recent advances in r-matrix data analysis · ian thompson nuclear data and theory group september...

LLNL-PRES-756668This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC

Recent Advances in R-matrix Data Analysis

CNR18 Conference,Berkeley, CA.

Ian ThompsonNuclear Data and Theory Group

September 28, 2018

LLNL-PRES-7566682

§ The theory to describe individual resonances in A+B scattering,and the non-resonant background between them.

§ Describes all the asymptotic properties of the A+B wave function outside some fixed radius, R ≥ a,in terms of pole energies epand reduced width amplitudes gnp for each channel n and pole p.— The gnp are calculated from structure theory, or— or the gnp are fitted to scattering data

§ The basis for making statistical approximations— Reich-Moore approximation— Hauser-Feshbach models, etc.

§ Basis for checking the accuracy of those models.

R-matrix Theory

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R-matrix Theorem: For Hermitian H = T+V + B with Bloch operator B,

with V ≠ 0 only for r ∈ [0,a] and E-independent, then the exact scattering solution HY=EY can be represented by a R-matrix at r=a with a set of pole energies ep and reduced width amplitudes gnp as

"#$(&) = ∑*+,- ./0.102340

R-matrix Practice: use finite number N of poles p: ∑*+,5We can obtain useful converged results with ‘background poles’.

§ Both exact and practical R-matrices:1. Yield unitary S-matrix at each energy2. Yield orthogonal scattering wave functions at different energies.

Foundation of R-matrix theory: ‘Lane and Thomas’

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§ Any proposed extension only accepted if at least

1. Yields unitary S-matrix at each energy

2. Yields orthogonal scattering wave functions at different energies.

§ Both conditions derive from a Hermitian and energy-independent

Hamiltonian.

§ Reich-Moore approximation: imaginary damping widths for

missing channels:

— Condition 1. not satisfied e.g. if imaginary damping terms: Reich-Moore

• Perhaps ok if a specific meaning is given to the missing flux, e.g. capture or fusion

§ Convenient changes to boundary conditions in the Bloch operator:

— Condition 2. not satisfied e.g. boundary condition B not constant: B=S(E)

— Conditions 1,2 are satisfied in the Brune basis (transformable to/from L&T!)

Approximations often used in R-matrix theory

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Verification of R-matrix Codes

§ IAEA (Vienna) has had a series of consultants meetings since 2015 to improve R-matrix modeling accuracy and ranges of applications.

§ Improve for neutron and charged-particle reactions.

§ Verify codesCompare search proceduresValidate R-matrix data fits.

§ Codes participating:— ORNL: SAMMY— LLNL: FRESCO

— LANL: EDA— JAEA: AMUR

— Notre Dame: AZURE2— Future confirmation:

• CEA: CONRAD

• Tsinghua: RAC

§ I have written ‘ferdinand.py’, to translate between most of the input-output formats.— Uses GNDS intermediate structure.

Recent meeting: https://www-nds.iaea.org/index-meeting-crp/CM_R-matrix2018/

LLNL-PRES-7566686

Examples: 4He+3He elastic scattering with p+6LiFind R-matrix fit.

Dominant channels: 4He+3He and p+6LiNext: p1+6Li* (p1=3+, p2=2+,etc) breakup.

Channels Data points(a,a) 1609(a,p) 120(p,a) 630(p,p) 420

7Be

Ex (MeV)

6Li + p

3He + α

3He + α

6Li + p

1

Jπ

0.00

1.59

7.79

6Li + p

05.61

7/2-

3/2-

4.57 7/2-

7.21 5/2-

9.90

6.73 5/2-

3/2-

0.43 1/2-

6Be + n

9.27

11.01 3/2-

10.68

9.17 6Li + p

2

9.926Li + p

3

Figure 1: Level diagram of 7Be. The low mass nucleus has only two bound states, the ground stateand the level at Ex = 0.429 MeV.

• The 9.206 MeV point at 36.999 deg in the Spiger aa data was erroneously digitized on EXFOR.The point has been removed.

• The Tombrello aa data have an additional digitization error of 0.5 mb/sr, this has been addedin quadrature to the percent errors for the A1295002 EXFOR data.

• The Lin pa data that James sent us was converted by him from cm data. Ian’s checks,however, show that in 3 cases of angles near 90 deg, James chose the wrong quadrant. Thishas been corrected.

• The Spiger aa data of A1094006 has a 0.08 MeV digitization error. Ian finds that the data aremuch more consistent with the other data sets if they are translated in energy by �0.1 MeV.This new energy scale has been adopted.

References

[1] A.C.L. Barnard, C.M. Jones, and G.C. Phillips. The scattering of He3 by He4. Nuclear Physics,50:629 – 640, 1964. ISSN 0029-5582. doi: http://dx.doi.org/10.1016/0029-5582(64)90235-4.URL http://www.sciencedirect.com/science/article/pii/0029558264902354.

Selected data to be fitup to 8.0 MeV 7Be* excitation

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3 4 5 6 7 80

500

1000

1500

2000

1: [email protected] - 1.0% @ 1.0







Search file: test1b-v9gL-xs2.sfrescoed+.sfresco;

Example: R-matrix fit 4He+3He scattering. 1. Barnard_aa data

4 5 6 7

1

1.5

2

2.5

data_Barnard_aa.dat_1-7-9gL-xs2.agr (spacing 0.2)

Experiment/R-matrix ratios

Dataset Chisq/pts av normBarnard_aa.dat 0.967 0.990

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0.990

1.000

1.010

AZURE2SAMMYFRESCOAMUR

1.00

1.05

RAC

0.980.99

11.01

0.996

0.998

1

1.002

3 4 5 6 70.99

1

1.01

3 4 5 6 7 80.99

1

1.01

1.02

Barnard, 3He(α,α)

3He

θlab

= 23.22° θlab

= 26.76°

θlab

= 30.93° θlab

= 37.01°

θlab

= 41.99° θlab

= 45.48°

Rat

io t

o A

MU

RE

cro

ss s

ecti

on

Laboratory Energy (MeV)

Code-Code comparisonsFor 4He+3He elastic scattering with p+6Li

Most codes agreewithin 1% to AMUR.

