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 ( or ) as particular cases.
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:
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.
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.


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Snapshot 1: hard-sphere case; only events of type B are possible: linear relationship between and characteristic of a hard-sphere collision
Snapshot 2: penetrable-sphere case; only events of types A and B are possible
Snapshot 3: square-well case; only events of types B and C are possible
Snapshot 4: penetrable square-well case; and ; only events of type B are possible
Snapshot 5: penetrable square-well case; but ; only events of types B and C are possible
Snapshot 6: penetrable square-well case; but ; only events of types A and B are possible
Snapshot 7: penetrable square-well case; and ; all three types of events A–C are possible
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