Atomic relaxation

Atomic relaxation processes can be induced by any ionisation process that leaves the interested atom in an excited state (i.e. with a vacancy in its electronic structure). Processes inducing atomic relaxation in Geant4 are photoelectric effect, Compton and ionisation (both Standard and Lowenergy).

Geant4 uses by default the Livermore Evaluation Atomic Data Library EADL [PCeal], that contains data to describe the relaxation of atoms back to neutrality after they are ionised. It is assumed that the binding energy of all subshells (from now on shells are the same for neutral ground state atoms as for ionised atoms [PCeal]).

Data in EADL includes the radiative and non-radiative transition probabilities for each sub-shell of each element, for Z =1 to 100. The atom has been ionised by a process that has caused an electron to be ejected from an atom, leaving a vacancy or “hole” in a given subshell. The EADL data are then used to calculate the complete radiative and non-radiative spectrum of X-rays and electrons emitted as the atom relaxes back to neutrality.

Non-radiative de-excitation can occur via the Auger effect (the initial and secondary vacancies are in different shells) or Coster-Kronig effect (transitions within the same shell).

Please see further detailed information on atomic deexcitation at http://geant4.web.cern.ch/node/1620.

Fluorescence

The simulation procedure for the fluorescence process is the following:

  1. If the vacancy shell is not included in the data, energy equal to the binding energy of the shell is deposited locally

  2. If the vacancy subshell is included in the data, an outer subshell is randomly selected taking into account the relative transition probabilities for all possible outer subshells.

  3. In the case where the energy corresponding to the selected transition is larger than a user defined cut value (equal to zero by default), a photon particle is created and emitted in a random direction in 4\(\pi\), with an energy equal to the transition energy, provided by EADL.

  4. the procedure is repeated from step 1, for the new vacancy subshell.

The final local energy deposit is the difference between the binding energy of the initial vacancy subshell and the sum of all transition energies which were taken by fluorescence photons. The atom is assumed to be initially ionised with an electric charge of \(+1e\).

Sub-shell data are provided in the EADL data bank [PCeal] for \(Z =1\) through 100. However, transition probabilities are only explicitly included for \(Z =6\) through 100, from the subshells of the K, L, M, N shells and some O subshells. For subshells O,P,Q: transition probabilities are negligible (of the order of 0.1%) and smaller than the precision with which they are known. Therefore, for the time being, for \(Z =1\) through 5, only a local energy deposit corresponding to the binding energy B of an electron in the ionised subshell is simulated. For subshells of the O, P, and Q shells, a photon is emitted with that energy B.

Since G4EMLOW8.0 a new X-ray fluorescence data library has been included with X-ray fluorescence transition probabilities calculated within the Hartree-Fock approach [Sco74], which is recognised to better reproduce PIXE experimental values when compared to the Hartree-Slater approach, adopted in the EADL data library [DDJS15]. This approach is called fluoANSTO in Geant4 and the transition probabilities are provided for :math: Z < 93`. For higher atomic numbers the EADL data libraries are used. The binding energies of all the subshells are derived from the EADL. This approach can be adopted to describe X-ray fluorescence in any application scenario, however, it is particularly recommended for PIXE applications. For more information the reader is referred to Bakr et al 2021 :cite:`em-bakr.

Auger process

The Auger effect is complementary to fluorescence, hence the simulation process is the same as for the fluorescence, with the exception that two random shells are selected, one for the transition electron that fills the original vacancy, and the other for selecting the shell generating the Auger electron.

Subshell data are provided in the EADL data bank [PCeal] for \(Z=6\) through 100. Since in EADL no data for elements with \(Z < 5\) are provided, Auger effects are only considered for \(5 < Z < 100\) and always due to the EADL data tables, only for those transitions which have a probability to occur \(> 0.1\%\) of the total non-radiative transition probability. EADL probability data used are, however, normalized to one for Fluorescence + Auger.

PIXE

PIXE (Particle Induced X-Ray Emission) can be simulated for ionisation continuous processes performed by ions. Ionised shells are selected randomly according the ionisation cross section of each shell once known the (continuous) energy loss along the step Mean Energy Loss.

Different shell ionisation cross sections models are available in different energy ranges:

  • ECPSSR [WBrandtGLapicki81][BL79] internal Geant4 calculation for K and L shells.

  • ECPSSR calculations from Factor Form according to Reis [eal11b] for K and L shells from 0.1 to 100 MeV and for M shells from 0.1 to 100 MeV.

  • ECPSSR calculations based on state of the art recommendations documented in Cohen et al. 2015 [DDJS15], called ANSTO ECPSSR. This approach provides the ionisation cross section of the K, L and M shells for incident protons and alpha particles up to :math:`5 MeV/amu`,

    in targets with :math:`Z < 93`.

    The proposed cross sections have been extensively validated against PIXE experimental measurements by many PIXE labs. For more information the reader is referred to [BDCohenS+18].

  • empirical “reference” K-shell values from Paul for protons [HP89] and for \(alpha\) [HP93]. Energies ranges are 0.1 - 10 MeV/amu circa, depending on the atomic number that varies between 4 and 32.

  • semi-empirical L-subshell values from Orlic [OST94]. Energy Range 0.1-10 MeV for Z between 41 and 92.

Outside Z and energy of limited shell ionisation cross sections, the ECPSSR internal calculation method is applied.

Please refer to Ref.[eal11a] and original papers to have detailed information of every model.

Bibliography

BDCohenS+18

Samer Bakr, David D.Cohen, Rainer Siegele, Sebastien Incerti, Vladimir Ivanchenko, Alfonso Mantero, Anatoly Rosenfeld, and Susanna Guatelli. Latest geant4 developments for pixe applications. Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms, 436:285–291, 2018. URL: https://doi.org/10.1016/j.nimb.2018.10.004.

BL79

W. Brandt and G. Lapicki. Phys. Rev. A, 20(N2):, 1979.

DDJS15(1,2)

Cohen D.D., Crawford J., and R. Siegele. K, l, and m shell datasets for pixe spectrum fitting and analysis. Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms, 363:7–18, 2015. URL: https://doi.org/10.1016/j.nimb.2015.08.012.

eal11a

A. Mantero et al. X-Ray Spec., 40():135–140, 2011.

eal11b

A. Taborda et al. X-Ray Spec., 40():127–134, 2011.

HP89

J. Sacher H. Paul. Atom. Dat. and Nucl. Dat. Tabl., 42(1):105–156, May 1989.

HP93

O. Bolik H. Paul. Atom. Dat. and Nucl. Dat. Tabl., 54(1):75–131, May 1993.

OST94

I. Orlic, C. H. Sow, and S. M. Tang. Semiempirical formulas for calculation of l subshell ionization cross sections. International Journal of PIXE, 4(4):217–230, 1994.

PCeal(1,2,3,4)

S.T. Perkins, D.E. Cullen, and M.H. Chen et al. Tables and graphs of atomic subshell and relaxation data derived from the llnl evaluated atomic data library (eadl), z=1-100. Technical Report UCRL-50400 Vol.30.

Sco74

J. H. Scofield. Relativistic hartree-slater values for k and l x-ray emission rates. At. Data Nucl. Data Tables, 14(2):121–137, 1974. URL: https://doi.org/10.1016/S0092-640X(74)80019-7.

WBrandtGLapicki81

W.Brandt and G.Lapicki. Phys. Rev. A, 23():, 1981.