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%****                Main Injector Department                    ****
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%****                Document Subject:  Activation               ****
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% Version       Author  Date          Description of modification,deficiencies
%  0.1          BCBrown 29-Jul-2011     Initial entry.
%  0.13         BCBrown 19-Aug-2011     Continue with equations
%  0.15         BCBrown 22-Aug-2011     Uniform Irradiation 
%  0.20         BCBrown 25-Aug-2011     Add weighted BLM option
%  0.25         BCBrown 26-Aug-2011     Formulas complete
%  0.3          BCBrown 06-Sep-2011     Tables
%  0.35         BCBrown 07-Sep-2011     Results
%  0.37         BCBrown 07-Sep-2011     formatting
%  0.40         BCBrown 11-Oct-2011     Text on decay correction, BLM Calib
%  0.42         BCBrown 14-Oct-2011     Getting results after corr for Quant Ave
%  0.45         BCBrown 19-Oct-2011     Formula for Absorbed Dose
%  0.5          BCBrown 21-Oct-2011     Update results table with correct values
%  0.55         BCBrown 24-Oct-2011     Absorbed Dose 
%  0.58         BCBrown 26-Oct-2011     More results, including 5 yr abs dose
%  0.95         BCBrown 27-Oct-2011     Ready for final review
%  0.96         BCBrown 02-Nov-2011     Fix small issues
%  1.00         BCBrown 21-Nov-2011     Final Version 1.0
%  1.05         BCBrown 05-Apr-2011     Revisions for notation from Beams-doc-4046
%  2.0          BCBrown 09-Apr-2011     Version 2.0
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\title{Analysis Procedures for Al Activation Studies}
\author{Bruce C. Brown \\ Accelerator Division, Main Injector Department\\
{\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  60510\\
}

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\begin{abstract}

Using a HPGe detector at the Fermilab Radiation Analysis Facility, we
measure the 1274 keV $\gamma$ ray from activation of $^{22}$Na induced
in Al tags in order to provide a measure of the hadron flux in the
collimation region of the Main Injector.  Using a constant to relate
flux to absorbed dose rate, the absorbed dose is measured.
Procedures, data and some results will be documented.

\end{abstract}

%%%\newpage
\section{Introduction}

\pgph The activation of $^{22}$Na in an $^{27}$Al target by a flux of
secondary hadrons is represented adequately by a constant cross
section of 10.1 millibarns per nucleus above a threshold of 30 MeV.
For most applications, the half life is sufficiently long (2.60 years)
that a measurement of the induced activation can be converted to a
fluence estimate (integrated hadron flux).  We will document that
procedure along with a more accurate measure of the fluence which uses
the history of the activation flux.

\pgph As a part of the Main Injector Collimation effort, a set of 15
locations in the MI300 collimator region were selected for using
aluminum tags to document the fluence of hadrons created by the
collimation of beam loss.  A set of 4 tags was placed at each location
prior to the commissioning of the collimators and a tag from each
location was removed at intervals of one year, two years and three
years.  The hadron fluence at shielded locations beside the secondary
collimators, on the aisle side of the steel and concrete masks (STCM)
and on the aisle side of the steel and marble masks (STMM) will be
compared with predictions of
MARS\cite{Mokhov:1995wa}\cite{Mokhov:2007sz} to help assure compliance
with environmental limits on radiation.  Tags at the upstream or
downstream end of main quadrupoles allow the same comparisons but also
provide a direct estimate of radiation damage (absorbed dose) and a
more refined dose estimate when compared using the MARS results.  Tags
were secured with `duct' tape but on the side of the STCM devices the
tape failed so the location of the tags during activation was less
well determined and didn't match fully the location simulated in MARS.

\section{Formulas}

\subsection{Activity and Activation}

\pgph For a sample with $N_I$ atoms of isotope $I$ which has a half life of
$t_{1/2}$ (or mean lifetime of $\tau$), the number of atoms changes by
decay in accordance with

\begin{equation}
  \mathcal{N}(t) = N_I \, e^{-\frac{t}{\tau}} =  N_I\, 2^{-\frac{t}{t_{1/2}}}
\end{equation}
The number of decays, $S$, also known as the activity of the sample,
is thereby

\begin{equation}
S(t)  =  - \frac{d\mathcal{N}}{dt} = \frac{N_I}{\tau} e^{-\frac{t}{\tau}}
\end{equation}
So the initial activity is 
\begin{equation}
S  =  \frac{N_I}{\tau} =  \frac{N_I \ln 2 }{t_{1/2}}
\end{equation}
This is the activity in becquerel (Bq).  To get the activity in
Curies, one divides by $3.7 \times 10^{10}$. For the activity in 
pico Curies (pCi) one divides by $3.7 \times 10^{-2}$.

\pgph  To obtain the specific activity, $S_A$, one expresses the number of
atoms in moles ($m$) or grams ($M$). For a material with atomic mass
$A$, the number of moles is $m = \frac{M}{A}$ .

\begin{equation}
S_A  (Bq/mole) =  \frac{S}{m} = \frac{N_A \ln 2 }{t_{1/2}}
\end{equation}

\begin{equation}
S_A  (Bq/gm) =  \frac{S}{M } = \frac{N_A \ln 2 }{A \, t_{1/2}}
\end{equation}

\begin{equation}
S_A  (pCi/gm) =  \frac{S}{ 3.7\times 10^{-2} M } = \frac{N_A \ln 2 }{ 3.7\times 10^{-2} A\, t_{1/2}}
\end{equation}
where $N_A$ is the Avogadro Constant.

