high beam energies, Mott scattering in combination with the nuclear one would occur, yielding diffraction patterns in the differential cross section values and leading to EFS (Elastic Forward Spectroscopy) and not to the classical energy regime suitable for ERDA. Moreover, as far as ultra-light ions are concerned (e.g., p-p, d-d or α-α scattering), when strong deviations from pure Coulomb scattering are expected, experimental differential cross-section datasets should be implemented when available, as e.g., those reported in IBANDL (Ion Beam Analysis Nuclear Data Library, https://www-nds.iaea.org/exfor/ibandl.htm) or EXFOR (https://www-nds.iaea.org/exfor/).
12
C on
12
C ions at 6 MeV in the CM system (c)
MottCalc were tested against the LISE + calculator, yielding practically identical results, with the exception of course of the screening factor contribution. In the case of odd-even or even-odd nuclei and more particularly for J = 1/2 ones, the oscillatory pattern is definitely less pronounced, yielding smaller differences from the classical approach, as shown in Fig. 2a–d for the corresponding case of 6.5 MeV (0.5 MeV/nucleon) 13C ions impinging on 13C nuclei, where the angular and energy distributions in the CM and lab systems are depicted. Following an oscillatory behavior, a minimum value occurs, centered around π/2 in the CM system and π/4 in the laboratory one, as expected for fermions. For higher spin fermions, namely for ones with J = 3/2, the oscillatory behavior persists in both the angular and energy distributions, but the obtained differences are even less pronounced and do not exceed 25% in the case of the widely used 35Cl beam at 35 MeV (1 MeV/ nucleon) and 33% for 197Au ions at 197 MeV (1 MeV/nucleon). Moreover, the effect of the ion mass and energy in the periodicity of the oscillatory behavior is evident, via the influence of the Sommerfeld parameter in the interference term. The resulting angular and energy distributions for the 35Cl and 197Au ions are depicted in Fig. 3a–d for the lab reference system. In the case of very high spin fermionic beams, such as 127I (J = 5/2), 45 Sc (J = 7/2) and 113In (J = 9/2), the results, obtained at energies corresponding to 1 MeV/nucleon and shown in Fig. 4a–f for the laboratory system, demonstrate that the oscillatory patterns in both the angular and energy distributions are severely dumped, and for the highest spin value they practically converge towards the classical limit, reducing the impact of the interference term, as expected by the presence of the (2J + 1) weighing factor in its denominator, shown in Eq. (2). At the other end, the case of high spin bosonic beams presents some interesting features, although their implementation for practical applications is severely limited, due to the fact that there exist only four
3. Results, discussion and comparisons The consequences of Mott scattering are more profound in the case of spinless bosons. This is crucial, because some of the low-Z, even-even stable nuclei, commonly implemented as beam ions in small accelerator labs, have J = 0 in their ground state. A typical example is shown in Fig. 1a–d for 6 MeV (0.5 MeV/nucleon) 12C ions impinging on 12C nuclei, where the angular and energy distributions in the CM and laboratory systems are depicted. For all the energy distributions the detection angle in the laboratory system was typically set to 30° (and 60° in the CM one accordingly). The differences from the classical case (denoted in the graphs as ‘Rutherford’, i.e., ignoring the interference term) are evident and the obtained values can be up to a factor of 2 higher than the classical ones. Following a strong oscillatory behavior, a maximum value occurs, centered around π/2 in the CM system and π/4 in the laboratory one, as expected for bosons. At the other extreme, very sharp minima can also be observed (with the cross section attaining values very close to zero). Random values obtained with the use of 46
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Fig. 2. a–d. (a) Angular distribution of 13C on 13C ions at 6.5 MeV in the laboratory system (b) angular distribution of 13C on 13C ions at 6 MeV in the CM system (c) energy distribution of 13C on 13C ions at θlab = 30° (d) energy distribution of 13C on 13C ions at θCM = 60o.
