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# 14.2: Born Approximation

Equation ([e15.17]) is not particularly useful, as it stands, because the quantity $$f({\bf k},{\bf k}')$$ depends on the, as yet, unknown wavefunction $$\psi({\bf r})$$. [See Equation ([e5.12]).] Suppose, however, that the scattering is not particularly strong. In this case, it is reasonable to suppose that the total wavefunction, $$\psi({\bf r})$$, does not differ substantially from the incident wavefunction, $$\psi_0({\bf r})$$. Thus, we can obtain an expression for $$f({\bf k},{\bf k}')$$ by making the substitution $$\psi({\bf r})\rightarrow\psi_0({\bf r}) = \sqrt{n}\,\exp(\,{\rm i}\, {\bf k}\cdot{\bf r})$$ in Equation ([e5.12]). This procedure is called the Born approximation .

The Born approximation yields $f({\bf k},{\bf k}') \simeq \frac{m}{2\pi\,\hbar^{\,2}} \int {\rm e}^{\,{\rm i}\,({\bf k}-{\bf k'})\cdot{\bf r}'}\,V({\bf r'})\,d^{\,3}{\bf r}'.$ Thus, $$f({\bf k},{\bf k}')$$ becomes proportional to the Fourier transform of the scattering potential $$V({\bf r})$$ with respect to the wavevector $${\bf q} = {\bf k}-{\bf k}'$$.

For a spherically symmetric potential, $f({\bf k}', {\bf k}) \simeq - \frac{m}{2\pi\, \hbar^{\,2}} \int\!\int\!\int \exp(\,{\rm i} \, q \,r'\cos\theta') \, V(r')\,r'^{\,2}\, dr'\,\sin\theta' \,d\theta'\,d\phi',$ giving $\label{e17.38} f({\bf k}', {\bf k}) \simeq - \frac{2\,m}{\hbar^{\,2}\,q} \int_0^\infty r' \,V(r') \sin(q \,r') \,dr'.$ Note that $$f({\bf k}', {\bf k})$$ is just a function of $$q$$ for a spherically symmetric potential. It is easily demonstrated that $\label{e17.39} q \equiv |{\bf k} - {\bf k}'| = 2\, k \,\sin (\theta/2),$ where $$\theta$$ is the angle subtended between the vectors $${\bf k}$$ and $${\bf k}'$$. In other words, $$\theta$$ is the scattering angle. Recall that the vectors $${\bf k}$$ and $${\bf k}'$$ have the same length, via energy conservation.

Consider scattering by a Yukawa potential ,

$\label{e10.35ee} V(r) = \frac{V_0\,\exp(-\mu \,r)}{\mu \,r},$ where $$V_0$$ is a constant, and $$1/\mu$$ measures the “range” of the potential. It follows from Equation ([e17.38]) that $f(\theta) = - \frac{2\,m \,V_0}{\hbar^{\,2}\,\mu} \frac{1}{q^{\,2} + \mu^{\,2}},$ because $\int_0^\infty \exp(-\mu \,r') \,\sin(q\,r') \, dr' = \frac{q}{q^{\,2}+ \mu^{\,2}}.$ Thus, in the Born approximation, the differential cross-section for scattering by a Yukawa potential is $\frac{d\sigma}{d {\mit\Omega}} \simeq \left(\frac{2\,m \,V_0}{ \hbar^{\,2}\,\mu}\right)^2 \frac{1}{[2\,k^{\,2}\, (1-\cos\theta) + \mu^{\,2}]^{\,2}},$ given that $q^{\,2} = 4\,k^{\,2}\, \sin^2(\theta/2) = 2\,k^{\,2}\, (1-\cos\theta).$

The Yukawa potential reduces to the familiar Coulomb potential as $$\mu \rightarrow 0$$, provided that $$V_0/\mu \rightarrow Z\,Z'\, e^{\,2} / (4\pi\,\epsilon_0)$$. In this limit, the Born differential cross-section becomes $\frac{d\sigma}{d{\mit\Omega}} \simeq \left(\frac{2\,m \,Z\, Z'\, e^{\,2}}{4\pi\,\epsilon_0\,\hbar^{\,2}}\right)^2 \frac{1}{ 16 \,k^{\,4}\, \sin^4( \theta/2)}.$ Recall that $$\hbar\, k$$ is equivalent to $$|{\bf p}|$$, so the previous equation can be rewritten $\label{e17.46} \frac{d\sigma}{d{\mit\Omega}} \simeq\left(\frac{Z \,Z'\, e^{\,2}}{16\pi\,\epsilon_0\,E}\right)^2 \frac{1}{\sin^4(\theta/2)},$ where $$E= p^{\,2}/2\,m$$ is the kinetic energy of the incident particles. Of course, Equation ([e17.46]) is identical to the famous Rutherford scattering cross-section formula of classical physics .

The Born approximation is valid provided that $$\psi({\bf r})$$ is not too different from $$\psi_0({\bf r})$$ in the scattering region. It follows, from Equation ([e15.9]), that the condition for $$\psi({\bf r}) \simeq \psi_0({\bf r})$$ in the vicinity of $${\bf r} = {\bf 0}$$ is $\label{e17.47} \left| \frac{m}{2\pi\, \hbar^{\,2}} \int \frac{ \exp(\,{\rm i}\, k \,r')}{r'} \,V({\bf r}')\,d^{\,3}{\bf r'} \right| \ll 1.$ Consider the special case of the Yukawa potential. At low energies, (i.e., $$k\ll \mu$$) we can replace $$\exp(\,{\rm i}\,k\, r')$$ by unity, giving $\frac{2\,m}{\hbar^{\,2}} \frac{|V_0|}{\mu^{\,2}} \ll 1$ as the condition for the validity of the Born approximation. The condition for the Yukawa potential to develop a bound state is $\frac{2\,m}{\hbar^{\,2}} \frac{|V_0|} {\mu^{\,2}} \geq 2.7,$ where $$V_0$$ is negative . Thus, if the potential is strong enough to form a bound state then the Born approximation is likely to break down. In the high-$$k$$ limit, Equation ([e17.47]) yields $\frac{2\,m}{\hbar^{\,2}} \frac{|V_0|}{\mu \,k} \ll 1.$ This inequality becomes progressively easier to satisfy as $$k$$ increases, implying that the Born approximation is more accurate at high incident particle energies.

$$\newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}$$ $$\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}$$ $$\newcommand {\btau}{\mbox{\boldmath\tau}}$$ $$\newcommand {\bmu}{\mbox{\boldmath\mu}}$$ $$\newcommand {\bsigma}{\mbox{\boldmath\sigma}}$$ $$\newcommand {\bOmega}{\mbox{\boldmath\Omega}}$$ $$\newcommand {\bomega}{\mbox{\boldmath\omega}}$$ $$\newcommand {\bepsilon}{\mbox{\boldmath\epsilon}}$$