\subsection{Cross Sections and Fluence}

\pgph 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$^{-2})$; the rate of
production is proportional to the flux, $\frac{d \Phi}{dt}$
(particles-cm$^{-2}$sec$^{-1}$).  In a material with $n_T$ target
atoms per unit volume, an interaction with cross section $\sigma$ will
produce $n_I$ atoms per unit volume of isotope $I$

\begin{equation}
    n_I = \Phi n_T \sigma.
\end{equation}
The activity, $S_A$ (Bq per cm$^3$), produced by $n_I$ atoms per cm$^3$

\begin{equation}
S_A  =  \frac{n_I \ln 2 }{t_{1/2}} = \frac{\Phi n_T \sigma \ln 2 }{t_{1/2}}
\end{equation}
We will want the specific activity per gram of target material, $S_A =S/\rho_T$
(Bq per gram). 

\begin{equation}
S_A (Bq/gm) = \frac{n_I \ln 2 }{\rho_T \, t_{1/2}} 
   = \frac{\Phi n_T \sigma \ln 2 }{\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) =  \frac{\Phi N_A\sigma \ln 2 }{ A_T \, t_{1/2}}
\end{equation}

\begin{equation}
S_A (pCi/gm) =  \frac{\Phi N_A\sigma \ln 2}{ A_T \, t_{1/2}\,3.7\times 10^{-2} }
\end{equation}
We calculate the fluence from the Specific Activity as
\begin{equation}
\Phi = \frac{S_A A_T  t_{1/2} 3.7\times 10^{-2} }{N_A \sigma \ln 2 }
\end{equation}

\subsection{Activation of  $^{27}$Al}
\pgph The reaction $^{27}$Al $\rightarrow$ $^{22}$Na has a total cross
section, $\sigma = 10.1 \times 10^{-27}$ cm$^2$ per atom with a
threshold at about 30 MeV (see Fig. IV.28 on Barbier\cite{Barbier:IR}, Page
194). Taking $N_A = 6.022 \times 10^{23}$, $A_T = 26.98$,
$t_{1/2} = 2.6027$ years or $82.135 \times 10^6$ seconds we find 

\begin{equation}
\Phi (hadrons/cm^2 )= 1.9448 \times 10^{10} S_A (pCi/gm)
\label{Eq:Na22fluence}
\end{equation}

\subsubsection{Particle Flux with Decay Correction from Exposure History}

\pgph For a hadron flux produced by 8 GeV proton beam losses, we assume that
the spatial distribution of shower particles remains relatively fixed
with time variations being due to beam quality (halo) and/or program
requirement changes\cite{Brown:TUO2C04}.  This implies that the
fluence of hadrons, $\Phi$, at a sampling point near a secondary
collimator, for example, will be proportional to the signal integral
on a nearby loss monitor, $L$.

\begin{equation}
  \Phi = \epsilon L
\end{equation}

\pgph When we relate activity of $^{22}$Na to the fluence we will correct for
decays.  For Main Injector operation, loss monitor sums, $L_i$, are recorded
for each acceleration cycle.  As developed in \cite{Brown:TUO2C04},
tools are available for adding these to provided loss history in
either 10 minute or 1 week intervals.  Weekly sums are sufficient for
correcting for $^{22}$Na decay (2.6027 year half life).  

\begin{equation}
LI_j = \sum_{t=t_j}^{t_j+T_s}  LI(t)
\label{Eq:BLMsum}
\end{equation}
To account for decays, we will weight these to provide an exponentially
weighted sum but express the life time using the half life

\begin{equation}
LW(I,T_M) = \sum_{j} LI_j  \,\frac{\ln 2}{t_{1/2}} \, 2^{-(T_M-T_j)/t_{1/2}}
\label{Eq:BLMdecaywsum}
\end{equation}
where $T_M$ is the radiation measurement time, $T_j$ is the
quanta time and $t_{1/2}$ is the half life for isotope $I$. With times
in seconds, $LW$ is in units of Rads/sec.  
We will also want the sum loss without half life weighting

\begin{equation}
L(I,T_M) = \sum_{j} LI_j \frac{\ln 2}{t_{1/2}}
\label{Eq:BLMUWsum}
\end{equation}

\pgph One can alternatively express these sums in Rads rather than in
Rads/sec by $LWu(I,T_M)$ and $Lu(I,T_M)$ where

\begin{equation}
LWu(I,T_M) = \sum_{j} LI_j   2^{-(T_M-T_j)/t_{1/2}}
\label{Eq:BLMdecaywsumu}
\end{equation}
 and 

\begin{equation}
Lu(I,T_M) = \sum_{j} LI_j 
\label{Eq:BLMUWsumu}
\end{equation}

\pgph We can provide the fluence (corrected for decays) by correcting the
uncorrected fluence in Eq. \ref{Eq:Na22fluence} using

\begin{equation}
     \Phi = \Phi_{total} = \Phi_{uncorr} \, \frac{L(I,T_M)}{LW(I,T_M)} 
                         = \frac{Lu(I,T_M)}{LWu(I,T_M)}
\label{Eq:Phi_corr}
\end{equation}
where $\Phi_{uncorr}$ (or $\Phi_{meas}$) is calculated 
using Eq.\ref{Eq:Na22fluence} from the measured activity.

\pgph It will be useful to have the ratio of activation divided by fluence.
The fluence or activity (corrected for decays) by

\begin{equation}
     \frac{\Phi_{total}} {\Phi_{weighted}}
          =  \frac{S_A }{S_{A(meas)}} = \frac{L(I,T_M)}{LW(I,T_M)}
\label{Eq:DecayCorr}
\end{equation}
where $\Phi_{weighted} = \Phi_{uncorr}$ is corrected for the isotope decay
to get $\Phi_{total}$, the fluence integrated over the observation time.

\begin{equation}
     \frac{\Phi_{weighted}}{LW(I,T_M)} = \frac{\Phi_{total}}{L(I,T_M)}
\label{Eq:FluxNorm}
\end{equation}

\begin{equation}
     \frac{S_A }{\Phi_{total}} = \frac{S_{A(meas)}}{\Phi_{uncorr}}
\label{Eq:ActivNorm}
\end{equation}

\begin{equation}
     \frac{S_A }{L(I,T_M)} = \frac{S_{A(meas)}}{LW(I,T_M)}
\label{Eq:ActivPerBLM}
\end{equation}

and/or

\begin{equation}
     \frac{S_A }{Lu(I,T_M)} = \frac{S_{A(meas)}}{LWu(I,T_M)}
\label{Eq:ActivPerBLMu}
\end{equation}

