The Effect of the E1 Strength Function on Neutron Capture Cross Sections

Transcription

1 The Effect of the E1 Strength Function on Neutron Capture Cross Sections Berkley J.T. Starks Brigham Young University-Idaho 009 March Introduction The myriad of phenomena that are observed throughout the universe stem from the interactions of 4 main forces. These forces in order of weakest to strongest are gravity, the weak nuclear force, the electromagnetic force, and the strong nuclear force. The electromagnetic force and the strong nuclear force are the primary factors as to the physical makeup of matter throughout the universe. The interplay of these forces is made clearly manifest within the atomic nucleus. Key to the study of nuclear physics is the nuclear cross section. Nuclear cross sections are a measure of the probability of certain reaction occurring. Dr. Krane defines the differential cross section, dσ/dω, as the probability that an incident particle, dσ, is scattered into the unit solid angle, dω. The probability of dσ being scattered into dω is the ration of the scattered current through dω to the incident current. The total cross section, σ, is the total probability of scattering in any direction. 1A. Nucleosynthesis Nucleosynthesis attempts to interpret the measured abundances of the nuclear species in terms of their nuclear properties and a set of environments in which nuclei can be synthesized by nuclear reactions. (Clayton, 69). Nucleosynthesis depends on networks of nuclear reactions/processes in order to produce all heavy elements. Key to these reaction networks is the cross section. It is theorized that most of the lighter elements (Z 6) were synthesized during the Big Bang. The rest of the elements with nuclear charge Z 6 are in fact the ashes of nuclear burning during stellar evolution. (Clayton, 71). The basic hypothesis of these studies as stated by Chandrasekhar is that: Apart from secondary effects of minor importance, the transmutation of of elements is the entire cause of the presence of all elements in the stars; they are all being synthesized continually in the stars which are assumed to have started as pure masses of hydrogen; (Chandrasekhar, 469) From this we can see the importance of nucleosynthesis studies in astrophysics. By using particle fluences and thermodynamic conditions in stars, we are able to understand the observed abundances of stable isotopes.

2 1B. Nuclear Cross Sections Nuclear cross sections determine the probability of of a process occurring. A cross section is highly dependent on the energy of the incident particle(s). As was discussed above, the differential cross section, dσ/dω, is the probability that an incident particle, dσ, is scattered into the unit solid angle, dω. From nucleosynthesis we can see the importance of cross sections in the creation of elements through the reaction networks. While we measure all cross sections that we are able to do, there are many unstable isotopes that it is impossible to test on. The problem lies in that these isotopes are produced in reaction networks, and will influence and interact with the other particles involved in the reaction. Because we are not able to test upon these isotopes, theory must provide cross sections for them if we are to fully understand the reaction networks. In measuring these cross sections there are several different reaction networks depending on the type of incident energy. One of the most critical is neutron capture cross sections. Neutron capture cross sections, n,, have several important ingredients. Most important at low energies is the photon transmission coefficient. Making up the photon transmission coefficient is the gamma ray strength function, f XL, the energy dependent width of the giant dipole resonance, GDR, T f, and the E1 strength function. In order to be able to increase the accuracy of theory, we must be able to reproduce measured cross sections on stable isotopes. The issue comes from that there are several ways to model the E1 strength function. Each of these models reproduces experimental cross sections in localized regions of the periodic table, but none are able to globally reproduce experimental cross sections. What we will be doing is modeling the E1 strength function using several different schemes and attempting to see if one model as a better global fit to the others. When calculating these models there are two ways to normalize the photon transmission coefficient, specifically to the model used for the E1 strength function. This can be normalized either to the average radiation widths,, at the neutron binding energy, Bn, or the E1 strength can be normalized to the Maxwellian averaged cross section (MACS). What we are trying to do is see what effect the E1 strength function has upon the cross section, and if by finding the appropriate f E1 we are endeavoring to be able to reproduce the experimental n, cross sections and be able to replicate both B n and 30 kev MACS. 1C. Some Basic Notation The following section will give a brief overview of the notation throughout this document. The notation used is the same as found in reference 4. Z - Proton number of the nucleus N - Neutron number of the nucleus A - mass of the nucleus in atomic mass units (generally A=Z+N)

3 J - angular momentum π - parity lower case letters - n, p, α, or γ particles capital letters nucleus (not n, p, α, or γ particles) - A further and more in-depth explanation of the notation will explained within chapter of this thesis under the section entitled The Hauser-Feshbach Statistical Model.