0.1% agreementwould be better.

Here only away from the 7 MeVresonance.(or maybe AMURcode is poor.)

RAC not agreeing.

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R-matrix fit of Spiger_aa data (more 4He+3He)

Dataset Chisq/pts av norm av shiftSpiger-A1094004-

lab_aa.dat 2.633 0.929 -0.048

8 10 12 14

1

1.5

2

2.5

3

3.5

data_Spiger-A1094004-lab_aa.dat_8-21-9gL-xs2.agr (spacing 0.2)

6 8 10 12 140

500

1000

1500

2000

2500

3000

8: [email protected] - 9.8% - 33 keV @ 1.0














Search file: test1b-v9gL-xs2.sfrescoed+.sfresco;

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10

§ Original fit done with B=-L boundary conditions. But can reversibly transform to the Brune basis.

Brune basis: matches resonances in phase shifts

0 2 4 6 8 10 12 144He incident energy (MeV, cm)

−180

−90

0

90

180

270

Dia

gonal p

hase

shift

(deg)

0.5+/ 0 1 0.5−/ 1 1 1.5+/ 2 1 1.5−/ 1 1 2.5+/ 2 1 2.5−/ 3 1 3.5+/ 4 1 3.5−/ 3 1 4.5+/ 4 1 4.5−/ 5 1

Jp=3.5- cm: -12.499 to 2.975 MeV

Jp=2.5- cm: 6.236 to 5.033 MeV

Jp=2.5- cm: 7.058 to 5.593 MeV

B=-L Brune basis

Historically, the B=S(E)approximation was usedto give R-matrix pole energiesclose to the observed resonances.But this approximation givesdifferent background between poles

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Extensions to higher energies?

§ At high incident energies, there are more and more inelastic or transfer two-body channels.— Numbers of partial waves

increase, but still standard theory.

§ At higher energies, breakup channels begin to open.— Normally three-body channels,

and more difficult to model— Maybe approximate by

cascaded two-body channels.

§ Alternatively, we could settle for damping widths Γ" to describe loss of flux to outside the model space (an ‘optical’ R-matrix model):#$%(') = ∑"+,- ./0.10

23405670/9

§ This could be allowed if there are specific physical channels missing from the model (never for bound states).

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§ Alternatively, we could settle for damping widths !Γ# to describe loss of flux to outside the model space like an optical model:

$%&(() = ∑#,-. /01/21345167891/;

— Inspired by “Reduced R-matrix” theory of Lane & Thomas (1958, ch. X).— Using approximations like Reich-Moore but for particles, as begun in RAC

§ This could be allowed if there are specific physical channels missing from the model (never for bound states).— Then the missing flux (from the unitarity defect) could (for example) be

fed into a Hauser-Feshbach decay model built only on the missing physics

§ But if the total width of a damped resonance is large, then the flux will start missing at lower energies, even below the known threshold for the excluded channels!

An ‘optical’ R-matrix model?

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Example of fitting damping widths:Redo 4He+3He elastic scattering without p+6Li § Fits data rather well,

— though c2 / df = 7.32 § But absorption still present

below 10 MeV: unphysical.

6 8 10 12 140

1000

2000

3000

Search file: test1b-v9gL-aa4.sfrescoed;

0 2 4 6 8 10 12 14 16 18 20

He4 energy (MeV, lab)

−100

0

100

200

300

Cro

ss s

ect

ion (

mb)

Absorption from damping widths(a,p) cross−section from full fit

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§ A simple proposal to include the known energy dependence of flux going to an excluded channel with known threshold !".

§ Make the damping width energy-dependent: #Γ%(!) — Make energy dependence behave as Γ = 2)*+,(!) like R-matrix widths— So choose

#Γ%(!) = .Γ% +,(! − !")/ +,(0% − !")— This cuts off the damping for ! ≤ !" , and gives #Γ%(0%) = .Γ% to be fitted.

§ Making this work depends on— Having good data for angular distributions above the !" threshold— May need to choose 0% energy in the Brune basis for best physics— Knowing physics of missing channels to estimate 2 and Coulomb barriers• For exit in 3-body hyper-spherical harmonic 4, use 2 = 4 + (33 − 6)/2

Energy-dependent Damping Widths?

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Example fit with Energy-dependent damping

Refit 4He + 3He data.Choose !" energy in the Brune basis, with # = 0, '( = 4.02 (MeV). Result: c2 / df = 4.25

0 2 4 6 8 10 12 14 16 18 20

He4 energy (MeV, lab)

0

50

100

150

200

250

Cro

ss s

ect

ion

(m

b)

Absorption from damping widths(a,p) cross−section from full fitAbsorption with E−dependent damping

First attempt at least gives average transfer cross-sections6 8 10 12 14

Incident He4 energy (MeV, lab)

0

500

1000

22: [email protected] + 8.0% - 36 keV @ 4.3








Search file: test1b-v9gL-aa10.sfrescoed;

recent advances in r-matrix data analysis · ian thompson nuclear data and theory group september...

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