\section{NEUTRINO RADIATION}
It has been shown\cite{ref41} that the surface radiation dose $D_B(Sv)$
at a time $t(s)$, in the plane of a bending magnet of field B(T), in
a circular collider with beam energy $E(TeV)$, average bending field
$**$, at a depth $d(m)$ (assuming a spherical earth), with muon
current (of each sign) of $I\textrm{(muons/sec/sign)}$ is given by:
$$
D_B\ \approx\ 4.4\ 10^{-24}\ {I_\mu \ E^3\ t \over d}\ {**** \over B}\ t
$$
and that the dose $D_S$ at a location on the surface, in line with a high
beta straight section of length $\ell$ is:
$$
D_S\ \approx\ 6.7\ 10^{-24}\ {I_\mu \ E^3\ t \over d}\ {\ell ****}\ t
$$
The first formula has been confirmed by a Monte Carlo simulation using
the MARS code\cite{mokhovrad}.
In all cases it is assumed that the average divergence angles satisfy the
condition: $\sigma_\theta << {1\over \gamma}.$
This condition is not satisfied in the straight sections approaching the IP,
and these regions, despite their length, do not contribute a significant dose.
For the 3 TeV parameters given in Tb.~\ref{sum} (muon currents
$I=6\times 10^{20}\,\mu^-/yr,$ $****= 6\, T,$ $B= 10\, T,$ and taking the federal limit on off site
radiation Dose/year, $D_{Fed}$ to be 1 mSv/year (100 mrem/year), then the
dose $D_B$ per year (defined as $10^7$ s), in the plane of a bending dipole
is:
$$
D_B = 1.07 \ 10^{-5}\ \ (\textrm{Sv})\ \approx 1 \% \ D_{Fed}
$$
and for a straight section of length 0.6 m is:
$$
D_B = 9.7 \ 10^{-5}\ \ (\textrm{Sv})\ \approx 10 \% \ D_{Fed}
$$
which may be taken to be a reasonable limit.
Special care will be required in the lattice design to assure that no
field free region longer than this is present. But it may be noted that the
presence of a field of even 1 T over any length, is enough to reduce the dose to the 10
\% Federal limit standard. For machines above 3 TeV, the muon current would probably have to be reduced.
For lower energy machines, the requirements get rapidly easier: a 0.5
TeV machine at 500 m depth could have 130 m straight sections, or if at 100 m depth
25 m lengths, for the same surface dose. For a 100 GeV machine the doses are
negligible.
**