\subsubsection{Particle Fluence for Uniform Irradiation}

\pgph To compare with some typical formulas for activation analysis, we will
consider the case where the fluence is delivered in a flux which is
uniform in time.  For $d\Phi / dt$ a constant for irradiation time from $0$
to $t_i$ (from Barbier\cite{Barbier:IR}, p 15).  This will produce 
$n_I$ nuclei of isotope $I$ per unit volume

\begin{equation}
    n_I (t) = n_T \sigma \frac{d\Phi}{dt} 
             \int_0^{t_i} e^{-(t_i-\tau)/\tau_I} d\tau
\end{equation}
\begin{equation}
     n_I (t) = n_T \sigma \frac{d\Phi}{dt}
             \tau_I (1 - e^{-t_i/\tau_I})
\end{equation}
After a cooling time, $t_c$, the number of atoms will have decayed to 
\begin{equation}
     n_I (t_c) =  n_T \sigma \frac{d\Phi}{dt}
           \tau_I (1 - e^{-t_i/\tau_I})e^{-t_c/\tau_I}
\end{equation}
So if $n_I (t_c)$ atoms per cm$^3$ of isotope, $I$, remain after a
uniform irradiation for $t_i$ and cool down $t_c$, we will have activity
of $S_A (Bq/gm) = n_I / (\tau_I \rho_T)$.  Again we can substitute for $n_T$

\begin{equation}
     S_A(t_c) (Bq/gm) = \frac{N_A}{A_T} \sigma \frac{d\Phi}{dt}
          (1 - e^{-t_i/\tau_I})e^{-t_c/\tau_I}
\end{equation}
alternatively, using $M/A_T = N_T/N_A$ we have 

\begin{equation}
     S_A(t_c) (Bq/gm) = \frac{N_T}{M} \sigma \frac{d\Phi}{dt}
          (1 - e^{-t_i/\tau_I})e^{-t_c/\tau_I}
\end{equation}
Now, using pCi rather than Bq, we calculate the flux

\begin{equation}
    \frac{d\Phi}{dt} = \frac{M \,\,3.7 \times 10^{-2} }{N_T \sigma }
        \frac{e^{t_c/\tau_I}}{(1 - e^{-t_i/\tau_I})}  S_A(t_c)(pCi/gm)
       = \frac{A_T \,\, 3.7 \times 10^{-2} }{N_A \sigma }
        \frac{e^{t_c/\tau_I}}{(1 - e^{-t_i/\tau_I})} S_A(t_c)(pCi/gm)
\end{equation}
We convert this corrected flux to a fluence by multiplying by the irradiation
time, $t_i$.

\begin{equation}
\Phi (hadrons/cm^2 ) =  \frac{d\Phi}{dt} t_i = \frac{A_T \,t_i\, 3.7 \times 10^{-2} }{N_A \sigma } \frac{e^{t_c/\tau_I}}{(1 - e^{-t_i/\tau_I})} S_A(t_c)(pCi/gm)
\end{equation}
For the uniform exposure assumption we find 

\begin{equation}
\Phi (hadrons/cm^2)  = \Phi_{total} = \Phi_{uncorr} \frac{t_i \,\ln 2}{t_{1/2}} 
            \frac{e^{t_c/\tau_I}}{(1 - e^{-t_i/\tau_I})}
\end{equation}


\subsubsection{Check Uniform Irradiation Against Irradiation History}

\pgph For long half life isotopes, the correction for irradiation exposure
in the Main Injector is relatively easy since the flux relation
between BLM locations and residual radiation locations has remained
sufficiently constant at most locations.  For $^{22}$Na with its
2.6027 year half life, corrections using weekly exposure sums are
adequate. To obtain fluence corrected for decays at the locations used
for Al Tags, we identify a nearby loss monitor and use
Eq~\ref{Eq:Phi_corr} for correction.  In Table~\ref{table:CorrCompare}
we show the correction for uniform exposure assumption along with the
exposure-weighted results for our six BLM's.  Al Tags were installed
on 10/12/2007.  

\begin{table}[tbhp]
\begin{center}
\caption{Comparison of Fluence Exposure Corrections }
\begin{tabular}{|l|c|c|c|c|c|}
\hline
BLM      & End         &Unweighted   & Weighted   &Weighted   & Uniform     \\       
         &Irradiation  &  Dose       & Dose       &Irradiation& Irradiation \\       
         &             &  (Rads)     & (Rads)     &Correction & Correction \\ \hline 
LI230    &10/08/2008   &939837.4272  & 857958.3111&1.095    &1.138   \\ \hline 
         &08/26/2009   &2804366.275  & 2334284.604&1.201    &1.270   \\ \hline 
         &08/12/2010   &3282460.675  & 2229548.281&1.472    &1.424   \\ \hline 
LI301    &10/08/2008   &258874.9632  & 236336.3585&1.095    &1.138   \\ \hline 
         &08/26/2009   &523647.936   & 420592.0558&1.245    &1.270   \\ \hline 
         &08/12/2010   &1339301.174  & 1056773.802&1.267    &1.424   \\ \hline 
LI302    &10/08/2008   &718681.4208  & 655374.0178&1.097    &1.138   \\ \hline 
         &08/26/2009   &1419595.632  & 1139964.994&1.245    &1.270   \\ \hline 
         &08/12/2010   &3369995.798  & 2652941.871&1.270    &1.424   \\ \hline 
LI303    &10/08/2008   &1266726.384  & 1158874.657&1.093    &1.138   \\ \hline 
         &08/26/2009   &2381237.309  & 1892791.13 &1.258    &1.270   \\ \hline 
         &08/12/2010   &4540779.734  & 3380354.205&1.343    &1.424   \\ \hline 
LI307    &10/08/2008   &363648.096   & 341515.3697&1.065    &1.138   \\ \hline 
         &08/26/2009   &1061289.734  & 878593.2369&1.208    &1.270   \\ \hline 
         &08/12/2010   &1761611.846  & 1290676.984&1.365    &1.424   \\ \hline 
LI309    &10/08/2008   &521118.0576  & 484220.0908&1.076    &1.138   \\ \hline 
         &08/26/2009   &1278976.608  & 1040719.564&1.229    &1.270   \\ \hline 
         &08/12/2010   &2124588.01   & 1540207.644&1.379    &1.424   \\ \hline 	 
\end{tabular}
\label{table:CorrCompare}
\end{center}
\end{table}

\pgph Note that we seek the fluence produced by a proton loss.  To get
the Activation/BLM with correction one increases the Activation by the
correction or decreases the unweighted BLM reading.  The uniform
irradiation correction is typically a few percent larger than the
weighted correction.  This is because the collimation system
commissioning during this period saw increased losses later in the
exposure.  The exception is for LI230 for the longer period since the
tuning eventually improved the loss at the primary collimator.