4 . Review of Theory A. Direct and Compound Reactions In a nuclear reaction there are two main types of processes that occur. These reactions types are called direct processes and compound processes. Direct reactions involve only very few nucleons...with the remaining nucleons of the target serving as passive spectators (Krane p. 379). These reactions only remove or insert a single nucleon from the target nucleus and are used to help explore and analyze the spectroscopic states in nuclei. Compound reactions happen when the incoming and target nuclei briefly merge and completely share the incident energy before the outgoing nucleons are ejected. One of the most important assumptions of the compound-nucleus model is that the relative probability for decay into any specific set of final products is independent of the means of formation of the compound nucleus (Krane pp ). An example of a compound process is the formation and then decay of 64Zn. 64Zn can be formed by either p + 63Cu or α + 60Ni. After 64Zn* is formed it can then decay into 63Zn + n, 6Cu + n + p, or 6Zn + n. The decay of 64Zn is completely independent of the way that it is formed. Whether it is through a proton capture, or and alpha capture, the formation does not affect the decay of the compound nucleus. As can be seen from the graphs below, the shape of the cross sections are relatively the same with only slight perturbations. The two experimentally observable differences between direct and compound reactions are (1) Direct processes occur very rapidly, in a time of the order of 10 - s, while compound-nuclear processes typically take much longer, perhaps to s...() The angular distributions of the outgoing particles in direct reactions tend to be more sharply peaked than in the case of compound-nuclear reactions (Krane pp ). This last point is due to the fact the since the nucleus remembers how it was made, the momentum of the incident particle is going to transfer its momentum in the same incident direction upon the target nucleons. One other difference between direct and compound reactions is summarized by Dr. Krane in that incident 1-MeV neutrons have a wavelength of ~ 4 fm. Because of this a 1-MeV neutron is most likely

5 to interact through compound nuclear reactions. Direct reactions involve higher energies (~0-MeV) have a de Broglie wavelength of 1 fm. Because of the higher wavelength the incident neutrons are more likely to interact with a single nucleon through direct nuclear processes. In stellar nucleosynthesis the energies involved are generally at most in the 5 MeV range, but typically far less. B. The Hauser-Feshbach Statistical Model With the formation of the compound nucleus at less energetic excitation levels the nuclear reactions proceed through narrow resonances. Resonances are a systematic excitation of the nucleus that go up in quantified steps instead of as a continuous spectrum. These resonances range in width from a few ev to several kev. These resonances correspond to nuclear states above the bound region. As the excitation energy increases, the spacings between the resonances decreases, eventually leading to the point where resonances are too tightly packed together to so that the specification of individual resonance properties is not possible. In order to analyze and predict what happens with the cross section, statistical models are used. These statistical models take the myriad of factors that influence the cross section and produce modeled cross sections. One of the statistical models used in statistical nuclear theory is the Hauser-Feshbach formula (see reference 8). This formula is given by: j k E j = ƛj T j J T k J g W J J g I g j J, T tot J where j is the incident particle, k is the outgoing particle, jk E j ƛj gi gj gj T j J T k J T tot J W J the average cross section for the reaction. The average is taken over an energy range that contains several compound nuclear resonances of spin and parity. 1 wavelength related to the wave number by ƛ j =. kj For n, p, particles, k j in the center of mass frame is k j= M U A j E j ℏ the statisical weight, J I 1 of the target. the statisical weight, J j 1 of the incident light particle. For photons g j =1. the statisical weight, J 1 of the compound state. the total transmission coefficient for forming the state J in the compound nucleus at energy E j. the total transmission coefficient for forming the state J in the compound nucleus at E k. the sum of all transmission functions related to decay of the state of interest. width fluctuation corrections (WFC). These define correlation factors with which all partial channels of incoming energy j and outgoing particle k, passing through excited state (E, J, ), should be multiplied.

