Parameterizations

From the simulation data presented in the last section we can now extract approximate expressions for the functions $Z(\theta,E)$ and $L(\theta,E)$ of Eqs. 3-4. To carry out the fit for $L(\theta,E)$, the Monte Carlo data for $p_e$ were histogrammed in bins of 200MeV $\times$ $1^0$. For each energy bin there is an independent histogram of $p_e$ vs $\theta$, as shown in Fig. 1. Each of these was then submitted to fit to a 4-parameter formula consisting of a monopole plus a linear term, as follows.

$$L(\theta) = (\theta_0-\theta)\left( a + \frac{b}{(\theta_0-\theta)^2 + \delta^2} \right)$$ (6)

where $L$ is measured in photoelectrons and the angles $\theta$, $\theta_0$ and $\delta$ are in degrees. The value of $\theta_0$ is the zero of the function, supposed to be the polar angle of the detector edge, and $\delta$ describes the width of the falling gain curve at the acceptance edge.

Values for each of the parameters $\theta_0$, $a$, $b$, and $\delta$ were obtained for the 9 energy bins shown in Figs. 1-2, and these in turn were fitted vs energy. The results are given in Eq. 7

$$\theta_0 = 27.5^0$$ (7)

$$\delta = 6.95 + 0.66 E$$

$$a = 7.4 E$$

$$b = 8900 E^{1.11}$$

The curves in Fig. 1 are the results of the above parameterization. Overall it gives a good description of the data. The points below $3^0$ have been excluded from the fit because these showers start inside the beam hole. The points above $26^0$ have also been excluded, which is about where one might guess from these data that the LGD acceptance ends. The biggest departure of the curve from the data occurs for the two highest energies in Fig. 1, where the data show a sharper maximum near $20^0$ than the curves do. One might invoke another parameter to patch up that region, but I consider the discrepancy to be negligible, especially in view of the small angular region and the fact that it is at larger angles where the photon spectrum is mostly soft.

The function $Z(\theta,E)$ was obtained by a similar procedure. The simulation data for shower mean $z$ were histogrammed on a grid of 200MeV $\times$ $1^0$ bins. The slices in constant $E$ were fitted to a function of $\theta$ and then the parameters were fitted as a function of $E$, as follows.

$$Z(\theta) = z_0 - q (\theta-\theta_m)^2$$ (8)

where $z_0$ is given by Eq. 5 and the other two parameters are described by the linear functions given in Eq. 9. $Z$ is measured in cm.

$$\theta_m = 7.5^0$$ (9)

$$q = 0.015 + 0.0040 E$$

This parameterization is shown by the curves in Fig. 2.