\begin{table}[tbhp]
\begin{center}
\caption{Comparison of One Year and Two Year Activation per Loss are
shown in Columns 3 and 4 where $Lu$ is total recorded loss and $LWu$ 
is loss weighted for $^{22}$Na decay. Comparisons including the 2010
tag analysis are shown in Columns 5 and 6, expressed as averages and 
standard deviations. }
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Tag Location       & BLM&  ${S_A}/{Lu}$& $ {S_A}/{LWu}$ & 3 Yr Average    & 3 Yr StDev/$<M>$\\ \hline
                   &    & 2009/2008   & 2009/2008 & $ {S_A}/{LW}$&\\ \hline
C301		& LI302	& 0.955	    &  1.085  & 0.000423 & 0.239 \\ \hline
C303		& LI303	& 0.945	    &  1.087  & 0.000770 & 0.163 \\ \hline
C307		& LI307	& 1.031	    &  1.170  & 0.001167 & 0.152 \\ \hline
STMM301		& LI302	& 0.853	    &  0.969  & 0.009192 & 0.193 \\ \hline
STMM303		& LI303	& 0.734	    &  0.845  & 0.005600 & 0.117 \\ \hline
STMM308		& LI309	& 0.864	    &  0.987  & 0.006968 & 0.167 \\ \hline
Q301DS		& LI301	& 0.834	    &  0.948  & 0.014152 & 0.047 \\ \hline
Q303DS		& LI303	& 0.888	    &  1.021  & 0.001838 & 0.122 \\ \hline
Q307DS		& LI307	& 0.981	    &  1.113  & 0.001912 & 0.136 \\ \hline
Q230DS		& LI230	& 1.009	    &  1.107  & 0.003552 & 0.199 \\ \hline
Q302US		& LI302	& 0.803	    &  0.912  & 0.013054 & 0.129 \\ \hline
		& 	& 	    &         &          &       \\ \hline
Average Ratio	& 	& 0.900	    &  1.022  &          &       \\ \hline
StDev Ratio	& 	& 0.093	    &  0.099  & Average StDev   & 0.151    \\ \hline
\end{tabular}
\label{table:ActCompare}
\end{center}
\end{table}

\pgph A step in understanding the precision of the activation analysis
can be obtained by comparing the activation (pCi/gm) with the BLM loss
(Rads) with decay correction for two successive measurements.
Table~\ref{table:ActCompare} shows the ratio $S_A/L$ and $S_A/LW$ for
collimator, quadrupole and some of the mask locations.  Unfortunately
the installation of samples secured to the side of the concrete mask
material by duct tape failed during the first or second year, yielding
no data for comparison at those locations.  The addition of masks
between C307 and C308 may have modified the flux at the C308 tag
location so we do not include that location in this comparison.  For
the (appropriately) corrected data, the average ratio for 2009/2008 is
1.022 with a Standard Deviation of 0.099.  Unsurprisingly, the
uncorrected ratio is only 0.900 corresponding to the difference in
cooling factors.  We consider the tags measured in 2010 by taking the
mean and standard deviation of $S_A/LW$ for sets of three tags at one
location.  Table~\ref{table:ActCompare} shows these values in its last
two columns.  The agreement is not as good with the average StDev = 0.151.

\subsection{Estimate of Energy Deposition}

\pgph In the Main Injector tunnel, the most common concern about
radiation damage is degradation of the epoxy insulated magnet coils.
Various components of the hadronic shower due to interaction of 8 GeV
protons contribute to radiation damage, an estimate of the total
energy deposit is a useful guide.  We will estimate this from an
estimate of the ionization energy deposited in a volume. Some of the
Al tags were placed for the purpose of evaluating the absorbed dose at
coils.  The fluence estimate provided by Al activation measurements
counts the hadrons above a threshold of about 30 MeV.  Although
neutrons dominate this spectrum, the shower process develops an
equilibrium distribution among charged and neutral components (as well
as between hadronic and electromagnetic components) such that energy
deposition estimates can simply relate the ionization loss to the
typical minimum ionization energy loss for a hadronic charged particle
of 2 MeV/gm-cm$^{-2}$.  For an absorbed dose, $d$, we use
the following:

\begin{equation}
    d = C \, \sum_j \int^\infty_0 dE \Phi_j (E) \frac{dE}{dx} (E,j)
\label{Eq:AbsDose}
\end{equation}
where where $\frac{dE}{dx}$ is in MeV-cm$^{2}$/gm-cm, $\Phi_j$ is the
flux of particles of type $j$ in particles/cm$^{2}$ and $C$ a constant
discussed below.  The approximation we will use is

\begin{equation}
    d \sim C \,  \Phi_j < \frac{dE}{dx}>
\label{Eq:AproxAbsDose}
\end{equation}
where we get d in Rad or Gray (Gy)  using
$C = $  1.602E-8 Rad-gm/Mev   or   $C =$ 1.6E-10 Gy-gm/Mev
and we take $< \frac{dE}{dx}> = 2$ MeV/gm-cm$^2$.  
Using Eq.~\ref{Eq:Na22fluence}, we find 

\begin{equation}
    d (Rad) \sim 1.602\times 10^{-8} Rad-gm/Mev \times < \frac{dE}{dx}>\times 
 \,1.9448 \times 10^{10} S_A (pCi/gm)  = 623 S_A (pCi/gm)
\label{Eq:AproxAbsDosefromFlux}
\end{equation}

\pgph This is only a crude estimate.  The MARS simulation will more
carefully relate the Al tag activation to the absorbed dose.

\section{Calibrations of BLM to Lost Protons}

\pgph Using the Al Tags activation to determine the fluence gives a ratio of
fluence at the tags to BLM signal at a nearby loss monitor.
Calibration of the BLM to the lost beam can be done based on either
sums from weekly or longer times or on measurements on one or a few
pulses of ordinary operation. See below for calibration using aperture
scans.  For this note we will calibrate to the distribution of
uncaptured beam loss on one pulse.  Using the uncaptured beam loss and
the signals of all the BLM's in the collimation region as recorded by
Application Program I129, we get a crude loss measurement.  Almost all
loss is from 8 GeV beam either before acceleration begins or at
uncaptured beam time.  We assume that the loss distribution and
thereby the BLM calibration will be similar for injection loss and
uncaptured beam loss.  Comparisons with other loss determinations will
be done later.  MARS results for the fluence can be compared with
these measurements based on the proton loss in each secondary
collimator.

