Phenomenological description of neutron capture cross sections at 30 keV
aarXiv:yymm.nnnn [nucl-ex]
Phenomenological description of neutron capturecross sections at 30 keV
Mikl´os Kiss
Berze N.J. Gimn´azium, Kossuth 33, H-3200 Gy¨ongy¨os, HungaryE-mail: [email protected] and
Zolt´an Tr´ocs´anyi
University of Debrecen and Institute of Nuclear Research of the Hungarian Academy ofSciences, H-4001 Debrecen P.O.Box 51, HungaryE-mail: [email protected]
Abstract
Studying published data of Maxwellian averaged neutron capture cross sections,we found simple phenomenological rules obeyed by the cross sections as afunction of proton and neutron number. We use these rules to make predictionsfor cross sections of neutron capture on nuclei with proton number above 83,where very few data are available.2011 a r X i v : . [ nu c l - e x ] A p r Introduction
Theoretical descriptions of nucleosynthesis in stars rely heavily on the knowledge of capturecross sections of slow neutrons on nuclei. The classical model of nucleosynthesis in weakneutron flux is based on slow neutron capture (the s process) that occurs along a path inthe stability valley of nuclei (see for instance, Refs. [1–4]). The s -process evolution codestake into account the most important processes (those with largest cross sections) alongthe stability valley. The necessary information on the neutron capture cross sections and β decay life times, needed to describe qualitatively the abundances of the s -process elements,is rather well known from laboratory experiments [5–8].The s -process model is capable to explain the observed abundance of heavy elementsfairly well [9]. The difference of observation and prediction is largely attributed to anotherprocess that occurs in stellar enviroment with high neutron flux, typically in supernovae.In such circumstances the neutron capture is very likely and neutron rich nuclei far fromthe stability valley build up very quickly due to repeated capture of neutrons. The nucleiproduced such a way are so unstable and short-lived that experimental information abouttheir capture cross sections and decay life times is not generally available.In a recent work we proposed a unified model of nucleosynthesis of heavy elements instars [10]. That approach takes into account all possible types of production and depletionmechanisms and solves the whole system of differential equations numerically. The resultof such an approach is that (instead of the s -process path) the evolution of the synthesisproceeds along a band in the valley of stable nuclei. The width of this band – and conse-quently the final abundances of nuclei – depends on the neutron flux and the capture crosssections on individual nuclei charactherized by both their proton and neutron numbers, σ ( Z, N ), which constitutes an essential input to the model calculations. Therefore, it isimportant to learn about these cross sections as much as possible.In this paper, we study the general features of Maxwellian averaged neutron capturecross sections collected in recent compilations of data [7, 8]. In section 2 we show somephenomenological observations. In the following section we use those to make some orderof magnitude predictions for the cature cross sections σ ( Z, N ) for proton numbers
Z >
Maxwellian averaged neutron capture cross sections (MACS) have been measured for manynuclei and made available in public data depositories. A comprehensive and completereview has been presented recently in Ref. [8]. Studying the available data, we can make1everal observations: (i) although cross sections of many nuclei have been measured, thereare still many missing, or rather uncertain data, especially for nuclei with
Z >
83 (seeFig. 1a); (ii) the cross sections vary over very large range of values (about four orders ofmagnitude); (iii) for any fixed neutron number N the the cross section is maximal for acorresponding value of the proton number Z max and decreases rapidly as | Z − Z max | increases(see Fig. 1b). The last point implies that in the Z − N plain for each N their is a unique value Z max ( N ) where the capture cross section attains its maximal value. The existence of such amaximum is qualitatively easily understood: for fixed N , increasing Z starting from a smallvalue of Z , the capture of an additional N stabilizes the nucleus in the strong repulsiveCoulomb field of the protons, the binding energy per nucleon increases. However, for Z above some value Z max ( N ) the nucleus developes a neutron skin and additional neutronsbecome more and more loosely bound and capturing any further neutrons becomes lesslikely. The quantitavie understanding is certainly more complex, which however, is beyondthe scope of the present paper.
