INSTRUCTIONS FOR USING REAZ_M
This is a finite difference program (Crank-Nicolson implicit scheme) for
modeling retrograde compositional zoning in garnet adjacent to biotite which has
behaved as a homogeneous infinite reservoir during cooling. The method is based
on a modification of Lasaga’s theory of stationary interface (Lasaga, 1983, Adv,
Phys Geochem. 2, 81-114) as given in Ganguly et al. (2000, EPSL, 183, 471-486),
plus the option to include the condition of moving interface as a consequence of
reaction during the diffusion process. The finite difference program has been
interfaced with an optimization program, MINUIT, to obtain the best fit to the
measured data from a set of input parameters which can be varied within
specified ranges. For further details, see Ganguly et al. (op. cit.) You have
only the executable version of the program, not the source code.
All three files, REAZ_M.exe, REAZ_M.INP and OUTPUT FILE should be in the
same directory (e.g. local disk (C:))
INPUT FILE
For the purpose of tutorial, the input file is broken up into three
segments, as shown below, followed by instructions for each segment. You would
need to edit the input file (REAZ_M.INP) to change the input parameters.
--------------------------Segment 1:-------------------------REZONE_M.INP
INITIAL
STEPSIZE
L_LIMIT
U_LIMIT
1.
B/
0.070
0.00100
0.0400
0.10
2.
PGAMA
5.0000
0.00100
0.1000
10.00
3.
CI
0.4400
0.00100
0.3000
0.50
4.
Rz
5.0000
0.00100
0.0000
20.00
--------------------------------------------------------------B/ (beta/gamma in Lasaga’s theory) is the ratio of the enthalpy change of
the Fe-Mg exchange reaction between garnet and biotite and Fe-Mg interdiffusion
coefficient The latter is calculated at the median position within the diffusion
zone, and is assumed to be independent of compositional change within this zone.
The estimated values of /within ~95% confidence interval is 0.07( 0.02). This
ratio controls the change of interface composition of garnet as a function of
time (Eq. 13, Ganguly et al., 1998). One should first try modeling of zoning
profile with the input value of /, as indicated in the above Table. If it does
not produce a satisfactory fit to the data, then / may be allowed to exceed
the specified limits.
PGAMA is Lasaga’s ’, but redefined in Ganguly, 2000, (Eq. 14)for a
nonlinear cooling model). The output file would provide the value of PGAMA that
produces the most satisfactory fit to the observed retrograde zoning profile
maintaining the constraints on the input parameters in the REAZ_M.INP file (see
the above Table).
CI is the initial garnet composition (Mg/(Mg + Fe)), which is taken as the
core composition showing a flat X(Mg) vs. distance relation.
Rz(micron) is the width of reaction zone at the rim.
For each parameter, you should enter an initial value to begin the
calculation. The last two columns in the above table give the lower and upper
limits within which these input parameters can be changed by the optimization
program, and the column STEPSIZE indicates the stepwise extent of these changes
during the computation. The L_LIMIT for CI must be the observed core
composition.
---------------------------Segment 2:---------------------------53
5
0.250 1
-5
-5
0.764 0.235
-4
-4
0.764 0.235
-3
-3
0.764 0.235
-2
-2
0.764 0.235
-1
-1
0.764 0.235
1
0
0.250 0.750
2
1
0.294 0.706
3
2
0.313 0.687
4
3
0.331 0.669
5
4
0.335 0.665
6
5
0.366 0.634
7
6
0.387 0.613
8
7
0.395 0.605
9
8
0.397 0.603
10
9
0.411 0.589
11
10
0.416 0.584
12
11
0.426 0.574
13
12
0.428 0.572
14
13
0.434 0.566
and so on …………………………………..
-----------------------------------------------------------------The two numbers in the first row indicate, respectively, the total No of
points in the analyses (in garnet and biotite) and the number of points for only
biotite. The two numbers in the second row represent, respectively, the edge
composition of garnet and the step size, in microns, in the microprobe analyses.
The step size must be constant. Since the edge composition of garnet cannot be
determined by microprobe spot analysis because of the convolution effect from
the adjacent biotite (see Ganguly et al., 1988, Amer Min 73, 901-909), it has to
be obtained by extrapolation from the trend of the measured composition within
garnet. There is obviously some room for flexibility in the determination of the
edge composition, and the input value may be varied to see if the fit between
the calculated and measured profiles improve.
The next set of numbers represents the mineral compositions with the
following order of the columns:
Point No.
Distance from the
X(Mg)
X(Fe)
Interface (microns)
The negative distance is for the biotite side, and the zero plus positive
distance is for the garnet side, with the zero being the distance of the
interface in the x-axis. X(Mg) = Mg/(Mg + Fe) so that X(Fe) = 1 – X(Mg).
---------------------------Segment 3 --------------------------------ERROR DEF
0.00001
PRINTOUT
0
SEEK
MIGRAD
SEEK
MIGRAD
SEEK
0.00001
0.00001
MIGRAD
0.00001
MIGRAD
0.00001
CALL FCN
3.
