Photon evaporation

Photon evaporation may be simulated as a continuum gamma transition using a dipole approximation or via discrete gamma transitions using an evaluated database of nuclear gamma transitions.

Computation of probability

As the first approximation we assume that dipole \(E1\)–transitions is the main source of \(\gamma\)–quanta from highly–excited nuclei [IMB+92]. The probability to evaporate \(\gamma\) in the energy interval \((\epsilon_{\gamma}, \epsilon_{\gamma}+d\epsilon_{\gamma})\) per unit of time is given

(256)\[W_{\gamma}(\epsilon_{\gamma}) = \frac{1}{\pi^2 (\hbar c)^3}\sigma_{\gamma}(\epsilon_{\gamma}) \frac{\rho(E^{*}-\epsilon_{\gamma})} {\rho(E^{*})}\epsilon^2_{\gamma},\]

where \(\sigma_{\gamma}(\epsilon_{\gamma})\) is the inverse (absorption of \(\gamma\)) reaction cross section, \(\rho\) is a nucleus level density is defined by Eq.(254).

The photoabsorption reaction cross section is given by the expression

\[\sigma _{\gamma}(\epsilon_{\gamma}) = \frac{\sigma_0 \epsilon^2_{\gamma} \Gamma^2_{R}} {(\epsilon^2_{\gamma} - E_{GDP}^2)^2 + \Gamma^2_R\epsilon^2_{\gamma}},\]

where \(\sigma_0=2.5A\) mb, \(\Gamma_R=0.3E_{GDP}\) and \(E_{GDP}= 40.3 A^{-1/5}\) MeV are empirical parameters of the giant dipole resonance [IMB+92]. The total radiation probability is

\[W_{\gamma} =\frac{1}{\pi^2 (\hbar c)^3}\int_{0}^{E^{*}} \sigma_{\gamma}(\epsilon_{\gamma}) \frac{\rho(E^{*}-\epsilon_{\gamma})} {\rho(E^{*})}\epsilon^2_{\gamma}d\epsilon_{\gamma}.\]

The integration is performed numerically. The energy of \(\gamma\)-quantum is sampled according to the Eq.(256) distribution.

Discrete photon evaporation

The last step of evaporation cascade consists of evaporation of photons with discrete energies. The competition between photons and fragments as well as giant resonance photons is neglected at this step. We consider the discrete E0, E1, M1, E2, M2, E3, M3 (including possible mixing) photon transitions from tabulated isotopes. Angular-correlated gamma-emission can be performed (if enabled). There are a large number of isotopes [had] with the experimentally measured exited level energies, spins, parities and relative transitions probabilities. This information is uploaded for each excited state in run time when the corresponding excited state is first created.

Transition information is also included for floating levels when available at [had]. Discrete photon evaporation also takes place between floating states from the same floating band. When the lowest floating state of a band is reached the nucleus is set to its ground state.

The list of isotopes included in the photon evaporation data base has been extended from \(A<=250\) to \(A<=294\). The highest atomic number included is \(Z=117\) (this ensures that Americium sources can now be simulated). For the heaviest of isotopes, in the so-called Super Heavy Elements region, the included files contain only placeholder values.

Internal conversion electron emission

An important competitive channel to photon emission is internal conversion. To take this into account, the photon evaporation database was extended to include internal conversion coefficients.

The database employ for this is the same as for discrete.photon.evaporation. If total internal conversion coefficient is not 0, partial conversion probabilities for K, L1, L2, L3, M1, M2, M3, M4, M5 and N+ respectively are included in the database. These coefficients are normalised to 1.0.

The calculation of the Internal Conversion Coefficients (ICCs) is done by a cubic spline interpolation of tabulated data for the corresponding transition energy. These ICC tables, which we shall label Band [BTL76][BT78], Rösel [RFAP78] and Hager-Seltzer [HS68], are widely used and were provided in electronic format by staff at LBNL. The reliability of these tabulated data has been reviewed in Ref. [RD00]. From tests carried out on these data we find that the ICCs calculated from all three tables are comparable within a 10% uncertainty, which is better than what experimental measurements are reported to be able to achieve.

