# Rutherford Differential Cross Section at Relativistic Energies

Physical constants used in computation of SR-NIEL code are from appendix A.1 (pp. 1108-1112) of [C. Leroy and P.G. Rancoita (2016)].

Atomic weights for target elements, used in SR-NIEL code, are from IUPAC (International Union of Pure and Applied Chemistry) and were published by [J. Meija et al. (2016)].

For 12 of these elements, as discussend in the caption of Table 2, the standard atomic weight is given as an atomic-weight interval with the symbol [a,b] to denote the set of atomic-weight values in normal materials; The average value of the range is taken for computation in SR-NIEL code. Atomic weights are complemented, when needed, by those listed at NIST (Standard Reference Database 144, last update: January 2015).

Atomic weight for projectile isotops, except for the proton and alpha particle masses, is the most abundand isotope (MAI). The isotopic compositions data are those from NIST and were published by [M. Berglund et al. (2009)]. The relative atomic masses of the isotopes data were published by [M. Wang et al. (2012)].

### References

Leroy and P.G. Rancoita (2016), Principles of Radiation Interaction in Matter and Detection - 4th Edition -, World Scientific. Singapore, ISBN-978-981-4603-18-8 (printed); ISBN.978-981-4603-19-5 (ebook). http://www.worldscientific.com/worldscibooks/10.1142/9167; it is also partially accessible via google books.

To be noted that, by quoting WSRID20 upon checking out the shopping cart, a 20% discount will be obtained.

It is also available in kindle edition.

Meija et al, Pure and Applied Chemistry. Vol. 88, 3, pp. 265–291, ISSN (Online) 1365-3075, ISSN (Print) 0033-4545, DOI: 10.1515/pac-2015-0305, February 2016

Michael Berglund and Michael E. Wieser, Pure Appl. Chem., Vol. 83, No. 2, pp. 397-410, 2011

Wang et al.,Chinese Physics, C 36, pp. 1287-1602, 2012

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As discussed in the Energy Partition Function webpage, the partition function gives the fraction of recoil energy that goes into further displacement and can be approximated with Robinson and Akkerman analytical treatment. Here is presented the comparison of the two models with respect experimental data.

In the following graph, experimental points of Ne, O, C recoils in Si can be found in Akkerman and Barak (2006) and are taken from H. O. Funsten et al. (2001) and H. O. Funsten et al. (2004). Data are superimposed to the Akkerman and Robison analytical approximation of the partition function.

K recoil in Ge data are found in Akkerman and Barak (2006) and come from I. A. Abroyan et al. (1962)

Data points for Si recoils in Si, along with the Lindhard curve, are taken from Sattler (1965). Data and Lindhard curve for Ge recoils in Ge come from Sattler (1966). Robinson and Akkerman analytical approximation are also superimposed:

### References

A. Akkerman and J. Barak (2006), IEEE Trans. on Nucl. Sci. vol. 53, 3667.

H. O. Funsten, S. M. Ritzau, R. W. Harper, and R. Korde, IEEE Trans. Nucl. Sci., vol. 48, pp. 1785–1789, 2001.

H. O. Funsten, S. M. Ritzau, and R. W. Harper, Appl. Phys. Lett., vol. 84, pp. 3552–3554, 2004.

I. A. Abroyan and V. A. Zborovskii, Dokl. Akad. Nauk, SSSR, vol. 144, pp. 531–534, 1962; Sov. Phys. Dokl., vol. 7, p. 417, 1962.

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The nuclear stopping power of electrons in matter can be obtained from the differential cross section describing the electron–nucleus scattering. For instance, in Chapter 2 and of [Leroy and Rancoita (2016)] (see also [Boschini et al. (2012)] and references therein), the differential cross section for electron–nucleus scattering was dealt to describe their interaction up to ultra high-energy and, in addition, it was accounting (above about 200 keV) for the effects due to the screening of Coulomb potentials, finite sizes and finite rest masses of target nuclei. In fact, it has to be remarked - as derived by Zeitler and Olsen ([Zeitler and Olsen (1956)]) - that spin and screening effects can be separately treated for small scattering angles; while at large angles (i.e., at large momentum transfer), the factorization is well suited under the condition that

(e.g., see [Zeitler and Olsen (1956), Idoeta and Legarda (1992)]). Zeitler and
Olsen suggested that for electron energies above 200 keV the overlap of spin
and screening effects is small for all elements and for all energies; for
lower energies the overlapping of the spin and screening effects may be
appreciable for heavy elements and large angles. Furthermore, to a first
approximation, the finite nuclear size effects can be accounted for by means of the
nuclear form factor (e.g., see Eqs. (2.262, 2.264, 2.265) of [Leroy and
Rancoita (2016)] and discussion in Sect. 2.4.2.1), for instance, the so called
exponential form factor (F_{exp}) expressed by Eq. (2.262) of [Leroy and Rancoita
(2016)].

