MICROPARTICLE AND ITS DIFFRACTION IMAGE

D. I. Blokhintsev

The structure of microparticles, molecules, atoms, and atomic nuclei is studied by means of the scattering by these particles of various waves: electromagnetic, electron, neutron, and others.

The distribution of intensities of the scattered waves, observed on a distant screen, forms the diffraction image of the particle.

In this case it is precisely elastic scattering, occurring without exchange of energy between the waves and the scattering particle, that yields a “picture” of the object in its unchanged initial state.

Such a picture, generally speaking, can be obtained only from a large ensemble of independent particles, since one and the same particle, when scattering is repeated, will change its state.

The distribution of intensities of the scattered waves on the screen is determined directly by the differential cross section \(Q(\theta)\,d\Omega\) for elastic scattering through an angle \(\theta\) into the solid angle \(d\Omega\). This cross section is expressed in terms of the amplitude \(A\) of the scattered wave \(u = A \dfrac{e^{ikr}}{r}\) (\(r\) is the distance from the particle to the screen, \(k\) is the wave number of the scattered waves) by the known relation: \(Q(\theta)=|A|^2\)*).

If the structure of the scattering particle and the forces acting between it and the particles belonging to the diffracting wave are known, then, using quantum mechanics, one can calculate the amplitude \(A\) of the scattered wave and at the same time find \(Q\) and, consequently, the distribution of intensities \(I\) on the screen of the scattered waves \(I \simeq Q\).

We, however, shall be interested in another question: what can be said about the structure of an object if the effective cross section is known, or, what is the same thing, the distribution of intensities \(I\) on the screen?

For definiteness, let us consider the case of weak scattering, when all the relations are especially simple.

*) See, for example, D. Blokhintsev, Introduction to Quantum Mechanics, §§ 74–75, or Mott and Massey, The Theory of Atomic Collisions, Ch. VII.

In this case the equation for the scattered wave \(u\) reads (“Born approximation”):

\[ \nabla^2 u + k^2 u = - 4\pi D(\mathbf{x})\psi_0(\mathbf{x}), \tag{1} \]

where the function \(D(\mathbf{x})\) is proportional to the difference \((n^2-1)\), \(n\) being the refractive index of the medium outside and inside the scattering particle, and \(\psi_0(\mathbf{x})=e^{i\mathbf{k}_0\mathbf{x}}\) is the primary incident wave, which we take to be plane, propagating in the direction \(\mathbf{k}_0\).

It follows from this equation*) that, as \(r\to\infty\), the amplitude of the scattered wave \(A\) is a function of the vector \(\mathbf{q}=\mathbf{k}-\mathbf{k}_0\), where \(\mathbf{k}\) is the wave vector of the primary particle scattered through an angle \(\theta\) (for elastic scattering \(|\mathbf{k}|=|\mathbf{k}_0|=k\)), and is equal to:

\[ A(\mathbf{q})=\int D(\mathbf{x}) e^{i\mathbf{q}\mathbf{x}}\,d\mathbf{x}\,**). \tag{2} \]

The function \(D(\mathbf{x})\), which determines the distribution of the refractive index inside the scattering particle, will be regarded by us as a quantity determining its structure. Thus, in a conventional sense, one may say that \(D(\mathbf{x})\) represents the object, and \(A(\mathbf{q})\) its diffraction images on the screen. From (2), using Fourier’s theorem, we find:

\[ D(\mathbf{x})=\frac{1}{(2\pi)^3}\int A(\mathbf{q}) e^{i\mathbf{q}\mathbf{x}}\,d\mathbf{q}. \tag{3} \]

If \(A(\mathbf{q})\) could be determined from experiment, then formula (3) would give an unambiguous answer to the question of the structure of the object.

In fact, when inverting the integral there arise two limitations relating to the empirical knowledge of the amplitudes of the scattered waves \(A(\mathbf{q})\).

The first of them is connected with the fact that the energy of the particles belonging to the primary wave is limited. If the momentum of these particles is equal to \(p\) (wavelength \(\lambda=\frac{2\pi\hbar}{p}\)), then the maximum value of the vector \(q=2k\sin\frac{\theta}{2}\) is limited and is equal to \(\frac{4\pi}{\lambda}\).

Because of this, since the amplitude \(A\) is known under these conditions only in the region \(q=\sqrt{q_x^2+q_y^2+q_z^2}\leq \frac{4\pi}{\lambda}\), instead of the true structure we can compute only

\[ \widetilde{D}(\mathbf{x})=\frac{1}{(2\pi)^3}\int_{q\leq \frac{4\pi}{\lambda}} A(\mathbf{q}) e^{i\mathbf{q}\mathbf{x}}\,d\mathbf{q}. \tag{3'} \]

The fine structure will be smoothed out, since in the expansion (3′) there are no—

) See the books cited above.
*) \(d\mathbf{x}=dx\,dy\,dz,\quad d\mathbf{q}=dq_x\,dq_y\,dq_z.\)

…there exist higher harmonics with \(q > \dfrac{4\pi}{\lambda}\). Therefore all details of the structure of the scattering particle that undergo a substantial change over a distance \(\Delta x < \dfrac{\lambda}{2}\) will not be taken into account in \(\widetilde{D}(\mathbf{x})\) (for \(\Delta x \ll \dfrac{\lambda}{2}\) the change of phase of the exponential function in the integral (3′) will be \(\ll 2\pi\)).

