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- Mesgun Sebhatu
- Winthrop University
- Rock Hill, SC 29733
- sebhatum@winthrop.edu
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- Identify the ultimate constituents of matter.
- Discover the fundamental forces through which the basic entities
interact.
- Unify the fundamental forces and come up with a theory of everything.
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- Historical background :
- The Yukawa Potential and OPEP (1935)
- Phenomenological potentials:
- e.g. The Reid Soft-core potential(1968)
- One Boson Exchange Potentials (OBEPs ):
- e.g. Bonn Potential (1970 – Present)
- QCD and/or Effective field theory
inspired potentials ( present)
- Solitary Wave Exchange Potentials (SWEPs):
- e.g. lF4 SWEP and SG SWEP (1975-?)
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- The field equations for spin-zero meson fields used in the derivation of
SWEPs are nonlinear generalization of the well known Klein-Gordon equation. They are of the
form1:
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- The simplest examples are:
- J = lF4 which leads to the cubic KG equation:
- ¶m¶mF + m2F +lF3
= 0
- And JsG = (sin lF) m2/l-m2 F
- which leads to the sine-Gordon equation:
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- A Pair of Quantized Solitary Wave Solution for the SGE from which
the SG SWEP is derived are :
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- Once the SG solution is expressed as
a tan-1 series (as shown below).
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- The lowest order NN interaction is represented by the 2nd
order Feynman diagrams shown below. Using Feynman rules an expression
for an NN scattering amplitude is written down. [See e.g. Bjorken and
Drell, Relativistic Quantum Mechanics (1964) ] The only change is that
the Feynman propagator is replaced by the SG propagator.
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- The momentum space SG SWEP obtained from the diagrams shown earlier with
leading non static terms is3:
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- In general, VNN(x) = VC
+ VT+ VLS + VLL
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- SWEPs yield good results with just the leading four terms n=0,1,2,3,
&4
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- 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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- For singlet NN states (S=0, T=1)
- S12=0, VLS=0
and <(t1 ¢ t2)(s1 ¢ s2)>=-3
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- CNS @ George Washington U.
- CNS maintains the world data base for experimental NN
etc. Phase shifts
- NN On-line from Netherlands
- They maintain NN Nijmegen Potentials, Phase shifts, Deuteron
Properties.
- U of Hamburg from Germany
- They have potentials obtained by inverting experimental phase shifts.
- The have also greatly extended my work on SWEPs they call them One
Solitary Boson Wave Exchange
Potentials (OSBEPs).
- Some general references:
- M. Sebhatu and E. W. Gettys, A
Least Squares Method for the Extraction of Phase Shifts, Computers in
Physics 3(5), 65 (1989)
- Calculation of Phase Shifts
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- 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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- 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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- V(3S1-3D1)=Vc+S12VT+L.SVLS
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- 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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- 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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