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% Modification History:
% Version Author Date Description of modification,deficiencies
% 0.1 BCBrown 09-Jan-2018 Initial entry.
% 0.11 BCBrown 11-Jan-2018 Add Figure for Toy Model Geometry
% 0.13 BCBrown 11-Jan-2018 Edit changes from 1/11/18 Mtg.
% 0.14 BCBrown 12-Jan-2018 Text about Mg, new references
% 0.15 BCBrown 19-Jan-2018 Built-in Concrete, Bateman Eq, Linearity
% 0.16 BCBrown 23-Jan-2018 Appendix A - MARS input files
% Details of Model Geometry in text
% 0.20 BCBrown 26-Dec-2018 Appendix B - Steel MARS Run plus misc.
% 0.22 BCBrown 11-Jan-2019 Add section on Mn-56 production...Plus
% 0.24 BCBrown 23-Jan-2019 review text
% 0.25 BCBrown 25-Jan-2019 Add Text on production and equilibrium
% 0.26 BCBrown 29-Jan-2019 Edits incl. Fix EN vs SAE table
% 0.40 BCBrown 15-Feb-2019 Begin Rewrite per Nikolai
% 0.41 BCBrown 20-Feb-2019 Continue, Fe isotope table
% 0.42 BCBrown 26-Feb-2019 Continue, material density table
% 0.43 BCBrown 15-Mar-2019 Return to formulas
% 0.45 BCBrown 20-Mar-2019 Perfecting formulas
% 0.50 BCBrown 04-Apr-2019 Restart on Text, Add Ca/Ca-40 Appen
% 0.51 BCBrown 11-Apr-2019 Review Ca/Ca-40 Results w/ comments
% 0.52 BCBrown 11-Apr-2019 Revisions after review
% 0.53 BCBrown 25-Apr-2019 More results from concrete run
% Put Material and Nuclear Tables in Appen
% 0.54 BCBrown 26-Apr-2019 Complete appendix on Concrete Activation
% 0.55 BCBrown 09-May-2019 Results section on summed activation
% Compare w/ Measurements, Start Conclusion
% 0.56 BCBrown 25-Jul-2019 Add Results for Cu MARS and Meas
% 0.57 BCBrown 26-Jul-2019 Review and repair notes from v0.55
% Will incorporate Vitaly Pronskikh
% Text and Graphics from 20190510
% 0.58 BCBrown 30-Jul-2019 Review and repair notes from v0.57
% Add logarithmic graph for concrete
% 0.60 BCBrown 31-Jul-2019 Additions to Conclusions, SS section...
% 0.61 BCBrown 08-Aug-2019 Corrections....
% 1.00 BCBrown 09-Aug-2019 For Beams Document Database
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\title{Radiation Studies for Main Injector Activation Using
MARS in a `Toy Model'}
\author{
Phil Adamson, Bruce C. Brown, Nikolai Mokhov, and Vitaly Pronskikh\\ Accelerator Division\\
{\em Fermi National Accelerator Laboratory }
\thanks{Operated by Fermi Research Alliance
under contract with the U. S. Department of Energy}
\\ \em P.O. Box 500 \\ \em Batavia, Illinois \\
}
\begin{document}
\bibliographystyle{unsrt}
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\newpage
\tableofcontents
\newpage
\begin{abstract}
MARS~\cite{MARS15-1995} \cite{MARS15-2014} \cite{MARS15-WEB}
\cite{Mokhov:2017klc} studies of energy deposition and activation in
the Main Injector tunnel are supplemented with studies with a
simplified geometry in which the loss of 8 GeV protons are modeled
in a cylindrical geometry. We use this to provide a more
computer-efficient source for understanding. We build this model to
compare with the Main Injector 20-Ton collimators and activation
studies of materials placed at the C307 Collimator. Using these
studies, we confirm basic activation issues for the Main Injector
collimator region while exploring issues with various elements in
order to determine which minor elements or isotopes may be important
while removing concern about materials for which detailed
specification can now be shown to not be of interest. We will
explore the activation of steel as measured in special studies,
examine activation of minor components of steel and document MARS
results for the activation of various compositions of concrete and
stainless steel. Results will be compared with some measurements
and implications for long term tunnel activation issues will be
examined.
\end{abstract}
%%\newpage
\section{Introduction}
\pgph The MARS model which was used for studies to gain approval for
the Main Injector Collimators~\cite{Rakhno:2007zz} is compute
intensive. While we update it for further analysis, we have a more
computationally efficient model (Toy Model) of the secondary
collimators which matches it well enough to allow us to gain insight
into the various physics issues with much better turn-around time on
the computers. With this model, we have explored several issues which
inform concerns to pursue further.
\pgph The study of activation of Steel and Copper in
Beams-doc-4046~\cite{BeamsDoc4046} employed `tags' of Main Injector
Steel, Copper, and Aluminum with an initial goal of discovering what
isotopes and thereby what half lifes would be of interest for
radiation monitoring in the tunnel. We had already
shown~\cite{Brown:TUO2C04} that the residual radiation levels in the
tunnel could be related to the loss monitor values at nearby loss
monitors by assuming a small number of isotope half lives. The tags
were in locations on the side (Shielded) of the collimator and on the
downstream end above the beam pipe (Unshielded) at a small angle from
the interacting beam.
\pgph A few surprises such as the significant activation of Antimony
(Sb) caused us to continue the MARS efforts in order to better
understand what was observed. Meanwhile we also want to explore any
long term issues which we might have failed to notice by predicting
any long half life isotopes and evaluating their impact on tunnel
activities. With this in mind, we will use this model to explore the
MARS predictions for the activation studies, explore expected
activation issues in the tunnel walls (concrete), beam pipe (316L
Stainless Steel and more recently some 2205 Stainless) and other
issues which will allow understanding of tunnel activation.
\pgph We remark that the results in Beams-doc-4046 were presented as
activation rates which were corrected to provide the equilibrium
activation to be achieved at a specified loss rate. We will provide
such rates from these studies. In getting these equilibrium rates we
will choose to first correct our results to a produced activation for
a given total loss as if it happened in a time short compared to the
half life. These are simply related by the half life (or life time)
but we call this out to reduce confusion.
\pgph We will introduce various concerns and describe their
significance for the radiation issues created by loss of 8 GeV
protons. We will then provide detailed results. Some concerns will
be addressed separately in appendices where we will also separate out
some of the studies.
\subsection{Elements in Concrete}
\label{Section:ConcreteStudy}
\pgph In review of the MI tunnel, one can consult Chapter 4 of the
Main Injector Technical Design Handbook\cite{Beams-doc-4294-v1}. The
concrete in the Main Injector tunnel arrived in three parts. The
floor was poured over `undisturbed' glacial till (clay). The
stairways and nearby features were formed and poured. The bulk of the
walls and ceilings were created with precast reinforced concrete.
Each of these are likely to have come from separate sources of
materials. If we describe concrete as a mixture of Portland cement,
sand, and aggregate, we can expect the aggregate to potentially be
different for the three tunnel construction types. We explored the
content using an x-ray fluorescence spectrometer but the paint on the
tunnel wall ($TiO_2$) limited the signals available on most walls. We
did get spectra from the floor and a few places on the walls and
conclude that detailed chemical results are likely to be difficult to
obtain with high reliability. We choose a different strategy. MARS
has a standard concrete description. We will using MARS to determine
whether likely differences from that description will impact the
expected activation patterns.
\pgph The native rock available for creating aggregate in Illinois is
mostly limestone (calcium carbonate - $CaCO_3$) and dolomite
($CaMg(CO_3)_2$) or mixtures thereof. Ca experiences very low
activation. We will use this Toy Model to explore what we might
expect for tunnel activation of Mg with many years of exposure for
tunnel walls near the Main Injector collimators. The MARS Build-in
ordinary concrete (Table~\ref{table:ConcreteComposition}) results will
be supplemented with Mg samples in the toy model. The fluorescence
spectroscopy has limited sensitivity to Mg so a different chemical
analysis would be required if this is important.
\pgph A result from the fluorescence spectroscopy is that several of
the measured spectra show Zirconium (Zr). After some discussion, we
conclude that the observed level (somewhat different in spectra from
different tunnel locations) is consistent with the natural abundance
of Zr (1.3 $\times 10^{-4}$). Since it is a heavier nucleus than most
of the materials in the tunnel, we will ask MARS whether it is likely
to be activated to a degree which is of interest.
\subsection{Minor Elements and Isotopes in Iron}
\pgph The activation study reported in
Beams-doc-4046~\cite{BeamsDoc4046} reported observation of Fe-59,
Sb-121, and Sb-123. Initial discussions of these results suggested
that they are due to neutron capture on Fe-58, Sb-120, and Sb-122.
The reported activation is not entirely negligible in the iron sample
measurements so we have exerted some effort to understand these
results. To better understand, we will also explore other steel
compositions and explore what the impact on activation might be.
\subsection{Elements in Various Stainless Steels}
\pgph The 304 stainless steel at the heart of the Main Injector and
Recycler collimators is similar to the 316L Stainless in the beam
pipes in the tunnel but its activation products are well shielded by
steel and marble so it has small impact for the radiation issues for
the collimation system with one exception. Some monitoring of the
residual radiation has been carried out by measuring at the downstream
end of the four collimators where the monitoring point is on the outer
surface of the stainless steel vacuum liner. Of additional interest
however, we have replaced some of the 316L beam pipe with 2205
(duplex) stainless steel beam pipe in high radiation areas of the Main
Injector. This alerted us to be interested in potential activation
issues with the many elements in our stainless steel. In particular,
the duplex stainless has a significantly higher Molybdenum (Mo)
content. Since it is a heavier nucleus, we will explore its
activation properties. For completeness, we will also document those
properties for other common elements in stainless steel including Cr,
Ni, and Mn. For P, S, and N we will assert that at the known
concentration, they do not activate for half life values of interest
to our studies.
\section{Formulas}
\pgph In examining the activation with a large range of half life
values, we will look at expectations for different time ranges. Let
us examine the formulas for production of activation. We will employ
notation similar to that in Beams-doc-4046~\cite{BeamsDoc4046}. In a
beam of particles, nuclear interactions produce new isotopes. The
number of new nuclei is proportional to the fluence, $\Phi$, measured
in particles per unit area (particles-cm$^{-2})$. In a material with
$n_S$ sample atoms, an interaction with cross section $\sigma_I$ will
produce $n_I$ atoms of isotope $I$ in the sample volume.
\begin{equation}
n_I = \Phi \, n_S \, \sigma_I.