The range in atomic number covered by these tables is Band: \(1 <= Z <= 80\); Rösel: \(30 <= Z <= 104\) and Hager-Seltzer: \(3, 6, 10, 14 <= Z <= 103\). For simplicity and taking into account the completeness of the tables, we have used the Band table for \(Z <= 80\) and Rösel for \(81 <= Z <= 98\).

The Band table provides a higher resolution of the ICC curves used in the interpolation and covers ten multipolarities for all elements up to \(Z=80\), but it only includes ICCs for shells up to M5. In order to calculate the ICC of the N+ shell, the ICCs of all available M shells are added together and the total divided by 3. This is the scheme adopted in the LBNL ICC calculation code when using the Band table. The Rösel table includes ICCs for all shells in every atom and for \(Z>80\) the N+ shell ICC is calculated by adding together the ICCs of all shells above M5. In this table only eight multipolarities have ICCs calculated for.

For the production of an internal conversion electron, the energy of the transition must be at least the binding energy of the shell the electron is being released from. The binding energy corresponding to the various shells in all isotopes used in the ICC calculation has been taken from the Geant4 file G4AtomicShells.hh.

The ENSDF data provides information on the multipolarity of the transition. The ICCs included in the photon evaporation data base refer to the multipolarity indicated in the ENSDF file for that transition. Only one type of mixed multipolarity is considered (M1+E2) and whenever the mixing ratio is provided in the ENSDF file, it is used to calculate the ICCs corresponding to the mixed multipolarity according to the formula:

fraction in \(M1 = 1/(1+\delta^{2})\)
fraction in \(E2 = \delta^{2}/(1+\delta^{2})\)

where \(\delta\) is the mixing ratio.

Bibliography

had(1,2): Evaluated nuclear structure data file (ensdf) - a computer file of evaluated experimental nuclear structure data maintained by the national nuclear data center, brookhaven national laboratory. http://www.nndc.bnl.gov/nudat2/. [Online; accessed 31-october-2017].
BT78: I.M. Band and M.B. Trzhaskovskaya. Tables of the gamma-ray internal conversion coefficients for the k-, l-, m- shells, 10 ≤ z ≤ 104. Special Report of Leningrad Nuclear Physics Institute, 1978.
BTL76: I.M. Band, M.B. Trzhaskovskaya, and M.A. Listengarten. Internal conversion coefficients for atomic numbers z ≤ 30. Atomic Data and Nuclear Data Tables, 18(5):433–457, nov 1976. URL: https://doi.org/10.1016/0092-640X(76)90013-9, doi:10.1016/0092-640x(76)90013-9.
HS68: R.S. Hager and E.C. Seltzer. Internal conversion tables part II: directional and polarization particle parameters for z = 30 to z = 103. Nuclear Data Sheets. Section A, 4(5-6):397–411, oct 1968. URL: https://doi.org/10.1016/S0550-306X(68)80017-5, doi:10.1016/s0550-306x(68)80017-5.
IMB+92(1,2): A.S. Iljinov, M.V. Mebel, N. Bianchi, E. De Sanctis, C. Guaraldo, V. Lucherini, V. Muccifora, E. Polli, A.R. Reolon, and P. Rossi. Phenomenological statistical analysis of level densities, decay widths and lifetimes of excited nuclei. Nuclear Physics A, 543(3):517–557, jul 1992. URL: https://doi.org/10.1016/0375-9474(92)90278-R, doi:10.1016/0375-9474(92)90278-r.
RD00: M Rysavý and O Dragoun. On the reliability of the theoretical internal conversion coefficients. Journal of Physics G: Nuclear and Particle Physics, 26(12):1859–1872, 2000. URL: http://stacks.iop.org/0954-3899/26/i=12/a=309.
RFAP78: F. Rösel, H.M. Fries, K. Alder, and H.C. Pauli. Internal conversion coefficients for all atomic shells. Atomic Data and Nuclear Data Tables, 21(2-3):91–289, feb 1978. URL: https://doi.org/10.1016/0092-640X(78)90034-7, doi:10.1016/0092-640x(78)90034-7.