APPENDIX
B
Table 4.4 by using quadruple precision
Caption of Table 4.4: Absolute errors of the first ten ERPs for the n-p system. The absolute error is written in normalized scientific notation as ε = εsig× 10b, where εsig
is the significand and b is the order of magnitude. Here we report b only. Values in parenthesis from Ref. [49]. Units are shown in Table 3.1.
Table B.1: Table 4.4 by using quadruple precision.
112
Table B.2: Table 4.4 by using quadruple precision. Continuation of Table B.1.
a (fm) 15 17 19 N 20 30 40 20 30 40 20 30 40 b(a0) (−4)−4 (−7)−7 −7 −3 −7 −9 −4 −7 −11 b(r0) (−6)−6 (−8)−8 −8 −5 −9 −9 −5 −8 −10 b(P0) (−7)−7 (−8)−7 −7 −7 −9 −9 −7 −10 −10 b(Q0,3) −5 −5 −5 −6 −6 −6 −6 −7 −7 b(Q0,4) −4 −4 −4 −5 −5 −5 −6 −6 −6 b(Q0,5) −3 −3 −3 −4 −4 −4 −4 −4 −4 b(Q0,6) −2 −2 −2 −3 −3 −3 −4 −4 −4 b(Q0,7) −1 −1 −1 −2 −2 −2 −3 −3 −3 b(Q0,8) −1 −1 −1 −1 −1 −1 −2 −2 −2 b(Q0,9) −1 −1 −1 −1 −1 −1 −1 −1 −1
Comparing Tables 4.4 and B.2 one sees how high order parameters become more accurate when the precision is increased from double to quadruple.
Note how the loss of precision for a = 19 fm in Table 4.4 is corrected by increasing the numerical precision as Table B.2 displays.
Around a = 10 fm, the “strange” behavior of b(Q0,8) in Table B.1 has drawn our
attention. We think that this effect comes from the numerical rounding. For a future research, results of Q0,8 for channel radii around 10 fm are shown in Table B.3
Table B.3: b(Q0,8) in Table B.1 by choosing channel radii around 10 fm.
a (fm) 9.4, 9.5 or 9.6 9.7, 9.8 or 9.9 10.1, 10.2 or 10.3
N 20 30 40 20 30 40 20 30 40
APPENDIX
C
Error propagation
In science, theory and experiment have different and complementary perspectives. Some-times theories guide one to propose new experimental techniques to corroborate predic-tions, estimapredic-tions, behaviors, etc; sometimes experiments provide results which indicate us whether the theory gives an acceptable description of the phenomena under con-sideration or not. This is one of several ways to understand the role of theories and experiments in science. The important fact to stress is that theory and experiment have a very especial point of meeting. In this thesis that point is the phase shift or more precisely a function depending on the phase shift (the effective-range function).
Experimental phase shifts, δexpl,i = δexpl (Ei), are those extracted from the
partial-wave decomposition of the scattering amplitude which is determined from the measured differential cross sections for a given energy Ei. This implies that the experimental
uncertainty of the differential cross section is propagated up to δl,iexp (δl,iexp± σexpl,i with
σl,iexp the phase-shift error).
From the theoretical perspective rather than phase-shift points, a function δtheo
l (E;{β})
is provided (via the ERF in this thesis), where the parameter set {β} = { ¯β} satisfy δl,itheo ≈ δexpl,i . Thus, the challenge is to find a family of sets around { ¯β} capable of describing the experimental data according to the uncertainties.
The way to face this challenge can be very controversial, and therefore, a special review about it is provided here.
C.1
Statistical uncertainties or statistical errors
Let us assume x as continuous variable that can be associated with a physical quantity. As the name indicates, the statistical uncertainty σ of x provides a valuable piece of information about the confidence level of the x-value in a statistical context. Thus, σ defines an interval around x = ¯x where x is expected to be measured or determined with a high enough confidence level, with ¯x the most probable value.
At this point, one is induced wrongly to think that ¯x is the correct value of x, which is a deterministic point of view. Here we should understand x in a probabilistic frame,
C.2. χ2 method 114 and only when σ→ 0, the deterministic perspective makes sense. Thus in a formal way, x has a distribution of probability with a maximum in ¯x as Fig. C.1 illustrates.
¯ x 2σ
Figure C.1: Example of a x-distribution.