\label{Eq:ProduceNI}
\end{equation}
Given a lifetime of $\tau_I$ or a half life of $t_{1/2} = \tau_I/\ln 2$,
the activity, $S$ (Bq), produced by $n_I$ atoms is
\begin{equation}
S (Bq) = \frac{n_I}{\tau_I} = \frac{n_I \ln 2 }{t_{1/2}}
= \frac{\Phi n_S \sigma_I}{\tau_I}
= \frac{\Phi n_S \sigma_I \ln 2 }{t_{1/2}}
\end{equation}
For these MARS calculations, we calculate the activity in Bq for the
sample and use the sample volume and density to convert to specific
activity while converting to pCi. For a volume, V, and density $\rho$
we will have $n_S/V$ target atoms per unit volume (cm$^3$) or $n_T =
n_s / \rho V$ atoms per gram. We will want the specific activity per
gram of target material
\begin{equation}
S_A (Bq/gm) = \frac{n_I }{V \rho_S \, \tau_I}
= \frac{n_I \ln 2 }{V \rho_S \, t_{1/2}}
= \frac{\Phi n_T \sigma_I \ln 2 }{V \rho_T \, t_{1/2}}
\end{equation}
Substituting for $n_T$ with $\rho_T N_A/A_T$ we have
\begin{equation}
S_A (Bq/gm) (Produced)= \frac{\Phi N_A\sigma_I}{ A_T \,\tau_I }
= \frac{\Phi N_A\sigma_I \ln 2 }{ A_T \, t_{1/2}}
\label{Eq:SpecificActivityBq_gm}
\end{equation}
which we will describe as the activity produced by fluence $\Phi$
\begin{equation}
S_A (pCi/gm)(Produced)
= \frac{\Phi N_A\sigma_I }{ A_T \,\tau_I\,3.7\times 10^{-2} }
= \frac{\Phi N_A\sigma_I \ln 2}{ A_T \, t_{1/2}\,3.7\times 10^{-2} }
\label{Eq:SpecificActivity}
\end{equation}
The particles (mostly hadrons) produced by $\Phi$ is proportional to
the number of interacting protons, p so let us describe it by $\Phi(p)
\times p = \frac{d\Phi}{dp} \times p $ with protons interacting at a
rate $\frac{dp}{dt}$ such that $\frac{d\Phi(t)}{dt} =
\frac{d\Phi}{dp}\frac{dp}{dt}$ or $\Phi = \frac{d\Phi(t)}{dp} \times
p$.
\begin{equation}
S_A (Produced) (Bq/gm)
= \frac{N_A\sigma_I}{ A_T \,\tau_I } \frac{d\Phi}{dp} p
\label{Eq:ActivationProducedBq}
\end{equation}
or
\begin{equation}
S_A (Produced) (pCi/gm)
= \frac{N_A\sigma_I}{ A_T \,\tau_I\,3.7\times 10^{-2} }\frac{d\Phi}{dp} p
\label{Eq:ActivationProducedpCi}
\end{equation}
Considering production for time $t_1$ with a uniform $\frac{dp}{dt}$
and cooldown for time $t_c$ we have the standard activation formula
(see Eq. 10 of Beams-doc-4046 or Eq. 3.9 of Barbier~\cite{Barbier:IR}
) reframed for $\Phi(p)$.
\begin{equation}
S_A(t_c)(Bq/gm)(observed)
= \frac{N_A \sigma_I}{A_T} \frac{d\Phi}{dp}\frac{dp}{dt}
(1 - e^{-t_i/\tau_I})e^{-t_c/\tau_I}
= S_A (Produced) \frac{\tau_I}{ p}\frac{dp}{dt}
(1 - e^{-t_i/\tau_I})e^{-t_c/\tau_I}
\label{Eq:ActivationEq}
\end{equation}
The number of interacting protons is $p = t_i \frac{dp}{dt}$ so
$(1/p)\frac{dp}{dt} = 1/t_i$.
For these MARS studies, the activation time, $t_i$, is 30 days and the
cooldown time, $t_c$, is 2 hours. For isotopes with lifetimes $t_i
\gg \tau_I$, the decays will match the production (equilibrium) and we
again consider Eq.~\ref{Eq:ActivationEq} but both decay corrections
are now zero if we consider a time while activation is ongoing (no
decay after activation). We find that
\begin{equation}
S_A(Bq/gm)(equilibrium)
= \frac{N_A \sigma_I}{A_T} \frac{d\Phi}{dp}\frac{dp}{dt}
= S_A (Produced) \frac{\tau_I}{ p}\frac{dp}{dt}
= S_A (Produced) \frac{\tau_I}{t_i}
\label{Eq:ActivationEqEquili}
\end{equation}
Using Eq.~\ref{Eq:ActivationProducedBq} for the Produced Activation,
Eq.~\ref{Eq:ActivationEq} for Observed Activation after uniform
exposure and a period of cooldown or Eq.~\ref{Eq:ActivationEqEquili}
for the Equilibrium activation for a given isotope I, we can relate
these quantities for a given MARS study. More generally, we will wish
to have these results normalized to a rate of delivery,
($ \frac{dp}{dt} $) or a total number of delivered protons, p.
\begin{equation}
S_A(\frac{(Bq/gm)}{(p/sec)})(equilibrium)
= \frac{N_A \sigma_I}{A_T} \frac{d\Phi}{dp}
= S_A (Produced) \frac{\tau_I}{ p}
\label{Eq:ActivationEqEquiliPerRate}
\end{equation}
where for this calculation we can use the $S_A (Produced)$ from our simulation
as corrected using Eq.~\ref{Eq:ActivationEq}. We note that the fluence,
$\Phi(p)$ is a property of the shielding configuration between the
proton interaction point and the sample. The MARS study will employ
as specified beam loss rate, $\frac{dp}{dt}$ which along with the
exposure time, $t_i$ and the cooldown time, $t_c$ are required to
understand the observed activation. Should we want the total produced
activation per proton (related by $\tau_I$ to the total number of produced
atoms, $n_I$), we use Eq.~\ref{Eq:ActivationEq}, correct
for decay during excitation and cooldown and divide by $\frac{\tau_I}{ p}
\frac{dp}{dt} = \frac{\tau_I}{t_i}$.
\pgph We can obtain the produced activation per proton as
\begin{equation}
S_A((Bq/gm)/p)(produced)= \frac{1}{\tau_I} S_A((Bq/gm)/(p/sec)(equilibrium)
\label{Eq:ActivationProducedRate}
\end{equation}
In review, we note that for equilibrium production for a given
isotope, the number of protons which contribute to the equilibrium
activation is lifetime $\times$ rate: $p = \tau_I \frac{dp}{dt}$.
\section{Toy Model Description}
\subsection{Toy Model Geometry}
\label{Sec:ToyModelGeometry}
\pgph
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{ToyModel.eps}
\caption{Geometry of Toy Model. Beam enters from the left to strike
the liner at the end of the tapered portion. MARS description is
in Appendix~\ref{Appen:MARSGeom} }
\label{Fig:ToyModelGeometry}
\end{figure}
\pgph In the Main Injector collimators, 8 GeV beam impinges on the
inside of a stainless steel collimation core. Our activation studies
(Beams-doc-4046) have been carried out on the outside of the marble
shielding at a large angle from the beam (shielded location) and at a
small angle from the beam on the front face of the collimator
(unshielded location). The Toy Model has been constructed to simulate
this configuration in an calculationally efficient manner.
\pgph The Toy Model collimator is a ~200 cm long cylinder with a 3.8
cm inner bore radius and a 63.5 cm outer radius. The first 36 cm of
the inner bore are tapered from 4.4 cm at the entrance to 3.8 cm. The
collimator main body in the model is made of the yoke steel (green),
with a stainless steel (blue) inner (1.9 cm thickness) and a marble
(yellow) outer layer (4.5 cm thickness). The pencil-like proton beam
of 1.25E12 p/s is parallel to the cylinder axis. It strikes the inner
surface of the bore at the end of the tapered part (30 cm from the
entrance to the collimator) on the bottom. The ``shielded'' samples
are 1 cm thick cylinders. They are placed onto the outer surface of
the marble layer at large angles ($52.5^o, 45^o, 39^o, and ~34.25^o$)
with respect to the proton beam. The ``unshielded'' ones - near the
exit at the downstream end of the collimator are 0.5 cm thick
cylinders stacked at small angles ($\approx50$ mRadians).
\subsection{Toy Model Execution}
\pgph For these studies, we employ runs with $1.25\times 10^{12}$
protons per second interacting at 36 cm into the collimator at the end
of the inner tapered portion of the toy model. Each calculation
describes a 30 day activation and a 2 hour cool down period. The
results use MARS to calculate the produced isotopes and the decay and
(potential) transmutation are calculated using DeTra which is called
through MARS. The output describes the activity which has been
produced and remains at the end of the 30 days plus 2 hours.
By using DeTra, the isotopes are ordered by the activation at the end
of the run. For a sample, the activity is calculated in becquerel
(Bq) which is decays per second in the sample. Isotopes are required
to have a half life $>$0.1 h and activities $>$ 1 Bq are reported.
The sample volume is noted. Using the sample density and the
conversion from becquerel (Bq) to curies (Ci) we examine results for
specific activity in pCi/gm\footnote{In some of the files of results,
the density used for this conversion is the density of Iron. The
correction for the actual density of the sample was done in the
spreadsheet for each sample.}.
\newpage
\subsection{Toy Model particle flux and dose distributions}
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{fnt-toy-1.eps}
\caption{Total neutron flux (0.001 eV $\le$ E$_n$ $\le$ 8 GeV) in the
Toy Model.}
\label{Fig:ToyModel-fnt-1}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{fpt-toy-2.eps}
\caption{Total proton flux (1 keV $\le$ E$_p$ $\le$ 8 GeV) in the Toy Model.}
\label{Fig:ToyModel-fpt-2}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{fgt-toy-3.eps}
\caption{Total photon flux (100 keV $\le$ E$_\gamma$ $\le$ 8 GeV)
in the Toy Model.}
\label{Fig:ToyModel-fgt-3}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{fle-toy-4.eps}
\caption{Total positron+electron flux (10 MeV $\le$ E$_{e^+e^-}$ $\le$ 8 GeV) in the Toy Model.}
\label{Fig:ToyModel-fle-4}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{dre-toy-5.eps}
\caption{Total residual dose on contact with the Toy Model after irradiation.}
\label{Fig:ToyModel-dre-5}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{fnt-toy-6.eps}
\caption{Total neutron flux (0.001 eV $\le$ E$_n$ $\le$ 8 GeV) in the Toy Model. XY view at Z=36 cm (beam interaction point).}
\label{Fig:ToyModel-fnt-6}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{fnt-toy-7.eps}
\caption{Total neutron flux (0.001 eV $\le$ E$_n$ $\le$ 8 GeV) in the Toy Model. XY view at Z=200 cm (downstream end of collimator).}
\label{Fig:ToyModel-fnt-7}
\end{figure}
\pgph Using elevation views of the Toy Model, we see the distribution
of neutrons (Fig.~\ref{Fig:ToyModel-fnt-1}), protons
(Fig.~\ref{Fig:ToyModel-fpt-2}), photons (Fig.~\ref{Fig:ToyModel-fgt-3}),
and positrons and electrons (Fig.~\ref{Fig:ToyModel-fle-4}). The
residual dose after a 1 day cooldown is shown in
Fig.~\ref{Fig:ToyModel-dre-5}. To gain insight into the angular
distributions, we see the neutrons in a cross section at the
interaction point (Fig.~\ref{Fig:ToyModel-fnt-6}) and a cm beyond the
downstream end of the collimator (Fig.~\ref{Fig:ToyModel-fnt-7}).
