As treated in [Boschini et al. (2011)], at small distances from the nucleus, the potential energy is a Coulomb potential, while - at distances larger than the Bohr radius - the nuclear field is screened by the fields of atomic electrons. The interaction between two nuclei is usually described in terms of an interatomic Coulomb potential (e.g., see Sections 2.2.1 and 2.2.2 of [Leroy and Rancoita (2016)] and Section 4.1 of [ICRU Report 49 (1993)]), which is a function of the radial distance r between the two nuclei

V (r) = r ΨI (rr),

where ez (projectile) and eZ (target) are the charges of the bare nuclei and ΨI is the interatomic screening function. This latter function depends on the reduced radius rr given by

rr = a , 

where aI is the so-called screening length (also termed screening radius). In the framework of the Thomas–Fermi model of the atom (e.g., see Chapters 1 and 2 of [Torrens (1972)]) - thus, following the approach of [ICRU Report 49 (1993)] -, a commonly used screening length for z = 1 incoming particles is that from Thomas–Fermi (e.g., see [Thomas (1927)Fermi (1928)])

 CT F a0 
aTF = ---1∕3--, 

and - for incoming particles with z 2 - that introduced by [Ziegler, Biersack and Littmark (1985)] (and termed universal screening length1)

a = --CT-F-a0--, 
 U z0.23 + Z0.23


a = ℏ--- 
 0 me2

is the Bohr radius, m is the electron rest mass and

 ( )2∕3 
C = 1- 3π- ≃ 0.88534 
 TF 2 4

is a constant introduced in the Thomas–Fermi model.  The simple scattering model due to [Wentzel (1926)] - with a single exponential screening-function ΨI(rr) {e.g., see [Wentzel (1926)] and Equation (21) in [Fernandez-Vera et al. (1993)]} - was repeatedly employed in treating single and multiple Coulomb-scattering with screened potentials (e.g., see [Fernandez-Vera et al. (1993)] - and references therein - for a survey of such a topic and also [Molière (1947, 1948)Bethe (1953)Butkevick et al. (2002)Boschini et al. (2010)]). The resulting elastic differential cross section differs from the Rutherford differential cross section by an additional term - the so-called screening parameter - which prevents the divergence of the cross section when the angle θ of scattered particles approaches 0. The screening parameter A s,M [e.g., see Equation (21) of [Bethe (1953)])] - as derived in [Molière (1947, 1948)] for the single Coulomb scattering using a Thomas–Fermi potential - is expressed2 as

 ( )2 [ ( )2] 
As,M = --ℏ--- 1.13 + 3.76 × αzZ-- 
            2p aI β

where aI is the screening length - from Eqs. (34) for particles with z = 1 and z 2, respectively; α is the fine-structure constant; p (βc) is the momentum (velocity) of the incoming particle undergoing the scattering onto a target supposed to be initially at rest; c and are the speed of light and the reduced Planck constant, respectively. When the (relativistic) mass - with corresponding rest mass m - of the incoming particle is much lower than the rest mass (M) of the target nucleus, the differential cross section - obtained from the Wentzel–Molière treatment of the single scattering - is:

dσW M (θ) (zZe2 )2 1 
--------- = ----- -------------------- (6) 
    dΩ pβc (2As,M + 1 - cosθ)2 
                ( zZe2 )2 1 
            = ------ [----------------]2 (7) 
                2 pβc As,M + sin2(θ∕2)
(e.g., see Section 2.3 of [Fernandez-Vera et al. (1993)] and references therein). Equation (7) differs from Rutherford’s formula - as already mentioned - for the additional term As,M to sin 2(θ∕2). The corresponding total cross section {e.g., see Equation (25) in [Fernandez-Vera et al. (1993)] } per nucleus is
 ( 2)2 
σW M = zZe-- -------π-------. 
            p βc As,M (1 + As,M )

Thus, for β 1 (i.e., at very large p) and with As,M 1, from Eqs. (58) one finds that the cross section approaches a constant:

 ( ) 
 W M 2-zZe2aI- 2----------π---------- 
σc ≃ ℏc 2. 
                     1.13 + 3.76 × (αzZ )

In case of a scattering under the action of a central potential (for instance that due to a screened Coulomb field), when the rest mass of the target particle is no longer much larger than the relativistic mass of the incoming particle, the expression of the differential cross section must properly be re-written - in the center of mass system - in terms of an “effective particle” with momentum (pr) equal to that of the incoming particle (pin) and rest mass equal to the relativistic reduced mass

μrel = ----, 

where M1,2 is the invariant mass; m and M are the rest masses of the incoming and target particles, respectively (e.g., see [Boschini et al. (2010)Starusziewicz and Zalewski (1977)Fiziev and Todorov (2001)] and references therein). The “effective particle” velocity is given by:

 ┌ ------------------ 
        ││ [ ( )2 ]-1 
βrc = c∘ 1 + μrelc- . 
                 p ′in

Thus, the differential cross section3 per unit solid angle of the incoming particle results to be given by

 WM ′ ( 2 )2 
dσ----(θ-) -zZe---- --------1--------- 
    dΩ ′ = 2p′inβrc [A + sin2(θ′∕2 )]2, 


 ( )2 [ ( )2] 
        ---ℏ---- αzZ-- 
As = 2 p′in aI 1.13 + 3.76 × βr

and θthe scattering angle in the center of mass system. 