That treatment allowed Boschini and collaborators (2012) to the nuclear
stopping poower of electrons using the Mott differential cross section (MDCS),
dσ_{sc,F,CoM}^{Mott}(T)∕dT dT, and its approximate expression, i.e., the McKinley and
Feshbach differential cross section (McFDCS), dσ_{sc,F,CoM}^{McF }(T)∕dT, so that also
the screened Coulomb fields, finite sizes and rest masses of nuclei were taken into
account, i.e.,

| (1) |

or

| (2) |

where T is the kinetic energy transferred to the target nucleus, T_{max} is the
maximum energy that can be transferred during a single collision process, n_{A} is
the number of nuclei (atoms) per unit of volume and, finally, the negative sign
indicates that energy is lost by electrons (thus, achieved by recoil targets). It has
to be remarked that, in the current treatment for the MDCS, Boschini and
collaborators (2013) derived an improved numerical approach and an interpolated
expression (e.g., see Sections 2.4.1–2.4.2 of [Leroy and Rancoita (2016)] and
[Boschini et al. (2013)]).

As discussed, for instance, in Sect. 2.4.3 of [Leroy and Rancoita (2016)] (see
also [Boschini et al. (2012)] and references therein), the large momentum transfers
- corresponding to large scattering angles - are disfavored by effects due to the
finite nuclear size accounted for by means of the nuclear form factor. For instance,
in Fig. 1 the ratios of nuclear stopping powers of electrons in silicon are shown as
a function of the kinetic energies of electrons from 200 keV up to 1 TeV. These
ratios are the nuclear stopping powers calculated neglecting i) nuclear size effects
(i.e., for ^{2} = 1) and ii) effects due to the finite rest mass of the target
nucleus both divided by that one obtained using Eq. (2). Above a few tens
of MeV, a larger stopping power is found assuming ^{2} = 1 and, in
addition, above a few hundreds of MeV the stopping power largely decreases
when effects due to the finite nuclear rest mass are not accounted for.

In Fig. 2 , the nuclear stopping powers in ^{7}Li, ^{12}C, ^{28}Si and ^{56}Fe are shown as
a function of the kinetic energy of electrons from 200 keV up to 1 TeV. These
nuclear stopping powers in MeV cm^{2}/g are calculated from Eq. (2) - using F_{
exp} -
and divided by the density of the medium. The flattening of the high
energy behavior of the curves is mostly due to the nuclear form factor
which prevents the stopping power to increase with increasing T_{max}. As
expected, the stopping power are slightly (not exceeding a few percent) varied
at large energies replacing F_{exp} with F_{gau} or F_{u} (e.g., see Eqs. (2.264,
2.265) of [Leroy and Rancoita (2016)], respectively). However, a further
study is needed to determine a most suited parametrization of the nuclear
form factor[Nagarajan and L. Wang (1974), Duda, Kemper and Gondolo
(2007), Jentschura and Serbo (2009)] particularly for high-Z materials.

### References

[Boschini et al. (2012)] M.J. Boschini, C. Consolandi, M. Gervasi, S. Giani, D. Grandi, V. Ivanchenko, P. Nieminem, S. Pensotti, P.G. Rancoita and M. Tacconi, Nuclear and Non-Ionizing Energy-Loss of electrons with low and relativistic energies in materials and space environment, Proc. of the 13th ICATPP Conference, October 3-7 2011, Villa Olmo, Como, Italy, World Scientific, Singapore (2012), 961-982, IBSN: 978-981-4405-06-5; http://www.worldscientific.com/doi/pdf/10.1142/9789814405072˙0147; http://arxiv.org/pdf/1111.4042v4.pdf