\subsection{Toy Model-2 unshielded particle spectra}
\pgph Toy Model-2 is the modification of Toy Model in which unshielded
samples are represented as four concentric nested rings, with the
thicknesses $R_{out}$ - $R_{in}$ = 2.5 cm. The innermost ring has
$R_{in}=5.7$ cm (radius of the inner bore), and the outermost has
$R_{out}=15.7$ cm. These samples are then at angles of 42.5, 57.8,
73.1 and 88.3 mRadians. The particle spectra in
Figures~\ref{Fig:ToyModel-neut-bcb}-\ref{Fig:ToyModel-phot-bcb} are
simulated for the particles crossing the upstream surfaces of each of
the rings.
\begin{figure}[bthp]
\centering
\includegraphics [height=0.6\textwidth, width=0.6\textwidth]{neut-bcb.eps}
\caption{Neutron spectra in Toy Model-2 unshielded sample locations. Black -- first (innermost), red -- second,
blue -- third, magenta -- fourth (outermost).}
\label{Fig:ToyModel-neut-bcb}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=0.6\textwidth, width=0.6\textwidth]{prot-bcb.eps}
\caption{Proton spectra in Toy Model-2 unshielded sample locations. Black -- first (innermost), red -- second,
blue -- third, magenta -- fourth (outermost).}
\label{Fig:ToyModel-prot-bcb}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=0.6\textwidth, width=0.6\textwidth]{phot-bcb.eps}
\caption{Photon spectra in Toy Model-2 unshielded sample locations. Black -- first (innermost), red -- second,
blue -- third, magenta -- fourth (outermost).}
\label{Fig:ToyModel-phot-bcb}
\end{figure}
\pgph The spectra in
Figures~\ref{Fig:ToyModel-neut-bcb}-\ref{Fig:ToyModel-phot-bcb}
indicate that while neutrons dominate at low energies, the fluxes of
protons and neutrons are comparable at 100 MeV (where inelastic
interactions and cascades dominate). However, at the latter energy,
proton fluxes in the innermost and outermost rings differ by a factor
of 5 (factor of 10 for photons). Therefore, angular differences in
production rates can be also significant.
\section{Analysis}
For uniform activation, the correction for decay during activation is
easily expressed. Similarly, the decay after excitation is also easy.
This has been described as the standard activation formula as shown in
Eq.~\ref{Eq:ActivationEq}. Dividing by the decay corrections, we
obtain the produced activation (Eq.~\ref{Eq:ActivationProducedBq}). At
this point we divide by the number of protons which interacted to
provide the activation per proton. Finally, we can multiply by the
lifetime to obtain the equilibrium activation per proton per second.
We discuss some results by comparing the equilibrium activation for
the MARS run in pCi/gm without dividing by the protons/sec.
\section{Results}
\label{Results}
\pgph This report is based on several MARS runs with various samples.
We choose to describe several of the studies in appendices where we
review results from one or a few MARS runs. In this section we will
describe some of the major topics for which we have conclusions.
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{Concrete_LogTimeDecay.eps}
\caption{Activation of Concrete shown with logarithmic cooling times.}
\label{Fig:ConcreteCooling}
\end{figure}
\pgph A typical MARS result allows the reader to examine the spectrum
on cooling in a single graph. We provide that for Concrete in
Figure~\ref{Fig:ConcreteCooling}
\subsection{Typical Activation Levels}
\label{Results:ActivationLevels}
\pgph When exploring any particular activation issue, the MARS toolset
provides many sorts of outputs including activation levels and cool
down curves. Our particular situation involving loss of 8 GeV protons
in the Main Injector Collimators will be further explored with
simulations of that system. This study is more general in that we
seek to gain knowledge of the sensitivity of our efforts to many
details for which we will decide how thoroughly we will explore to
determine the exact configuration of our devices. With that in mind,
we will summarize some of the overall activation properties of some
interesting materials. Using our 30 Day activation with 2 hour cool
down MARS studies, arbitrarily, we will provide the equilibrium
activation for the samples in pCi/gm for the specific activation rate
of our study. The results in pCi/gm/(p/sec) are available in the
analysis spreadsheets. While we occasionally want results for 2 hours
of cool down since we can hurry into the tunnel and make measurements
then, more typically we want results for days or more with potential
cool down up to 12 weeks (2016 hours or $7.26 \times 10^{6}$ seconds).
To explore personnel exposure, we need answers to about 20\% and will
attempt to arrive at 5\% accuracy. We could re-sort the MARS output
to provide the equilibrium activation properly ordered. Instead, we
will keep the list as sorted by activation after 30 days/2 hrs and sum
activation over about 99\% of the activation. Some unimportant
activation is calculated for long lifetimes. We choose to label half
lives longer than 20 years and declare that we will not expect to
reach equilibrium for those isotopes.
\begin{table}[tbhp]
\begin{center}
\caption{MARS Results on Summed Equilibrium Activation at $1.25\times 10^{12}$
protons interacting per second.}
\begin{tabular}{|l|c|c|c|}
\hline
Material &Isotopes & Isotopes &Activation \\
&$>$1 Bq & in sum &pCi/gm \\ \hline
\multicolumn{3}{c|}{Unshielded} \\ \hline
MI Steel &75 &33(inc $^3$H)&2.68E+07 \\ \hline
SS 316L &152 &49(inc $^3$H)&2.98E+07 \\ \hline
Concrete & 43 &33 &6.13E+06 \\ \hline
Copper (uns6)& 71 &53 &3.31E+08 \\ \hline
Magnesium &9 &9 &1.32E+07 \\ \hline
Calcium &27 &22 &3.83E+07 \\ \hline
Titanium & 28 &46 &1.35E+07 \\ \hline
Chromium &53 &49 &2.27E+07 \\ \hline
Manganese &56 &42 &1.82E+08 \\ \hline
Iron & 59 &46 &2.24E+07 \\ \hline
Nickel &68 &41 &2.79E+07 \\ \hline
Zirconium &151 &116 &9.90E+07 \\ \hline
Molybdenum &178 &61(inc $^3$H)&1.01E+08 \\ \hline
\multicolumn{3}{c|}{Shielded} \\ \hline
MI Steel &83 &31(inc $^3$H)&5.79E+06 \\ \hline
SS 316L &153 &48 (arb) &1.36E+07 \\ \hline
Concrete & 52 & 40 &7.36E+05 \\ \hline
Copper (shi10)&66 & 45 &5.66E+07 \\ \hline
Magnesium &9 &9 &1.96E+06 \\ \hline
Calcium &31 &20 &7.70E+05 \\ \hline
Titanium & 42 &28 &1.84E+06 \\ \hline
Chromium &49 &49 &4.17E+06 \\ \hline
Manganese &53 &40 &1.48E+08 \\ \hline
Iron &56 &44 &2.71E+06 \\ \hline
Nickel &62 &24 &2.76E+06 \\ \hline
Zirconium &116 &63 &3.32E+06 \\ \hline
Molybdenum &140 &50 &9.26E+07 \\ \hline
\end{tabular}
\label{table:summedEquilibrium}
\end{center}
\end{table}
\pgph In Table~\ref{table:summedEquilibrium} we have shown the
calculated equilibrium activation for our nominal loss rate. The
Unshielded location sees a higher energy and more intense flux than
the shielded location. The limit of 1 Bq in the sample is for the
sample volume at each location but the shielded samples have nearly
$\times$50 more volume. This accounts for the otherwise surprising
results that the total number of isotopes with $>$1 Bq is larger in
shielded samples for many of the lighter elements and materials.
Despite this volume difference the list of produced isotopes in
unshielded samples is longer for chromium and heavier elements. The
choice of isotopes to include in the activation sum was not precise.
The total activation is nearly unaffected but a careful sorting would
result in a different number of isotopes to be included. Frequently
isotopes with half lives shorter than 2 hours were included and rarely
ones with half life longer than 20 year were also added in but in
neither case were they added if a substantial change would be made in
the sum. If the reader is particularly interested in this aspect of
these results, the spreadsheet results may be of interest (or one can
re-sort the MARS outputs) since for several of the samples, one or two
isotopes may contribute a large fraction of the activation.
\pgph We also would note that the entire exercise is based on the
decay rate in Bq. For this reason, the radiological impact of some of
the most significant activation products by decay rate are not really
that important for energy deposited or biological impact. For
example, Fe-55 produces only keV gammas (or do you say x-rays) and
they are not readily detected by the Geiger counters we employ for
radiation monitoring.
\subsection{Manganese Activation}
\label{Results:Manganese}
\pgph Since the studies which correlated loss to activation
(``Measuring Correlations Between Beam Loss and Residual Radiation in
the Fermilab Main Injector'')~\cite{Brown:TUO2C04}, we have known that
Mn-56 (half life of 2.5789 hours) was an important contributor to the
radiation for early access to the accelerator tunnel. By examining
the activation of steel samples and pure Fe as reported in
Appendix~\ref{Appen:SteelStudy} we gain an appreciation for what is
involved. Two facts stand out when examining those results:
\begin{itemize}
\item The activation ratio for Mn-56 for unshielded to shielded is
small (0.783 - 1.292) for the steel samples whereas it is large
(8.77) for the Fe sample. This small ratio is not typical of the
products of spallation reactions. For the measurements in
Beams-doc-4046~\cite{BeamsDoc4046} the ratio of Unshielded/Shielded
as 4.545 which is also very low among the measured samples.
\item For Mn-56, the ratio of Fe/(steel) is very small for all three
steel materials for both the shielded and unshielded samples.
\end{itemize}
Together, these suggested that much of the Mn-56 activation was likely
to be due to neutron capture. Since each of these samples contain
some Mn which is nearly 100\% Mn-55, we will examine the analysis
results for consistency with that speculation.
\begin{table}[tbhp]
\begin{center}
\caption{MARS results on Mn-56 Production. MARS results are corrected
for decays to provide the produced activation of the target material.
Corrected column accounts for the weight fraction of Mn in the sample.}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
Target & Wt Frac of Mn & pCi/gm/p & pCi/gm/p \\ \hline
Material& & MARS & Corrected \\ \hline
Unshielded& & & \\ \hline
MI Steel& 0.0052 & 1.07E-10& 2.05E-08 \\ \hline
SS316 & 0.02 & 2.90E-10& 1.45E-08 \\ \hline
CAST Iron& 0.0018 & 4.20E-11& 2.34E-08 \\ \hline
Mn & 1.00 & 1.31E-08& 1.31E-08 \\ \hline
Shielded & & & \\ \hline
MI Steel & 0.0052 & 1.06E-10 &2.04E-08 \\ \hline
SS316 & 0.02 & 3.70E-10 &1.85E-08 \\ \hline
CAST Iron & 0.0018 & 3.26E-11 &1.81E-08 \\ \hline
Mn & 1.00 & 1.17E-08 &1.17E-08 \\ \hline
\end{tabular}
\label{table:Mn-55Production}
\end{center}
\end{table}
\pgph We note that the production of Mn-56 in iron is much smaller
but not negligible. MARS predicts that 2.36E-12 pCi/g/p is produced
in the shielded location while 2.30E-11 pCi/g/p is produced in the
unshielded location. The ratio of Unshielded/Shielded for Fe is
8.78 while the Fe/MI Steel is 0.0248 for shielded but 0.216 for
shielded. These are suggestive of a spallation reaction to produce
Mn-56 in Iron. It also is in the appropriate direction to explain
the difference in the unshielded results for CAST Iron.
