**************************************************************************
* BremsLib: a library of shape functions and singly differential cross
* sections of bremsstrahlung at electron energies from 10 eV to 3 MeV
**************************************************************************
Version 1.0.5
Andrius Poškus
Institute of Chemical Physics, Vilnius University,
Saulėtekio al. 9, build. III, 10222 Vilnius, Lithuania
E-mail: andrius.poskus@ff.vu.lt or andrius.poskus@live.com
March 4, 2019
The data library BremsLib contains the values of the singly
differential cross sections (CS) and shape functions (SF) of atomic-field
bremsstrahlung. The SF is defined as the ratio of the doubly differential
cross section (DCS) and the angular integral of the same DCS, which is
evaluated by numerical quadrature. The CS and SF have been calculated for
100 values of the atomic number Z from 1 to 100, 56 values of the incident
electron energy T1 from 10 eV to 3 MeV, and 12 values of the ratio k/T1
from 0 to 0.999983, where k is the energy of the bremsstrahlung photon (a
total of 67200 combinations of the three physical parameters Z, T1, k/T1).
The first 50 values of T1 (from 10 eV to 800 keV) are equal to
T1= (1.0, 1.2, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 8.0) * 10^n keV,
where n = -2, -1, 0, 1, 2. The remaining 6 values of T1 are the first six
values corresponding to n = 3 (i.e., 1.0 MeV, 1.2 MeV, 1.5 MeV, 2.0 MeV,
2.5 MeV, and 3.0 MeV). The first 10 values of k/T1 are 0, 0.1, ..., 0.9,
the next value of k/T1 is 0.95, and the last value of k/T1 is 0.99 when
T1 <= 5 keV, or 1 - (50 eV / T1) when T1 > 5 keV.
The shape functions corresponding to T1 < 1 MeV are tabulated for 181
equidistant values of the photon emission angle ("theta") from 0 to
180 deg. The shape functions corresponding to T1 >= 1 MeV have been
calculated for 221 values of the angle: 51 equidistant values from 0 to
10 deg (in increments of 0.2 deg) and 170 equidistant values from 11 deg
to 180 deg (in increments of 1 deg).
*** OPERATION INSTRUCTIONS:
Uncompress the downloaded file. This will create folder "BremsLib".
Folder "BremsLib" contains the ASCII files with the bremsstrahlung
cross section library. Those data were generated using BREMS v1.4.12.0.
The ZIP archive with the distribution package of the current version
of BREMS may be downloaded from the following websites:
http://web.vu.lt/ff/a.poskus/brems/
http://www.apsoftware.org/BREMS/
The calculations were performed using the BREMS control option
"accuracy_level=2". At T1 >= 1 MeV, the density of the angular grid in the
interval 0 < theta < 10 deg was increased by a factor of 5 in comparison
with the default set of angles. Those were the only non-default control
options affecting the values of the differential cross sections, except
for 54 combinations of the physical parameters Z, T1 and k/T1 when the
values of the BREMS control parameters "Kmax1" and "Kmax2" were increased
in comparison with the default values (see below).
*** LIBRARY DATA FORMAT AND CALCULATION OF THE RELATIVE UNCERTAINTIES:
The BremsLib library data are stored in three sets of ASCII files:
I. The files with the final (extrapolated) values of the singly
differential cross sections (CS) and shape functions (SF).
II. The files with the values of the relative uncertainties of the CS
and SF. Those files are located in the subfolder "relErr" of the main
BremsLib library data folder.
III. The files with the values of the original (unfitted) CS and SF.
Those files are located in the subfolder "unfitted" of the main
BremsLib library data folder.
Each of those three sets consists of 100 files: one file for each value of
the atomic number Z from 1 to 100. The file name starts with the string
"ShapeFn_", or "relErr_", or "unfitted_", followed by a three-digit
integer number indicating the value of Z, and by the file name extension
".txt". In each file, there is one table for each value of the initial
electron energy T1 (the value of T1 is indicated before each table).
The first line in each table contains the values of k/T1 ("k/T1"),
where k is the energy of the bremsstrahlung photon.
The next three lines contain the maximum values of the quantum
numbers |kappa[1]|, |kappa[2]| and l in the truncated series defining the
CS and the doubly differential cross section (DCS). Those maximum values
are denoted "K1max", "K2max", and "lmax", respectively.
The fifth line of the table contains the number of values of
max(|kappa[1]|,|kappa[2]|) used for the nonlinear fitting (extrapolation)
of the singly and doubly differential cross sections ("Nfit").
The sixth line contains the final (extrapolated) values of the scaled
singly differential cross section ("CS", in mb), or its relative
uncertainties ("CSerr"), or the original (unfitted) scaled singly
differential cross section ("CS_0", in mb). The scaling factor is equal
to k / Z^2.
The remainder of the table contains the values of the final
(extrapolated) shape functions (in units of 1/sr), or their relative
uncertainties, or the original (unfitted) shape functions (in units of
1/sr), along with the corresponding values of the photon emission angle
"theta" (in degrees).
The value of the fitted or unfitted doubly differential cross section
(DCS) can be obtained by multiplying the corresponding SF and the CS,
because the CS is approximately equal to the angular integral of the DCS
(this equality is not exact due to the numerical errors, especially in the
case of the fitted (D)CS). The relative uncertainty of the DCS is equal to
the square root of the sum of the squared relative uncertainties of the SF
and of the CS.
