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9.8: Low Energy Scattering

At low energies (i.e., when $ 1/k$ is much larger than the range of the potential) partial waves with $ l>0$ , in general, make a negligible contribution to the scattering cross-section. It follows that, at these energies, with a finite range potential, only $ s$ -wave scattering is important.

As a specific example, let us consider scattering by a finite potential well, characterized by $ V=V_0$ for $ r<a$ , and $ V=0$ for $ r\geq a$ . Here, $ V_0$ is a constant. The potential is repulsive for $ V_0>0$ , and attractive for $ V_0<0$ . The external wavefunction is given by [see Equation (986)]

 

$\displaystyle A_0(r) = \exp(\,{\rm i}\, \delta_0)\,\left[ j_0(k\,r) \cos\delta_...
...elta_0\right] = \frac{ \exp(\,{\rm i} \,\delta_0)\, \sin(k\,r+\delta_0)}{k\,r},$ (1008)

 

 

where use has been made of Equations (962)-(963). The internal wavefunction follows from Equation (991). We obtain

 

$\displaystyle A_0(r) = B \,\frac{\sin (k'\,r)}{r},$ (1009)

 

 

where use has been made of the boundary condition (992). Here, $ B$ is a constant, and

 

$\displaystyle E - V_0 = \frac{\hbar^2 \,k'^{\,2}}{2\,m}.$ (1010)

 

 

Note that Equation (1009) only applies when $ E>V_0$ . For $ E<V_0$ , we have

 

$\displaystyle A_0(r) = B \,\frac{\sinh(\kappa\, r)}{r},$ (1011)

 

 

where

 

$\displaystyle V_0 - E = \frac{\hbar^2 \kappa^2}{2\,m}.$ (1012)

 

 

Matching $ A_0(r)$ , and its radial derivative at $ r=a$ , yields

 

$\displaystyle \tan(k\,a+\delta_0) = \frac{k}{k'} \,\tan (k'\,a)$ (1013)

 

 

for $ E>V_0$ , and

 

$\displaystyle \tan(k\,a+ \delta_0) = \frac{k}{\kappa} \,\tanh (\kappa\, a)$ (1014)

 

 

for $ E<V_0$ .

Consider an attractive potential, for which $ E>V_0$ . Suppose that $ \vert V_0\vert\gg E$ (i.e., the depth of the potential well is much larger than the energy of the incident particles), so that $ k' \gg k$ . It follows from Equation (1013) that, unless $ \tan (k'\,a)$ becomes extremely large, the right-hand side is much less that unity, so replacing the tangent of a small quantity with the quantity itself, we obtain

 

$\displaystyle k\,a + \delta_0 \simeq \frac{k}{k'}\,\tan (k'\,a).$ (1015)

 

 

This yields

 

$\displaystyle \delta_0 \simeq k\,a \left[ \frac{\tan( k'\,a)}{k'\,a} -1\right].$ (1016)

 

 

According to Equation (1006), the scattering cross-section is given by

 

$\displaystyle \sigma_{\rm total} \simeq \frac{4\pi}{k^2} \sin^2\delta_0 =4\pi \,a^2\left[\frac{\tan (k'\,a)}{(k'\,a)} -1\right]^{\,2}.$ (1017)

 

 

Now,

 

$\displaystyle k'a = \sqrt{ k^2 \,a^2 + \frac{2 \,m \,\vert V_0\vert\, a^2}{\hbar^2}},$ (1018)

 

 

so for sufficiently small values of $ k\,a$ ,

 

$\displaystyle k' a \simeq \sqrt{\frac{2\, m \,\vert V_0\vert\, a^2}{\hbar^2}}.$ (1019)

 

 

It follows that the total ($ s$ -wave) scattering cross-section is independent of the energy of the incident particles (provided that this energy is sufficiently small).

Note that there are values of $ k'a$ (e.g., $ k'a\simeq 4.49$ ) at which $ \delta_0\rightarrow \pi$ , and the scattering cross-section (1017) vanishes, despite the very strong attraction of the potential. In reality, the cross-section is not exactly zero, because of contributions from $ l>0$ partial waves. But, at low incident energies, these contributions are small. It follows that there are certain values of $ V_0$ and $ k$ that give rise to almost perfect transmission of the incident wave. This is called the Ramsauer-Townsend effect, and has been observed experimentally.

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