\pgph When we explore the results of fitting Bar-coded residual
radiation measurements to half-life weighted loss monitor (BLM)
measurements, we had been concerned that the fraction attributed to
Mn-56 was more variable than we had expected. Now we conclude that
the different materials should have different Mn-56 production.
We successfully describe the residual radiation at about the 20\%
level by assuming that dominant isotopes other than Mn-56 are
Mn-54 and Mn-52. Their production from various materials appears
to be more 'ordinary.'
\subsection{Production of Fe-59 and Fe-55}
\pgph The observation of Fe-59 in the MI-Steel sample~\cite{BeamsDoc4046}
was assumed to be due to neutron capture on Fe-58. MARS upgrades
have allowed this to be explored in these studies. The measurements
failed to observe Fe-59 in the unshielded results (but this is
likely due to limited effort as this study was terminated). The
ratio of Unshielded/Shielded for Fe-59 is consistent with being
dominated by neutron capture (smaller ratios).
\pgph MARS predicts significant activation of Fe-55 but the resulting
5.9 keV x-rays have limited penetration and low energy such as to
produce little radio-logical impact. MARS results will show up in
spreadsheets but we will not discuss them further. The HPGe detector
used for the activation studies in Beams-doc-4046 does not provide a
useful measurement for such low energies.
\subsection{Compare MARS Toy Model Activation to Measurements for MI Steel}
\pgph The MARS simulations have provided guidance for comparisons of
the activation in the Main Injector Collimator region. The Toy Model
provides a similar geometry to understand important issues but we do
not expect a complete match to the measured results reported in
Beams-doc-4046-v3 \cite{BeamsDoc4046}. At this point we will show
that we are agreeing with the measurements at the level we might have
expected. Plots shown in
Figures~\ref{Fig:CompareMeastoMARS_Shielded}~and~\ref{Fig:CompareMeastoMARS_Unshielded}
show that various isotopes have similar rates for Measured and MARS
with Measured/MARS = 3.489 for the unshielded samples suggesting that
the Toy Model geometry is too long (162 cm in Model vs. 127 cm in 20-T
Collimators) between the interaction point and the samples whereas the
Measured/MARS = 0.553 for the Shielded samples would indicate that
there is not enough shielding in the model for those samples (due to
different longitudinal position (detection angle) and less marble).
Table~\ref{table:ActivationResults} allows the reader to examine this
information for many produced isotopes.
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{EQActivationShi.eps}
\caption{Equilibrium Activation of MI Steel at Shielded Location. Isotopes
are shown in order provided in Table~\ref{table:ActivationResults}.}
\label{Fig:CompareMeastoMARS_Shielded}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=3in]{EQActivationUns.eps}
\caption{Equilibrium Activation of MI Steel at Unshielded Location. Isotopes
are shown in order provided in Table~\ref{table:ActivationResults}. }
\label{Fig:CompareMeastoMARS_Unshielded}
\end{figure}
\begin{table}[p]
%\begin{table}[tbhp]
\begin{center}
\caption{Activation of Main Injector Steel from Measurement and MARS}
\begin{tabular}{|l|l|c|c|c|c|c|c|c|}
\hline
&Element &Half Life&\multicolumn{2}{c|}{Measured} &\multicolumn{2}{c|}{MARS}&\multicolumn{2}{c|}{Measured$/$MARS} \\ \hline
& &Days&\multicolumn{2}{c|}{pCi/gm/(p/sec)}&\multicolumn{2}{c|}{pCi/gm/(p/sec)}& & \\ \hline
& & &Shielded &Unshielded &Shielded &Unshielded &Shi&Unshi \\ \hline
1 &Cr-48 &0.89833 & &4.715E-08 &3.198E-09 &4.474E-08 & &1.054\\ \hline
2 &Cr-51 &27.7 &5.960E-08 &5.994E-06 &1.689E-07 &1.338E-06 &0.353 &4.481\\ \hline
3 &Fe-52 &0.344792 &2.657E-10 &5.365E-08 &3.280E-09 &3.422E-08 &0.081 &1.568\\ \hline
4 &Fe-59 &44.5 &6.866E-08 & &6.299E-08 &1.837E-07 &1.090 & \\ \hline
5 &K-43 &0.9292 & &6.801E-08 &7.783E-10 &1.826E-08 & &3.724\\ \hline
6 &Mn-52 &5.591 &1.658E-08 &1.700E-06 &8.587E-08 &7.256E-07 &0.193 &2.343\\ \hline
7 &Mn-54 &312.2 &1.341E-07 &1.332E-05 &4.214E-07 &2.510E-06 &0.318 &5.305\\ \hline
8 &Mn-56 &0.1074 &1.420E-06 &6.455E-06 &9.848E-07 &9.90E-07 &1.442 &6.520\\ \hline
9 &Na-24 &0.62329 &8.607E-10 &5.109E-08 &6.053E-10 &1.064E-08 &1.422 &4.803\\ \hline
10 &Sb-122 &2.7 &1.159E-07 &2.218E-07 &1.265E-07 &4.454E-08 &0.916 &4.981\\ \hline
11 &Sb-124 &60.2 &4.240E-08 & &9.555E-08 &3.568E-08 &0.444 & \\ \hline
12 &Sc-44m &2.44 &1.129E-09 &2.945E-07 & & & & \\ \hline
13 &Sc-46 &83.83 &3.534E-09 &7.611E-07 &9.060E-09 &1.605E-07 &0.390 &4.741\\ \hline
14 &Sc-47 &3.341 &2.171E-09 &2.859E-07 &5.895E-09 &8.868E-08 &0.368 &3.224\\ \hline
15 &Sc-48 &1.82 &4.296E-10 &4.655E-08 &1.934E-09 &2.762E-08 &0.222 &1.685\\ \hline
16 &V-48 &15.98 &9.934E-09 &1.480E-06 &3.627E-08 &4.821E-07 &0.274 &3.070\\ \hline
17 &Sc-44 &0.1654 &2.177E-09 &6.054E-07 &9.502E-09 &1.985E-07 &0.229 &3.050\\ \hline
18 &K-42 &0.515 & &8.309E-08 &1.568E-09 &4.656E-08 & &1.785\\ \hline
\end{tabular}
\label{table:ActivationResults}
\end{center}
\end{table}
\subsection{MARS Toy Model Results for Copper with Comparison to Measurements}
\pgph For a Toy Model run to study copper samples, we uses four
shielded and four unshielded samples. We gain an understanding of the
sensitivity to the geometry specific to the four stacked unshielded
samples and the different angles/absorption depths for the shielded
samples. We also have measurements of these samples reported in
Beams-doc-4046 and we compare Meas/MARS. The sample ratio comparisons
for the most important 10 isotopes are shown in
Figures~\ref{Fig:CompareCuShielded}~and~\ref{Fig:CompareCuUnshielded}.
A sense of the expected relative activation for the four shielded
locations is provided by the flux plots in
Figures~\ref{Fig:ToyModel-fnt-1}~thru~\ref{Fig:ToyModel-fgt-3}. The
results on the four shielded samples are consistent with expectations
from the flux plots. Table~\ref{table:CuActivationResults} documents
the Measured and MARS results for isotopes for which Beams-doc-4046
has results.
\begin{figure}[bthp]
\centering
\includegraphics [height=5in,angle=-90]{ActivationCu_ShieldedSamples.eps}
\caption{Ratio of sample activation to mean of samples for shielded
Cu samples. These samples differ in angle but also in absorption
depth. Isotopes are shown in order employed in Table~\ref{table:CuActivationResults}. }
\label{Fig:CompareCuShielded}
\end{figure}
\begin{figure}[bthp]
\centering
\includegraphics [height=5in,angle=-90]{ActivationCu_UnshieldedSamples.eps}
\caption{Ratio of sample activation to mean of samples for unshielded
Cu samples. These samples span a range of angles near the beam. Isotopes
are shown in order employed in Table~\ref{table:CuActivationResults}. }
\label{Fig:CompareCuUnshielded}
\end{figure}
\begin{table}[p]
%\begin{table}[tbhp]
\begin{center}
\caption{Activation of Copper from Measurement and MARS.\newline Average Meas/MARS is 0.401 for Shielded (shi8) and
3.4949 for Unshielded (uns7)}
\begin{tabular}{|l|l|c|c|c|c|c|c|c|}
\hline
&Element &Half Life&\multicolumn{2}{c|}{Measured} &\multicolumn{2}{c|}{MARS}&\multicolumn{2}{c|}{Measured$/$MARS} \\ \hline
& &Days&\multicolumn{2}{c|}{pCi/gm/(p/sec)}&\multicolumn{2}{c|}{pCi/gm/(p/sec)}& & \\ \hline
& & &Shielded &Unshielded &Shielded &Unshielded &Shi &Unshi \\ \hline
1 &Co-55 &0.7304 & & 8.15E-08 & 2.592E-09 & 4.069E-08 & & 2.002\\ \hline
2 &Co-56 &77.233 & 4.812E-09 & 9.74E-07 & 1.691E-08 & 1.834E-07 & 0.285& 5.308\\ \hline
3 &Co-57 &271.74 & 2.52E-08 & 2.83E-06 & 8.244E-08 & 6.919E-07 & 0.305& 4.092\\ \hline
4 &Co-58 &70.86 & 4.048E-08 & 4.61E-06 & 1.360E-07 & 9.280E-07 & 0.298& 4.970\\ \hline
5 &Co-60 &1925.8 & 2.758E-08 & & 8.843E-08 & 4.936E-07 & 0.312& \\ \hline
6 &Cr-51 &27.7 & 4.261E-09 & & 1.247E-08 & 2.571E-07 & 0.342& \\ \hline
7 &Cu-61 &0.139 & & 4.19E-06 & 1.761E-07 & 1.002E-06 & & 4.181\\ \hline
8 &Cu-64 &0.5291 & 5.471E-05 & 1.04E-04 & 5.061E-05 & 4.774E-05 & 1.081& 2.172\\ \hline
9 &Fe-59 &44.5 & 2.909E-09 & & 7.115E-09 & 4.580E-08 & 0.409& \\ \hline
10 &K-43 &0.9292 & & 4.39E-08 & 8.042E-11 & 7.949E-09 & & 5.520\\ \hline
11 &Mn-52 &5.591 & 1.816E-09 & 3.36E-07 & 6.259E-09 & 1.223E-07 & 0.290 & 2.748\\ \hline
12 &Mn-54 &312.2 & 1.024E-08 & & 2.890E-08 & 3.581E-07 & 0.354 & \\ \hline
13 &Mn-56 &0.1074 & & 5.75E-07 & 7.773E-09 & 8.677E-08 & & 6.628\\ \hline
14 &Na-24 &0.62329 & & 1.28E-08 & & 2.929E-09 & & 4.379\\ \hline
15 &Ni-57 &1.4833 & & 7.77E-08 & 2.686E-09 & 3.240E-08 & & 2.399\\ \hline
16 &Sc-44m &2.44 & & 7.90E-08 & & & & \\ \hline
17 &Sc-47 &3.341 & & 4.65E-08 & 8.372E-10 & 3.388E-08 & & 1.373\\ \hline
18 &Sc-48 &1.82 & & 2.74E-08 & 2.680E-10 & 1.004E-08 & & 2.727\\ \hline
19 &V-48 &15.98 & 1.114E-09 & 1.73E-07 & 3.375E-09 & 1.003E-07 & 0.330 & 1.729\\ \hline
20 &Sc-44 &0.1654 & & 1.16E-07 & 9.562E-10 & 5.272E-08 & & 2.198\\ \hline
\end{tabular}
\label{table:CuActivationResults}
\end{center}
\end{table}
\section{Conclusion}
\pgph As outlined above, our purpose was to draw conclusions about the
required specification detail for materials in the region of the Main
Injector Collimation System. As we developed this information, we
found that we have also a more general picture of the activation
properties of a variety of materials and elements. Among conclusions
we draw from there studies:
\begin{itemize}
\item For activation of concrete, we will wish to suggest some care
with regard to the production of Na-24 but that has only a 15 hour
half life so it has little impact when planning tunnel activities.
