Scattering Length

 Scattering Length

[ 1 ] Scattering Length :

 

- At Low Energy N-P Scattring Cross Section Can Be Written In

  The Form Of Scattring Length

- For S Wave Scattring

  Ψ = sin (kr + δo)/kr

- From Asymptotic Behaviour

  Ψ = u(r)/r

- u(R) = rΨ(r)

  u(r) = sin (kr + δo)/kr.......(1)

- At Low Energy Both K And Phase Differance So Are Small So We

  Define A Lenght Such That

  lim (k → 0) [ -sin δo / k ] ≈ a (Scattring Length).......(2)

  Scattring Cross Section -

- For S Wave (l=0)

  σ = (4π/k^2) sin^2 δo

     = 4π(sin δo/k)^2

     = 4π (-a)^2

  σ = 4πa^2......(3)

- It Is N-P Scattring Cross Sction At Law Energy Which Is

  Independant Of Energy.

- Wave Function

  u(r) = sin (kr + δo)/k

- at low energy

  u(r) ≈ r + (δo/k)........(4)

- From Equation (2)

  -sinδo/k = a

  -δo/k ≈ a

   δo/k ≈ -a

- Put Value In Equation (4)

  u(r) = r - a

- So Asyptotic Wave Function Is Linear Equation So Value Of

   Scattring Length Can Be Find

- Using Plot Of Wave Function.

(I) Where +Ve Value Of Scattring Length Represent Bound State

     Of (N-P) System.

(II)Where -Ve Value Of Scattring Length Represent Unbound

      State Of (N-P) System.

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