a
b
c
d
Fig. 3. a–d. (a) Angular distribution of 35Cl on 35Cl ions at 35 MeV in the laboratory system (b) angular distribution of 197Au on 197Au ions at 197 MeV in the laboratory system (c) energy distribution of 35Cl on 35Cl ions at θlab = 30° (d) energy distribution of 197Au on 197Au ions at θlab = 30o. 47
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Fig. 4. a–f. (a) Angular distribution of 127I on 127I ions at 127 MeV in the laboratory system (b) energy distribution of 127I on 127I ions at θlab = 30° (c) angular distribution of 45Sc on 45Sc ions at 45 MeV in the laboratory system (d) energy distribution of 45Sc on 45Sc ions at θlab = 30° (e) angular distribution of 113In on 113In ions at 113 MeV in the laboratory system (f) energy distribution of 113In on 113In ions at θlab = 30o.
stable odd-odd nuclei in nature yielding integer values for J greater than zero (namely 2H, 6Li, 10B and 14N). Angular distributions in the laboratory system have been generated with the use of MottCalc, at energies corresponding to 0.5 MeV/nucleon, more specifically for 6Li (J = 1), 10B (J = 3), as well as, for the exotic radioactive nucleus 134Cs (J = 4), commonly used as radioactive tracer and are shown in Fig. 5a–c. The effect of the ion mass in the periodicity of the oscillatory behavior is again evident, while the differences from the classical case are increasingly dumped at higher J values. The main difference from the fermionic case, however, is the occurrence of the interesting phenomenon of ‘transverse isotropy’ at typical accelerator energies suitable for analytical applications [7], meaning the relative invariance (small variation) of the obtained differential cross section values and the practical convergence towards the classical limit over a large angular range, after a critical value of the Sommerfeld parameter has been 3J + 2 . This typically extends between surpassed, namely for s ~30 and 60° in the laboratory system, as depicted in Fig. 5a–c.
4. Conclusions In the present work a new software tool has been presented, named MottCalc, which has been developed for the calculation of Mott scattering of heavy ions (elastic Coulomb scattering of identical nuclei), particularly useful for educational purposes and in the case of specific ERDA applications. The results obtained with the use of MottCalc (which include the screening effect and can be recorded in R33 format) have been tested against the simpler online LISE + calculator. Angular and energy differential cross-section distributions generated by the code for various fermionic and bosonic beams revealed the impact of the ion mass, energy and ground state properties in the obtained results. Both versions of the code, namely as Excel spreadsheet (written in Visual Basic) and as stand-alone application (written in C++, along with Qt libraries for the graphical interface) can be easily accessed and downloaded from the web site of NTUA (http://nuclearphysics.ntua.gr/ downloads.php), following the corresponding hyperlinks. MottCalc will 48
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Fig. 5. a–c. (a) Angular distribution of 6Li on 6Li ions at 3 MeV in the laboratory system (b) angular distribution of 10B on 10B ions at 5 MeV in the laboratory system (c) angular distribution of 134Cs on 134Cs ions at 67 MeV in the laboratory system.
also be included in SIMNRA v. 7.03 in the near future. The current version of MottCalc has been tuned and tested for 32 bit and 64 bit MS Windows 10, but the source code is also readily available, should a user wishes to adapt the program to other operating systems (e.g., Linux, or MacOS).
References [1] I. Bogdanović-Radović, M. Jakšić, F. Schiettekatte, J. Anal. At. Spectrom. 24 (2009) 194–198. [2] C.J. Joachain, Quantum Collision Theory, North Holland Publishing Company, 1975, pp. 148–160. [3] M. Mayer, Nucl. Instrum. Meth. B332 (2014) 176–180. [4] O.B. Tarasov, D. Bazin, Nucl. Instrum. Meth. B376 (2016) 185–187. [5] R. Bass, Nucl. Phys. A 231 (1974) 45–63. [6] D.A. Bromley, J.A. Kuehner, E. Almqvist, 123, Phys. Rev. 3 (1974) 878–893. [7] M.S. Hussein, L.F. Canto, R. Donangelo, W. Mittig, Few Body Syst. 57 (2016) 195–203.
Declaration of Competing Interest The authors declared that there is no conflict of interest.
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