For Na-22, the activation is not large while aggregate of pure
dolomite (mineral) would add 50\% to 100\% to the production of
Na-22 so some care may be required for very careful activation
studies. The contribution of zirconium will be negligible.
\item The activation of Mn-56 is consistent with being dominated by
neutron capture on Mn-55 for many materials in our tunnel.
\item We observe that the equilibrium activation of various materials
is different, confirming the understanding that we can avoid high
activation with some materials. Molybdenum and Manganese are more
highly activated but typical concentrations are not high. While
Copper is also more highly activated, the produced isotopes for
Copper typically have shorter half lives.
\end{itemize}
\pgph We have employed the activation for a 30 day exposure and 2 hour
cooldown which we converted to a total produced activation and an
equilibrium activation. While this is a convenient analysis tool, we
did not find that the use of produced activation provided new insight.
\pgph For the measurements in Beams-doc-4046, we found two classes for
the Ratio Uns/Shi which seemed to correlate with spallation (high
ratio) and neutron activation (low ratio). In searching for this
correlation in the MARS Toy Model results, we found that the geometry
explored failed to reproduce the very large ratios. We attribute that
to the specifics of the study geometry and find that looking at
Uns/Shi ratios in this study was sometimes suggestive of production
mechanisms but not sufficient to provide useful guidance.
\pgph A presentation on the Toy Model at SATIF12 \cite{bcbrown-SATIF12} used
older MARS code and a different tactic for the comparison. In addition
to the improved agreement due to detailed specification of the sample
isotopes, we find that comparing equilibrium activation explicitly
allows better matching to the exposure history of the measurements.
\pgph With the study of sensitivities in this work, we believe we
are ready to resume MARS studies of the activation of the Main Injector
Collimation region.
% \section{Acknowledgments}
\pgph
\appendix
\newpage
\section{Typical MARS Geometry Description}
\label{Appen:MARSGeom}
\begin{verbatim}
A typical set of MARS Input Files:
File: GEOM.INP
Detector Solenoid geometry
!OPT
collim1 2 0 1 0.0 0.0 36.0 3.8 5.7 163.
collim2 2 0 2 0.0 0.0 36.0 5.7 59. 163.
collim3 2 0 3 0.0 0.0 0.0 59. 63.5 199.
taper1 4 0 2 0.0 0.0 0.0 4.4 59. 3.8 59. 36.
uns1 2 0 4 0.0 0.0 199. 5.7 10.7 0.5
uns2 2 0 5 0.0 0.0 199.5 5.7 10.7 0.5
uns3 2 0 6 0.0 0.0 200.0 5.7 10.7 0.5
uns4 2 0 7 0.0 0.0 200.5 5.7 10.7 0.5
shi1 2 0 8 0.0 0.0 70. 63.5 64.5 15.0
shi2 2 0 9 0.0 0.0 85. 63.5 64.5 15.0
shi3 2 0 10 0.0 0.0 100. 63.5 64.5 15.0
shi4 2 0 11 0.0 0.0 115. 63.5 64.5 15.0
TR1 0.0 0.0 0.0 0.0 90.0 0.0
stop
_________________________________________________________
fILE: MATER.INP
m1516 BCB with MCNP6 neutron libraries 11/28/17
1 'STST'
2 'YOKE'
3 'MRBL'
4 'STEEL' 7.85 9
5.5845E+01 2.6000E+01 0.987484
1.2011E+01 6.0000E+00 0.000033
5.4938E+01 2.5000E+01 0.0052
3.0974E+01 1.5000E+01 0.00051
3.2066E+01 1.6000E+01 0.00006
2.8085E+01 1.4000E+01 0.0036
2.6982E+01 1.3000E+01 0.00276
1.4007E+01 7.0000E+00 0.000023
1.2176E+02 5.1000E+01 0.00033
5 'Fe-58' 7.874 1
58.00000 26.00000 1.000
6 'SB'
7 'ZR'
8 'STEEL' 7.85 9
5.5845E+01 2.6000E+01 0.987484
1.2011E+01 6.0000E+00 0.000033
5.4938E+01 2.5000E+01 0.0052
3.0974E+01 1.5000E+01 0.00051
3.2066E+01 1.6000E+01 0.00006
2.8085E+01 1.4000E+01 0.0036
2.6982E+01 1.3000E+01 0.00276
1.4007E+01 7.0000E+00 0.000023
1.2176E+02 5.1000E+01 0.00033
9 'Fe-58' 7.874 1
58.00000 26.00000 1.000
10 'SB'
11 'ZR'
\end{verbatim}
\section{Simplifications for Bateman Equations for Main Injector Radiation}
\pgph The measurements of activation and residual radiation in the
Main Injector tunnel such as those reported in
Beams-doc-4046\cite{BeamsDoc4046} involve activation created by 8 GeV
protons with activation times up to decades but measurement after
activation following 2 hours or more of cooldown. Only a few
materials contain nuclei heavier than iron which limits the decay
chains of interest. The total loss at any location is much less than
$1 \times 10^{21}$ protons. We will demonstrate that this can be
discussed while discarding most terms in the Bateman Equation with no
non-linear terms which would involve striking nuclei produced in the
activation process. We will discuss this using the presentation of
the Bateman Equations from the DeTra documentation\cite{Aarnio:DeTra}.
We will copy this directly from Section 2 of that document.
\begin{minipage}[l]{\textwidth}
\rule[0mm]{\textwidth}{1mm}
Decay and transmutation of nuclides can be expressed as
\begin{equation}
\frac{dN_i}{dt} = S_i + \sum_j b_{ji} \lambda_j N_j(t)
+ \sum_j g_{ji} \sigma_j \phi N_j(t)
- \lambda_i N_i(t) - \sigma_i \phi N_i(t)
\label{Eq:Bateman}\end{equation}
where
\begin{itemize}
\item $N_i$ is the abundance of nuclide $i$,
\item $S_i$ is the external production rate of nuclide $i$,
\item $b_{ji}$ is the branching ratio of the decay of nuclide $j$ to nuclide $i$,
\item $\lambda_i$ is the decay constant of nuclide $i$,
\item $g_{ji}$ is the fraction of absorptions in nuclide $j$ leading to nuclide $i$,
\item $\sigma_i$ is the spectrum averaged absorption cross section of nuclide i, and
\item $\phi$ is the transmutating flux.
\end{itemize}
We note that
\begin{itemize}
\item The first term on the right is the source term. It is any
external production rate of nuclides i.e. it does not depend on any
of the abundances $N$. Below we divide the source term in two
components - one is due to fission and the other due to accelerator
production.
\item The second term describes the build-up rate of the nuclide $i$
due to decay of other nuclides, $N_j$.
\item The third term describes the transmutation production rate of
nuclide $i$ from other nuclides, $N_j$. This term includes both the
transmutation of unstable isotopes and activation of the stable
ones. Note that, even though the accelerator production of the
source term above could also be called transmutation or activation,
we have reserved the word transmutation for this reaction only.
\item The fourth term gives the decay rate of nuclide $i$, and the
last term defines the depletion of the nuclide $i$ due to its
transmutation.
\end{itemize}
\pgph In the solution of Eq.~\ref{Eq:Bateman} we assume that all the
quantities, except N, above are constant i.e. do not depend on
time. The assumption is usually well satisfied.
\rule[0mm]{\textwidth}{1mm}
\end{minipage}
\pgph As prescribed by Aarnio, we use the first term to describe the
activation produced by loss of 8 GeV protons. There will be no
fission in the materials we need to consider. In Beams-doc-4046, we
considered two isotopes which decay to isotopes which were found in
our measurements and conclude that K-42 is not from decay of Ar-42 and
Sc-44 is not from decay of Ti-44 based on measurements with different
exposure/cooldown times. We conclude that the second term can be
eliminated from our considerations. In similar fashion, we can
dismiss the third term since the production rate is insufficient
to produce sufficient target nuclides, $N_j$. Finally, the fifth
term can be neglected by the same rate considerations.
\pgph With the remaining first and fourth terms in this expression, we
conclude that the analysis of activation when the beam exposure time
is well-measured can be analyzed using the time-weighting procedure
described in Beams-doc-4046 (and Beams-doc-3980~\cite{BeamsDoc3980}).
\section{Examination of Linearity of Activation with Beam Delivered}
\pgph We also wish to explore any non-linear buildup of activation
in our situation. We examine Equation~\ref{Eq:ProduceNI}.
In a beam of particles, nuclear interactions produce new
isotopes. The number of new nuclei is proportional to the fluence,
$\Phi$, measured in particles per unit area (particles-cm$^{-2})$. In
a material with $n_T$ target atoms per unit volume, an interaction
with cross section $\sigma_I$ will produce $n_I$ atoms per unit volume
of isotope $I$
\begin{equation}
n_I = \Phi n_T \sigma_I.
\label{Equ:ProduceNI}
\end{equation}
\begin{equation}
\frac{n_I}{n_T} = \Phi \sigma_I.
\label{Eq:ConvFrac}
\end{equation}
The fraction $n_I/n_T$ will remain small compared to 1 provided the
fluence $\Phi$ does not exceed a limit set by relevant cross sections,
$\sigma$. We study losses of $1.25 \times 10^{12}$ protons/second.
This leads to $3.95 \times 10^{19}$ protons for a year of continuous operations.
If these would be lost on an area of 1 mm x 1 mm the fluence would be
$3.95 \times 10^{21}$. For $\frac{n_I}{n_T}= 0.01$ would require
$\sigma = 2.53 \time 10^{-24}$ $cm^2$ (2.53 barns) whereas the cross
sections of interest are at most many millibarns. As the protons
produce a shower of hadrons, the peak number of particles will increase
by a factor of up to 10 for 8 GeV showers but they will spread such
that the fluence in particles per $cm^2$ will hardly grow. We argue
that calculations of this activation will remain linear in the total
proton loss for all issues in our list of concerns. Note that
extending the time frame to 10 years will increase the total fluence
but then many of the produced isotopes will have decayed. The
most significant isotope in our study is Mn-54 with a half-life of
312 days.