\pgph Using a single pulse of \$23 Cycle Beam (PBar plus NuMI beam), a
measurement of the collimator loss was used for a crude calibration of
the beam loss monitor response.  The uncaptured beam loss from a pulse
in which 34.83E12 protons was injected was 1.12E12 (3.2\%).  This was
allocated to the 4 secondary collimators in accordance with the sum of
BLM signal in the two adjacent loss monitors (LI301 + LI302 for C301);
(LI303 + LI304 for C303); (LI307 + LI308 for C307); (LI309 for C308).
These loss monitors accounted for 78\% of the total loss.  The other
22\% of the loss includes BLM signal from protons which interacted in
the secondary collimators but had leakage particle which escaped to
nearby regions.  It is then assumed that either of the pair of loss
monitors can be used to determine the protons lost on that collimator.
Applying this to LI308 would be OK for a fixed geometry but masks were
added twice since the initial collimator installation so we will avoid
further use of LI308 for this study.  Data was recorded in file
pfl-11June13-143510-amc01.csv and analyzed in same file name with
extension .xlxs.  Table~\ref{table:BLMCalib} shows the calibration
values used.


\begin{table}[tbhp]
\begin{center}
\caption{Calibration of BLM readings }
\begin{tabular}{|l|c|}
\hline
BLM    &  Rads/E12Protons\\ \hline
LI230(Aperture Scan)&  0.34 \\ \hline
LI301  &  0.11465\\ \hline
LI302  &  0.33161\\ \hline
LI303  &  0.36167\\ \hline
LI304  &  0.08459\\ \hline
LI307  &  0.24201\\ \hline
LI308(skip)  &  0.20425\\ \hline	
LI309  &  0.44626\\ \hline
\end{tabular}
\label{table:BLMCalib}
\end{center}
\end{table}

\subsection{Calibration Using Aperture Scan}

\pgph An alternative method to calibrate the Beam Loss Monitors to the
local beam loss is to move the beam with a 3-bump until the local loss
monitor shows a sufficient signal and note the change in transmission
which resulted.  Performing such a calibrated loss measurement at many
or all of the loss monitors would be helpful in further understanding
of the BLM's, the apertures and the loss around the ring.  For this
study, we have crude calibrations using the method above.  For the
loss monitor at the primary collimator, the loss due to striking the
primary collimator is mostly distributed to the various secondary
collimators.  Using the I79 MI Aperture Scan console application
program, the beam was moved radially outward with the I:H230:3 bump
while a fast time plot recorded I:BEAM, I:BMLOST, I:HP230, and
I:LI230.  With an injection bump of +30 mm, about 10\% of the beam was
lost (\$29 cycle at 8E12).  The result is a signal for radial outward
loss of 0.34 Rad/1E12 protons lost.