In physics, σ represents the error of x and is quantified by the standard deviation for symmetrical x-distributions. Thus, the convention x = ¯x± σ means that the interval [¯x− σ; ¯x + σ] is the most probable to find the true value of x (for a normal distribution this interval defines the 68.2% of probability).
For the moment, let me suppose that x represents δl,iexp and that we can determinate it experimentally. If we perform in the laboratory a first experiment, we can record our estimation as δl,iexp-1. Suppose now that we repeat the experiment and record our second estimation as δl,iexp-2. Clearly, there is no reason to think that both values are identical. We can keep repeating the experiment and recording our estimations. At the end, we shall see that our records show a range where δexpl,i usually falls, and then, we report δexpl,i = ¯δexpl,i ± σexpl,i which gives us an idea about the distribution in Fig. C.1.
This idea can be extended to an arbitrary energy set. The question is then, how can we describe consistently all the data set{¯δexpl,1 ± σl,1exp, ¯δl,2exp± σexpl,2 ,· · · , ¯δl,Nexp± σexpl,N}? This is the main goal of error propagation. Here we shall explore two proposals of error propagation: the χ2 method and the Monte Carlo technique.
C.2
χ
2method
This is the most usual method to propagate errors (see for instance Ref. [75]). The assumptions of the method make it powerful in practice and limited in theory, which means that its application should be carried out carefully.
For a quantity y(x) [for example δl(E)] with an uncertainty σ, the assumptions are:
1. A set of N points ¯yi = ¯y(xi) together with their uncertainties σi are known.
2. Every point is normally distributed with mean ¯yi and standard deviation σi.
3. The errors are independent.
Under these assumptions the χ2 method finds a function f (x;{β}) (with {β} a set of free parameters) which maximizes the probability of finding{fi = f (xi;{β})} around
{¯yi}. This leads to minimize
χ2= N � n=1 (yi− fi)2 σ2 i (C.1) with respect to{β}.
C.3. Monte Carlo technique 115 function is expanded around { ¯β} which allows us to obtain the covariance matrix and the correlation matrix [75]. The latter has a form
1 c1,2 . . . c1,Nβ c2,1 1 . . . c2,Nβ .. . ... . .. ... cNβ,1 cNβ,2 . . . 1 , (C.2)
with−1 < ci,j= cj,i< 1. If ci,j ≈ −1 (1), one says that βi and βj have a linear negative
(positive) relationship. If ci,j ≈ 0 there is no a linear relationship between them.
χ2-analyses for the potential model of the d-wave of12C+α (see Chapter 5) provide correlation matrices like
p2,1 q2,0 q2,1 q2,2 q2,3 p2,1 q2,0 q2,1 q2,2 q2,3 1 0.82 0.79 −0.77 0.69 0.82 1 0.30 −0.28 0.19 0.79 0.30 1 −0.99 0.98 −0.77 −0.28 −0.99 1 −0.99 0.69 0.19 0.98 −0.99 1
In this example, the function to fit is a [1/3]-Pad´e approximant. Note that the non-diagonal elements are not close to zero in the sense |ci,j| << 1, and the function
is nonlinear on the parameters, which means that the standard χ2-method is not appropriate to propagate errors.
Box C.1: Example of a correlation matrix for fits in Chapter 5.
The correlation matrix provides a valuable information to apply error propagation by using the covariance matrix. If the model is linear on{β}, then χ2expansion up to second
order is exact and the errors are propagated correctly. If the model is approximately linear (or if ci,j ≈ 0 and |σi/yi| < 0.1, rule of thumb), the propagation can be accepted
as a valid approximation. Otherwise, the covariance matrix is not enough to provide a consistent error propagation and the relevant information is given by the correlation matrix and the diagonal elements of the covariance matrix.
C.3
Monte Carlo technique
Monte Carlo (MC) methods have a wide application in different fields. Here we shall see how this method is used to propagate uncertaintes [56, 76], or more precisely, distribu-tions.
As we have mentioned at the beginning of this appendix, by repeating an experiment we can determine the distribution of probability, and therefore, the uncertainty, of a given quantity x. This distribution provides all the information required to obtain the distribution of a function f (x) (note that we are in the field of functions of random variables). Let me illustrate this with an example. Suppose the typical case in physics where a variable x follows a normal distribution with standard deviation σx(x = ¯x±σx).