\section{Activation of Natural Ca and Ca-40}
\label{Appen:Ca-0}
\pgph Calcium is an important component of concrete and we employ
marble (CaCO$_2$) for shielding the secondary collimators in the Main
Injector Collimation System. In light of the special activation
properties of Ca-40 due to the nuclear shell model feature that it is
a 'magic' (closed shell) nucleus, we have taken the opportunity
presented by the Toy Model to explore the special nature of Ca-40
compared with natural Ca. The isotopic abundance of natural Ca is
shown in Table~\ref{table:CaIsotopes}. In determining how to present
activation studies, we have chosen this comparison to develop
presentation methods for other parts of this study. As discussed in
the Section on Formulas, we provide formulas for produced activation
and equilibrium activation production and relate them to the
activation calculated for 30 Days of activation and 2 hours of
cooldown with an interaction rate of $1.25 \times 10^{12}$
protons/second. To compare with measurements or other studies, we
will get produced activation per interacting proton and equilibrium
activation per interacting protons per second.
\pgph We have explored equilibrium activation in
Beams-doc-3980~\cite{BeamsDoc3980} and
Beams-doc-4046~\cite{BeamsDoc4046} but here we have the results for 30
Days of activation and 2 hours of cool down so we have decided to
compare these various representations of the calculated results. In
Figure~\ref{Fig:CaActivation} we have plotted the 30 Day/ 2 hr
activation with the isotopes in the order of decreasing activation for
each sample as sorted by DeTra. We then also plot the produced
activation and the equilibrium activation for that same calculation.
We noted that the short half life produced activities can be quite
large and for very long half lives the equilibrium activation would be
very large so we have marked the points on the equilibrium curve with
solid green points for the half lives longer than 20 years while we
use open blue circles to identify the equilibrium values for isotopes
with half life values less than 2 hours (The list includes ones down
to 0.1 hr).
\pgph Since the equilibrium activation for Shielded and Unshielded
locations are proportional to the proton loss rate,
Table~\ref{table:CaActivationShielded} and
Table~\ref{table:CaActivationUnshielded} provide the equilibrium
activation for this study and the equilibrium activation per proton
per second. These tables are ordered with the activity from the
Ca-shi sample 30 Day / 2 Hr study.
\pgph The results and analysis of these samples are available in the
spreadsheet
\newline
Analysis\_Concrete\_Feb2019.xlsx where the samples with
Ca-40 are labeled ca20 (for 20 neutrons and 20 protons).
\begin{figure}[t]
%\begin{figure}[bthp]
\centering
%% \includegraphics {CompareCaActivation.eps}
\includegraphics [height=4.3in]{CompareCaActivation.eps}
\caption{Produced activation in Ca-40 and natural Ca for Shielded and
Unshielded locations. Activation after 30 Days of beam on and 2
hours of cooling (black lines and circles) is compared with
activation produced by the interacting protons in that run (red
lines and circles) and with the equilibrium activation produced at
that proton loss rate ((green lines and circles). To clarify the
lifetime effects, the equilibrium points for half life values longer
than 20 years are shown with solid green circles while those with
half life values less than 2 hours are shown with open blue
circles.}
\label{Fig:CaActivation}
\end{figure}
\begin{table}[thp]
%\begin{table}[tbhp]
\begin{center}
\caption{Equilibrium Activation of Ca and Ca-40 -- Shielded}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
& & & &\multicolumn{2}{c|}{Shielded Ca}& \multicolumn{2}{c|}{Shielded Ca-40}\\ \hline
Element & Z &A &HalfLife &\multicolumn{2}{c|}{Equilibrium Activation}&\multicolumn{2}{c|}{Equilibrium Activation}\\ \hline
& & &sec &pCi/gm &pCi/gm/(p/sec) &pCi/gm &pCi/gm/(p/sec)\\ \hline
Ar &18 &37 &3.03E+06 &5.19E+05 &4.15E-07 &4.34E+05 &3.47E-07\\ \hline
P &15 &32 &1.23E+06 &4.63E+04 &3.70E-08 &4.00E+04 &3.20E-08\\ \hline
K &19 &43 &8.03E+04 &1.13E+04 &9.02E-09 &9.62E+03 &7.69E-09\\ \hline
K &19 &42 &4.45E+04 &9.00E+03 &7.20E-09 &8.99E+03 &7.19E-09\\ \hline
P &15 &33 &2.19E+06 &1.32E+04 &1.06E-08 &1.23E+04 &9.84E-09\\ \hline
S &16 &35 &7.56E+06 &2.94E+04 &2.35E-08 &2.72E+04 &2.18E-08\\ \hline
Si &14 &31 &9.44E+03 &4.36E+03 &3.49E-09 &3.54E+03 &2.83E-09\\ \hline
Ar &18 &41 &\textcolor{blue}{6.58E+03} &1.81E+03 &1.45E-09 &1.39E+03 &1.11E-09\\ \hline
Na &11 &24 &5.39E+04 &9.13E+02 &7.30E-10 &5.87E+02 &4.70E-10\\ \hline
Be &4 &7 &4.60E+06 &1.96E+03 &1.57E-09 &1.17E+03 &9.39E-10\\ \hline
H &1 &3 &3.89E+08 &1.23E+05 &9.85E-08 &9.64E+04 &7.71E-08\\ \hline
Cl &17 &38 &\textcolor{blue}{2.23E+03} &5.27E+03 &4.22E-09 &4.14E+03 &3.31E-09\\ \hline
Sc &21 &44 &1.43E+04 &5.51E+02 &4.41E-10 &1.44E+02 &1.15E-10\\ \hline
Sc &21 &43 &1.40E+04 &1.96E+02 &1.57E-10 &2.60E+02 &2.08E-10\\ \hline
F &9 &18 &\textcolor{blue}{6.59E+03} &2.61E+02 &2.09E-10 &3.26E+02 &2.61E-10\\ \hline
S &16 &38 &1.02E+04 &1.30E+02 &1.04E-10 & & \\ \hline
Cl &17 &39 &\textcolor{blue}{3.34E+03} &3.17E+02 &2.53E-10 &3.26E+02 &2.61E-10\\ \hline
Mg &12 &28 &7.53E+04 &6.52E+01 &5.22E-11 & & \\ \hline
Na &11 &22 &8.21E+07 &1.76E+03 &1.41E-09 &1.43E+03 &1.15E-09\\ \hline
K &19 &44 &\textcolor{blue}{1.33E+03} &9.27E+02 &7.42E-10 &8.32E+02 &6.65E-10\\ \hline
Ar &18 &39 &\textcolor{green}{8.49E+09} &7.77E+04 &6.22E-08 &6.20E+04 &4.96E-08\\ \hline
Ca &20 &41 &\textcolor{green}{3.22E+12} &1.11E+07 &8.85E-06 &9.21E+06 &7.37E-06\\ \hline
K &19 &38 &\textcolor{blue}{4.58E+02} &2.56E+05 &2.05E-07 &2.32E+05 &1.85E-07\\ \hline
C &6 &11 &\textcolor{blue}{1.22E+03} &6.52E+01 &5.22E-11 & & \\ \hline
Ar &18 &42 &\textcolor{green}{1.04E+09} &4.36E+02 &3.49E-10 &4.57E+02 &3.65E-10\\ \hline
Si &14 &32 &\textcolor{green}{4.17E+09} &8.13E+02 &6.50E-10 &3.26E+02 &2.61E-10\\ \hline
Mg &12 &27 &\textcolor{blue}{5.67E+02} &7.17E+02 &5.74E-10 &1.04E+03 &8.35E-10\\ \hline
Ar &18 &44 &\textcolor{blue}{7.12E+02} &6.52E+01 &5.22E-11 & & \\ \hline
Cl &17 &36 &\textcolor{green}{9.50E+12} &2.72E+05 &2.18E-07 &2.31E+05 &1.85E-07\\ \hline
Cl &17 &34m &\textcolor{blue}{1.92E+03} &2.41E-01 &1.93E-13 &1.21E-01 &9.65E-14\\ \hline
Al &13 &29 &\textcolor{blue}{3.94E+02} &2.25E+03 &1.80E-09 &1.17E+03 &9.39E-10\\ \hline
\end{tabular}
\label{table:CaActivationShielded}
\end{center}
\end{table}
\begin{table}[thp]
%\begin{table}[tbhp]
\begin{center}
\caption{Equilibrium Activation of Ca and Ca-40 -- Unshielded}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
& & & &\multicolumn{2}{c|}{Unshielded Ca}& \multicolumn{2}{c|}{Unshielded Ca-40}\\ \hline
Element & Z &A &HalfLife &\multicolumn{2}{c|}{Equilibrium Activation}&\multicolumn{2}{c|}{Equilibrium Activation}\\ \hline
& & &sec &pCi/gm &pCi/gm/(p/sec) &pCi/gm &pCi/gm/(p/sec)\\ \hline
Ar &18 &37 &3.03E+06 &3.40E+06 &2.72E-06 &3.45E+06 &2.76E-06\\ \hline
P &15 &32 &1.23E+06 &5.05E+05 &4.04E-07 &4.97E+05 &3.98E-07\\ \hline
K &19 &43 &8.03E+04 &5.54E+04 &4.43E-08 &7.02E+04 &5.62E-08\\ \hline
K &19 &42 &4.45E+04 &5.93E+04 &4.74E-08 &4.47E+04 &3.58E-08\\ \hline
P &15 &33 &2.19E+06 &1.69E+05 &1.36E-07 &1.74E+05 &1.39E-07\\ \hline
S &16 &35 &7.56E+06 &2.22E+05 &1.77E-07 &1.99E+05 &1.60E-07\\ \hline
Si &14 &31 &9.44E+03 &3.08E+04 &2.46E-08 &4.28E+04 &3.42E-08\\ \hline
Ar &18 &41 &\textcolor{blue}{6.58E+03} &1.23E+04 &9.85E-09 &9.16E+03 &7.33E-09\\ \hline
Na &11 &24 &5.39E+04 &2.15E+04 &1.72E-08 &3.05E+04 &2.44E-08\\ \hline
Be &4 &7 &4.60E+06 &5.54E+04 &4.43E-08 &7.64E+04 &6.11E-08\\ \hline
H &1 &3 &3.89E+08 &1.32E+06 &1.06E-06 &1.38E+06 &1.10E-06\\ \hline
Cl &17 &38 &\textcolor{blue}{2.23E+03} &2.77E+04 &2.22E-08 &2.14E+04 &1.71E-08\\ \hline
Sc &21 &44 &1.43E+04 &8.33E+03 &6.67E-09 &6.54E+03 &5.23E-09\\ \hline
Sc &21 &43 &1.40E+04 &3.90E+03 &3.12E-09 &9.80E+03 &7.84E-09\\ \hline
F &9 &18 &\textcolor{blue}{6.59E+03} &5.85E+04 &4.68E-08 &4.89E+04 &3.91E-08\\ \hline
S &16 &38 &1.02E+04 & & & & \\ \hline
Cl &17 &39 &\textcolor{blue}{3.34E+03} &3.08E+03 &2.46E-09 & & \\ \hline
Mg &12 &28 &7.53E+04 & & & & \\ \hline
Na &11 &22 &8.21E+07 &8.93E+04 &7.14E-08 &1.17E+05 &9.36E-08\\ \hline
