**4. Neutrino interactions related to effective dose**

In a muon collider, muon decays arise principally from Eqs. 1 and 2 that produce neutrinos and antineutrinos. The neutrinos interact through a variety of complex processes. A neutrino interaction discussion is simplified by following the methodology of Cossairt et al. (1997) and defining four processes (A, B, C, and D) to describe neutrino interactions with matter. The deposition of energy into tissue defines the effective dose (Bevelacqua, 2009, 2010a).

Process A involves neutrino scattering from atomic electrons. Electrons that recoil from elastic neutrino scattering deposit their energy in tissue and produce a neutrino effective dose. Process A occurs over a wide range of energy and the electron tissue interaction may involve multiple scattering of electrons.

In Process B, neutrinos interact coherently with nuclei. This process is only effective for low neutrino energies where the neutrino wavelength is too long to resolve the individual nucleons within the nucleus. At higher energies, Processes C and D become more important. Process B leads to low-energy ions having large linear energy transfer values. These ions deposit their energy into tissue according to their ranges, which are typically << 1 cm. Although Process B is independent of the neutrino generation, the cross section for neutrinos is about twice the antineutrino cross section (King 1999a).

Process C involves neutrino scattering from nucleons without shielding between the neutrinos and tissue. At energies below about 500 MeV, tissue dose is due to recoil nucleons. As the neutrino energy increases above about 0.5 GeV, secondary particle production increases. Eventually, these secondary particles produce particle showers or cascades in tissue. Process C is independent of the neutrino generation, affecting all three generations in the same manner.

Process D is similar to Process C with the exception that the neutrinos are shielded before striking tissue. Neutrinos with energy greater than about 0.5 GeV, emerging from a layer of

more peaked in the beam direction as the muon energies increase. In addition, the neutrino

where θ is called the characteristic angle, opening half-angle, or half-divergence angle of the

<sup>2</sup> *m c E*

In Eqs. 3 and 4, L is the distance to the point of interest such as the distance from the muon decay location to the earth's surface, θ is given in radians, E is the muon beam energy, and <sup>2</sup> *mc* is the rest mass of the muon (105.7 MeV). As the muon energy increases, the neutrino beam radius and size of the resultant hadronic showers tend to be smaller than the size of a

> <sup>4</sup> 10 *E TeV* [ ]

Therefore, the emergent neutrino beam will consist of a narrow diverging beam that is

Table 1 summarizes straw-man muon collider parameters (King, 1999b). It should be noted that the straw-man muon colliders are constructed below the earth's surface to provide muon shielding. However, the neutrino attenuation length is too long for the beam to be appreciably attenuated by any practical amount of shielding, including the expanse of ground between the collider and its exit from the surface of the earth. Therefore, the effective dose reduction principle as applied to neutrinos will no longer include shielding as an element. In fact, shielding the neutrino beam will produce hadronic showers and increase the effective dose. This peculiar behavior has its basis in the nature of the weak interaction, the uncharged nature of the neutrino, and the TeV energies that will be encountered in

> E (TeV) 2 5 50 L (km) 62 36 36 r (m) 3.3 0.8 0.08 Collider depth (m) 300 100 100

The neutrinos exiting a muon collider will not only have a narrow conical shape, but will also have an extent that is quite long. The long, narrow plume of neutrinos will produce secondary muons and hadronic showers at a significant distance from the muon collider.

This distance will be greater than tens of kilometers for TeV muon energies.

The characteristic angle varies inversely with energy. If E is expressed in TeV:

r = θ L (3)

(4)

(5)

beam radius (r) will be relatively small and is given by (King, 1999b):

muon decay cone

person.

conical in shape.

proposed muon colliders.

Table 1. Straw-Man Muon Collider Parameters.

material (e.g., earth shielding), result in a larger effective dose than unshielded neutrinos. The increase in effective dose arises from the fact that the tissue is exposed to the secondary particles produced by neutrino interactions in the shielding material as well as the neutrino beam. Process D is also independent of the neutrino generation.

A process that involves an increase in effective dose with added shielding is unique. One of the basic tenants for reducing effective dose for most radiation types (e.g., alpha and beta particles, heavy ions, muons, neutrons, photons, pions, and protons) is shielding the radiation source (Bevelacqua, 2009 and 2010a). The unique nature of Process D has a significant impact on the evaluation and control of neutrino effective dose.
