**7. Neutrino effective dose**

A muon collider provides a platform for colliding beams of muons ( ) and antimuons ( ) (Geer, 2010). The collider may involve a pair of linear accelerators with intersecting beams or a storage ring that circulates the muons and antimuons in opposite directions prior to colliding the two beams. The accelerator facility energy is usually expressed as the sum of the muon and antimuon energies. For example, a 100 TeV accelerator consists of a 50 TeV muon beam and a 50 TeV antimuon beam. Since muon colliders produce large muon currents, neutrinos will be copiously produced from the decay of both muons and antimuons (See Eqs. 1 and 2).

Neutrino effective dose calculations are performed for two potential muon collider configurations. The first configuration utilizes the intersection of the beams of two muon linear colliders. The linear collider effective dose model incorporates an explicit representation of the neutrino cross section and evaluates the effective dose assuming specific values for the muon energy, number of muon decays per year, and accelerator 98 Particle Physics

Neutrinos can interact directly with tissue or with intervening matter to produce charged particles that result in a biological detriment. The radiation environment is complex and simulations (e.g., Monte Carlo methods) can be used to model the dynamics of the neutrino

simulation is too dependent on specific accelerator characteristics and will not add to the health physics presentation. Rather than performing a Monte Carlo simulation, we follow the analytical approach of Cossairt et al. (1997) and King (1999b) to quantify the neutrino effective dose. This approach is acceptable in view of the current uncertainties in muon collider technology and the nature of the neutrino interaction for both charged current (CC)

Following King (1999c), the dominant interaction of TeV-scale neutrinos is deep inelastic scattering with nucleons that include CC and NC components. In the NC process, the neutrino is scattered by a nucleon (N) and loses energy with the production of hadrons (X)

cross section. This NC process can be interpreted as elastic scattering off one of the quarks

CC scattering is similar to NC scattering except that the neutrino is converted into its

 *NlX* where l is an electron/muon for electron/muon neutrinos. At the quark level, a charged W boson is exchanged with a quark to produce another quark (q′) whose charge

The final state quarks produce hadrons on a nuclear distance scale that contribute to the effective dose. The CC and NC scattering processes are included in the Process A –D

 ) (Geer, 2010). The collider may involve a pair of linear accelerators with intersecting beams or a storage ring that circulates the muons and antimuons in opposite directions prior to colliding the two beams. The accelerator facility energy is usually expressed as the sum of the muon and antimuon energies. For example, a 100 TeV accelerator consists of a 50 TeV muon beam and a 50 TeV antimuon beam. Since muon colliders produce large muon currents, neutrinos will be copiously produced from the decay of both muons and

Neutrino effective dose calculations are performed for two potential muon collider configurations. The first configuration utilizes the intersection of the beams of two muon linear colliders. The linear collider effective dose model incorporates an explicit representation of the neutrino cross section and evaluates the effective dose assuming specific values for the muon energy, number of muon decays per year, and accelerator

, e, μ, τ, and hadrons) involved in the interaction. Performing a neutrino

*N X* reaction. This NC reaction contributes about 25 percent of the total

*q l q* and

 *q q* ).

*q l q*.

*NlX* and

) and antimuons

 *<sup>e</sup>* , *<sup>e</sup>* , ,

interaction including the energy and angular dependence of each particle (e.g.,

and neutral current (NC) weak processes (King, 1999c).

(q) inside the nucleon through the exchange of a virtual Z0 boson (

corresponding charged lepton (l). This includes reactions such as

A muon collider provides a platform for colliding beams of muons (

**6. Neutrino interaction model** 

 , , 

through a

(   

differs by one unit through processes such as

descriptions noted in previous discussion.

**7. Neutrino effective dose** 

antimuons (See Eqs. 1 and 2).

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 muon collider's operating parameters.

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 accelerator parameters and operational assumptions.

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

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 assumed to be well-collimated.