K &19 &44 &\textcolor{blue}{1.33E+03} &1.35E+04 &1.08E-08 &1.56E+04 &1.24E-08\\ \hline
Ar &18 &39 &\textcolor{green}{8.49E+09} &3.97E+05 &3.17E-07 &3.78E+05 &3.02E-07\\ \hline
Ca &20 &41 &\textcolor{green}{3.22E+12} &3.00E+07 &2.40E-05 &2.91E+07 &2.33E-05\\ \hline
K &19 &38 &\textcolor{blue}{4.58E+02} &1.87E+06 &1.50E-06 &1.94E+06 &1.55E-06\\ \hline
C &6 &11 &\textcolor{blue}{1.22E+03} &3.08E+03 &2.46E-09 &9.16E+03 &7.33E-09\\ \hline
Ar &18 &42 &\textcolor{green}{1.04E+09} & & &6.11E+03 &4.89E-09\\ \hline
Si &14 &32 &\textcolor{green}{4.17E+09} &9.23E+03 &7.39E-09 &6.11E+03 &4.89E-09\\ \hline
Mg &12 &27 &\textcolor{blue}{5.67E+02} &6.16E+03 &4.93E-09 &6.11E+03 &4.89E-09\\ \hline
Ar &18 &44 &\textcolor{blue}{7.12E+02} & & &3.05E+03 &2.44E-09\\ \hline
Cl &17 &36 &\textcolor{green}{9.50E+12} &1.70E+06 &1.36E-06 &1.71E+06 &1.37E-06\\ \hline
Cl &17 &34m &\textcolor{blue}{1.92E+03} & & &1.13E+01 &9.04E-12\\ \hline
Al &13 &29 &\textcolor{blue}{3.94E+02} &4.00E+04 &3.20E-08 & & \\ \hline
N &7 &13 &\textcolor{blue}{5.98E+02} &9.23E+03 &7.39E-09 &1.83E+04 &1.47E-08\\ \hline
\end{tabular}
\label{table:CaActivationUnshielded}
\end{center}
\end{table}
\begin{table}[thp]
%\begin{table}[tbhp]
\begin{center}
\caption{Activation Ratio of Ca-40/Natural Ca}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
Element & Z &A &Shielded &$\sigma$ &Unshielded &$\sigma$ \\ \hline
Ar &18 &37 &0.835711427 &0.019590378 &1.015286456 &0.043269619 \\ \hline
P &15 &32 &0.865260494 &0.057160137 &0.983974388 &0.109091963 \\ \hline
K &19 &43 &0.85235126 &0.110222173 &1.267781778 &0.30670135 \\ \hline
K &19 &42 &0.999445538 &0.096296458 &0.754205899 &0.341441102 \\ \hline
P &15 &33 &0.929568489 &0.103568808 &1.026991349 &0.190049992 \\ \hline
S &16 &35 &0.925560457 &0.071494469 &0.899492028 &0.188275921 \\ \hline
Si &14 &31 &0.812189308 &0.181102752 &1.389482424 &0.408982581 \\ \hline
Ar &18 &41 &0.76869496 &0.218022456 &0.743994688 &0.756263672 \\ \hline
Na &11 &24 &0.643073795 &0.43944213 &1.417542743 &0.519764199 \\ \hline
Be &4 &7 &0.599972235 &0.314954261 &1.378064231 &0.320793584 \\ \hline
H &1 &3 &0.783065674 &0.038665736 &1.046219894 &0.070338053 \\ \hline
Cl &17 &38 &0.785110731 &0.17499904 &0.771812138 &0.514820267 \\ \hline
Sc &21 &44 &0.261406676 &0.784150472 &0.784271397 &1.223948392 \\ \hline
Sc &21 &43 &1.325668363 &0.581851063 &2.510831708 &0.79824397 \\ \hline
F &9 &18 &1.249518617 & &0.835382718 & \\ \hline
S &16 &38 &small & & & \\ \hline
Cl &17 &39 &1.030069498 & & & \\ \hline
Mg &12 &28 &small & & & \\ \hline
Na &11 &22 &0.814821774 & &1.310011964 & \\ \hline
K &19 &44 &0.897333907 & &1.148617835 & \\ \hline
Ar &18 &39 &0.797134169 & &0.951756311 & \\ \hline
Ca &20 &41 &0.833292574 & &0.969894709 & \\ \hline
K &19 &38 &0.905385649 & &1.038875314 & \\ \hline
C &6 &11 &small & &2.977177031 & \\ \hline
Ar &18 &42 &1.046496913 & & & \\ \hline
Si &14 &32 &0.401355603 &0.532929412 &0.661499198 &0.907767685 \\ \hline
Mg &12 &27 &1.454608853 & &0.992190498 & \\ \hline
Ar &18 &44 &small & &large? & \\ \hline
Cl &17 &36 &0.850375273 & &1.002754098 & \\ \hline
Cl &17 &34m & & & & \\ \hline
Al &13 &29 & & & & \\ \hline
N &7 &13 & & &1.98398702 & \\ \hline
\end{tabular}
\label{table:CaToCa-40Ratio}
\end{center}
\end{table}
\pgph When we look at the Ratio Table
(Table~\ref{table:CaToCa-40Ratio}) we are forced to realize that some
of the potential interesting differences between natural Ca and Ca-40
will have to be explored with higher statistics. We have the $\sigma$
(standard deviation) for the ratios which are calculated as the
root-sum-squares of the errors in each activation and find that the
error is too large to draw interesting conclusions. Obviously if the
ratio Ca-20/Ca is larger than 1, we must be limited by statistical
errors. When the ratio is smaller than 1, we need to look at the RMS
error and we see that for no unshielded isotope is there a
significantly larger production in the Ca sample. For the shielded
samples, we see for Ar-37, P-32, K-43 there are statistically
significant excess production in the Ca samples and some others show
deviations which might indicate an interesting excess but overall,
these are the significant isotopes for Ca excess. If we sum the
equilibrium activation (using the isotope order of the 30 Day/2 Hr
activation) and compare Ca to Ca-40 we find that there is essentially
no excess for the unshielded (forward angle) samples. For the
shielded (large angle) samples, the Ar-37 is 67\% of the activation
for both shielded and unshielded and the shielded Ca/Ca-40 ratio is
1.20 for that isotope. Comparing the running sums we find that for
the shielded samples, Ca/Ca-40 remains at 1.18 - 1.20 for all sums
down to the first very long lived ($>$20 yr) halflife.\footnote{ When
studying the measured activation for Beams-doc-4046, Vernon Cupps
noted the possibility of 'Secular Equilibrium' for observing
K-42 and Sc-44. After comparing activation with various exposure
times, we concluded that we were not seeing Ar-42 $\rightarrow$ K-42
nor Ti-44 $\rightarrow$ Sc-44. These studies are consistent with
that for Ca. One can also look for this in the other samples.}
\pgph In reviewing this study of Ca and Ca-40, we conclude that
emphasizing the study of equilibrium activation is adequate. Using
the results from MARS/DeTra for 30 days of activation and 2 hr of
cooldown and exploring half life values from 2 hours to 20 years is
entirely adequate for the Main Injector activation considerations.
\pgph We would now like to quantify the reputation of Ca for low activation.
We will compare it to various other materials we have studied in Table~\ref{table:CaVsMISteel}.
\begin{table}[tbhp]
\begin{center}
\caption{Comparison of Equilibrium Activation of Various Samples}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
Element & Z &A &HalfLife & Sample &Activity & Activity \\ \hline
& & &sec & &pCi/gm & pCi/gm/(p/sec)\\ \hline
& & & & Shielded & & \\ \hline
Ar &18 &37 &3.03E+06 & Ca &5.19E+05 &4.15E-07\\ \hline
P &15 &32 &1.23E+06 & Ca &4.63E+04 &3.70E-08\\ \hline
Na &11 &24 &5.39E+04 & Concrete &2.85E+05 &2.28E-07\\ \hline
Ar &18 &37 &3.03E+06 & Concrete &7.25E+04 &5.80E-08\\ \hline
Si &14 &31 &9.44E+03 & Concrete &4.15E+04 &3.32E-08\\ \hline
K &19 &42 &4.45E+04 & Concrete &1.65E+04 &1.32E-08\\ \hline
Be &4 &7 &4.60E+06 & Concrete &2.84E+04 &2.27E-08\\ \hline
Cu &29 &64 &4.57E+04 & Copper &6.326E+07 &5.061E-05\\ \hline
Cu &29 &61 &1.20E+04 & Copper &2.202E+05 &1.761E-07\\ \hline
Co &27 &58 &6.12E+06 & Copper &1.700E+05 &1.360E-07\\ \hline
Mn &25 &56 &9.28E+03 &MI Steel &1.23E+06 &9.85E-07\\ \hline
Mn &25 &52 &4.83E+05 &MI Steel &1.07E+05 &8.59E-08\\ \hline
Mn &25 &54 &2.70E+07 &MI Steel &5.27E+05 &4.21E-07\\ \hline
& & & & Unshielded & & \\ \hline
Ar &18 &37 &3.03E+06 & Ca &3.40E+06 &2.72E-06\\ \hline
P &15 &32 &1.23E+06 & Ca &5.05E+05 &4.04E-07\\ \hline
Na &11 &24 &5.39E+04 & Concrete &1.01E+06 &8.08E-07\\ \hline
Ar &18 &37 &3.03E+06 & Concrete &7.38E+05 &5.90E-07\\ \hline
Be &4 &7 &4.60E+06 & Concrete &6.02E+05 &4.81E-07\\ \hline
Cu &29 &64 &4.57E+04 & Copper &5.96E+07 &4.77E-05\\ \hline
Cu &29 &61 &1.20E+04 & Copper &1.25E+06 &1.00E-06\\ \hline
Co &27 &58 &6.12E+06 & Copper &1.16E+06 &9.28E-07\\ \hline
Mn &25 &56 &9.28E+03 &MI Steel &1.24E+06 &9.90E-07\\ \hline
Mn &25 &52 &4.83E+05 &MI Steel &9.07E+05 &7.26E-07\\ \hline
Mn &25 &54 &2.70E+07 &MI Steel &3.14E+06 &2.51E-06\\ \hline
\end{tabular}
\label{table:CaVsMISteel}
\end{center}
\end{table}
%\newpage
\section{Toy Model Study with Various Steel Targets}
\label{Appen:SteelStudy}
\pgph A high statistics study of activation was carried out using a
set of four steel varieties as targets. Included were Natural Fe, Main
Injector Steel, Cast Iron, and 316 Stainless Steel. The composition
used for these materials is shown in
Table~\ref{table:SteelComposition}. As in other studies reported in
this document, a beam of 1.25$\times 10^{12}$ protons/sec struck the
Toy Model collimator for 30 days after which a cool down of 2 hours
was imposed. MARS results were processed with DeTra to provide
isotopes produced which were sorted by activity. The files of results
were then explored in Excel (file:Analysis\_Steels\_Oct2018.xlsx).