EXIT
---------------------------------------------------------------------These are instructions for the optimization program, MINUIT. DO NOT CHANGE
ANYTHING.
You would ultimately get a combination of the best values for B/r, CI and
PGAMA that best fit the measured data. If you want to keep B/r and/or CI
constant, then you would need to enter 0 for the step size for these parameters
in the input file.
TO RUN THE PROGRAM:
It is better to run the program from the DOS instead of Windows (it will,
however, run in Windows.XP). Go to DOS, and the subdirectory where the input
file (REAZ_M.INP) and the executable program (REAZ_M.EXE) are stored. At the
prompt, just type REAZ_M, and the program will start running. It will ask for an
input file name, to which the answer is REAZ_M.INP. It will then ask for an
output filename in Italian (NOME DEL FILE DI USCITA DEI DATI). Give any name
where you want the results to be stored: xxx.out. At the end of computation, you
will see the output file name on the screen. Click on the file name to open the
file.
OUTPUT FILE
As the program starts computing, you would notice that it is moving
through different optimization schemes, from SEEK to MIGRAD to SIMPLEX. An
example of the numbers that appear at the end of the output file is shown below.
-------------------------------------------------------------------------------START MIGRAD MINIMIZATION.
TO MINIMUM (EDM) .LT. 0.10E-10
CONVERGENCE CRITERIA
--
ESTIMATED DISTANCE
OR EDM .LT. 0.10E-05 AND FRACTIONAL CHANGE
IN VARIANCE MATRIX .LT. 0.10E-01
MIGRAD FAILS TO FIND IMPROVEMENT
MIGRAD TERMINATED WITHOUT CONVERGENCE
FCN VALUE CALLS EDM
HINT.EXT.PARAMETER VALUE
ERROR INTERN.VALUE INT.STEP SIZE
0.15300E+08 928 0.34E-05 1
1
B/r
0.800E-01 0.256E-15 0.15708E+01
0.73780E-08
WARNING - ABOVE PARAMETER IS AT LIMIT.
2
2
PGAMA 0.9425E+00 0.8144E-08 -0.94982E+00 -0.11357E-08
3
3
CI
0.48000E+00 0.1665E-14 0.15708E+01 0.84267E-08
WARNING - ABOVE PARAMETER IS AT LIMIT.
4
RZ
0.10000E-01
ERRORS CORRESPOND TO FUNCTION CHANGE OF
*********
*** 14****CALL
**********
0.0000
1.0000
2.0000
3.0000
4.0000
FCN
0.250000
0.294000
0.313000
0.331000
0.335000
3.00000
0.250000
0.328122
0.362007
0.379943
0.391362
0.0000
5.0000
6.0000
7.0000
8.0000
9.0000
10.0000
11.0000
12.0000
13.0000
14.0000
15.0000
16.0000
17.0000
18.0000
0.366000
0.387000
0.395000
0.397000
0.411000
0.416000
0.426000
0.428000
0.434000
0.437000
0.444000
0.439000
0.443000
0.446000
0.399656
0.406186
0.411577
0.416160
0.420136
0.423633
0.426743
0.429533
0.432054
0.434344
0.436434
0.438350
0.440112
0.441737
and so on
---------------------------------------------------------------The output file shows the value of PGAMA that yields the best fit to the
measured profile, along with the values of other parameters that the
optimization program has used to calculate the profile. There is a “warning” for
a parameter if it has to be pushed to the limiting value specified in the input
data file. Recalculation of the profile with relaxation of the limiting value
may improve the fit. For example, excellent fit to the measured data was
obtained in the above example by increasing the input value of / to 0.15 (the
input file entry is / = 0.10 (L_limit = 0.08 U_Limit = 0.15)
The numbers in the three columns are as follows:
Distance from
the interface
(microns)
X(Mg)_Grt
measured
X(Mg)_Grt
calculated
You can copy and paste these data in an excel file and compare the measured and
calculated profiles.
The T-t path and cooling rate at any temperature can be calculated from
the value of PGAMA (’) and Eq. (14) of Ganguly et al., 2000, EPSL paper):
’ = (a2Q)/(RD(To)
a is the effective grain size (i.e. the distance from the interface to the end
of data in the input file), Q is the activation energy of Fe-Mg interdiffusion
in garnet, D(To) is the diffusion coefficient at the peak temperature and  is a
a cooling time constant (K-1t-1). Here you would need to keep track of the units
so that ’comes out as a dimensionless parameter. For example, If D(To) is in
cm2/s, then a must be in cm, and if Q is in Joules/mol then R (gas constant)
must be 8.314 J/mol-K. The cooling time constant is then in unit of K-1s-1, which
can be converted to the unit of K-1Myr-1 upon multiplication by 3.154(1013).
The value of the cooling time constant, , is calculated from the last
equation. The T-t path is then calculated from the following relation (Ganguly
et al. 2000, EPSL):
1/T(K) = 1/To(K) + t
so that the cooling rate at any temperature, T’, is given by
dT/dt = - (T’)2