# Energy Loss (d2n)

We can divide the energy loss problem into two parts: energy loss for incoming electrons (before the primary scattering interaction), and energy loss for outgoing electrons (after the primary scattering interaction). The tables below, with reference to Chiranjib's dissertation, show the materials in the path of each set of electrons, plus a mean energy loss calculation for a nominal scattering angle of $45^{\circ}$.

Please note that the actual energy loss curve follows a Landau distribution, and the most likely energy loss value is NOT the same as the mean energy loss reported below.

Chiranjib Dutta's dissertation (especially the appendices) is a good resource.

# Mean Energy Loss for Incoming Electrons

All thicknesses taken from Chiranjib's dissertation.

 Material $X_0$ (cm) Thickness (cm) Thickness ($X_0$) Mean Energy Loss $(\langle E_f \rangle - E_i)/E_i$ Beam pipe window (Be) 35.28 0.0254 0.000719 0.00072 He-4 in target enclosure 528107.5 22.86 0.000043 0.000043 Target entrance window (glass) 7.038 0.01 0.00142 0.001420 19.8 cm (half cell length) of He-3 43423 19.8 0.000456 0.000456

# Mean Energy Loss for Outgoing Electrons

• For the He-3 and target side wall, starred quantities (thickness and mean energy loss) assume a scattering angle of 45 degrees.
• The He-4 thickness and mean energy loss assumes that the target enclosure is spherically symmetric around the interaction point. In actuality this depends on the vertex position.
 Material $X_0$ (cm) Thickness (cm) Thickness ($X_0$) Mean Energy Loss $(\langle E_f \rangle - E_i)/E_i$ He-3 in cell 43423 1.34* 0.000031* 0.000031* Side wall of target cell (glass) 7.038 0.156* 0.022165* 0.021922* He-4 in target enclosure 528107.5 79.05 0.000150 0.000150 Air (distance is for LHRS) 30423 51.23 0.001684 0.001682 Kapton entry window (LHRS) 28.6 0.0254 0.000888 0.000887

## Yellow Cover

Since the composition and thickness of the yellow cover on the target enclosure is not quite known, the table below gives sample values, assuming a thickness of 35 mils = 0.0889 cm and spherical symmetry about the interaction point, for several different types of plastics listed in the PDG table of Atomic and Nuclear Products and Materials.

 Material $X_0$ (cm) Thickness (cm) Thickness ($X_0$) Mean Energy Loss $(\langle E_f \rangle - E_i)/E_i$ Nylon 35.525 0.0889 0.00250 0.00250 Polycarbonate 34.583 0.0889 0.00257 0.00257 Polyethylene 50.303 0.0889 0.00177 0.00176 Mylar 28.536 0.0889 0.00312 0.00311 Kapton 28.577 0.0889 0.00311 0.00311 Acrylic 34.076 0.0889 0.00261 0.00260 Polypropylene 49.744 0.0889 0.00179 0.00178 Polystyrene 41.311 0.0889 0.00215 0.00215 Teflon 15.836 0.0889 0.00561 0.00560 Polyvinyltoluene 42.621 0.0889 0.00208 0.00208

# 3He Glass Cell Characteristics

• Material: Aluminosilicate (GE180)
• $\rho$: 2.76 $\textrm{g}/\textrm{cm}^3$
• $X_0$: 19.4246 $\textrm{g}/\textrm{cm}^2$
• $Z_{\textrm{eff}} = 19.56$
• $A_{\textrm{eff}} = 40.51$ $\textrm{g/mol}$
• Composition by weight:
 Material Composition by Weight (%) $Z_{\textrm{eff}}$ $Z/A$ SiO$_2$ 60.3 11.56 0.4993 BaO 18.2 53.52 0.4174 Al$_2$O$_3$ 14.3 11.14 0.4904 CaO 6.5 17.99 0.4993 SrO 0.25 35.64 0.4439 Totals/Weighted Averages 99.55 19.56 0.4829
• $Z_{\textrm{eff}}$ is calculated as: $Z_{\textrm{eff}} = \left(\sum_i f_i Z_i^{2.94}\right)^{1/2.94}$, where $f_i$ is the fraction of electrons associated with each element in the compound.
• $A_{\textrm{eff}}$ is calculated as: $A_{\textrm{eff}} = \frac{Z_{\textrm{eff}}}{Z/A}$