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 accelerate to the beam energy Eo:

$$H = \bigcap\_{0}^{E\_{\overline{v}}} dE \,\delta H(E) \tag{6}$$

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

$$
\delta H(E) = H' \frac{1}{f\_{ss}} \frac{df(E)}{dE} \tag{7}
$$

where *df E*( ) *dE dE* is the fraction of muons that decay via Eqs. 1 and 2 in the energy interval E to E + dE, which may be written as:

$$\frac{df\left(E\right)}{dE} = \frac{1}{\gamma \,\beta \, c \, \tau \, g} \tag{8}$$

where

$$\gamma = \frac{E\_o}{mc^2} \tag{9}$$

2 expressions (1.453, 1.323, 1.029, 0.512, and 0.175) are the total summed neutrino-nucleon and antineutrino-nucleon cross sections divided by energy at neutrino energies of 0.1, 1, 10, 100, and 1000 TeV, respectively, given in units of 10-38 cm2/GeV. As an approximation, the muon energies in Table 2 are set equal to the corresponding neutrino energies. Following Quigg (1997), the cross section factor is a dimensionless number and is normalized such that

Eq. 13 may be approximated by replacing the energy-weighted integral of X(E) by its value at E = Eo /2. This choice is acceptable given the energy dependence of the cross section and the associated uncertainties in the collider design parameters. With this selection, the annual

> <sup>2</sup> / 2 <sup>2</sup> *o o K N H EE g*

> > Muon Beam Energy (TeV)

0.1 0.05 5.7x10-5 1 0.5 5.2x10-3 10 5 0.45 100 50 30 500 250 440 1,000 500 1.4x103 5,000 2,500 1.5x104 10,000 5,000 4.2x104 50,000 25,000 4.8x105

Table 3. Annual Neutrino Effective Doses for a Linear Muon Collider Using the Narrow

As a practical example (Zimmerman, 1999), consider a 1,000 TeV muon linear accelerator assuming Eo = 500 TeV (i.e., two, 500 TeV linear muon accelerators) and N = 6.4 x 1018 muon decays per year. Using these values in Eq. 14 with a g = 1 GeV/m value leads to an annual effective neutrino dose of 1.4 Sv/y, which is a significant value that cannot be ignored. Health physicists at a linear muon collider will need to contend with large neutrino effective doses within and outside the facility. Table 3 provides expected annual neutrino effective doses for a variety of accelerator energies using the same N and g values noted above and

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H (mSv/y)

E < 1 (-1.453 α + 1.323 (α + 1)) / 1.453 1 < E < 10 (1.323 (1- α) + 1.029 α) / 1.453 10 < E < 100 (1.029 (2- α) + 0.512 (α-1)) / 1.453 100 < E < 1,000 (0.512 (3- α) + 0.175 (α-2)) / 1.453 E > 1,000 (0.175/1.453) 33- <sup>α</sup>

Muon Energy Range (TeV) X(E)

α = log10(E) where E is the muon energy expressed in TeV.

neutrino effective dose (mSv/y) becomes:

the narrow beam approximation.

Beam Approximation.

Table 2. Cross Section Factor X(E) as a Function of Muon Energy.

Accelerator Facility Energy (TeV)

X(E = 0.1 TeV) = 1.0.

In Eq. 8, β = v / c, τ is the muon mean lifetime (2.2 x 10-6 s), and g is the accelerator gradient (dE/dl). The other parameters appearing in Eq. 7 include fss (the ratio of the straight section length to the ring circumference) and H΄ (the effective dose that is applicable as the muon energy reaches the TeV energy range), where

$$f\_{ss} = \frac{l\_{ss}}{C} \tag{10}$$

In Eq. 10, C is the ring circumference:

$$C = \frac{2\,\pi\,E\_o}{0.3\,\overline{B}}\tag{11}$$

In Eqs. 9 – 11, v is the muon velocity, lss is the straight section length, Eo is the muon energy, *B* is the ring's average magnetic induction, and N is the number of muon decays in a year.

In the narrow beam approximation, the effective dose is independent of distance (L) for L < 5 Eo (King, 1999b) where L is expressed in km and Eo in TeV. Using this approximation,

$$H' = K' \ N \ l\_{ss} \ \overline{B} \to X \tag{12}$$

where K' is a constant that depends on the units used to express the various quantities appearing in Eq. 12, and X = X(E ) is the cross section factor defined in subsequent discussion.

Combining these results leads to the annual neutrino effective dose (H) in mSv/y:

$$H = \frac{NK}{\mathcal{S}} \int\_0^{\mathcal{E}} EX(E) dE \tag{13}$$

where K = 6.7 x 10-21 mSv-GeV /m-TeV2 if g is expressed in GeV/m, N is expressed in muon decays per year, E is the muon energy in TeV, and the cross section factor is dimensionless (Bevelacqua, 2004).