[Boschini et al. (2013)] M.J. Boschini, C. Consolandi, M. Gervasi, S. Giani, D. Grandi, V. Ivanchenko, P. Nieminem, S. Pensotti, P.G. Rancoita, M. Tacconi (2013), An expression for the Mott cross section of electrons and positrons on nuclei with Z up to 118, Rad. Phys. Chem. 90, 39-66; doi: 10.1016/j.radphyschem.2013.04.020, http://www.sciencedirect.com/science/article/pii/S0969806X13002454; http://arxiv.org/pdf/1304.5871v1.pdf

[Consolandi et al. (2006)] C. Consolandi, P.D’Angelo, G. Fallica, R. Modica, R. Mangoni, S. Pensotti and P.G. Rancoita (2006), Systematic Investigation of Monolithic Bipolar Transistors Irradiated with Neutrons, Heavy Ions and Electrons for Space Applications, Nucl. Instr. and Meth. in Phys. Res. B 252 (2006), 276, doi:10.1016/j.nimb.2006.08.018; http://www.sciencedirect.com/science/article/pii/S0168583X0600913X.

[Duda, Kemper and Gondolo (2007)] G. Duda, A. Kemper and P. Gondolo (2007), J. Cosm. Astrop. Phys. 04, 012, doi:10.1088/1475-7516/2007/04/012

[Idoeta and Legarda (1992)] R. Idoeta and F. Legarda (1992, Nucl. Instr. and Meth. in Phys. Res. B 71), 116–125.

[Jentschura and Serbo (2009)] U.D. Jentschura and V.G. Serbo (2009), E. Phys. J. C 64, 309–317.

[Leroy and Rancoita (2016)] C. Leroy and P.G. Rancoita (2016), Principles of Radiation Interaction in Matter and Detection - 4th Edition -, World Scientific. Singapore, ISBN-978-981-4603-18-8 (printed); ISBN.978-981-4603-19-5 (ebook); http://www.worldscientific.com/worldscibooks/10.1142/9167; it is also partially accessible via google books. To be noted that, by quoting WSRID20 upon checking out the shopping cart, a 20% discount will be obtained. It is also available in kindle edition.

[Nagarajan and L. Wang (1974)] M.A. Nagarajan and L. Wang (1974), Phys. Rev. C 10, 2206-2209.

[Zeitler and Olsen (1956)] E. Zeitler and A. Olsen (1956), Phys. Rev. 136, A1546-A1552.

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The relationship of the Rutherfor differential cross section between laboratory and center of mass systems at relativistic energies is treated in Section 1.6 of Chapter 1 in [Leroy and P.G. Rancoita (2016)]. Chapter 1 is availble for downloading at http://www.worldscientific.com/worldscibooks/10.1142/9167.

Leroy and P.G. Rancoita (2016), Principles of Radiation Interaction in Matter and Detection - 4th Edition -, World Scientific. Singapore, ISBN-978-981-4603-18-8 (printed); ISBN.978-981-4603-19-5 (ebook).http://www.worldscientific.com/worldscibooks/10.1142/9167; it is also partially accessible via google books.

To be noted that, by quoting WSRID20 upon checking out the shopping cart, a 20% discount will be obtained.

The book is also available in kindle edition.

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The electron–nucleus scattering used in the sr-niel approach is treated, for
instance, in Chapter 2 and of [Leroy and Rancoita (2016)] (see also [Boschini et
al. (2012), Boschini et al. (2013)] and references therein). In this framework, the
Mott differential cross section (MDCS), dσ_{sc,F,CoM}^{Mott}(T)∕dT dT, is derived and
employed accounting for effects due to the screened Coulomb fields, finite sizes
and rest masses of nuclei.

The scattering of electrons by unscreened atomic nuclei was treated by Mott (1929) (see also Sections 4–4.5 in Chapter IX of [Mott and Massey (1965)]) extending a method of Wentzel (1927) (see also [Born (1926)]) and including effects related to the spin of electrons. Wentzel’s method was dealing with incident and scattered waves on point-like nuclei. The differential cross section (DCS) - the so-called Mott (unscreened) differential cross section (MDCS) - was expressed by Mott (1929) as two conditionally convergent infinite series in terms of Legendre expansions (see also [Bartlett and Watson (1940)] and Equation 46 in Chapter IX of [Mott and Massey (1965)]). In Mott–Wentzel treatment, the scattering occurs on a field of force generating a radially dependent Coulomb - unscreened [screened] in Mott (1929) [Wentzel (1927)] - potential. Mott equation - computed using Darwin’s solution to the Dirac equation - is also referred to as an exact formula for the differential cross section, because no Born approximation of any order is used in its determination.