C.3. Monte Carlo technique 116 Fig. C.2 shows, the f -distribution is also a normal distribution with standard deviation σf = aσx. Note that the uncertainty of f is exactly the result obtained by the standard
error propagation. ¯ x 2σx ¯ f f (x) = ax σf = aσx 2σ f
Figure C.2: Schematic representation of error propagation via distributions. The example in Fig. C.2 is a simple one that can be proved formally. In general the function f (x) is more complicated than a straight line, and the f -distribution can be very difficult of finding theoretically. For these cases, the MC technique provides an accurate statistical method to overcome the difficulties of a formal deduction.
The idea behind MC is the same of repeating an experiment to determine the value and uncertainty of a quantity x. Specifically, MC generates x-values according with the experimental expectations, i.e., MC makes a set of x-values experimentally equiva-lent. Thus, if we know (or at least, if we can suppose or infer) the x-distribution, we can simulate the experiment via MC an arbitrary number of times. This allows us to propagate the x-distribution through an arbitrary function f (x) and therefore the error propagation can be carried out accurately. The only price to pay is that the set of x-values generated by MC should be large enough (usually several thousands) to rebuild the x-distribution. As an example, Fig. C.3 shows how MC recovers the distributions in Fig. C.2 by choosing a = 0.5 and x = 10± 1.
In this case, it is evident that the three: the formal deduction, the standard error propagation and the MC technique, provide the same result for σf. Unfortunately, this
is not valid in general and restricts the standard error propagation considerably as Fig. C.4 shows.
Note how MC provides a consistent f -distribution for all cases, and the standard error propagation only achieves that when the function is linear (or approximately linear around x = ¯x± σx). The reason is simple, the fact that x is normally distributed does
not mean that f (x) follows a normal distribution.
Up to now, we have explored the case of a function of a single random variable. This can be generalized for a multivariable function, for instance, χ2 or fi in Eq. (C.1), where
C.3. Monte Carlo technique 117 100 1000 10 000 100 000 x 4 6 8 10 12 14 16 4 6 8 10 12 14 16 4 6 8 10 12 14 16 4 6 8 10 12 14 16 f 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8
Figure C.3: Example of distribution and error propagation by using MC technique. x = 10± 1 normally distributed and f(x) = 0.5x. Formal or exact distribution (dashed lines), MC technique (histograms) and σf-estimation by the standard error propagation
(solid lines = 2σf). The first row indicates the number of equivalent data sets generated
C.3. Monte Carlo technique 118 Σx 0.5 1 1.5 2 x 4 6 8 10 12 14 16 4 6 8 10 12 14 16 4 6 8 10 12 14 16 4 6 8 10 12 14 16 f1�x� 10 15 20 25 10 15 20 25 10 15 20 25 10 15 20 25 f2�x� 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 f3�x� 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
Figure C.4: f -distributions for x = 10± σx normally distributed and f1(x) = 1.5x + 3,
f2(x) = 10x−2 and f3(x) = tan(x/12). Exact (dashed lines), MC technique for 104 sets
APPENDIX
D
Fits and extrapolations
In this appendix we shall see how sensitive is an extrapolation according to the fit under several conditions. This will provide clear arguments to trust or not in a given estimation obtained from an extrapolation after a fit.
To start, Fig. D.1 shows a set of point according to a smooth function f (x)1. The
open circle will be the extrapolation point and our goal is to determinate it by using fits according to different sets of filled circles.
� � x0 a b c d f0 x0 a b c d f0
Figure D.1: Behavior (line) and set of data points (filled circles) for a given function f (x). a, b, c and d define the x-ranges in Fig. D.2.
In practice, one can understand the filled circles as data points coming from the ex-periment, which of course include uncertainties. Therefore we can analyze two scenarios for experimental data: with and without high precision as Fig. D.2 shows.
By analyzing Fig. D.2 from top to bottom, it is clear that2: First, with a few points at “small” x-values (close to a) the fit cannot describe correctly the curvature and therefore the extrapolation is compromised. Second, with the full input data set the fit provides a
1Note that this is general, i.e., f (x) represents an arbitrary quantity such as phase shift, distance,
temperature, etc.