In deriving the linear muon collider effective dose relationship, a number of assumptions were made (Bevelacqua, 2004). These assumptions are explicitly listed to ensure the reader clearly understands the basis for Eq. 13. The relevant assumptions include applicability of the narrow beam approximation. The individual receiving the effective dose is assumed to be: (1) uniformly irradiated, (2) within the footprint of the neutrino beam, (3) within the footprint of the hadronic particle shower that results from the neutrino interactions, and (4) irradiated by only one of the linear muon accelerators whose energy is one-half the total linear muon collider energy. Given the TeV muon energies and the earth shielding present, charged particle equilibrium exists and Process D dominates the neutrino effective dose. In addition, the muon beam is well-collimated, the neutrino effective dose calculation assumes a 100% occupancy factor, and the neutrino effective dose is an annual average based on the number of muon decays in a year.

The cross section factor is a parameterization of the neutrino cross section (See Table 2) in terms of a logarithmic energy interpolation (Quigg, 1997). The numerical factors in the Table 100 Particle Physics

In Eq. 8, β = v / c, τ is the muon mean lifetime (2.2 x 10-6 s), and g is the accelerator gradient (dE/dl). The other parameters appearing in Eq. 7 include fss (the ratio of the straight section length to the ring circumference) and H΄ (the effective dose that is applicable as the muon

*ss ss*

2 0.3 *Eo <sup>C</sup> B* 

In Eqs. 9 – 11, v is the muon velocity, lss is the straight section length, Eo is the muon energy, *B* is the ring's average magnetic induction, and N is the number of muon decays in a year. In the narrow beam approximation, the effective dose is independent of distance (L) for L < 5 Eo (King, 1999b) where L is expressed in km and Eo in TeV. Using this approximation,

where K' is a constant that depends on the units used to express the various quantities appearing in Eq. 12, and X = X(E ) is the cross section factor defined in subsequent

0

where K = 6.7 x 10-21 mSv-GeV /m-TeV2 if g is expressed in GeV/m, N is expressed in muon decays per year, E is the muon energy in TeV, and the cross section factor is dimensionless

In deriving the linear muon collider effective dose relationship, a number of assumptions were made (Bevelacqua, 2004). These assumptions are explicitly listed to ensure the reader clearly understands the basis for Eq. 13. The relevant assumptions include applicability of the narrow beam approximation. The individual receiving the effective dose is assumed to be: (1) uniformly irradiated, (2) within the footprint of the neutrino beam, (3) within the footprint of the hadronic particle shower that results from the neutrino interactions, and (4) irradiated by only one of the linear muon accelerators whose energy is one-half the total linear muon collider energy. Given the TeV muon energies and the earth shielding present, charged particle equilibrium exists and Process D dominates the neutrino effective dose. In addition, the muon beam is well-collimated, the neutrino effective dose calculation assumes a 100% occupancy factor, and the neutrino effective dose is an annual average based on the

The cross section factor is a parameterization of the neutrino cross section (See Table 2) in terms of a logarithmic energy interpolation (Quigg, 1997). The numerical factors in the Table

*Eo N K H EX E dE*

( )

Combining these results leads to the annual neutrino effective dose (H) in mSv/y:

*<sup>l</sup> <sup>f</sup> <sup>C</sup>* (10)

(11)

*H K N l BE ss* (12)

*<sup>g</sup>* (13)

energy reaches the TeV energy range), where

In Eq. 10, C is the ring circumference:

discussion.

(Bevelacqua, 2004).

number of muon decays in a year.

2 expressions (1.453, 1.323, 1.029, 0.512, and 0.175) are the total summed neutrino-nucleon and antineutrino-nucleon cross sections divided by energy at neutrino energies of 0.1, 1, 10, 100, and 1000 TeV, respectively, given in units of 10-38 cm2/GeV. As an approximation, the muon energies in Table 2 are set equal to the corresponding neutrino energies. Following Quigg (1997), the cross section factor is a dimensionless number and is normalized such that X(E = 0.1 TeV) = 1.0.


Table 2. Cross Section Factor X(E) as a Function of Muon Energy.

