Fitting and calculation of photon strength functions

This project is based on a ROOT [1] script (essentially C++) to fit a two-component Lorentzian-type function to the Giant Dipole Electric Resonance (GDER) observed in photonuclear reactions. The data used in this script are from ²⁸Si(γ,xn) photonuclear absorption measurements [2], [3].

This script requires ROOT to be installed and has been tested with various versions of ROOT-5 and ROOT-6. To run the script, first launch a ROOT session:

$ root -l

Then exceute the following commands in the C++ interpreter and follow the prompts:

> .L psfSiGDER.C
> reading()

The figure below is an example of the resulting Lorentzian fit to the GDER in ²⁸Si(γ,xn) and the calculated photon strength functions (PSF) utilizing the parametrizations obtained from the fit. The functional form of the fit is known as the Brink-Axel (BA) model (also referred to as the Standard Lorentzian (SLO) model):

$$ f_{E1}(E_{\gamma}) = \frac{1}{3(\pi \hbar c)^{2}} \sum\limits_{i=1}^{N=2} \frac{\sigma_{G_{i}} E_{\gamma} \Gamma_{G_{i}}^{2}}{(E^{2}{\gamma}-E{G_{i}}^{2})^{2} + E_{\gamma}^{2}\Gamma_{i}^{2}}, $$

where the resonance-shape driving parameters $E_{G}$, $\Gamma_{G}$, and $\sigma_{G}$ correspond to the resonance centroid (MeV), width (MeV), and cross section (mb), respectively, and are obtained from the fitting procedure. This strength function contains the constant $\frac{1}{3(\pi \hbar c)^{2}} = 8.68 \times 10^{-8}~\text{mb}^{-1}\text{MeV}^{-2}$, and is singularly dependent on the $\gamma$-ray energy $E_{\gamma}$. The other models shown on the plot are the Generalized Lorentzian (GLO), Enhanced Generalized Lorentzian (EGLO), and the Kadmenski-Markushev-Furman (KMF) model. These models are widely adopted and explained in the literature, for example, refer to the articles bundled with this project for explicit definitions of the GLO [4], [5], EGLO [4], and KMF [5] models that have been utilized in the statistical modeling of the tungsten [4] and lanthanum [5] isotopes to describe decay-scheme observables following thermal neutron capture.

Selecting the source data

In the figure above, the photonuclear data of Goryachev et al. [3] are used to obtain the fit. The appropriate filename string is declared in the reading function of the psfSiGDER.C script:

const std::string iFile = "si28_gxn_goryachev.dat";

The results of the fitted parametrizations are also defined in the MyParameters() constructor so that these parameters are then used by the PSF models when this constuctor gets called to create an object of the class:

//Deduced fitted parametrizations to Goryachev data
MyParameters::MyParameters(){ EG1 = 20.24; GammaG1 = 4.03; SigmaG1 = 13.45;  
  EG2 = 27.68; GammaG2 = 6.46; SigmaG2 = 6.92; EK0 = 1.0; }

To use the photonuclear data of Caldwell et al. [2] instead, change the string filename declaration to:

const std::string iFile = "si28_gxn_caldwell.dat";

and then comment out the Goryachev-deduced parameters and uncomment the Caldwell-deduced parametrization of the BA (SLO) model:

//Deduced fitted parametrizations to Caldwell data
MyParameters::MyParameters(){ EG1 = 20.47; GammaG1 = 3.73; SigmaG1 = 9.43;  
  EG2 = 27.56; GammaG2 = 7.98; SigmaG2 = 2.95; EK0 = 1.0; }

References

[1] ROOT: Data Analysis Framework; https://root.cern.ch/

[2] J.T. Caldwell et al., (γ,n) cross sections of ¹⁶O and ²⁸Si, Phys. Lett. 6, 213 (1963); http://dx.doi.org/10.1016/0031-9163(63)90548-1

[3] B.I. Goryachev et al., Structure of (γ,n) Cross Sections in ²⁸Si, ³²S, and ⁴⁰Ca, Yadern. Fiz. 7, 1168 (1968); Soviet J. Nucl. Phys. 7, 698 (1968).

[4] A.M. Hurst et al., Investigation of the tungsten isotopes via thermal neutron capture, Phys. Rev. C 89, 014606 (2014); http://dx.doi.org/10.1103/PhysRevC.89.014606

[5] A.M. Hurst et al., Radiative-capture cross sections for the ¹³⁹La(n,γ) reaction using thermal neutrons and structural properties of ¹⁴⁰La, Phys. Rev. C 99, 024310 (2019); http://dx.doi.org/10.1103/PhysRevC.99.024310