\begin{table}[ht]
{ \small
\begin{center}
\caption{Results for Activation Analysis for Al Tags installed 10/12/07 \newline 
* indicates tag not found at installation position.  }
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Location    & Tag ID& Date	& Activ	& BLM	& UnWt BLM	& fluence & Lost  & Fluence per\\ 
	& 	&Removed  	& pCi/gm	& 	& Rads	& hadrons/cm$^2$ &Protons 	&Proton Lost \\ \hline
C301	& 6510	&  10/08/08	& 230	& LI302	& 7.187E+05& 4.90523E+12 & 2.167E+18 	&2.263E-06\\ \hline
	& 6198	&  08/26/09	& 434	& 	& 1.420E+06& 1.05111E+13 & 4.281E+18 	&2.455E-06\\ \hline
	& 6187	&  08/12/10	& 1430	& 	& 3.370E+06& 3.53282E+13 & 1.016E+19    &3.476E-06\\ \hline
C303	& 6368	&  10/08/08	& 777	& LI303	& 1.267E+06& 1.65178E+13 & 3.502E+18 	&4.716E-06\\ \hline
	& 6503	&  08/26/09	& 1380	& 	& 2.381E+06& 3.37647E+13 & 6.584E+18 	&5.128E-06\\ \hline
	& 6548	&  08/12/10	& 3080	& 	& 4.541E+06& 8.04643E+13 & 1.369E+19    &5.876E-06\\ \hline
C307	& 6325	&  10/08/08	& 339	& LI307	& 3.636E+05& 7.02029E+12 & 1.503E+18 	&4.672E-06\\ \hline
	& 6168	&  08/26/09	& 1020	& 	& 1.061E+06& 2.39624E+13 & 4.385E+18 	&5.464E-06\\ \hline
	& 6559	&  08/12/10	& 1740	& 	& 1.762E+06& 4.61877E+13 & 5.312E+18    &8.695E-06\\ \hline
C308	& 6318	&  10/08/08	& 1690	& LI309	& 5.211E+05& 3.53724E+13 & 1.168E+18 	&3.029E-05\\ \hline
	& 6833	&  08/26/09	& 5210	& 	& 1.279E+06& 1.24523E+14 & 2.866E+18 	&4.345E-05\\ \hline
	& 6534	&  08/12/10	& 6020	& 	& 2.125E+06& 1.61501E+14 & 6.407E+18    &2.521E-05\\ \hline
STCM301	& 6820*	&  10/08/08	& 2310	& LI302	& 7.187E+05& 4.92656E+13 & 2.167E+18 	&2.273E-05\\ \hline
	& 6834*	&  08/26/09	& 14300	& 	& 1.420E+06& 3.46333E+14 & 4.281E+18 	&8.090E-05\\ \hline
	& 6507*	&  08/12/10	& 28800	& 	& 3.370E+06& 7.11506E+14 & 1.016E+19    &7.001E-05\\ \hline
STCM303	& 6842	&  10/08/08	& 7430	& LI303	& 1.267E+06& 1.5795E+14  & 3.502E+18 	&4.510E-05\\ \hline
	& 6167*	&  08/26/09	& 6660	& 	& 2.381E+06& 1.62952E+14 & 6.584E+18 	&2.475E-05\\ \hline
	& 6811*	&  08/12/10	& 26600	& 	& 4.541E+06& 6.94919E+14 & 1.369E+19    &5.075E-05\\ \hline
STCM308	& 6828*	&  10/08/08	& 1210	& LI309	& 5.211E+05& 2.53258E+13 & 1.168E+18 	&2.169E-05\\ \hline
	& 6822*	&  08/26/09	& 10300	& 	& 1.279E+06& 2.46179E+14 & 2.866E+18 	&8.590E-05\\ \hline
	& 6319*	&  08/12/10	& 25800	& 	& 2.125E+06& 6.92149E+14 & 6.407E+18    &1.080E-04\\ \hline
STMM301	& 6841	&  10/08/08	& 5440	& LI302	& 7.187E+05& 1.16019E+14 & 2.167E+18 	&5.353E-05\\ \hline
	& 6821*	&  08/26/09	& 9170	& 	& 1.420E+06& 2.22089E+14 & 4.281E+18 	&5.188E-05\\ \hline
	& 6110*	&  08/12/10	& 29800	& 	& 3.370E+06& 7.36211E+14 & 1.016E+19    &7.244E-05\\ \hline
STMM303	& 6502	&  10/08/08	& 6690	& LI303	& 1.267E+06& 1.42219E+14 & 3.502E+18 	&4.061E-05\\ \hline
	& 6538	&  08/26/09	& 9230	& 	& 2.381E+06& 2.25832E+14 & 6.584E+18 	&3.430E-05\\ \hline
	& 6659	&  08/12/10	& 20800	& 	& 4.541E+06& 5.43395E+14 & 1.369E+19    &3.968E-05\\ \hline
STMM308	& 6169	&  10/08/08	& 3070	& LI309	& 5.211E+05& 6.42563E+13 & 1.168E+18 	&5.503E-05\\ \hline
	& 6335	&  08/26/09	& 6510	& 	& 1.279E+06& 1.55594E+14 & 2.866E+18 	&5.429E-05\\ \hline
	& 6523	&  08/12/10	& 12800	& 	& 2.125E+06& 3.43392E+14 & 6.407E+18    &5.360E-05\\ \hline
Q301DS	& 6195	&  10/08/08	& 3510	& LI301	& 2.589E+05& 7.4774E+13  & 2.258E+18 	&3.312E-05\\ \hline
	& 6605	&  08/26/09	& 5920	& 	& 5.236E+05& 1.43346E+14 & 4.567E+18 	&3.139E-05\\ \hline
	& 6200	&  08/12/10	& 14300	& 	& 1.339E+06& 3.52465E+14 & 4.039E+18    &8.727E-05\\ \hline
Q303DS	& 6831	&  10/08/08	& 1960	& LI303	& 1.267E+06& 4.16665E+13 & 3.502E+18 	&1.190E-05\\ \hline
	& 6330	&  08/26/09	& 3270	& 	& 2.381E+06& 8.00077E+13 & 6.584E+18 	&1.215E-05\\ \hline
	& 6511	&  08/12/10	& 7090	& 	& 4.541E+06& 1.85225E+14 & 1.369E+19    &1.353E-05\\ \hline
Q307DS	& 6516	&  10/08/08	& 573	& LI307	& 3.636E+05& 1.18662E+13 & 1.503E+18 	&7.897E-06\\ \hline
	& 6515	&  08/26/09	& 1640	& 	& 1.061E+06& 3.85278E+13 & 4.385E+18 	&8.786E-06\\ \hline
	& 6344	&  08/12/10	& 2830	& 	& 1.762E+06& 7.51213E+13 & 5.312E+18    &1.414E-05\\ \hline
Q230DS	& 6327	&  10/08/08	& 2570	& LI230	& 9.398E+05& 5.47525E+13 & 2.764E+18    &1.981E-05\\ \hline
	& 6513	&  08/26/09	& 7740	& 	& 2.804E+06& 1.80845E+14 & 8.248E+18    &2.193E-05\\ \hline
	& 6521	&  08/12/10	& 9690	& 	& 3.282E+06& 2.77454E+14 & 9.898E+18    &2.803E-05\\ \hline
Q302US	& 6275	&  10/08/2008	& 8320	& LI302	& 7.187E+05& 1.77441E+14 & 2.167E+18 	&8.187E-05\\ \hline
	& 6504	&  08/26/2009	& 13200	& 	& 1.420E+06& 3.19692E+14 & 4.281E+18 	&7.468E-05\\ \hline
	& 6508	&  08/12/2010	& 39500	& 	& 3.370E+06& 9.7585E+14  & 1.016E+19    &9.603E-05\\ \hline
\end{tabular}
\label{table:ActResult}
\end{center}
}
\end{table}