Furthermore (e.g., see Section 2.2.2 of [Leroy and Rancoita (2016)]), assuming an isotropic azimuthal distribution one can re-write Eq. (10) in terms of the kinetic energy transferred from the projectile to the recoil target as:

 ( ) 
dσW-M-(T)- -zZe2-- 2 ----Tmax------- 
    dT = π p′ β c 2. 
                 in r [Tmax As + T]

Furthermore, since

 p c2 
 βrc = ---- 
    p′ = -pM-- (13) 
    in M1,2 
Tmax = 2p-M-- 
            M 21,2
with p and E the momentum and total energy of the incoming particle in the laboratory, then one finds
 Tmax 2E2 
--′-----2 = -2---4. 
(pinβrc) p M c

Therefore, Eq. (12) can be re-written as

 W M 2 
dσ----(T-)= 2π (zZe2 )2--E-----------1-------. 
    dT p2M c4[Tmax As + T ]2

Equation (14) expresses - as already mentioned - the differential cross section as a function of the (kinetic) energy T achieved by the recoil target.


[Agostinelli et al., Geant4 (2003)]   S. Agostinelli et al., Geant4 a simulation toolkit, Nucl. Instr. and Meth. in Phys. Res. A 506 (2003), 250-303; see also the website:

[Boschini et al. (2010)]   M.J. Boschini, C. Consolandi, M. Gervasi, S.Giani, D.Grandi, V. Ivanchenko and P.G. Rancoita, Geant4-based application development for NIEL calculation in the Space Radiation Environment, Proc. of the 11th ICATPP Conference, October 5–9 2009, Villa Olmo, Como, Italy, World Scientific, Singapore (2010), 698–708, IBSN: 10-981-4307-51-3.

[Boschini et al. (2011)]   M.J. Boschini, C. Consolandi, M. Gervasi, S. Giani, D. Grandi, V. Ivantchenko, S. Pensotti, P.G. Rancoita, M. Tacconi, Nuclear and Non-Ionizing Energy-Loss for Coulomb Scattered Particle from Low Energy up to relativistic regime in Space Radiation Environment, Proc. of the 12th ICATPP Conference, October 7-8 2010, Villa Olmo, Como, Italy, World Scientific, Singapore (2011), 9-23, IBSN: 978-981-4329-02-6; or

[Fiziev and Todorov (2001)]   P.P. Fiziev and I.T. Todorov, Phys. Rev. D 63 (2001), 104007-1–104007-9.

[Bethe (1953)]   H. A. Bethe, Phys. Rev. 89 (1953), 1256–1266.

[Butkevick et al. (2002)]   A. V. Butkevick et al., Nucl. Instr. and Meth. in Phys. Res. A 488 (2002), 282-194.

[Fermi (1928)]   E. Fermi, Z. Phys. 48 (1928), 73–79.

[Fernandez-Vera et al. (1993)]   J.M. Fernandez-Vera et al., Nucl. Instr. and Meth. in Phys. Res. B 73 (1993), 447–473.

[ICRU Report 49 (1993)]   ICRU, ICRU Report 49, Stopping Powers and Ranges for Protons and Alpha Particles (1993).

[Kalbitzer and Oetzmann (1976)]   S. Kalbitzer and H. Oetzmann, Phys. Lett. A 59 (1976), 197–198.

[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);; 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.

[Lindhard and Sharff (1961)]   J. Lindhard and M. Sharff, Phys. Rev. 124 (1961), 128–130.

[Lindhard and Sharff (1961)]   J. Lindhard and M. Sharff, Phys. Rev. 124 (1961), 128–130.

[Molière (1947, 1948)]   von G. Molière, Z. Naturforsh. A2 (1947), 133–145; A3 (1948), 78.

[Starusziewicz and Zalewski (1977)]   A. Starusziewicz and K. Zalewski Acta Phys. Pol. B 8 (no. 10) (1977), 815–817.

[Thomas (1927)]   L.H. Thomas, Proc. Cambridge Phil. Soc. 23 (1927), 542.

[Torrens (1972)]   I.M. Torrens, Interatomic Potentials, Academic Press (New York) 1972.

[Wentzel (1926)]   G. Wentzel, Z. Phys. 40 (1926), 590–593.

[Ziegler, Biersack and Littmark (1985)]   J.F. Ziegler, J.P. Biersack and U. Littmark, The Stopping Range of Ions in Solids, Vol. 1, Pergamon Press (New York) 1985.

1Another screening length commonly used is that from [Lindhard and Sharff (1961)] (see also [Kalbitzer and Oetzmann (1976)] and references therein):

 ---CT-F a0---- 
aL = (2∕3 2∕3)1∕2. 
     z + Z

2It has to be remarked that the screening radius originally used in [Molière (1947, 1948), Bethe (1953)] was that from Eq. (3).

3By inspection of Eqs. (5, 7, 10, 11), one finds that for βr 1 the cross section is given by Eq. (9).