2Here we wish to discuss the effect of an extrapolation in general terms. At the end of this appendix
120 global good description without any preference for “small” x-values, leading therefore to inaccurate extrapolations. Third, increasing arbitrarily the number of free parameters for finding a better extrapolation can provide the contrary effect and include unexpected behaviors for the extrapolation. Moreover, when there is not high precision, the extra flexibility introduced by more free parameters will try to explain the statistical fluctua-tions leading to inconsistent descripfluctua-tions and extrapolafluctua-tions. Fourth, the balance among all the previous effects (not too small/large x-range, not overestimate the flexibility for the fit) leads us to a consistent extrapolation.
� � x0 a b f0 x0 a b f0 �� x0 a b f0 x0 a b f0 � � x0 a d f0 x0 a d f0 �� x0 a d f0 x0 a d f0 � � x0 a d f0 x0 a d f0 �� x0 a d f0 x0 a d f0 � � x0 a c f0 x0 a c f0 �� x0 a c f0 x0 a c f0
121
The function f (x) is assumed as a polynomial of order 10 to provide a smooth (but not too simple) variation. Its coefficients are (from order 0 to 10): -1.000, 0.0833, -0.6319, 0.0700, -0.1375, 0.0261, -0.0308, 0.0080, -0.0071, 0.0023 and -0.0017.
The randomized data are generated by adding a random number between −0.1 and 0.1. The set of points used are listed below.
From top to bottom, the fits in Fig. D.2 correspond to polynomials of first, fourth, twelve and third order respectively.
x f (x) frand(x) 0.50 -1.12 -1.15 0.55 -1.15 -1.07 0.60 -1.18 -1.26 0.65 -1.22 -1.20 0.70 -1.26 -1.24 0.75 -1.31 -1.23 0.80 -1.36 -1.30 0.85 -1.41 -1.33 0.90 -1.48 -1.43 0.95 -1.54 -1.51 1.00 -1.62 -1.67 1.05 -1.70 -1.60 1.10 -1.79 -1.74 1.15 -1.89 -1.80 1.20 -2.00 -2.07 1.25 -2.12 -2.03 1.30 -2.25 -2.33 1.35 -2.40 -2.38 1.40 -2.56 -2.63 1.45 -2.75 -2.70 1.50 -2.95 -2.99 1.55 -3.18 -3.24 1.60 -3.44 -3.48 1.65 -3.74 -3.64 1.70 -4.08 -3.99 1.75 -4.48 -4.57 1.80 -4.95 -4.91 1.85 -5.50 -5.50 1.90 -6.16 -6.09
APPENDIX
E
Phase shift data from Ref. [1] for l = 2
Files README.TXT and table2.dat of Ref. [1] are shown in Boxes E.1 and, E.2, E.3 and E.4 respectively.
ARTICLE INFORMATION
Document Number: E-PRVCAN-79-032904 Journal: Phys. Rev. C 79, 055803 (2009)
All Authors: P. Tischhauser, A. Couture, R. Detwiler, J. Gorres, C. Ugarde, E.Stech, M. Wiescher, M. Heil, F. Kappeler, R. E. Azuma, L. Buchmann.
Title: Measurement of elastic12C+α scattering: Details of the experiment, analysis, and discussion of phase
shifts.
DEPOSIT INFORMATION
Description: The submitted files contain the phase shift information of the12C+α elastic scattering experiment
by P. Tischhauser et al. There are seven files according to the angular momenta l = 0 . . . 6. In each of these files the first column is the alpha energy, the second column the phase shift as derived from the globalized Monte Carlo simulations, the third column is the same phase shift randomized by the error from the Monte Carlo simulations and the fourth column is the error of the phase shifts from the Monte Carlo simulations. The files included are:
table0.dat - l = 0 phase shift information; table1.dat - l = 1 phase shift information; table2.dat - l = 2 phase shift information; table3.dat - l = 3 phase shift information; table4.dat - l = 4 phase shift information; table5.dat - l = 5 phase shift information; table6.dat - l = 6 phase shift information. Total No. of Files: 8
Filenames: README.TXT, table0.dat, table1.dat, table2.dat, table3.dat, table4.dat, table5.dat, table6.dat Filetypes: ASCII and .dat
Contact Information: Dr. Lothar R. Buchmann
TRIUMF - 4004 Wesbrook Mall - Vancouver, B.C. - CANADA V6T 2A3 Phone: 1+604-222-7403 / Fax: 1+604-222-1074 / Email: lothar@triumf.ca
Box E.1: File README.TXT of Ref. [1].
