Development of Software for Visualizing
Correlations Between Proton-Deuteron
Scattering Parameters
Tristan Swartz
Dr. Timothy Black, Faculty Advisor
Overall Study
Analysis of Scattering States for
Proton-Deuteron Collisions
Extract Parameters
Use to Answer Mysteries of Few Nucleon
Systems
„
2S
1/2
Singularity
Proton-Deuteron Scattering Study
Highlights
Extensive Experimental Data Collected
Initial Expectation Parameter Values Extracted
Parameter and Expansion Coefficients Derived
Theoretical VS Experimental Values
Parameters
Two Major Groups
Scattering (δ)
Different Spins
Different Spatial Angular Momentum
Both Different
Mixing (ε,η,ξ)
Mixes Spin
Mixes Both
Mixes Orbital
Effective Range Expansion
System is Charged
Describe Parameters with Expansion
Scattering:
δ(k) = (1/ao) + (1/2)r k2 + φ k4 + ….
Mixing:
Θ(k) = β0 + β1k2 + β2k4 +…
Energy and J Reliance
Constraints
Many Parameters
76 x 13
Energy Level
Total Angular Momentum Conservation
(Spin Limit)
Parity Conservation
Constraint Example
J = 5/2
Parity = (-1)L =1(+)
J = |L-S|, |L-S| + 1, …, L+S
Allowed Values: L=2, S=3/2; L=2, S=1/2;
L=4, S=3/2
Analysis
Derive Initial Parameters from Outcome
Data
Posterior Distribution (Bayesian
Approach):
P(Sj|D) = (P(D|Sj)*P(Sj))/(P(D))
Analysis
Solutions are Expectation Values
<S> = Σ Sj * P(Sj|D)
Discrete
Gibbs Sampler
Numerical Algorithm
S1(1) ~ P(S1|D, S2(0), S3(0),…, Sn(0) )
S2(1) ~ P(S2|D, S1(1), S3(0) ,…, Sn(0))
…..
Sn(1) ~ P(Sn|D, S1(1), S2(1) ,…, Sn-1(1))
S1(2) ~ P(S1*|D, S2(1), S3(1) ,…, Sn(1))
…and so forth
„
Phenomenological Values for Scattering Parameters Found with
S-Matrix and Seyler Formulation
Correlation Complications
Small Variation
Possible Inaccuracies
Meaning of Results
Objective
Design a Program to Graphically Illustrate
the Degree of Correlation Between a Pair
of Parameters at the Same J and Parity
Utilizing Dr. Black’s Parameter Value Data
Sets
Methodology
Matlab®
Fast Processing
Handles Large Data Sets
Relatively Inexpensive
Superior Graphic Manipulation Capability
Methodology
Data Set Format
J = 1/21/2- EFFECTIVE RANGE PARAMETERS
SET # 2P 1/24P 1/21/21/20 -.4596E0.5512E.4596E-03 0.3228E0.3228E-01 4.527
0.5512E-03 -.8989E.8989E-01 6.677
1 -.3407E0.5102E.3407E-03 0.2118E0.2118E-01 4.998
0.5102E-03 -.7280E.7280E-01 5.492
2 -.1938E0.5687E.1938E-03 -.3820E.3820E-01 8.714
0.5687E-03 -.9626E.9626E-01 7.131
3 -.3768E03
0.1707E01
5.310
0.5636E03
.9362E.3768E
0.1707E
0.5636E
.9362E-01 7.000
4 -.3843E0.1191E.3843E-03 0.9070E0.9070E-02 5.877
0.1191E-03 0.1845E0.1845E-01 1.283
5 -.2737E0.4779E.2737E-03 -.1183E.1183E-01 6.911
0.4779E-03 -.6555E.6555E-01 4.961
6 -.1180E-4.641
-.2873E-3.204
.1180E-02 0.2059
.2873E-03 0.1143
7 -.2022E-.3897E.2022E-03 -.4026E.4026E-01 8.944