\begin{table}[ht]
{ \small
\begin{center}
\caption{Further Results for Activation Analysis for Al Tags installed 10/12/07 \newline 
* indicates tag not found at installation position.  }
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Location &Tag ID&Activity& BLM ID&Wt BLM(LWu)	&$ {S_A}/{LWu}$& AbsDose         &AD/BLM&AbsDose/$p_{lost}$ \\
	&	&pCi/gm	&	&  Rads		&pCi/gm/Rad	& Rads          &Rads/Rad&Rads/proton	 \\ \hline
C301	& 6510	& 230	& LI302	& 655374.0178	& 3.509E-04	& 1.433E+05	& 0.219	& 6.613E-14 \\ \hline
	& 6198	& 434	& 	&   1139964.994	& 3.807E-04	& 2.704E+05	& 0.237	& 6.317E-14 \\ \hline
	& 6187	& 1430	& 	&   2652941.871	& 5.390E-04	& 8.911E+05	& 0.336	& 8.768E-14 \\ \hline
C303	& 6368	& 777	& LI303	& 1158874.657	& 6.705E-04	& 4.842E+05	& 0.418	& 1.382E-13 \\ \hline
	& 6503	& 1380	& 	&   1892791.13	& 7.291E-04	& 8.599E+05	& 0.454	& 1.306E-13 \\ \hline
	& 6548	& 3080	& 	&   3380354.205	& 9.111E-04	& 1.919E+06	& 0.568	& 1.402E-13 \\ \hline
C307	& 6325	& 339	& LI307	& 341515.3697	& 9.926E-04	& 2.112E+05	& 0.619	& 1.406E-13 \\ \hline
	& 6168	& 1020	& 	&   878593.2369	& 1.161E-03	& 6.356E+05	& 0.723	& 1.449E-13 \\ \hline
	& 6559	& 1740	& 	&   1290676.984	& 1.348E-03	& 1.084E+06	& 0.840	& 2.041E-13 \\ \hline
C308	& 6318	& 1690	& LI309	& 484220.0908	& 3.490E-03	& 1.053E+06	& 2.175	& 9.018E-13 \\ \hline
	& 6833	& 5210	& 	&   1040719.564	& 5.006E-03	& 3.246E+06	& 3.119	& 1.133E-12 \\ \hline
	& 6534	& 6020	& 	&   1540207.644	& 3.909E-03	& 3.751E+06	& 2.436	& 5.855E-13 \\ \hline
STCM301	& 6820*	& 2310	& LI302	& 655374.0178	& 3.525E-03	& 1.439E+06	& 2.196	& 6.642E-13 \\ \hline
        & 6834*	& 14300	& 	&   1139964.994	& 1.254E-02	& 8.911E+06	& 7.817	& 2.082E-12 \\ \hline
	& 6507*	& 28800	& 	&   2652941.871	& 1.086E-02	& 1.795E+07	& 6.765	& 1.766E-12 \\ \hline
STCM303	& 6842	& 7430	& LI303	& 1158874.657	& 6.411E-03	& 4.630E+06	& 3.995	& 1.322E-12 \\ \hline
	& 6167*	& 6660	& 	&   1892791.13	& 3.519E-03	& 4.150E+06	& 2.193	& 6.303E-13 \\ \hline
	& 6811*	& 26600	& 	&   3380354.205	& 7.869E-03	& 1.658E+07	& 4.903	& 1.210E-12 \\ \hline
STCM308	& 6828*	& 1210	& LI309	& 484220.0908	& 2.499E-03	& 7.540E+05	& 1.557	& 6.457E-13 \\ \hline
	& 6822*	& 10300	& 	&   1040719.564	& 9.897E-03	& 6.418E+06	& 6.167	& 2.239E-12 \\ \hline
	& 6319*	& 25800	& 	&   1540207.644	& 1.675E-02	& 1.608E+07	& 10.438& 2.509E-12 \\ \hline
STMM301	& 6841	& 5440	& LI302	& 655374.0178	& 8.301E-03	& 3.390E+06	& 5.172	& 1.564E-12 \\ \hline
	& 6821*	& 9170	& 	&   1139964.994	& 8.044E-03	& 5.714E+06	& 5.013	& 1.335E-12 \\ \hline
	& 6110*	& 29800	& 	&   2652941.871	& 1.123E-02	& 1.857E+07	& 6.999	& 1.827E-12 \\ \hline
STMM303	& 6502	& 6690	& LI303	& 1158874.657	& 5.773E-03	& 4.169E+06	& 3.597	& 1.190E-12 \\ \hline
	& 6538	& 9230	& 	&   1892791.13	& 4.876E-03	& 5.751E+06	& 3.039	& 8.736E-13 \\ \hline
	& 6659	& 20800	& 	&   3380354.205	& 6.153E-03	& 1.296E+07	& 3.834	& 9.465E-13 \\ \hline
STMM308	& 6169	& 3070	& LI309	& 484220.0908	& 6.340E-03	& 1.913E+06	& 3.951	& 1.638E-12 \\ \hline
	& 6335	& 6510	& 	&   1040719.564	& 6.255E-03	& 4.057E+06	& 3.898	& 1.415E-12 \\ \hline
	& 6523	& 12800	& 	&   1540207.644	& 8.311E-03	& 7.976E+06	& 5.179	& 1.245E-12 \\ \hline
Q301DS	& 6195	& 3510	& LI301	& 236336.3585	& 1.485E-02	& 2.187E+06	& 9.255	& 9.687E-13 \\ \hline
	& 6605	& 5920	& 	&   420592.0558	& 1.408E-02	& 3.689E+06	& 8.771	& 8.077E-13 \\ \hline
	& 6200	& 14300	& 	&   1056773.802	& 1.353E-02	& 8.911E+06	& 8.432	& 2.206E-12 \\ \hline
Q303DS	& 6831	& 1960	& LI303	& 1158874.657	& 1.691E-03	& 1.221E+06	& 1.054	& 3.487E-13 \\ \hline
	& 6330	& 3270	& 	&   1892791.13	& 1.728E-03	& 2.038E+06	& 1.077	& 3.095E-13 \\ \hline
	& 6511	& 7090	& 	&   3380354.205	& 2.097E-03	& 4.418E+06	& 1.307	& 3.226E-13 \\ \hline
Q307DS	& 6516	& 573	& LI307	& 341515.3697	& 1.678E-03	& 3.571E+05	& 1.045	& 2.376E-13 \\ \hline
	& 6515	& 1640	& 	&   878593.2369	& 1.867E-03	& 1.022E+06	& 1.163	& 2.330E-13 \\ \hline
	& 6344	& 2830	& 	&   1290676.984	& 2.193E-03	& 1.763E+06	& 1.366	& 3.320E-13 \\ \hline
Q230DS	& 6327	& 2570	& LI230	& 857958.3111	& 2.995E-03	& 1.601E+06	& 1.867	& 5.793E-13 \\ \hline
	& 6513	& 7740	& 	&   2334284.604	& 3.316E-03	& 4.823E+06	& 2.066	& 5.847E-13 \\ \hline
	& 6521	& 9690	& 	&   2229548.281	& 4.346E-03	& 6.038E+06	& 2.708	& 6.100E-13 \\ \hline
Q302US	& 6275	& 8320	& LI302	& 655374.0178	& 1.270E-02	& 5.184E+06	& 7.911	& 2.392E-12 \\ \hline
	& 6504	& 13200	& 	& 1139964.994	& 1.158E-02	& 8.225E+06	& 7.215	& 1.921E-12 \\ \hline
	& 6508	& 39500	& 	& 2652941.871	& 1.489E-02	& 2.461E+07	& 9.278	& 2.422E-12 \\ \hline
\end{tabular}
\label{table:ActResultII}
\end{center}
}
\end{table}

\newpage
\clearpage

\section{Results}

\pgph In order to provide flexibility in employing this data, we will
supply raw data with only fundamental corrections and various ratios
using the BLM's and calibrations of protons lost in terms of the BLM's
which are not yet determined with good accuracy.  We will expect to
compare with MARS calculations in terms of ratios to both the BLM's
and to protons lost.