Furthermore, the MDCS was derived in the laboratory reference system for
infinitely heavy nuclei initially at rest with negligible spin effects and must be
numerically evaluated for any specific nuclear target. Effects related to the recoil
and finite rest mass of the target nucleus (M) were neglected. Thus, in this
framework the total energy of electrons has to be smaller or much smaller than
Mc^{2}.

It has to be remarked that Mott’s treatment of collisions of fast electrons with atoms - accounting for screening effects - involves the knowledge of the wave function of the atom and uses the first Born approximation (e.g., see Sections 2–5 of Chapter XVI of [Mott and Massey (1965)]); thus, as discussed by many authors (for instance, see [Idoeta and Legarda (1992), Lijian, Quing and Zhengming (1995), Boschini et al. (2012), Boschini et al. (2013)] and also references therein), in most cases the computation of the cross section depends on the application of numerical methods (see a further discussion in Sect. 2.4.2 in [Leroy and Rancoita (2016)]). Particularly, in calculations for electron transport in materials or in the determination of induced radiation damage due to atomic displacements resulting from Coulomb interaction on nuclei, this treatment may require an excessive time-consuming procedure for accounting the effect of nuclear screening by atomic electrons.

In practice, for the above mentioned calculations a factorization of the elastic screened cross section is often employed. It involves the unscreened differential cross section on point-like nuclei and a factor which takes into account the screening of the nuclear charge by the atomic electrons. Expressions for this term - which is also employed in the treatment of nucleus–nucleus interactions - were derived and discussed by many authors. Furthermore, in electron scattering on nuclei above 10 MeV, as discussed by Fernandez-Varea, Mayol and Salvat (1993a) (see also [Boschini et al. (2012)] and references therein), the effects due to the finite nuclear size have to be taken into account and are usually expressed by a multiplicative term, the so-called nuclear form factor.

Among the spherically symmetric form factors treated in literature, one finds
those for i) the exponential charge distribution (_{exp}) (e.g., see Equation (6)
of [Butkevick (2002)], Equation (93) at page 252 of [Hofstadter (1957)] and
references therein), ii) the Gaussian charge distribution (_{gau}) (e.g., see
Equation (6) of [Butkevick (2002)] and references therein) and, finally, iii) the
uniform–uniform folded charge distribution over spheres with different radii (_{u})
(e.g., see Equation (22) of [Fernandez-Varea et al. (1993a)], [Helm (1956)] and
references therein). The above mentioned form factors _{exp}, _{gau} and _{u} are
expressed as:

| (1) |

(e.g., see Equation (6) of [Butkevick (2002)]), where r_{n} is the nuclear radius,
approximated by

| (2) |

with A the atomic weight (e.g., see Equation (7) of [Butkevick (2002)]),

| (3) |

(e.g., see Equation (6) of [Butkevick (2002)]) and, finally,

| (4) |

with q the value of the momentum transfer and

_{n}usually in agreement within 5% with the root mean square radius of the nuclear charge distribution, (see, for instance, in Table 1 of [De Vries, De Jager and De Vries (1987)]).

Furthermore, one can remark (e.g., see footnote at page 186 of [Leroy and
Rancoita (2016)]) that the Mott differential cross sections - calculated using
sr-treatment with _{exp}, _{gau} and _{u} form factors - were capable to describe the
large angle decreasing found in experimental data of electron scattering on He (at
2.091 and 4.048 GeV from [Camsonne et al. (2014)]), C (at 375 and 750 MeV
from [Sick and McCarthy (1970)]), O (at 375 and 750 MeV from [Sick
and McCarthy (1970)]), Al (at 250 and 500 MeV from [Li and Yearian
(1974)]), Si (at 250 and 500 MeV from [Li and Yearian (1974)]), In (at
153 and183 MeV from [Hahn et al. (1956)]), Au (at 153 and183 MeV
from [Hahn et al. (1956)]) - slightly favoured were _{gau} or _{u} form factors -;
as expected, the data largely disagreed with the calculated values using
||^{2} = 1. Furthermore, using ||^{2} = 1 the stopping powers resulted larger from
about 13.8% for Al at 250 MeV up to about 28.3% for Au at 183 MeV
with respect to those derived with _{exp}. Finally, it is worth to note that
the Mott cross section resulted marginally affected by the form factor,
because electron scatterings at large angles are extremely improbable.
at large energies replacing _{exp} with _{gau} or _{u}. However, a further
study is needed to determine a most suited parametrization of the nuclear
form factor particularly for high-Z materials (e.g., see [Nagarajan and
Wang (1974), Duda, Kemper and Gondolo (2007), Jentschura and Serbo
(2009)]).