6 Looking at the Hauser-Feshbach formula in parts we see that: ƛ j I k g J T j J is g g j J, T J T tot J for forming the compound nucleus in state J with incident paricle j. And is the probability that the compound nucleus in state J will decay by emitting particle k. This formulation of Hauser-Feshbach is for the binary reaction (j,k). The Hauser Feshbach formula can be modified such that it can allow for multiple particle emission. This requires multiple compound steps. Of prime importance to our studies at present is the T factor found within. At the lower energies when particle transmission is impossible/highly improbable the T tot is dominated by the photon transmission coefficient. Also, for capture reactions, T is in the numerator of Hauser-Feshbach, so ~ T. (See references 4 and 6 for further information on Hauser-Feshbach.) C. Nuclear Level Density There are several different models for nuclear structure. A few of these models are the shell model, the liquid-drop model, and the interacting boson model. (Nuclear Physics) The shell model at its simplest level predicts that nuclei having closed (completely occupied) shells of protons and neutrons should be unusually stable-as is, in fact, observed (Nuclear Physics p. 37). This fact is observed in nature. This trait is analogous with the chemical analysis of the noble gases that have filled outer electron shells and are non reactive. The nuclear level density, tot E x, J,, is defined as the number of excited nuclear states with spin, J, and parity,, per MeV around the excitation energy E x. The total level density is the sum of ll nuclear states around a given energy, i.e. tot E x = E x, J, J When these level densities are analytically expressed they are generally factorized by: E x, J, = P E x, J, R E x, J tot E x with P E x, J, being the parity distribution, and generally is taken to be ½, and with R E x, J being representing the spin distribution. The specific model that is incorporated into the TALYS code and that we will be using is the Fermi Gas Model. The Backshifted Fermi Gas Model (FGM) is based on the assumption of evenly spaced single particle excitations. Collective excitations, or the excitations of multiple particles are not considered.

7 Assuming that the projections of total angular momentum are randomly coupled, the FGM level density is: F E x, J, = 1 J 1 J 1 e 3 1 e au 1 4 a U 5 4 with the ½ in the front being P E x, J,, and being the spin-cutoff parameter. a is the level density parameter, and U is the and U = E with being the backshift related to the pairing energy. Using the FGM the Fermi gas distribution is given by: R F E x, J = 1 J J 1 e using this definition to sum over all F E x, J, we get the total Fermi gas level to be: tot F E x = 1 e au 1 a 14 U 54 showing that tot, the spin-cutoff F E x is dependent on the level density parameter a, and parameter. Energy-dependent shell effects are taken into account by making the level density parameter, a, energy dependent. This formulation is: a=a E x = a 1 W U 1 e U where a is the asymptomatic level density value obtained in the absence of any shell effects (i.e. a=a E x, and W is the shell corrrection energy. The asymptotic value a is typically parameterised by a= A A /3 where A is the atomic mass number of the element in question, and and are parameters that that have been chosen to give the best average level density description over a large range of nuclei. can be related to the undeformed moment of inertia, I 0, and the thermodynamic temperature, t. This is related by = = I 0 t. In this is the parallel spin cut-off parameter, or the projection of the angular momentum onto the spin axis. Due to the fact that /t suffers from shell effects and is not constant, is rewritten as: m0 R A a 5 = = F E x = I 0 t, or substituting in a : I 0= a ℏ c

8 R=1.A 1/3 is the radius, and m0 is the neutron mass in AMU's. This can be rewritten as: F E x = D. 5 /3 A a au Particle and Photon Transmission Coefficients At sufficiently low incident neutron energies, the average radiative capture width, Γ γ, is due entirely to the s-wave interaction, and it is Γγ at the neutron separation energy Sn that is often used to normalize gamma-ray transmission coefficients. (Gardner, p. 6) The gamma-ray strength function is related to the photon transmission coefficient by: T XL = L 1 f XL because T XL is directly dependent on f XL the normalization of f XL is of prime importance to the photon transmission coefficient. When normalizing the gamma-ray strength function, there are several ways to model the strength function. The 5 models that we will be considering are the Blatt-Weisskopf method, using a BrinkAxel Lorentzian, a Kopecky-Uhl generalized Lorentzian, the Hartree-Fock BCS tables, or the HartreeFock Bogolyubov tables. D-1. Blatt-Weisskopf Use of the Blatt-Weisskopf is f E1 =constant. While at higher other levels, E, M, etc., the BlattWeisskopf may be a sufficient model for f E, f M and so on, but it is not a sufficient model at f E1. D-. Brink-Axel Lorentzian The Brink-Axel model describes the standard Lorentzian form of the transmission coefficient at the giant dipole resonance and is calculated by: f Xl E = K Xl Xl E Xl E E Xl E Xl where Xl is the strength of the dipole resonance, E Xl is the energy, and Xl is the width of the dipole resonance. And K Xl is 1 K Xl = l 1 ℏ c Use of the Brink-Axel Lorentzian is the commonly used model for all other transmissions besides for M1.