The activation ratios were explored for Unshielded/Shielded for each
material. In addition, the activity for each isotope in the
unshielded and shielded targets was compared to the production in the
natural Fe sample.
\section{Toy Model Study for Activation of Various Stainless Steel}
\label{Appen:StainlessSteelStudy}
\pgph Stainless steel is a critical component of accelerators and
since we employ it for beam pipes, beam loss will almost always
activate some stainless steel. We have replaced some 316L stainless
pipe with 2205 duplex stainless in a high radiation area near
Collimator C301. To acquire some understanding of potential
activation issues for these materials, we have toy model runs with
samples in both the shielded and unshielded locations for Fe, 316L,
chromium (Cr), manganese (Mn), nickel (Ni), and molybdenum (Mo). In
Section~\ref{Results:Manganese} we show results on the activation of
manganese. Let us here examine the activation of all of these
materials to explore what concerns we should have about how much care
in specification of materials is required for the tunnel activation
studies. Additionally, we should see what understanding will be
helpful in selecting materials for future accelerator components.
\pgph Using Toy Model results for the various components of stainless
steel, we can simulate the activation of 304, 316, and 2205 alloys.
We separately simulated a sample of 316. We find that the
results have no surprises. For some produced isotopes like Mn-56, the
2205 is predicted to achieve a lower activation while for many
isotopes it 2205/316 is 1.448 so we conclude that use of 2205 alloy
should not be restricted due to activation concerns. For those
unfamiliar with 2205, we note that is is not suitable for beam
pipe through magnets due to its magnetic properties.
\section{Toy Model Study For Concrete}
\label{Appen:ConcreteStudy}
\pgph Studies with samples of concrete in both shielded and unshielded
locations were supplemented with samples of Ca, Mg and Zr.
Table~\ref{table:ConcreteComposition} shows the MARS built-in concrete
composition. As discussed in Section~\ref{Section:ConcreteStudy}, we
have not determined the relative amount of Ca which might be replaced
with Mg when the aggregate includes some Dolomite. In spreadsheet
Analysis\_Concrete\_Feb2019.xlsx, we have included the results for
comparing concrete with Ca and Mg. In the worksheet:Analysis, we
explore the activation of 15.1\% of Ca or 15.1\% of Mg (the weight
fraction of Mg in Dolomite and/or the reduction of Ca) The results are
available in the spreadsheet. We choose to look instead at the most
significant contributions to the activity in concrete. For both
shielded and unshielded concrete samples, the highest activation
(sorted by 30 day activation/ 2 hour cooldown) is Na-24 followed by
Ar-37.
For Na-24, in the shielded (unshielded) sample the contribution from
Ca is only 4.85$\times 10^{-4}$ (3.23$\times 10^{-3}$) and the Mg
contribution is x300 (x74) larger however, that Mg contribution is
only 14.5\% (24\%) of the Na-24 in concrete. The Ar-37 in the
concrete comes largely from the Ca and no Na-24 is produced in Mg.
For shielded concrete, the next two activated isotopes are Si-31 and
K-42 where the Ca contribution is modest (or small) and there is no
Mg contribution. For the unshielded concrete, the next isotope on
the list is Be-7 for which the contribution of either Ca or Mg is
very small and Si-31 is next on that list and again Mg would not
contribute. Nearby on the list is F-18 where for the unshielded
sample, Mg would be quite significant but we declare it not very
interesting due to the 1.83 hour half life. We also note that
Na-22 falls further down the list with this ordering but would be
much higher if ranked by equilibrium activation. For Na-22, the
activation from Mg would roughly double the Concrete for shielded
samples and provide about 50\% more for the unshielded. This
might be of interest for very careful activation studies.
In looking through all the MARS/DeTra output, only Mg-27 with
a half life of 567 seconds shows a very high production on Mg
which would be interesting only for pure dolomite in the concrete
and for very early observation after excitation.
\pgph The Toy Model study of zirconium (Zr) included shielded and
unshielded samples. We expected and found that the list of isotopes
with activation greater than 1 Bq in the sample was long with 151
isotopes in unshielded zirconium and 116 in the shielded sample. The
zirconium equilibrium activation is higher than than concrete with a
sum of 3.87$\times 10^{6}$ pCi/gm for shielded ( 1.0$\times 10^{8}$ pCi/gm for
unshielded) even when including a few very long half life isotopes
which will not reach equilibrium. We compare this with the same
total equilibrium activation of shielded concrete (2.54$\times 10^{6}$
pCi/gm) or unshielded concrete (1.28$\times 10^{7}$ pCi/gm) and
we see that at a weight fraction of 1.3$\times 10^{-4}$ of zirconium
in typical materials on the earth's surface, the contribution
of zirconium to our activation studies can be ignored.
\section{Tables of Selected Material and Nuclear Properties}
\pgph The reader can find here various items which define the
properties of the materials which we study with the Toy Model.
\pgph The Wikipedia articles on steels provides most of its
information with regard to the EN specification of stainless steel
whereas most specifications at Fermilab employ the ASTM designation.
Let us have the correlation conveniently available by providing
Table~\ref{table:SSDesignation} here. See
\begin{verbatim}
https://en.wikipedia.org/wiki/Stainless_steel
or
https://en.wikipedia.org/wiki/Steel_grades
\end{verbatim}for a source)
\begin{table}[htb]
\begin{center}
\caption{Stainless Steel Designations}
\begin{tabular}{|c|c|}
\hline
EN & ASTM \\
&or SAE\\ \hline
1.4301& 304 \\ \hline
1.4306& 304L \\ \hline
1.4311& 304NL \\ \hline
1.4948& 304H \\ \hline
1.4401& 316 \\ \hline
1.4436& 316 \\ \hline
1.4404& 316L \\ \hline
1.4406& 316LN\\ \hline
1.4462& 2205 \\ \hline
\end{tabular}
\label{table:SSDesignation}
\end{center}
\end{table}
\begin{table}[tbhp]
\begin{center}
\caption{MARS15 Built-in Ordinary Concrete (density = 2.35 g/cm$^3$)}
\begin{tabular}{|l|c|}
\hline
Element & Weight Fraction \\ \hline
H & 0.006 \\ \hline
C & 0.03 \\ \hline
O & 0.50 \\ \hline
Na & 0.01 \\ \hline
Al & 0.03 \\ \hline
Si & 0.2 \\ \hline
K & 0.01 \\ \hline
Ca & 0.2 \\ \hline
Fe & 0.014 \\ \hline
\end{tabular}
\label{table:ConcreteComposition}
\end{center}
\end{table}
\begin{table}[tbhp]
\begin{center}
\caption{Isotopic Content of Iron (from Wikipedia)}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
Isotope & Abundance \\ \hline
&mole fraction \\ \hline
Fe-54 &0.05845\\ \hline
Fe-56 &0.91754\\ \hline
Fe-57 &0.02119\\ \hline
Fe-58 &0.00282\\ \hline
\end{tabular}
\label{table:FeIsotopicComposition}
\end{center}
\end{table}
\begin{table}[tbhp]
\begin{center}
\caption{Composition (weight fraction) of some common steels. We
acknowledge that the specification of most stainless steel and
other common steels define a range of elemental content. We will
do some calculations with these results so we choose a description
which specifies a definite content and will explore whether the
differences allowed will become of interest. X-ray fluorescence
measurement of Duplex Stainless 2205 beam pipe at MI301 was
carried out on 1/22/2015.}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
Material & Natural Fe & Main Inj Steel & Cast Iron & 316 SS & 304 SS & 2205 SS \\ \hline
Source & & Measured & MARS & MARS & MARS & Measured
\\ \hline
Fe & 1 & 0.9875 & 0.9347 & 0.655 & 0.695 & 0.682 \\ \hline
Mn & 0 & 0.0052 & 0.0018 & 0.02 & 0.02 & 0.00736 \\ \hline
Cr & 0 & 0 & 0 & 0.17 & 0.19 & 0.217 \\ \hline
Ni & 0 & 0 & 0 & 0.12 & 0 & 0.0549 \\ \hline
Mo & 0 & 0 & 0 & 0.025 & 0 & 0.0362 \\ \hline
Cu & 0 & 0 & 0.002 & 0 & 0 & 0.00199 \\ \hline
Sb & 0 & 0.00033 & 0 & 0 & 0 & 0 \\ \hline
P & 0 & 0.00051 & 0 & 0 & 0 & 0 \\ \hline
S & 0 & 0.00006 & 0 & 0 & 0 & 0 \\ \hline
Si & 0 & 0.0036 & 0.025 & 0.01 & 0 & 0 \\ \hline
N & 0 & 0.000023 & 0 & 0 & 0 & 0 \\ \hline
C & 0 & 0.000033 & 0.0365 & 0 & 0 & 0 \\ \hline
\end{tabular}
\label{table:SteelComposition}
\end{center}
\end{table}
\begin{table}[tbhp]
\begin{center}
\caption{MARS15 Material Density}
\begin{tabular}{|l|c|}
\hline
Element & Density (gm/cm$^3$) \\ \hline
Mg & 1.738 \\ \hline
Ca & 1.55 \\ \hline
Conc & 2.35 \\ \hline
marble & 2.7 \\ \hline
Ni & 8.902 \\ \hline
Cr & 7.19 \\ \hline
Mn & 7.33 \\ \hline
Mo & 10.22 \\ \hline
S316 & 7.92 \\ \hline
Cast & 7.31 \\ \hline
Fe & 7.87 \\ \hline
\end{tabular}
\label{table:SampleDensities}
\end{center}
\end{table}
\begin{table}[tbhp]
\begin{center}
\caption{MARS built-in Isotopic Abundance (Mole Fraction) for Calcium}
\begin{tabular}{|l|c|}
\hline
Isotope &Mole Fraction \\ \hline
Ca-40 &0.96941 \\ \hline
Ca-42 &0.00647 \\ \hline
Ca-43 &0.00135 \\ \hline
Ca-44 &0.02086 \\ \hline
% Ca-46 &$4\times 10^{-5}$ \\ \hline
Ca-46 &$4(3)\times 10^{-5}$ \\ \hline
Ca-48 &0.00187 \\ \hline
\end{tabular}
\label{table:CaIsotopes}
\end{center}
\end{table}
\begin{table}[p]
%\begin{table}[tbhp]
\begin{center}
\caption{MARS built-in Marble composition}
\begin{tabular}{|l|c|c|}
\hline
Element & Weight Fraction &Atomic Fraction\\ \hline
Ca & 0.400431 &0.2\\ \hline
C & 0.120005 &0.2\\ \hline
O & 0.479564 &0.6\\ \hline
\end{tabular}
\label{table:marbleComposition}
\end{center}
\end{table}
%\newpage
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\end{document}