124
Line Eα(MeV) δ2 (◦) δ2rand(◦) σ2 (◦)
1 2.607 -0.353 -0.327 0.099 2.657 -0.389 -0.397 0.109 2.707 -0.427 -0.359 0.119 2.755 -0.464 -0.526 0.130 ! 2.808 -0.508 -0.448 0.142 ! 2.808 -0.508 -0.353 0.142 2.858 -0.551 -0.550 0.154 ! 2.903 -0.591 -0.409 0.165 ! 2.903 -0.591 -0.600 0.165 10 2.907 -0.594 -0.466 0.166 2.958 -0.641 -0.644 0.179 3.007 -0.687 -0.814 0.192 3.009 -0.689 -0.589 0.192 ! 3.049 -0.727 -0.689 0.203 ! 3.049 -0.727 -0.757 0.203 ! 3.049 -0.727 -0.801 0.203 3.098 -0.774 -0.780 0.217 3.110 -0.785 -0.762 0.220 3.134 -0.808 -0.909 0.227 20 3.200 -0.871 -0.764 0.246 3.202 -0.873 -0.999 0.247 3.203 -0.874 -1.043 0.247 3.205 -0.876 -0.926 0.247 3.248 -0.916 -1.217 0.260 3.307 -0.966 -1.117 0.278 ! 3.309 -0.967 -0.993 0.278 ! 3.309 -0.967 -1.087 0.278 3.346 -0.994 -1.294 0.290 3.396 -1.020 -0.657 0.305 30 3.398 -1.021 -0.889 0.306 3.411 -1.024 -1.420 0.310 3.435 -1.024 -1.098 0.318 ! 3.472 -0.997 -1.319 0.330 ! 3.472 -0.997 -0.775 0.330 3.510 -0.888 -1.192 0.343 ! 3.536 -0.649 -0.773 0.354 ! 3.536 -0.649 -0.760 0.354 3.547 -0.409 -0.133 0.360 3.548 -0.378 -0.139 0.361 40 3.558 0.119 -0.149 0.368 3.561 0.388 0.417 0.371 3.563 0.631 0.135 0.373 3.566 1.156 0.969 0.377 ! 3.568 1.691 1.575 0.380 ! 3.568 1.691 1.741 0.380 3.571 3.119 2.962 0.386 3.573 5.136 5.433 0.390 3.576 18.058 17.864 0.401 ! 3.578 136.425 136.394 0.411 50 ! 3.578 136.425 136.496 0.411 ! 3.578 136.425 136.239 0.411 3.581 170.468 170.076 2.424 3.583 173.367 173.321 0.887 3.586 -4.812 -4.517 4.049 ! 3.588 -4.172 -5.204 2.698 ! 3.588 -4.172 -7.721 2.698 3.591 -3.567 -4.068 0.823 ! 3.593 -3.294 -2.983 1.547 ! 3.593 -3.294 -1.220 1.547 60 3.596 -2.995 -2.956 0.338
Line Eα (MeV) δ2 (◦) δrand2 (◦) σ2(◦)
61 ! 3.598 -2.845 -2.904 0.316 ! 3.598 -2.845 -2.898 0.316 3.603 -2.574 -2.695 0.334 3.608 -2.394 -2.121 0.343 3.628 -2.041 -1.942 0.361 3.647 -1.907 -2.032 0.370 3.648 -1.903 -1.999 0.371 3.697 -1.793 -1.611 0.388 3.711 -1.785 -1.520 0.393 70 3.746 -1.782 -2.057 0.404 3.748 -1.782 -1.920 0.405 ! 3.796 -1.802 -2.218 0.419 ! 3.796 -1.802 -2.009 0.419 3.811 -1.811 -1.536 0.423 3.847 -1.836 -2.028 0.433 3.861 -1.847 -2.361 0.437 3.897 -1.876 -1.487 0.447 3.911 -1.888 -1.911 0.451 3.948 -1.920 -2.131 0.461 80 3.958 -1.928 -1.836 0.463 ! 4.012 -1.976 -2.057 0.477 ! 4.012 -1.976 -2.038 0.477 ! 4.012 -1.976 -2.327 0.477 ! 4.045 -2.004 -1.783 0.485 ! 4.045 -2.004 -1.744 0.485 ! 4.045 -2.004 -2.057 0.485 ! 4.059 -2.016 -2.341 0.515 ! 4.059 -2.016 -1.525 0.515 4.060 -2.017 -2.401 0.515 90 4.062 -2.019 -2.616 0.515 4.094 -2.046 -2.672 0.523 4.112 -2.061 -2.750 0.526 ! 