9 D-3. Kopecky-Uhl generalized Lorentzian The general way that many models depict the photon transmission for E1 radiation is through the use of a Kopecky-Uhl generalized Lorentzian. This is calculated by: f E1 E, T = K E1 [ E E1 E 0.7 E1 4 T ] E1 E1 E E E1 E E1 E E 3E1 where E1, E1, and E E1 are the peak, width, and energy of the giant dipole resonance of the nucleus E1 is the energy dependent damping and is given by: in question. E1 = E1 E 4 T E E1 and T, the nuclear temperature: T= E n S n E a S n E n is the incident neutron energy, S n is the neutron seperation energy, and is the pairing connection. D-4. Hartree-Fock BCS & Hartree-Fock-Bogolyubov tables The last two ways of modeling the E1 strength function is to allow for microscopic corrections (Goriely et al). Using the Hartree-Fock Method the E1 strength function is calculated by: GDR / E f L E, E i = E E i GDR / E where GDR is the giant dipole resonance width and is taken from experimental data when experimental data is to be had, otherwise is taken for empirical statistics. Using the BCS tables with the Hartree-Fock Method or combining Hartree-Fock with the Bogolyubuv method leads to sight perturbations in the calculations of the transmission coefficient, but as will be seen in the data, the perturbations are are only slight. D-5. Normalization of E1 When normalizing the gamma ray transmission coefficient it can be either normalized to the average radiation widths, or to the Maxwellian averaged cross section. D-5a. Radiation Widths When normalizing to the radiation widths, on most isotopes the radiative widths, GDR, is measured. For unmeasured radiation widths on unstable isotopes, the average radiation widths are calculated by:

10 J 1 1 J 1 B n, J B n, J 0= J 1 J 1 where J is the spin of the target nucleus and F. 1 B n, J =. Maxwellian Averaged Cross Section Another way of normalizing the E1 strength function is to normalize to the Maxwellian-averaged cross section. The Maxwellian-averaged cross section is the reaction rate < > divided by the mean velocity v T = kt/ at a given temperature T. (Hoffman p. 18). μ is the reduced mass and k is Boltzman's constant. What the Maxwellian-averaged cross sections endeavor to do is to analyze a thermally averaged cross section set for a group of atoms. This becomes of prime importance in astrophysical studies. The Maxwellian-averaged cross section reduces to: < > = vt n d 0 vt n E W E, kt de kt 0 where W E, kt =E e E/ kt and E is the center of mass energy. =

11 3. Code 3A. TALYS In order to calculate the cross sections relevant to the study found herein, the TALYS nuclear reaction code was used. TALYS gives us a wide range of control over the inputs into the Hauser-Feshbach statistical model. The TALYS code can either be used with a variety of different options, or can be used in a black-box mode where standardized global inputs are used. In this latter mode, it is possible to generate a cross section very easily with minimal input by the user. For the studies contained herein we used TALYS in a primarily black-box mode, altering only the E1 strength function. A basic TALYS input file is as follows: projectile n element Fe mass 56 energy 14 These four lines are all that is needed to run up a basic cross section. The line projectile tells TALYS what the incident light particle is (n, p, alpha, etc...). On the element line we define what element the target is by using the nomenclature found on the periodic table. Mass is the target's mass where we define which isotope of the element we are using. Finally the energy line lets us know what energies we are using for our reaction. These energies are given in MeV. For our studies we wished to calculate our cross sections over a range of energies ranging from 10 ev to 0 MeV. Also, we wanted to test the effects of the different models for the photon transmission coefficient. In order to do this our input file looked like: projectile n element ca mass 40 energy erange strength 1 Here all the inputs are the same as above, except that the file erange contained our range of energies and the line strength. Even though on the line energy we do not contain on specific energy, but rather a file that contains a range of energies, these values are still considered blackbox making the only change being that of the E1 strength function. What the strength key does is it allows us to define which strength function for the E1 transmission coefficient we will use. A value of 1 specifies a Kopecky-Uhl generalized Lorentzian. Value specifies a Brink-Axel Lorentzian. Value 3 uses the Hartree-Fock BCS tables, and value 4 uses the Hartree-Fock-Bogolyubov tables. (See reference 6 for a complete listing of all TALYS input commands).