Some of the relative uncertainties in the files "relErr_.txt"
(where "" denotes a three-digit integer indicating the value of the
atomic number Z) are negative. A negative value of the relative
uncertainty indicates that the nonlinear fitting was not performed,
because the relative range of variation of the (D)CS inside the fitting
interval was less than 0.001 (i.e., 0.1%). In such a case, the relative
uncertainty is replaced by the mentioned relative range of variation with
the minus sign.
The positive values of the relative uncertainty ("Err") are equal to
|Err| = max(|Err0|, Err1, Err2, |Diff_01|, C*|Diff02|, |Diff12|),
where "Err0", "Err1", "Err2" are the estimates of the relative uncertainty
of the CS or SF calculated in three runs of BREMS:
Err0 is the default estimate of the relative uncertainty,
Err1 is obtained after processing the "..._test.txt" files,
Err2 is obtained after processing the "..._test2.txt" files,
"Diff_01", "Diff_02", "Diff_12" are relative differences between three
estimates of the CS or SF - the default one ("value No. 0"), the one
obtained after processing the "..._test.txt" files ("value No. 1"), and
the one obtained after processing the "..._test2.txt" files ("value No.
2"):
Diff_01 - relative difference between the values No. 0 and No. 1,
Diff_02 - relative difference between the values No. 0 and No. 2,
Diff_12 - relative difference between the values No. 1 and No. 2,
and C is the adjustment factor, which is equal to
C = (max|kappa1| + max|kappa2| - lmax) / (2*Delta[lmax]),
where lmax is the maximum value of the quantum number "l" used in the sum,
and Delta[lmax] is the decrement of lmax used for testing the sensitivity
of the (D)CS to the truncation of the sum with respect to l.
If the relative uncertainty of the fitted CS or SF tabulated in the
library is greater than 0.05 (i.e., 5 %), then it is reliable enough.
However, if it is less than 0.005 (and positive), then it is frequently
too small by an order of magnitude (this has been determined empirically).
Consequently, it is recommended to modify the estimate of the relative
uncertainty additionally when 0 < Err < 0.05: if 0 < Err < 0.005, then it
should be increased by a factor of 10, and if 0.005 < Err < 0.05, then it
should be set equal to 0.05. Mathematically, this replacement of Err is
defined as follows:
Err' = max(Err,min(0.05, 10*Err)) (Err > 0).
For 28 (out of 67200) combinations of the three physical parameters
Z, T1 and k/T1, the relative uncertainty of CS, max[SF], or
max[sin(theta)*SF], which is obtained with the default values of
max|kappa1| and max|kappa2| corresponding to the option
"accuracy_level=2", is greater than 5 % (with the absolute maximum of
those three relative uncertainties equal to 10 %). In those 28 cases, as
well as in the other 22 cases when similarly large relative deviations had
been obtained with the previous version of BremsLib (v1.0.4), the
calculations were redone with max|kappa1| and max|kappa2| (BREMS control
parameters "Kmax1" and "Kmax2", respectively) greater by 40 than the
default values. In addition, max|kappa1| and max|kappa2| were increased
in four cases corresponding to Z <= 2, T1 = 3MeV, k <= 0.3 MeV, for the
purpose of testing the accuracy of the extrapolation procedure implemented
in BREMS (the values of max|kappa1,2| at k = 0 and k = 0.3 MeV were
increased to 500 and 300, respectively). All these 54 combinations of Z,
T1 and k are listed in the table below.
Z T1(MeV) k(MeV)
---------------------
1 1.5 0
1 2 0
1 2 0.4
1 2.5 0
1 2.5 0.25
1 2.5 0.5
1 2.5 0.75
1 3 0
1 3 0.3
1 3 0.6
1 3 0.9
2 3 0
2 3 0.3
3 0.003 0
3 0.004 0
3 2 0.2
3 2.5 0.5
4 2 0.2
8 3 0.3
9 2.5 0.25
9 3 0.3
9 3 0.6
10 2.5 0.25
10 3 0.3
11 2 0.2
11 3 0.3
12 3 0.3
13 3 0.3
18 2.5 0.25
18 3 0.3
19 2.5 0.25
19 3 0.3
20 3 0.3
21 3 0.3
21 3 0.6
22 3 0.3
23 3 0.3
24 2.5 0
24 3 0.3
25 3 0.3
26 3 0.3
27 3 0.3
28 3 0.3
29 3 0.3
30 3 0.3
31 3 0.3
32 3 0.3
33 3 0.3
34 3 0.3
35 3 0.3
36 3 0.3
33 3 0.6
34 3 0.6
36 3 0.6
---------------------
After redoing the calculations for the 54 combinations of the physical
parameters listed above, the relative uncertainty of CS and
max[sin(theta)*SF] is always less than 5 %, whereas the relative
uncertainty of max[SF] is greater than 5 % in only two cases (when Z = 1,
T1 = 2.5 MeV, k = 0.75 MeV, and when Z = 19, T1 = 3 MeV, k = 0.3 MeV, the
relative uncertainty of the SF at theta=0 is equal to 5.3 % and 6.9 %,
respectively). In addition, the relative uncertainty of the SF is
sometimes greater than 5 % for the values of the angle "theta" that do not
correspond to the maximum SF or maximum sin(theta)*SF.