\pgph The RAF Reports which provide the measured activities and their
error (corrected to the sampling time when the samples were delivered
to RAF) are provided as separate files in this Beams Document Database
document.  The uncertainty on the activity measurement is about 15\%
plus a small contribution for counting error.  The data and analysis
of these aluminum tags activations are carried out with a spreadsheet,
AlActData.xls which is another file incorporated in the Beams Document
Database with the document.  An analysis which provides flux and
fluence estimates by RAF are included as a worksheet.  BLM data is
incorporated in the spreadsheet and the $^{22}$Na decay correction is
calculated there.  The main body of analysis is carried out in
worksheet ColAlTags.

\pgph The raw activation and BLM data are recorded here and used in the 
analysis.  The decay correction for cooling time has been applied at RAF.
We correct for decay during irradiation using the weekly BLM summed data.

\pgph Using BLM calibration values above for RADS per protons lost ,
we now provide the Fluence at the Al Tag locations normalized to the
lost protons inferred from nearby loss monitors.
Table~\ref{table:ActResult} shows the data from this activation study.
We include the tag and BLM locations, activation, BLM sum, fluence,
lost protons and fluence per lost proton.  Note that the fluence is
calculated after correcting the activation for decay during
irradiation.

\pgph Additional results including the activity (again), absorbed dose
and ratios: Activity/BLM, Absorbed dose /BLM and Absorbed Dose/Protons
lost are provided in Table~\ref{table:ActResultII}.  For these
ratios, we used the measured activity and $LW$, the loss monitor sum
weighted for decay.  We note that the calibration of the BLM in Rads
is accepted from the BLM system based on a calibration of the
ionization monitor from two decades ago whereas we use a fluence to
dose constant for calibration of the Al Tag results.  We find that
tags shielded by the C301, C303 and C307 collimators see less dose
than the BLM's which are placed on the wall of the MI tunnel.  We
presume that the tag at C308 was exposed to losses not fully absorbed
in C307 since it was not protected by masks (STCM nor STMM) during the
time of this sampling.  Similarly the tags on the masks and the
quadrupoles see more absorbed dose than the BLM but all these results
fall in the range of 0.2 to 10.  If better calibrations of the BLM's
in terms of lost protons are obtained, new calibrations should be
applied to this data.


\begin{table}[b]
\begin{center}
\caption{5 Year Absorbed Dose at Some Collimator Region Quadrupoles
based on Using Fluence to Dose Constant. BLM losses summed from Oct 2006
through Oct 2011.}
\begin{tabular}{|l|c|c|c|c|}
\hline
Location & BLM	&$<$Abs Dose$>$/BLM & 5 Yr BLM	& 5 Yr AbsDose \\
         &      & Rads/Rad	& Rads	        & Rads \\ \hline
Q301DS	 & LI301	& 8.82	& 2.361E+06	& 2.082E+07 \\ \hline
Q303DS	 & LI303	& 1.15	& 6.959E+06	& 7.973E+06 \\ \hline
Q307DS	 & LI307	& 1.19	& 2.288E+06	& 2.726E+06 \\ \hline
Q230DS	 & LI230	& 2.21	& 3.614E+06	& 8.000E+06 \\ \hline
Q302US	 & LI302	& 8.13	& 6.303E+06	& 5.127E+07 \\ \hline
\end{tabular}
\label{table:ActResultIII}
\end{center}
\end{table}

\pgph With these measurements, we are able to address one of the
concerns about the collimation installation.  Radiation damage to
accelerator components in the collimation region has the extra
penalty, beyond the usual issue of spares and downtime due to the
higher radiation levels in this region.  Radiation damage due to
interactions in the primary collimator create dose at the downstream
end of Q230.  Shine downstream of C301 produces loss on the upstream
end of Q302. Backscatter from C301, C303, and C307 produces radiation
at the downstream end of Q301, Q303, and Q307 respectively.
Polyethylene blocks were installed on the upstream end of the
collimators to mitigate some of this problem but we installed Al tags
to measure activation which we relate to the absorbed dose.  While the
uncertainties in the fluence to dose calculation are substantial, our
knowledge of the radiation hardness of the epoxy used in the
insulation of the magnet coils is also uncertain.  In TM-2391, a
damage concern at 4 MGy (400 MRad) absorbed dose is quoted based on
information from the Proton Driver design study.  In the Main Injector
Design Handbook (Chapter 3.1 p. 25), studies were reported which
assure resistance to 100 MRad and limited measurements to 1 GRad
showed retention of good physical properties.  In Table~\ref{table:ActResultIII}
we use the average of our measurements of absorbed dose normalized to
the BLM reading and the full 5 years of loss data at these locations
since the current BLM electronics provided recorded data.  The collimation
system has been in use for about 3.5 years of this interval.  We find
that Q301DS has an absorbed dose of 20 MRad, Q302US has 50 MRad and 
others have less absorbed does.  We should still be several years from
problems with radiation damage to coils. 
 
\section{Conclusions}

\pgph The activation of aluminum tags by production of $^{22}$Na
provides a measure of the hadronic flux at fifteen locations in the
Main Injector Collimation region.  A high purity germanium
spectrometer is used to measure the activation.  Using detailed loss
history in nearby beam loss monitors (BLM's), we have corrected for
decay during the irradiation period.  Samples removed after
irradiation for 1 year, 2 years and 3 years show activation
proportional to the loss at the nearby loss monitor with uncertainties
of 15\%.  We convert the activation to a fluence of hadrons using the
known spallation cross section on Al of 10.1 microbarns.  Using a
crudely determined BLM sensitivity to proton losses, we provide the
results as a ratio to the BLM reading or to the lost protons.  These
results are suitable for comparison with simulations of the collimator
system using the MARS code. Additional activation studies on Al, Cu
and steel at collimator C307 which were carried out in 2011 are
reported in Beams-doc-4046\cite{BeamsDoc4046}.

\section{Acknowledgments}

\pgph This note is intended only to document this effort.  The
contributions from Vernon Cupps, Gary Lauten and others are the basis
for this report.

\section{Appendix A - List of Symbols}

\begin{description}
\item[A] Atomic Mass Number or Atomic Weight
\item[$N_A$] Avogadro Constant (Avogadro's number), $6.02214179(30)\times10^{23}$ mol$-1$ (gram).

\item[$S_A$] Specific activity (in Bq) -- number of decays per second per amount of substance
\item[$\Phi$] fluence in particles per unit area (particles-cm$^{-2}$)

\item[$\frac{d\Phi}{dt}$] flux in particles per unit area per second (particles-cm$^{-2}$-s$^{-1}$)

\end{description}


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