4.148 -2.090 -1.993 0.534 ! 4.148 -2.090 -2.065 0.534 ! 4.162 -2.101 -1.596 0.537 ! 4.162 -2.101 -1.994 0.537 ! 4.175 -2.111 -1.776 0.540 ! 4.175 -2.111 -2.095 0.540 4.185 -2.118 -1.574 0.542 100 4.187 -2.120 -1.943 0.542 4.198 -2.128 -2.682 0.544 ! 4.212 -2.138 -2.264 0.547 ! 4.212 -2.138 -2.416 0.547 4.227 -2.148 -1.900 0.549 ! 4.228 -2.149 -2.291 0.550 ! 4.228 -2.149 -1.760 0.550 ! 4.228 -2.149 -2.476 0.550 4.237 -2.155 -2.934 0.551 ! 4.238 -2.156 -2.388 0.551 110 ! 4.238 -2.156 -2.112 0.551 4.248 -2.163 -1.969 0.553 ! 4.250 -2.164 -2.419 0.553 ! 4.250 -2.164 -2.274 0.553 4.258 -2.169 -1.542 0.555 4.262 -2.172 -2.075 0.556 4.263 -2.172 -1.988 0.556 4.268 -2.175 -2.585 0.557 ! 4.275 -2.180 -1.702 0.558 ! 4.275 -2.180 -1.693 0.558 120 ! 4.278 -2.182 -1.974 0.559
125
Line Eα (MeV) δ2 (◦) δrand2 (◦) σ2(◦)
121 ! 4.278 -2.182 -2.972 0.559 ! 4.288 -2.188 -2.896 0.560 ! 4.288 -2.188 -1.965 0.560 ! 4.288 -2.188 -2.480 0.560 4.297 -2.193 -2.558 0.561 ! 4.300 -2.195 -2.148 0.562 ! 4.300 -2.195 -2.156 0.562 4.312 -2.201 -2.660 0.563 4.313 -2.202 -1.663 0.564 130 4.323 -2.207 -2.385 0.565 4.328 -2.210 -2.442 0.566 4.332 -2.212 -2.094 0.566 4.338 -2.215 -2.640 0.567 4.342 -2.217 -1.899 0.568 4.347 -2.220 -2.081 0.568 4.353 -2.223 -1.879 0.569 4.359 -2.225 -2.393 0.570 ! 4.363 -2.227 -1.997 0.571 ! 4.363 -2.227 -2.749 0.571 140 ! 4.388 -2.238 -2.042 0.574 ! 4.388 -2.238 -2.382 0.574 ! 4.413 -2.247 -2.862 0.576 ! 4.413 -2.247 -2.372 0.576 4.419 -2.249 -1.612 0.577 4.447 -2.258 -2.138 0.580 ! 4.448 -2.258 -2.430 0.580 ! 4.448 -2.258 -2.336 0.580 4.463 -2.262 -2.394 0.581 4.464 -2.262 -2.745 0.581 150 4.512 -2.269 -2.432 0.585 ! 4.514 -2.270 -2.956 0.585 ! 4.514 -2.270 -2.554 0.585 4.549 -2.270 -2.893 0.586 4.565 -2.270 -2.438 0.587 4.615 -2.261 -2.186 0.587 4.649 -2.250 -2.852 0.586 4.650 -2.250 -1.394 0.586 4.665 -2.243 -2.212 0.585 4.715 -2.214 -1.488 0.570 160 4.737 -2.198 -2.084 0.072 4.765 -2.173 -1.763 0.602 4.816 -2.117 -2.358 0.587 4.851 -2.069 -2.103 0.582 4.865 -2.047 -1.665 0.579 4.915 -1.957 -2.025 0.571 4.948 -1.886 -1.798 0.564 4.962 -1.853 -1.930 0.561 5.016 -1.705 -1.692 0.522 ! 5.048 -1.601 -1.406 0.514 170 ! 5.048 -1.601 -1.453 0.514 5.066 -1.537 -1.557 0.509 ! 5.117 -1.326 -0.889 0.493 ! 5.117 -1.326 -1.541 0.493 5.141 -1.211 -1.231 0.485 5.167 -1.074 -0.819 0.475 5.171 -1.051 -0.680 0.474 5.189 -0.946 -0.523 0.466 5.207 -0.832 -0.458 0.459 ! 5.216 -0.772 -0.570 0.455 180 ! 5.216 -0.772 -1.180 0.455
Line Eα(MeV) δ2(◦) δ2rand(◦) σ2 (◦)
