Notes
Slide Show
Outline
1
The Padua Algorithm
for the Computation of
Deuteron Binding  Energy
 and Wave Functions
  • Mesgun Sebhatu
  •  (sebhatum@winthrop.edu)
  • Winthrop University
  •  Rock Hill, SC 29733


  • NSBP/NSHP 2005
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N-N STATES
  • L= O, 1, 2,  3,  4, 5,…
  • L = S, P, D, F, G, H,…
  • J = L+S;   S= O  or 1


  • 2S+1Lj   is how NN states are specified
  • When S = 0, 2S+1 =1, Singlet States
  • When S =1,  2S+1= 3,  triplet States


  • L= 0, 2, 4, … Even & L= 1, 3, 5, …  Odd States
  • 1S0,1D2 ,1G4, …are leading even singlet states
  • 1P1 , 1F3 , 1H5 , …are leading odd singlet states


  • 3S1-3D1  is the most interesting example of a coupled triple state.
  • It has the only bound NN State—the deuteron.
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Introduction
  • A deuteron is is the only two-nucleon bound state.
  • It is the simplest nucleus. It plays a role in nuclear physics that that resembles  the role the hydrogen atom plays in atomic  physics.
  • It is a nucleus of deuterium (21H)--an isotope of hydrogen with an abundance of 0.015%.
  • It consists of a proton and neutron with total spin S=1 (parallel), a total angular momentum J=1, and angular momentum  ℓ =0 ,2.
  • It is a coupled state of mostly 3S1 and a small  admixture of 3D1.
  • Using a novel algorithm by a Padua Group a coupled Schrödinger Equation (AKA Rarita-Schwinger Equation[RSE]) can be solved to obtain a pair of wave functions u(3S1) and w(3D1) as eigenvectors and the binding energy as an eigenvalue.
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The Rarita-Schwinger Equation for a Deuteron—An 3S1-3D1 NN State
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The Rarita-Schwinger equation can be rewritten as as a two-point boundary matrix eigenvlaue equation of the form


  • A is matrix that contains the RS equation
  • Y is an eigenvector that consists u(x) and w(x)
  • λ= α2 is an eigenvalue that contains the binding energy
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The 3S1-3D1 state Reid Soft Core  Potential   is chosen for simplicity for testing the
Padua Algorithm
  • V(3S1-3D1)=Vc+S12VT+L.SVLS
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SWEPs vs REID SC
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Central (VC), Tensor (VT) and Spin-Orbit (VLS) Reid Soft Core Potentials
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The ODE describing the deuteron (RSE) is now a  two-point boundary eigenvalue matrix equation
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The key element in the Padua Algorithm the transformation t=tan-1(x). Since tan(π/2)= ∞.
 This transforms r =∞ to t= π/2
 i.e.,  0<x<∞ is transformed to  0<t<π/2
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  The RS Equation in Discrete Form
  • After the 1st and 2nd derivatives in the RSE are replaced by central difference  approximations.
  •                                h=Δt


  • The  RSE can be cast into the standard form
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The A(2N x2N) Matrix
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The Matrix Elements a(n,n)
of the square matrix A of order 2N are zero except the following:
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The eigenvectors u(nh) and w(Nh) are:
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A is  an 2N by 2N Matrix
  • For the purposes of this presentation a short FORTRN 90 program
  • to fill up the matrix elements of the A matrix was written.
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Deuteron wave functions u(x) and w(x) form Reid Soft Core NN potential
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Deuteron Parameters
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Deuteron wave functions u(x) and w(x)
The dashed line are from Reid SC and the solid lines form SG-SWEP