Approximate expressions for the Mott (unscreened) differential cross section were derived as early as in the 1940s and 1950s (e.g., see [Bartlett and Watson (1940), McKinley and Feshbach (1948), Feshbach (1952), Curr (1955), Doggett and Spencer (1956)), Sherman (1956)]).

Idoeta and Legarda (1992) (as suggested, for instance, in [Sherman (1956)])
evaluated the MDCS exploiting recursion relationships of the gamma functions
showing that the ratio - appearing in the MDCS - fulfills the condition for the
application of the Stirling’s formula. In addition, they applied the trasformation
of Yennie, Ravenhall and Wilson (1954) to the infinite series of Legendre
polynomials. Finally, they obtained tabulated values for electrons and positrons
scattering on a few nuclei with kinetic energies from 5 keV and 10 MeV and a
maximum error of less than 10^{-3} %. Moreover, they discussed the good agreement
of their results with those previously found - within the sensitivity of the used
approximations - in [McKinley and Feshbach (1948), Curr (1955), Yadav
(1955), Motz, Olsen and Koch (1964), Doggett and Spencer (1956)), Sherman
(1956)].

Subsequently, Lijian, Quing and Zhengming (1995) developed a fitting
procedure for the numerical values determined following the approach of Idoeta
and Legarda (1992), then expressing the ratio (^{Mott}) of the MDCS to
Rutherford differential cross section (RDCS) as an analytical formula
depending on 30 parameters with a maximum error of less than 1 % only for
electrons with kinetic energies from 1 keV up to 900 MeV. Above 900 MeV,
no further energy dependence was exhibited by the parameters. These
parameters depend on the nuclear target and were tabulated for nuclei
with Z up to 90. More recently, Boschini and collaborators (2013) also
discussed an improved numerical approach - employed in the current sr-niel
framework and web-calculators - for determining the ratio, ^{Mott}, of the
unscreened MDCS with respect to Rutherford’s formula. These results were
compared and found in excellent agreement with those from Idoeta and
Legarda (1992). Moreover, the calculated numerical values of ^{Mott} - obtained
from that improved numerical solution - were used to provide an interpolated
practical expression for both electrons and positrons scattering on nuclei with
1 ≤ Z ≤ 118 in the kinetic-energy range from 1 keV up to 900 MeV. The latter
expression was also compared with that found previously - when available and
usable - for electrons (on nuclei with 1 ≤ Z ≤ 90) by Lijian, Quing and
Zhengming (1995). The two expressions exhibited a very good agreement for low
and high-Z nuclei.

Furthermore, Boschini and collaborators (2012) extended the treatment of the electron–nucleus interaction based on the Mott differential cross section to account for effects due to screened Coulomb potentials, finite sizes and finite rest masses of nuclei for electrons above 200 keV and up to ultra high energies. This extended treatment allows one to determine both the total and differential cross sections, thus, subsequently to calculate the resulting nuclear and non-ionizing stopping powers (or non-ionizing energy-loss, NIEL, e.g., see Chapter 7 of [Leroy and Rancoita (2016)]).

In the sr-niel framework, the unscreened and screened Coulomb electron–nucleus scattering are dealt following the approach of Boschini and collaborators (2012, 2013). The unscreened MDCS is discussed in Sects. 2.4.1–2.4.1.2 of [Leroy and Rancoita (2016)] in which the analytical approximate equation from [McKinley and Feshbach (1948)] and the interpolated expressions from [Lijian, Quing and Zhengming (1995), Boschini et al. (2013)] are discussed for the scattering on a target nucleus with a rest mass much larger than the mass corresponding to the (total) energy (E) of the incoming particle. In addition, the Coulomb elastic scattering with screened nuclear potentials and effects due to both the finite nuclear size and rest mass are discussed in Sects. 2.4.2-2.4.2.2 of [Leroy and Rancoita (2016)]. While, the calculation of the Coulomb nuclear stopping power of electrons in materials is treated in Sect. 2.4.3 of [Leroy and Rancoita (2016)].

