Approximation of Photonuclear Cross Sections.

The photonuclear cross sections parameterized in the G4PhotoNuclearCrossSection class cover all incident photon energies from the hadron production threshold upward. The parameterization is subdivided into five energy regions, each corresponding to the physical process that dominates it.

• The Giant Dipole Resonance (GDR) region, depending on the nucleus, extends from 10 Mev up to 30 MeV. It usually consists of one large peak, though for some nuclei several peaks appear.

• The ``quasi-deuteron'' region extends from around 30 MeV up to the pion threshold and is characterized by small cross sections and a broad, low peak.

• The $\Delta$ region is characterized by the dominant peak in the cross section which extends from the pion threshold to 450 MeV.

• The Roper resonance region extends from roughly 450 MeV to 1.2 GeV. The cross section in this region is not strictly identified with the real Roper resonance because other processes also occur in this region.

• The Reggeon-Pomeron region extends upward from 1.2 GeV.

In the GEANT4 photonuclear data base there are about 50 nuclei for which the photonuclear absorption cross sections have been measured in the above energy ranges. For low energies this number could be enlarged, because for heavy nuclei the neutron photoproduction cross section is close to the total photo-absorption cross section. Currently, however, 14 nuclei are used in the parameterization: $^1$H, $^2$H, $^4$He, $^6$Li, $^7$Li, $^9$Be, $^{12}$C, $^{16}$O, $^{27}$Al, $^{40}$Ca, Cu, Sn, Pb, and U. The resulting cross section is a function of $A$ and $e = log(E_\gamma)$, where $E_\gamma$ is the energy of the incident photon. This function is the sum of the components which parameterize each energy region.

The cross section in the GDR region can be described as the sum of two peaks,

$$GDR(e) = th(e,b_1,s_1)\cdot exp(c_1-p_1\cdot e) + th(e,b_2,s_2)\cdot exp(c_2-p_2\cdot e) .$$ (15.1)

The exponential parameterizes the falling edge of the resonance which behaves like a power law in $E_\gamma$. This behavior is expected from the CHIPS model, which includes the nonrelativistic phase space of nucleons to explain evaporation. The function

$$th(e,b,s) = \frac{1}{1+exp(\frac{b-e}{s})} ,$$ (15.2)

describes the rising edge of the resonance. It is the nuclear-barrier-reflection function and behaves like a threshold, cutting off the exponential. The exponential powers $p_1$ and $p_2$ are

$$\begin{eqnarray*} p_1 = 1, p_2 = 2 \mbox{\hspace*{1mm} for \hspace*{7mm} $A < 4... ...= 4, p_2 = 8 \mbox{\hspace*{1mm} for \hspace*{6mm} $A \ge 12$} . \end{eqnarray*}$$

The $A$-dependent parameters $b_i$, $c_i$ and $s_i$ were found for each of the 14 nuclei listed above and interpolated for other nuclei.

The $\Delta$ isobar region was parameterized as

$$\Delta (e,d,f,g,r,q)=\frac{d\cdot th(e,f,g)}{1+r\cdot (e-q)^2},$$ (15.3)

where $d$ is an overall normalization factor. $q$ can be interpreted as the energy of the $\Delta$ isobar and $r$ can be interpreted as the inverse of the $\Delta$ width. Once again $th$ is the threshold function. The $A$-dependence of these parameters is as follows:

• $d=0.41\cdot A$ (for $^1$H it is 0.55, for $^2$H it is 0.88), which means that the $\Delta$ yield is proportional to $A$;

• $f=5.13-.00075\cdot A$. $exp(f)$ shows how the pion threshold depends on $A$. It is clear that the threshold becomes 140 MeV only for uranium; for lighter nuclei it is higher.

• $g = 0.09$ for $A \ge 7$ and 0.04 for $A < 7$;

• $q=5.84-\frac{.09}{1+.003\cdot A^2}$, which means that the ``mass'' of the $\Delta$ isobar moves to lower energies;

• $r=11.9 - 1.24\cdot log(A)$. $r$ is 18.0 for $^1$H. The inverse width becomes smaller with $A$, hence the width increases.

The $A$-dependence of the $f$, $q$ and $r$ parameters is due to the $\Delta+N\rightarrow N+N$ reaction, which can take place in the nuclear medium below the pion threshold.

The quasi-deuteron contribution was parameterized with the same form as the $\Delta$ contribution but without the threshold function:

$$QD(e,v,w,u)=\frac {v}{1+w\cdot (e-u)^2}.$$ (15.4)

For $^1$H and $^2$H the quasi-deuteron contribution is almost zero. For these nuclei the third baryonic resonance was used instead, so the parameters for these two nuclei are quite different, but trivial. The parameter values are given below.

• $v = \frac {exp(-1.7+a\cdot 0.84)}{1+exp(7\cdot (2.38-a))}$, where $a=log(A)$. This shows that the $A$-dependence in the quasi-deuteron region is stronger than $A^{0.84}$. It is clear from the denominator that this contribution is very small for light nuclei (up to $^6$Li or $^7$Li). For $^1$H it is 0.078 and for $^2$H it is 0.08, so the delta contribution does not appear to be growing. Its relative contribution disappears with $A$.

• $u = 3.7$ and $w = 0.4$. The experimental information is not sufficient to determine an $A$-dependence for these parameters. For both $^1$H and $^2$H $u = 6.93$ and $w = 90$, which may indicate contributions from the $\Delta$(1600) and $\Delta$(1620).

The transition Roper contribution was parameterized using the same form as the quasi-deuteron contribution:

$$Tr(e,v,w,u)=\frac {v}{1+w\cdot (e-u)^2}.$$ (15.5)

Using $a=log(A)$, the values of the parameters are

• $v = exp(-2.+a\cdot 0.84)$. For $^1$H it is 0.22 and for $^2$H it is 0.34.

• $u = 6.46+a\cdot 0.061$ (for $^1$H and for $^2$H it is 6.57), so the ``mass'' of the Roper moves higher with $A$.

• $w = 0.1+a\cdot 1.65$. For $^1$H it is 20.0 and for $^2$H it is 15.0).

The Regge-Pomeron contribution was parametrized as follows:

$$RP(e,h)=h\cdot th(7.,0.2)\cdot (0.0116\cdot exp(e\cdot 0.16)+0.4\cdot exp(-e\cdot 0.2)),$$ (15.6)

where $h=A\cdot exp(-a\cdot (0.885+0.0048\cdot a))$ and, again, $a=log(A)$. The first exponential in Eq. 15.6 describes the Pomeron contribution while the second describes the Regge contribution.