10 30 50 70 90 110 130 150 170102030405060708090100 − − σ [ m b ] NZ n-capture cross sections -3 -1 n [ m b ] Z n-capture cross sections for fixed N ............................... N=30N=31 . N=60N=61N=90N=91
Figure 1: (a) MACS (at 30 keV) on nuclei as a function of the proton and neutron number.(b) Dependence on the proton number Z of the MACS on nuclei with fixed neutron number N = 30, 31, 60, 61, 90 and 91 (indicated by vertical lines on Fig. 1a).If we plot the Z max ( N ) function then a rather simple picture emerges: it appears thata simple, almost linear function can describe the data, especially for small N . This featurebecomes even more salient if we devide the nuclei into four groups according to the even/oddnumber of protons and neutrons: (i) Z (ee)max for Z even, N even, (ii) Z (oo)max for Z odd, N odd,(iii) Z (eo)max for Z even, N odd, and (iv) Z (oe)max for Z odd, N even, as shown in Fig. 2.In Fig. 2 crosses mark the values of Z max where the n-capture cross section is maximalfor a fixed value of the neutron number N as taken from Ref. [7]. The solid lines represent2 Z ee m a x N Position in Z of highest n-capture cross section for fixed N
Z-even N-even nuclei(N+2.47)/(1 + 0.013 N )stability valley Z e o m a x N Position in Z of highest n-capture cross section for fixed N
Z-even N-odd nuclei(N+0.42)/(1 + 0.013 N )stability valley Z o e m a x N Position in Z of highest n-capture cross section for fixed N
Z-odd N-even nuclei(N+0.07)/(1 + 0.013 N )stability valley Z oo m a x N Position in Z of highest n-capture cross section for fixed N
Z-odd N-odd nuclei(N-0.38)/(1 + 0.013 N )stability valley
Figure 2: The function Z max ( N ) for even-even, odd-odd, even-odd and odd-even nuclei.The crosses are the experimental values and the solid line represents the fit to the functionin Eq. (1). The dashed line runs through the bottom of the stability valley.fits of simple functions to these points in the form of f ( N ; a x , b, c ) = N + a x bN c , (1)with a x , b and c being fitted parameters, and x = ee, oo, eo, or oe. We determined thevalues of these parameters in two steps. First, we minimized the function χ ( a, b, c ) = (cid:88) i =1 (cid:16) Z max ( N i ) − f ( N ; a, b, c ) (cid:17) , (2)i.e. nuclei belonging to all four groups are taken into account and all points are assumedto have weight σ i = 1. The upper limit in each group was chosen the largest value forwhich Z max can be identified. With such a choice the we find N ( x )max = 50, 56, 53 and 55maxima in the groups of even-even, odd-odd, even-odd and odd-even nuclei, respectively(50 + 56 + 53 + 55 = 214). This fit gives a = 0 . , b = 0 . , c = 0 . , (3)3ith correlation index i = (cid:118)(cid:117)(cid:117)(cid:116) − χ ( a, b, c ) (cid:80) i =1 (cid:16) Z max ( N i ) − Z (cid:17) = (cid:114) − . . , (4)i.e. the coefficient of determination is almost one, i = 0 .
994 ( Z = (cid:80) i =1 Z max ( N i ) =45 . χ ( a x ) = N ( x )max (cid:88) i =1 (cid:16) Z ( x )max ( N i ) − f ( N ; a x , . , . (cid:17) (5)separately for each group ( x = ee, oo, eo, oe). These fits result are a ee = 2 . , a oo = − . , a eo = 0 . , a oe = 0 . , (6)with coefficient of determination above 0.99 in all cases.We also exhibit the line of the stablity valley in Fig. 2, as a function of N (instead ofthe usual A = Z + N ) Z stab = N + a s b s N c s , (7)with parameters a s = 0 . , b s = 0 . , c s = 0 . . (8)We see clearly that the highest n-capture cross sections lie above the stability valley andthe separation grows with N .We can also observe regularity in the Z-dependence of the cross section at fixed N (seeFig. 1b). We can extrapolate this regularity as well as the Z max values to the region in thenuclide chart where very few data available for n-capture cross sections on nuclei (nucleiwith proton number above 83, see Fig. 1a.The first observation is a simple trend in the behaviour of the function σ max ( N ) ≡ σ (cid:16) Z max ( N ) (cid:17) . Putting σ max ( N ) on a double logarithmic plot as shown in Fig. 3a (leftpanel), we find that the general trend is well described by a fourth-order power function, σ max ( N ) = (cid:18) N (cid:19) mb . (9)This general trend is slightly modulated with some oscillatory behaviour, with minimaaround magic numbers, as seen on Fig. 3b, where the ratios of the measured cross sectionsto σ max ( N ) are shown. 4 m a x ( N )[ m b ] N(a)
