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Classical Scattering with a Penetrable Square-Well Potential
Classical Scattering with a Penetrable Square-Well Potential
The penetrable square-well (PSW) interaction potential is defined by a repulsive energy barrier of height and range , surrounded by an attractive well of depth and width . The PSW interaction model includes the square-well potential and the penetrable-sphere potential (0 or ) as particular cases.
ϵ
r
σ
ϵ
a
λ-σ
(∞)
ϵ
r
ϵ
a
λ-σ0
A classical projectile particle of mass moving with speed and impact parameter is scattered an angle by a fixed target that interacts with the projectile via the PSW potential, analogously to the classical Rutherford scattering problem. Three types of collisions are possible:
μ
g
b
χ(b,g)
A. the particle penetrates into the inner repulsive core;
B. the particle is reflected by the core; and
C. the particle misses the core.
Events of type A occur if and , while events of type C take place if and ; otherwise, the collision is of type B.
μ2>
2
g
ϵ
r
b<
1-2/μ
σϵ
r
2
g
μ>/2>-1
2
g
ϵ
a
2
λ
2
σ
λ>b>
1-2/μ
σϵ
a
2
g
This Demonstration plots versus for different possible values of the potential parameters and of the particle speed. The control panel also includes the PSW potential as a visual aid in the identification of the control variables.
cosχ
2
(b/λ)