Eq. 13 may be approximated by replacing the energy-weighted integral of X(E) by its value at E = Eo /2. This choice is acceptable given the energy dependence of the cross section and the associated uncertainties in the collider design parameters. With this selection, the annual neutrino effective dose (mSv/y) becomes:

$$H = \frac{KN}{2g} \mathbf{X} \{ E\_o / \mathcal{D} \} \, E\_o^2 \tag{14}$$

As a practical example (Zimmerman, 1999), consider a 1,000 TeV muon linear accelerator assuming Eo = 500 TeV (i.e., two, 500 TeV linear muon accelerators) and N = 6.4 x 1018 muon decays per year. Using these values in Eq. 14 with a g = 1 GeV/m value leads to an annual effective neutrino dose of 1.4 Sv/y, which is a significant value that cannot be ignored. Health physicists at a linear muon collider will need to contend with large neutrino effective doses within and outside the facility. Table 3 provides expected annual neutrino effective doses for a variety of accelerator energies using the same N and g values noted above and the narrow beam approximation.


Table 3. Annual Neutrino Effective Doses for a Linear Muon Collider Using the Narrow Beam Approximation.

all secondary radiation. Accordingly, the neutrino radiation is in equilibrium with its

Using the equilibrium condition and averaging over all production angles, provides the following differential fluence relationship for the neutrino radiation from a circular muon

() 2 <sup>1</sup>

number of neutrinos per unit area) following the muon decays.

relationship of Cossairt et al. (1997) is used:

consider the effects of the third lepton generation.

neutrino attenuation length is briefly examined.

neutrino energy is expressed in TeV.

*dN E E dE E E*

*o o*

where N(E) is the number of neutrinos per unit area, E is the neutrino energy, Eo is the energy of the primary muons before decay, and Φ is the integral neutrino fluence (total

For secondary particle equilibrium, the fluence to effective dose conversion factor

Eq. 17 was derived for the neutrino energy range of 0.5 GeV to 10 TeV. In deriving the muon neutrino effective dose to fluence conversion factor of Eq. 17, Cossairt et al. (1997) did not

In Eq. 17, K = 10-15 μSv-cm2/GeV2. In view of the trend in the neutrino data (Particle Data Group, 2010; Quigg, 1997), Eq. 17 is used at energies beyond those considered by Cossairt et al. (1997). This is reasonable because increasing energy and increasing number of secondary shower particles (hadrons) is the main reason for the rising fluence to effective dose conversion factor with increasing neutrino energy for the equilibrium (shielded neutrino) case or process D described earlier. It is also reasonable because the neutrino attenuation length (λ) decreases with increasing energy of the primary neutrinos. Although TeV energy units are used in the final result, GeV units are used in the derivation of the neutrino effective dose to facilitate comparison with Silari & Vincke (2002) and Johnson et al. (1998). Prior to developing the neutrino effective dose relationship for a circular muon collider, the

The neutrino attenuation length is written in terms of the neutrino interaction cross section σν:

*A A N N* 

These results permit the neutrino attenuation length to be written as (Johnson et al. ,1998):

<sup>6</sup> 1 3/ 0.5 10 *TeV g cm x km E*

where A and ρ are the atomic number and density of the shielding medium, NA is Avogadro's number, N is the number density of atoms of the shielding medium per unit volume, and σν is on the order of 10-35 cm2 (E / 1 TeV ) (Johnson et al. ,1998) where the

1

 

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3

<sup>2</sup> *C E KE* ( ) (17)

 

(16)

(19)

secondary radiation.

collider (Silari & Vincke, 2002):

The values of Table 3 suggest that the annual effective dose limit for occupational exposures of 20 mSv/y and the annual effective dose limit to the public (1 mSv/y) can be exceeded by TeV energy muon accelerators (ICRP 103, 2007). The values in Table 3 also exceed the emergency effective dose limit of 250 mSv set for the Fukushima Daiichi accident that is based on ICRP 60 (1991).

A TeV - PeV scale muon collider will also challenge the acute lethal radiation dose (LD50, 30) of about 4 Gy (Bevelacqua 2010a). Although the feasibility of TeV - PeV scale machines remains to be determined, the significant radiation hazards associated with their operation merits careful attention to the effects of neutrino effective doses at offsite locations.

Selecting an accelerator location will be an issue for TeV energy muon linear colliders due to public radiation concerns arising from neutrino interactions. Given these radiation concerns, a muon collider location may be restricted to low population or geographically isolated areas to minimize the public neutrino effective dose.