181 ! 5.216 -0.772 -0.794 0.455 5.224 -0.716 -0.310 0.452 5.228 -0.688 -0.600 0.450 5.232 -0.659 -0.966 0.449 5.235 -0.637 -0.764 0.447 5.237 -0.622 -0.507 0.446 5.238 -0.615 -0.829 0.446 ! 5.239 -0.607 -0.165 0.445 ! 5.239 -0.607 -0.598 0.445 190 5.240 -0.600 -0.435 0.445 5.241 -0.592 -0.580 0.444 5.242 -0.584 -0.687 0.444 5.243 -0.577 -0.448 0.444 5.244 -0.569 -1.120 0.443 5.245 -0.561 -0.677 0.443 5.246 -0.554 -0.240 0.442 ! 5.247 -0.546 -0.643 0.442 ! 5.247 -0.546 -0.779 0.442 5.249 -0.530 -0.497 0.441 200 5.250 -0.523 -0.516 0.440 ! 5.251 -0.515 0.006 0.440 ! 5.251 -0.515 -0.379 0.440 ! 5.254 -0.491 -0.049 0.439 ! 5.254 -0.491 -0.260 0.439 ! 5.254 -0.491 -0.560 0.439 ! 5.254 -0.491 -0.638 0.439 ! 5.256 -0.475 -1.108 0.438 ! 5.256 -0.475 -0.563 0.438 ! 5.259 -0.450 -0.413 0.436 210 ! 5.259 -0.450 -0.413 0.436 ! 5.259 -0.450 -0.424 0.436 ! 5.261 -0.434 -0.369 0.435 ! 5.261 -0.434 -0.374 0.435 5.263 -0.417 -0.402 0.434 5.274 -0.324 -0.137 0.429 ! 5.276 -0.306 0.265 0.428 ! 5.276 -0.306 -0.100 0.428 ! 5.276 -0.306 -0.303 0.428 ! 5.286 -0.216 -0.110 0.423 220 ! 5.286 -0.216 -0.623 0.423 5.297 -0.112 0.004 0.418 5.315 0.069 0.292 0.408 ! 5.316 0.080 0.000 0.408 ! 5.316 0.080 0.388 0.408 5.319 0.112 0.284 0.406 5.347 0.432 0.433 0.390 5.364 0.649 0.881 0.380 5.411 1.354 1.136 0.351 5.415 1.423 1.368 0.348 230 5.461 2.332 2.376 0.315 ! 5.518 3.880 3.951 0.267 ! 5.518 3.880 3.896 0.267 5.569 5.905 5.998 0.216 5.609 8.215 8.421 0.167 5.610 8.284 8.289 0.166 5.611 8.354 8.414 0.164 5.643 11.028 11.046 0.117 5.668 13.910 13.852 0.073 5.692 17.722 17.663 0.072 240 5.716 23.193 23.112 0.082
126
Line Eα(MeV) δ2 (◦) δrand2 (◦) σ2(◦)
241 5.719 24.043 23.911 0.083 5.743 32.812 32.907 0.111 5.763 44.011 43.908 0.160 5.768 47.582 47.429 0.168 ! 5.773 51.513 51.581 0.171 ! 5.773 51.513 51.404 0.171 5.784 61.472 61.365 0.161 5.794 71.963 70.808 2.002 5.799 77.578 77.625 0.642 250 5.804 83.316 83.183 0.263 ! 5.819 100.143 100.168 0.103 ! 5.819 100.143 100.171 0.103 ! 5.819 100.143 100.113 0.103 5.824 105.293 105.333 0.155 5.834 114.544 114.548 0.246 5.835 115.388 115.804 0.253 5.839 118.611 118.530 0.281 5.844 122.311 122.293 0.310 5.845 123.009 123.467 0.315 260 5.855 129.274 129.246 0.351 5.859 131.448 131.411 0.359 5.863 133.455 133.381 0.365 5.864 133.932 134.158 0.365 5.876 138.985 138.833 0.368 5.879 140.077 140.176 0.366 5.895 144.995 144.873 0.350 5.899 146.024 146.560 0.344 5.919 150.286 150.174 0.312 5.943 153.994 154.051 0.272 270 5.944 154.125 154.150 0.270 5.969 156.910 156.847 0.229 5.989 158.637 158.754 0.199 5.996 159.161 159.110 0.189 6.020 160.716 160.810 0.155 ! 