Finally, it has to be remarked that the currently described treatment of the MDCS for elastic scattering on nuclei was implemented into the current Geant4 distribution (e.g., see [Agostinelli et al. (2003), Geant4: Class G4eSingleScatteringModel (2015)]) (see, also, the discussion in [Boschini et al. (2012)] and references therein).

### References

[Agostinelli et al. (2003)] S. Agostinelli et al. (2003), Geant4 a simulation toolkit, Nucl. Instr. and Meth. in Phys. Res. A 506, 250–303.

[Bartlett and Watson (1940)] J.H. Barlett and R.E. Watson (1940), Proc. Am. Acad. Arts. Sci. 74, 53.

[Born (1926)] M. Born (1926), Z. Phys. 38, 803.

[Boschini et al. (2012)] M.J. Boschini, C. Consolandi, M. Gervasi, S. Giani, D. Grandi, V. Ivanchenko, P. Nieminem, S. Pensotti, P.G. Rancoita and M. Tacconi, Nuclear and Non-Ionizing Energy-Loss of electrons with low and relativistic energies in materials and space environment, Proc. of the 13th ICATPP Conference, October 3-7 2011, Villa Olmo, Como, Italy, World Scientific, Singapore (2012), 961-982, IBSN: 978-981-4405-06-5; http://www.worldscientific.com/doi/pdf/10.1142/9789814405072˙0147; http://arxiv.org/pdf/1111.4042v4.pdf

[Boschini et al. (2013)] M.J. Boschini, C. Consolandi, M. Gervasi, S. Giani, D. Grandi, V. Ivanchenko, P. Nieminem, S. Pensotti, P.G. Rancoita, M. Tacconi (2013), An expression for the Mott cross section of electrons and positrons on nuclei with Z up to 118, Rad. Phys. Chem. 90, 39-66; doi: 10.1016/j.radphyschem.2013.04.020, http://www.sciencedirect.com/science/article/pii/S0969806X13002454; http://arxiv.org/pdf/1304.5871v1.pdf

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[Duda, Kemper and Gondolo (2007)] G. Duda, A. Kemper and P. Gondolo (2007), J. Cosm. Astrop. Phys. 04, 012, doi:10.1088/1475-7516/2007/04/012

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[Feshbach (1952)] H. Feshbach (1952), Phys. Rev. 88, 295–297.

[Geant4: Class G4eSingleScatteringModel (2015)] Class Description: single scattering interaction of electrons on nuclei ([Boschini et al. (2012)]), Software Reference Manual Source Code Version: Geant4 10.n avaliable at the website http://geant4.cern.ch/support/userdocuments.shtml. See also Sections 6.7–6.8 of Physics Reference Manual for version 10.n available at the website: http://geant4.cern.ch/support/userdocuments.shtml

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[Idoeta and Legarda (1992)] R. Idoeta and F. Legarda (1992, Nucl. Instr. and Meth. in Phys. Res. B 71), 116–125.

[Jentschura and Serbo (2009)] U.D. Jentschura and V.G. Serbo (2009), E. Phys. J. C 64, 309–317.

[Leroy and Rancoita (2016)] C. Leroy and P.G. Rancoita (2016), Principles of Radiation Interaction in Matter and Detection - 4th Edition -, World Scientific. Singapore, ISBN-978-981-4603-18-8 (printed); ISBN.978-981-4603-19-5 (ebook); http://www.worldscientific.com/worldscibooks/10.1142/9167; it is also partially accessible via google books. To be noted that, by quoting WSRID20 upon checking out the shopping cart, a 20% discount will be obtained. It is also available in kindle edition.

[Li and Yearian (1974)] G.C. Li and M.R. Yearian (1974), Phys. Rev. C 9, 1861.

[Lijian, Quing and Zhengming (1995)] T. Lijian, H. Quing and L. Zhengming (1995), Radiat. Phys. Chem. 45, 235–245.

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