Z evenZ odd(N/10) -1 m ea s m a x ( N ) / m a x ( N ) N(b)
Z evenZ odd
Figure 3: (a) Largest neutron capture cross sections as a function of the neutron number.(b) Ratio of the measured largest cross sections to σ max given in Eq. (9).The second observation is that if we normalize the cross sections σ ( Z, N ) for a fixedneutron number N with the largest cross section σ max ( N ), then the profile of the dependenceon the proton number is rather similar for all neutron numbers. This similarity is best seenif the position of the largest cross section is shifted by − Z max to zero, therefore, we definethese normalized and shifted cross section values, ρ N ( z ) = σ ( z + Z max , N ) σ max ( N ) ≡ σ ( Z, N ) σ (cid:16) Z max ( N ) (cid:17) , (10)for all values of N , where data are available. Then we define the average by ρ ( z ) = 1 N z N z (cid:88) N =1 ρ N ( z ) , (11)with squared standard deviation σ ( z ) = 1 N z ( N z − N z (cid:88) N =1 (cid:104) ρ N ( z ) − ρ ( z ) (cid:105) , (12)where N z is the number of available data for fixed z . This average is shown in Fig. 4. Asseen from Fig. 4b this function is well approximated with an almost exponential functionin both positive and negative directions, but with different exponents. More precisely, wefit the logarithm of the average with quadratic functions of the form a i z + b i z + c i withsubscript of the coefficients refering to three regions in z : (i) i = 1 for z < −
26, (ii) i = 2for − ≤ z <
0, and (iii) i = 3 for 0 < z . For i = 2 and 3 we fix c i = 0. This form ensures5able 1: Result of the fit to the average function ρ ( z ). i a i b i c i χ /d.o.f1 0.0044 1.135 17.95 5.29/52 -0.0025 0.2658 0 6.15/93 -0.0058 -0.3948 0 7.14/4the constraint ρ (0) = 1. We also require the continuity of the fitted function at z = − χ (cid:39) (cid:88) z (cid:104) ln ρ ( z ) − ( a i z + b i z + c i ) (cid:105) (cid:16) σ ( z ) ρ ( z ) (cid:17) , (13)summed over values of z in the three regions separately. The result of these fits is presentedin Table 1 and shown in Fig. 4. ( z ) -20 -15 -10 -5 0 5 10 15 z (a) -3 -2 -1 ( z ) -20 -15 -10 -5 0 5 10 15 z (b) exp(0.266 z-0.003 z )exp(-0.395 z-0.006 z ) Figure 4: Average of the normalized neutron capture cross sections as a function of z = Z − Z max . The errorbars represent the standard deviation σ ( z ).Each function ρ N ( z ) differs from the average in two ways: (i) tipically the larger N thewider ρ N ( z ) (as seen on Fig. 1a), (ii) in addition there are seemingly random fluctuations.The origin of the latter could be either a small physical effect, or simply error of themeasurement: there are published values for cross sections σ ( Z, N ) that differ by a factorof two. While it is difficult to consider the effect of the latter, the first effect can be takeninto account by a simple appropiate scaling of the width of the average to those of thefunctions ρ N ( z ), which we discuss in the next section.6 Predictions
The phenomenological observations made in the previous section can be used to makepredictions for the order of magnitude of neutron capture cross sections in regions of thenuclide chart where experimental data are not available. We make these predictions in twosteps. First we validate our procedure by comparing our predictions to measured crosssections. Then we use our procedure to make predictions.
