**5. Neutrino beam characteristics at a muon collider**

Neutrinos are produced when the muon beam particles decay (See Eqs. 1 and 2). Weak interactions of muon neutrinos can be described in terms of two broad categories: charged current and weak current interactions. Charged current interactions involve the exchange of W-bosons to form secondary muons. Neutral current interactions produce uncharged particles through the exchange of Z-bosons. Both types of interactions produce hadron particle showers. Therefore, the neutrino induced radiation hazard will include secondary muons and hadronic showers. The hadronic showers have a much shorter range than the muons, but the number of particles in a hadronic shower can be quite large. The neutrino radiation hazard arises from these penetrating charged particle showers (Bevelacqua, 2008).

For TeV energy neutrinos, direct neutrino interactions in man account for less than 1% of the total effective dose because the primary hadrons from the neutrino interactions will typically exit the person before producing a charged particle shower (King, 1999b; Cossairt et al., 1996, 1997). Most of the neutrino effective dose is derived from particle showers produced in the shielding material.

The muon beam and subsequent neutrino beam are assumed to be well-collimated and to have a minimum divergence angle. For practical situations, the muons in the accelerator beam will have a small divergence angle and will be periodically focused using electromagnetic fields to ensure their collimation. No beam divergence is assumed in the subsequent calculations. Therefore, the actual beam will be somewhat more diffuse than assumed in the neutrino effective dose calculations. The neutrino beam will still produce particle showers, but they will be somewhat broader and less intense than the assumed well-collimated result. The beam divergence is analogous to the divergence of a laser beam as it exits an aperture (Bevelacqua, 2009, 2020).

The magnitude of the effective dose from a particle shower is dependent on the material in the interaction region lying directly upstream of the individual being irradiated. Calculation of the neutrino effective dose considers the configuration where a person is (1) completely bathed in the neutrino beam, and (2) is surrounded by material that will produce particle showers from neutrino interactions. These requirements lead to a bounding set of effective dose predictions.

These assumptions are too conservative for the TeV energies that will be encountered in mature muon colliders, but they provide a bounding neutrino effective dose result given the current level of design. Basic physics principles suggest that the neutrino interactions will be 96 Particle Physics

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

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

Neutrinos are produced when the muon beam particles decay (See Eqs. 1 and 2). Weak interactions of muon neutrinos can be described in terms of two broad categories: charged current and weak current interactions. Charged current interactions involve the exchange of W-bosons to form secondary muons. Neutral current interactions produce uncharged particles through the exchange of Z-bosons. Both types of interactions produce hadron particle showers. Therefore, the neutrino induced radiation hazard will include secondary muons and hadronic showers. The hadronic showers have a much shorter range than the muons, but the number of particles in a hadronic shower can be quite large. The neutrino radiation hazard arises from these penetrating charged particle showers (Bevelacqua, 2008). For TeV energy neutrinos, direct neutrino interactions in man account for less than 1% of the total effective dose because the primary hadrons from the neutrino interactions will typically exit the person before producing a charged particle shower (King, 1999b; Cossairt et al., 1996, 1997). Most of the neutrino effective dose is derived from particle showers

The muon beam and subsequent neutrino beam are assumed to be well-collimated and to have a minimum divergence angle. For practical situations, the muons in the accelerator beam will have a small divergence angle and will be periodically focused using electromagnetic fields to ensure their collimation. No beam divergence is assumed in the subsequent calculations. Therefore, the actual beam will be somewhat more diffuse than assumed in the neutrino effective dose calculations. The neutrino beam will still produce particle showers, but they will be somewhat broader and less intense than the assumed well-collimated result. The beam divergence is analogous to the divergence of a laser beam

The magnitude of the effective dose from a particle shower is dependent on the material in the interaction region lying directly upstream of the individual being irradiated. Calculation of the neutrino effective dose considers the configuration where a person is (1) completely bathed in the neutrino beam, and (2) is surrounded by material that will produce particle showers from neutrino interactions. These requirements lead to a bounding set of effective

These assumptions are too conservative for the TeV energies that will be encountered in mature muon colliders, but they provide a bounding neutrino effective dose result given the current level of design. Basic physics principles suggest that the neutrino interactions will be

beam. Process D is also independent of the neutrino generation.

**5. Neutrino beam characteristics at a muon collider** 

produced in the shielding material.

dose predictions.

as it exits an aperture (Bevelacqua, 2009, 2020).

significant impact on the evaluation and control of neutrino effective dose.

more peaked in the beam direction as the muon energies increase. In addition, the neutrino beam radius (r) will be relatively small and is given by (King, 1999b):

$$\mathbf{r} = \boldsymbol{\theta} \text{ L} \tag{3}$$

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

$$
\theta = \frac{m \ c^2}{E} \tag{4}
$$

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 person.

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

$$\theta \approx \frac{10^{-4}}{E[TeV]} \tag{5}$$

Therefore, the emergent neutrino beam will consist of a narrow diverging beam that is conical in shape.

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 proposed muon colliders.


Table 1. Straw-Man Muon Collider Parameters.

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.

operational characteristics (e.g., accelerator gradient or the increase in muon energy per unit accelerator length). The operational parameter approach is more familiar to high-energy physicists, but it serves to illustrate the sensitivity of the neutrino effective dose to the key

The second configuration is a circular muon collider. The neutrino effective dose for the circular muon collider involves an integral over energy of the differential fluence and fluence to dose conversion factor. This approach is more familiar to health physicists, but much of the muon collider's operating parameters are absorbed into other parameters and are not explicitly apparent. Using both approaches yields not only the desired neutrino effective dose, but also illustrates the sensitivity of the effective dose to a number of

The bounding neutrino effective dose from a linear muon collider is derived following King (1999b) and is based on the effective dose from a straight section (ss) of a circular muon collider. This derivation incorporates a limiting condition from a circular accelerator with a number of straight sections as part of the facility. Parameters unique to the circular collider such as the ring circumference and straight section length appear in intermediate equations, but cancel in the final effective dose result. In the linear muon collider, the muon beam is

In a linear muon collider, the total neutrino effective dose (H) is defined in terms of an effective dose contribution δH(E) received in each energy interval E to E + dE as the muons

( )

(6)

*<sup>g</sup>* (8)

(9)

(7)

0

*df E* 1 *dE c* 

 <sup>1</sup> *ss df E HE H*

*f dE*

*dE* is the fraction of muons that decay via Eqs. 1 and 2 in the energy interval E

2 *Eo mc*

*Eo H dE H E* 

muon collider's operating parameters.

assumed to be well-collimated.

accelerate to the beam energy Eo:

to E + dE, which may be written as:

where *df E*( ) *dE*

where

accelerator parameters and operational assumptions.

**7.1 Bounding neutrino effective dose – linear muon collider** 

The effective dose contribution δH(E) is written as (King, 1999b):