12 In order to quickly and efficiently run our code a wrapper was used to generate the individual input files from Z=1 to Z=84. The different strength functions were used on all the isotopes tested in order to see the affect that they had upon the calculations. These cross sections were later turned into Maxwellian-averaged cross sections and compared to experimental data. The other use for the wrapper is that most calculations using TALYS are done for one incident energy and take very little time. Because we are calculating cross sections over a range of energies, the code takes a lot longer to run. What the wrapper does is it distributes the jobs over a series of Beowulf cluster nodes and allows for multiple cross sections to be calculated at the same time. 3B. MACS+ After TALYS has calculated the cross sections, it is necessary to convert these cross sections into the thermally averaged Maxwellian averaged cross sections. To do this a code written by Dr. Kevin Kelley is used. This code is called macs+. This code was developed during Dr. Kelley's time at Lawrence Livermore National Laboratory (LLNL). This code as originally made to be used with the STAPRE nuclear reaction code, but in conjunction with the wrapper, can be used to convert cross sections produced with TALYS into Maxwellian averaged cross sections. In order to produce a MACS, an input file looks like: macs+ n tot.int 10 where macs+ calls the code, n tot.int is the interpolated cross section produced by the TALYS wrapper, and 10 is the atomic mass of the initial heavy particle. When the code runs it produces a *.ps graph of the file with experimental data points graphed along with the systematic calculation.

13 4. Effect of the Photon Transmission Coefficient Provided below will be an analysis of the effect of the photon transmission coefficient on the Maxwellian averaged cross section. We will be looking at the overall systematic and how well each model globally fits to experimental data. In the plots that follow the data for Nitrogen 14 has been left out because on a global basis, using all 4 models for the E1 strength function the MACS for Nitrogen 14 was off by over a factor of 80 when compared to experimental data. For a full listing of all data and graphs the reader is referred to the appendixes. 4A. Brink-Axel Lorentzian Figure 1: Global Comparison of calculated MACS vs. experimental MACS Brink-Axel Lorentzian As can be seen from the plot above of the ratio of calculated MACS over the experimental MACS for the use of a Brink-Axel Lorentzian, the data is reasonably good, with a few outliers. But even the data points that are close to a ratio of 1, there are still many that are off by a factor of or 3. Looking at a specific cross section, in this case (and the case for al strength functions that we will be

14 comparing at this time) we will use Nickel 61. The MACS for Nickel 61 using the Brink-Axel Lorentzian is: The plot here is a logarithmic plot of the cross section (in millibarns) versus the energy (in MeV). By observing the graph we can see that the systematic is close to the measured data, but is still high, at some points by a factor of or 3. 4B. Kopecky-Uhl Generalized Lorentzian The Kopecky-Uhl generalized lorentzian is similar in calculation to the Brink-Axel Lorentzian, except that it allows for energy and width fluctuations and corrections, but it still has a general Lorentzian shape. Looking at the data for Nickel 61 using the Kopecky-Uhl Lorentzian (see figure 3 on next page) it can be seen that the data is roughly the same as when the Brink-Axel Lorentzian was used. Looking at an overall systematic (figure 4) it can be seen that like is brother, the Brink-Axel Lorentzian, it is still off by several orders of magnitude.