6.041 161.829 161.944 0.127 ! 6.041 161.829 161.808 0.127 ! 6.041 161.829 161.792 0.127 ! 6.041 161.829 161.923 0.127 ! 6.042 161.877 161.856 0.125 280 ! 6.042 161.877 161.945 0.125 ! 6.042 161.877 161.901 0.125 ! 6.042 161.877 161.791 0.125 ! 6.042 161.877 161.844 0.125 ! 6.042 161.877 162.082 0.125 ! 6.042 161.877 161.783 0.125 ! 6.042 161.877 161.896 0.125 ! 6.042 161.877 161.960 0.125 ! 6.043 161.925 161.899 0.124 ! 6.043 161.925 161.899 0.124 290 ! 6.043 161.925 162.034 0.124 ! 6.043 161.925 161.938 0.124 ! 6.043 161.925 161.853 0.124 ! 6.043 161.925 161.801 0.124 ! 6.043 161.925 161.997 0.124 ! 6.043 161.925 161.842 0.124 ! 6.043 161.925 161.978 0.124 ! 6.043 161.925 162.150 0.124
Line Eα(MeV) δ2 (◦) δrand2 (◦) σ2(◦)
! 6.043 161.925 161.864 0.124 ! 6.043 161.925 161.752 0.124 300 ! 6.043 161.925 161.979 0.124 ! 6.043 161.925 161.900 0.124 ! 6.043 161.925 161.975 0.124 ! 6.043 161.925 161.758 0.124 ! 6.043 161.925 162.049 0.124 ! 6.043 161.925 161.851 0.124 ! 6.043 161.925 161.900 0.124 ! 6.043 161.925 161.965 0.124 ! 6.043 161.925 162.006 0.124 ! 6.043 161.925 161.925 0.124 310 ! 6.043 161.925 161.918 0.124 ! 6.043 161.925 162.029 0.124 ! 6.043 161.925 161.935 0.124 6.099 164.109 164.112 0.103 6.119 164.707 164.733 0.094 6.149 165.478 165.481 0.089 6.170 165.946 165.932 0.087 6.199 166.514 166.507 0.061 6.219 166.862 166.790 0.083 6.249 167.329 167.362 0.116 320 6.282 167.781 167.766 0.151 ! 6.298 167.980 168.097 0.168 ! 6.298 167.980 167.864 0.168 6.370 168.757 168.715 0.243 6.401 169.045 168.944 0.275 6.422 169.227 168.925 0.296 ! 6.474 169.644 169.801 0.348 ! 6.474 169.644 169.714 0.348 ! 6.485 169.727 169.606 0.359 ! 6.485 169.727 169.845 0.359 330 6.490 169.765 170.089 0.364 6.495 169.802 169.525 0.369 6.500 169.838 169.597 0.374 6.501 169.845 169.776 0.375 6.506 169.882 169.839 0.380 6.507 169.889 169.991 0.381 ! 6.511 169.918 170.085 0.385 ! 6.511 169.918 170.451 0.385 6.516 169.954 169.996 0.389 6.520 169.982 170.041 0.393 340 6.521 169.989 169.845 0.394 6.525 170.017 169.933 0.398 6.526 170.025 169.470 0.399 6.531 170.060 170.257 0.404 6.532 170.067 170.164 0.405 6.536 170.094 170.237 0.409 6.540 170.122 169.761 0.412 6.542 170.136 170.080 0.414 6.545 170.157 169.969 0.417 ! 6.571 170.333 170.979 0.442 350 ! 6.571 170.333 170.412 0.442 6.595 170.492 170.563 0.464 ! 6.620 170.655 171.328 0.487 ! 6.620 170.655 170.558 0.487 ! 6.620 170.655 170.628 0.487
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