.3897E-04 0.5528E0.5528E-01 -.4504
8 -.6724E03
0.9828E01
.8033
0.5820E03
-.1039
7.690
.6724E
0.9828E
0.5820E
9 -.5198E0.4711E.5198E-03 0.6087E0.6087E-01 1.598
0.4711E-03 -.4428E.4428E-01 1.768
10 -.1131E0.2444E.1131E-03 -.6068E.6068E-01 9.792
0.2444E-03 -.1012E.1012E-01 2.255
11 -.8537E-.4445
0.3123E.8537E-03 0.1226
0.3123E-03 -.3075E.3075E-01 3.747
12 -.4511E0.5677E.4511E-03 0.2826E0.2826E-01 4.861
0.5677E-03 -.9541E.9541E-01 7.111
13 -.9325E-1.283
0.5413E.9325E-03 0.1433
0.5413E-03 -.8474E.8474E-01 6.215
-1.408
0.3571E14 -.8574E.8574E-03 0.1304
0.3571E-03 -.3911E.3911E-01 3.904
15 -.2599E0.5045E.2599E-03 -.2204E.2204E-01 7.841
0.5045E-03 -.7483E.7483E-01 5.660
16 -.2512E0.5548E.2512E-03 -.1603E.1603E-01 7.097
0.5548E-03 -.9092E.9092E-01 6.756
17 -.9464E-.7492
0.2704E.9464E-03 0.1379
0.2704E-03 -.1378E.1378E-01 2.408
18 -.1327E-3.680
0.1513E.1327E-02 0.2065
0.1513E-03 0.1040E0.1040E-01 1.364
19 -.6190E0.3990E.6190E-03 0.6710E0.6710E-01 2.482
0.3990E-03 -.4437E.4437E-01 3.842
Methodology
Data Loading
„
Initially Manually
„
File Browser
„
Verification Screen
„
Vector Designator
Verification Screen
Vector Designator
Methodology
Auto-Generate Code for Loading
„
Multiple Simultaneous Loads Possible
„
Doesn’t Cover Import
Auto-Generated M-Code
Methodology
Create Bivariate 3-D Histogram
„
Hist3 Function
„
[X] = [ParameterVector1, ParameterVector2]
„
Possible Permutations
„
Uploads in a GUI
Methodology
hist3(X,[7 7],'FaceAlpha',.65);xlabel('MPG');
ylabel('Weight');set(gcf,'renderer','opengl');
Methodology
Covariance Coefficients
„
Defined Using ‘corrcoef(X) = Matrix’
„
Read as Matrix
„
No Graph
Methodology
Enhanced Insight from Multiple Viewpoints
Rotation Tool
Use ‘view ([azimuth, elevation])’
Methodology
Cosmetic Touches
„
„
„
„
„
„
Axis Labels: ‘xlabel’, ‘ylabel’, ‘zlabel’
Nbins
Ctrs
Edges
Title
‘FaceAlpha’, #<1
Methodology
Write and Run Program as a Whole
Integrate Loading with Plots and
Covariance Calculations
Two Programs Based on Parameter
Permutations
Results
Programs ‘loadnplot2’ and ‘loadnplot3’
Many Input Arguments
Creates Graphs
Results
Results
Results
Results
Results
Future Direction
Prototype can be Improved
„
Customized Sequential Program
„
Data Set Formatting
„
Iterative Power Creates All Permutations
„
Coding Independent of Matlab®
Works Cited
Bayesian Data analysis, Black, T. and Thompson, W..
Determination of Proton-Deuteron Scattering Lengths, Black, T et al.
Bayesian Computation Via the Gibbs Sampler and Related Markov Chain
Monte Carlo Methods. A. F. M. Smith and G. O. Roberts. Journal of the
Royal Statistical Society. Series B (Methodological), Vol. 55, No. 1 (1993),
pp. 3-23
Matlab® Software (Produced graphs)
www.mathworks.com/access/helpdesk/help/toolbox (All code pictures from
here)