Our procedure relies on three pieces of information concluded from the analysis of the shapeof ridge of Maxwellian averaged neutron capture cross sections :1. position of Z max as a function of the neutron number (location of the ridge top onthe nuclide chart) obeys the simple function Eq. (1);2. values of σ max ( N ) (height of the ridge for given value of Z max ( N )) obey the simplefunction Eq. (9);3. characteristic behaviour of the average function ρ ( z ) (slope of the ridge) is as givenby Fig. 4.In order to predict the cross section values for fixed neutron number, we proceed along thefollowing steps:1. Given N , find the position of Z max from Eq. (1), which gives two maxima, one foreven proton numbers ( Z (e)max ) and one for odd proton numbers ( Z (o)max ).2. Given Z max (either Z (e)max , or Z (o)max ), position the maximum location of the averagefunction ρ ( z ) to Z max .3. Scale the height and width of the function ρ ( z ) to the available measured data byperforming a two-parameter fit: (i) the scale factor of the height, (ii) the scale factorof the width.The third step is hampered by the discrepancies in the measured cross section values, whichcan sometimes be quite significant as shown in Table 2 for heavy elements. Discrepanciesexist among data for lighter elements, but generally within a factor of two [8].7able 2: Ratios of largest and smallest measured neutron captured cross sections forelements beyond bismuth [8].nucleus σ max /σ min nucleus σ max /σ min nucleus σ max /σ min20482 Pb U Cm Pb U Cm Pb U Cm Ac Np Cm Ac Np Bk Th Np Bk Th Np Bk Th Np Bk Th Pu Cf Pa Pu Cf Pa Pu Cf Pa Pu Cf Pa Pu Cf Pa Am Cf U Am Es U Am Es U Cm Es U Cm Es U Cm Es U We can compare the values of the predicted cross sections to those measured experimentallyover the regions of the nuclide chart where data are abundantly available ( Z ≤ -3 -2 -1 n [ m b ]
16 18 20 22 24 26 28 30 32 34 Z n-capture cross sections for N=30 experimentprediction -3 -2 -1 n [ m b ]
16 18 20 22 24 26 28 30 32 34 Z n-capture cross sections for N=31 experimentprediction -1 n [ m b ]
35 40 45 50 55 60 Z n-capture cross sections for N=60 experimentprediction -1 n [ m b ]
35 40 45 50 55 60 Z n-capture cross sections for N=61 experimentprediction n [ m b ]
55 60 65 70 75 80 Z n-capture cross sections for N=90 experimentprediction n [ m b ]
55 60 65 70 75 80 Z n-capture cross sections for N=91 experimentprediction Figure 5: Dependence on the proton number Z of MACS (at 30 keV) on nuclei with fixedneutron number N = 30, 31, 60, 61, 90 and 91: comparison of the predictions of thephenomenological model to measure data. 9 .3 Predictions of unkown cross sections Our procedure can be used to make prediction for cross sections in regions of the nuclidechart where some experimental information are available, such as
Z >
83. In this regionthe general trend can be fitted to the measured data to complete the ridge. With such aprocedure we obtain cross section values shown in Table 3. We can now use those preditionsto complete the picture exhibited on Fig. 1. The result of such completion is shown in Fig. 6.
10 30 50 70 90 110 130 150 170102030405060708090100 − − σ [ m b ] NZ n-capture cross sections
Figure 6: Ridge of MACS (at 30 keV) on nuclei as a function of the proton and neutronnumber.
We studied the dependence of the published MACS data on the proton and neutron number.We found a simple characteristic behaviour that we call the shape of the ridge of MACS in10able 3: Predictions for neutron capture cross sections (in mbarns) as a function of theproton and neutron number for elements beyond bismuth.Z (cid:31)
N 132 133 134 135 136 137 138 139 140 141 14284 3 28 3 216 53 331 99 121 90 157 8285 74 577 99 772 534 229 530 171 385 29 21586 6 59 5 330 3 512 156 205 150 259 14687 135 1024 163 1276 875 437 792 302 693 74 37688 13 118 8 601 227 653 448 343 6 421 25889 242 1786 266 2083 1560 2020 1366 523 1229 184 65090 26 233 15 751 252 1400 429 1400 433 1550 48491 428 3063 428 3359 2269 600 1770 695 2140 1213 225092 51 450 25 1118 412 1790 427 492 770 425 155093 743 5168 681 5347 3590 2717 2514 1506 3692 600 102094 98 849 41 1648 666 2667 861 1496 1036 1693 75095 1267 8577 1069 8406 5613 4816 3635 2499 6256 2345 309396 184 1568 68 2407 1063 3938 1289 2383 1631 2635 219197 2124 14000 1657 13048 8675 8379 5212 4087 10442 5132 506598 336 2831 111 3485 1675 5761 1913 3748 2538 4055 361199 3500 6718 2536 20000 13250 14312 7411 6584 17168 10876 8185100 600 5000 178 5000 2605 8353 2813 5820 3904 6172 5873the nuclide chart. This shape can be described by the position and height of the ridge andthe decrease of the slope. Quantifying these characteristics, we made predictions for crosssections in regions of the nuclide chart where only few data are available. Such predictionsare vital for computer programs aimed at simulating the formation of heavy elements instars.
Acknowledgments
This research was supported by the T ´AMOP 4.2.1./B-09/1/KONV-2010-0007 project. Weare grateful to I. Angeli for useful discussions.11 eferences [1] M. E. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoyle,
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