15 Figure 3: MACS for Nickel 61 using a Kopecky-Uhl Generalized Lorentzian Figure 4: Global Comparison of calculated MACS vs. experimental MACS Kopecky-Uhl Generalized Lorentzian

16 By looking at these two global systematics it can be seen that while in some cases a Lorentzian is an adequate model for the E1 strength function, globally it does not fit very well. In order to better model the E1 strength function, other models must be looked at. One of these models is the Hartree-Fock model. 4C. Hartree-Fock BCS Tables When calculating the E1 strength function using the Hartree-Fock method, it is customary to have preciously calculated the E1 strength functions and to have tabulated them. This is because of the myriad of different inputs that go into the Hartree-Fock method, it would considerably lengthen calculation time to calculate the E1 strength function every time. What can be seen from us of the Hartree-Fock method is a better global fit for the overall Maxwellian averaged cross section. Figure 5: Global Comparison of calculated MACS vs. experimental MACS Hartree-Fock BCS Tables First off, it can be seen that the Hartree-Fock BCS tables are more closely centered around a ratio of 1. Also, the outliers in data tend to have a lower calculated over experimental ratio than that of using a Lorentzian. Looking at the MACS for Nickel 61 we see:

17 Figure 6: MACS for Nickel 61 using the Hartree-Fock BCS Tables As can be immediately seen, the Hartree-Fock BCS tables are a lot more accurate on Nickel 61. Also, it can be seen that unlike the Lorentzian's, the BCS tables actually fall within the error of the measured cross sections. This is a vast improvement. 4D. Hartree-Fock-Bogolyubov Tables Lastly for our considerations is the use of the Hartree-Fock-Bogolyubuv Tables. This table is another set of calculated E1 strength functions using the Hartree-Fock method, but with slight alterations made by Bogolyubuv. Looking at Figure 7 we can see the global fit is approximately the same as the BCS tables with some minor differences between them. In the BCS tables we can see that the ratio of calculated over experimental never grow larger than a factor of about 13 or 14 while the Bogolyubuv tables have a data point at a factor of 0 times larger. But even with that, it can be seen that the rest of the outliers are tightly packed close to a ratio of 1.

18 Figure 7: Global Comparison of calculated MACS vs. experimental MACS Hartree-Fock Bogolyubuv Tables Looking at the data for Nickel 61 using the Bogolyubuv tables: Figure 8: MACS for Nickel 61 using the Hartree-Fock-Bogolyubuv Tables

19 It can be seen from figure 8, that in this case the Bogolybuv tables provided a closer fit to experimental data than the BCS tables did. While this may be the case for Nickel 61, other targets are not the same. 4E. Conclusions and Future Research From the data provided above, and from the data seen in the appendixes it can be seen that the HartreeFock Tables provide a better fit for the normalization of the photon transmission coefficient. While in many cases a generalized or even an enhanced Lorentzian may be adequate, globally speaking the Hartree-Fock method provides a better model for the E1 strength function. The differences between these four models presented show us that the E1 strength function strongly impacts the cross section for neutron capture cross section. But even with the better global fit of the E1 strength function using the Hartree-Fock method it is important to note that other models can be used for the E1 strength function, and that there are a myriad of other inputs to the Hauser-Feshbach statistical model that could possible effect the calculation of the nuclear cross section. While the research presented herein has been extremely comprehensive for an analysis of these 4 models on neutron capture cross sections, future research should be done on the influence of the other factors that go into the HF model. Also, the actual calculation of BCS and Bogolybuv tables would warrant merit in that it could be used to better calculate these tables for an improved cross section. While a lot as been learned about neutron capture cross sections, more work is still needed to better understand the calculation of nuclear cross sections.

20 Appendix A Calculated Cross Sections Z=1 to Z=84 #Z A Element Recommended MACS dmacs Strength1 Strength Strength3 Strength n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E+01

21 #Z A Element Recommended MACS dmacs Strength1 Strength Strength3 Strength n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E+00

22 #Z A Element Recommended MACS dmacs Strength1 Strength Strength3 Strength n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E+0.850E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E+0

23 #Z A Element Recommended MACS dmacs Strength1 Strength Strength3 Strength n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E+01

24 n E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E+03

25 #Z A Element Recommended MACS dmacs Strength1 Strength Strength3 Strength n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E n n E E E E E E E E+0