Let us now turn to the second limitation. Its essence is that the experiment in general determines not the amplitude \(A(\mathbf{q})\) itself, but only the differential cross section \(Q(\mathbf{q})\). If the amplitude of the scattered wave is represented in the form:

\[ A(\mathbf{q}) = a(\mathbf{q}) e^{i\alpha(\mathbf{q})}, \tag{4} \]

where \(\alpha(\mathbf{q})\) is the phase, \(a(\mathbf{q}) = +\sqrt{Q(\mathbf{q})}\), then the observed quantity is \(a(\mathbf{q})\). On the contrary, the phase \(\alpha(\mathbf{q})\) remains entirely arbitrary, since it drops out of the expression for the cross section.

Let us now see what consequences follow from this circumstance.

From the reality of the quantity \(D(\mathbf{x})\) it follows that \(A(\mathbf{q}) = A(-\mathbf{q})\). It is not difficult to see that from this condition there follows \(a(\mathbf{q}) = a(-\mathbf{q})\), \(\alpha(\mathbf{q}) = -\alpha(-\mathbf{q})\). Therefore the integral (3) may be represented in the form:

\[ D(\mathbf{x}) = D_s(\mathbf{x}) + D_a(\mathbf{x}), \tag{5} \]

where

\[ D_s(\mathbf{x}) = \frac{1}{(2\pi)^3} \int a(\mathbf{q}) \cos \alpha(\mathbf{q}) \cos(\mathbf{q}\mathbf{x})\, dq, \tag{6} \]

\[ D_a(\mathbf{x}) = \frac{1}{(2\pi)^3} \int a(\mathbf{q}) \sin \alpha(\mathbf{q}) \sin(\mathbf{q}\mathbf{x})\, dq. \tag{6′} \]

\(D_s(\mathbf{x})\) represents the part of the structure symmetric with respect to the inversion transformation (replacement of \(\mathbf{x}\) by \(-\mathbf{x}\)), while \(D_a(\mathbf{x})\) is the antisymmetric part.

From (6′) it is seen that \(D_a(\mathbf{x}) = 0\) only in the case when \(\alpha(\mathbf{q}) = 0\), i.e. if the amplitude of the scattered wave \(A(\mathbf{q})\) is real. Taking \(\alpha(\mathbf{q}) = 0\), we obtain the unique and quite definite value \(D_s(\mathbf{x})\), corresponding to the effective cross section \(Q(\mathbf{q})\) found from experiment. Therefore we may state the following proposition: to a given diffraction pattern there corresponds a unique object symmetric with respect to the inversion group and an innumerable multitude of nonsymmetric ones.

Let us now give an example illustrating this proposition. Suppose we have a particle with the structure:

\[ D_a(\mathbf{x}) = \frac{\varepsilon}{\pi^{3/2}\Delta^3} \left[ e^{-\frac{(x-l)^2+\rho^2}{\Delta^2}} - e^{-\frac{(x+l)^2+\rho^2}{\Delta^2}} \right]. \tag{7} \]

Here \(\rho^2 = y^2 + z^2\); \(x=\pm l\), \(\rho=0\) determines the position of the maximal deviations of \(D_a(\mathbf{x})\) from zero (for small \(\Delta\)), \(\dfrac{\varepsilon}{\pi^{3/2}\Delta^3}\) gives the magni-

reason for these deviations. In Fig. 1 the graph of \(D_a(\mathbf{x})\) in the plane \(\rho=0\) is shown. \(D_a(\mathbf{x})\), as is seen, represents an asymmetric dipole-like object. Computing \(A(\mathbf{q})\) for this case, for which we substitute (7) into (2), we easily find:

\[ A(\mathbf{q})=2i\varepsilon e^{-\frac{q^2\Delta^2}{4}}\sin(q_xl), \tag{8} \]

i.e.

\[ a(q)=2\varepsilon e^{-\frac{q^2\Delta^2}{4}}\left|\sin(q_xl)\right|,\qquad \alpha(q)=(-1)^m\frac{\pi}{2}, \]

when \(m\pi<q_xl<(m+1)\pi\). The phase \(\alpha(q)\) therefore changes discontinuously, within the limits from \(-\dfrac{\pi}{2}\) to \(+\dfrac{\pi}{2}\).

In Fig. 2 the scattering intensity \(I(\theta)\) is shown as a function of the scattering angle \(\theta\), for the structure (7) (the primary beam is here assumed parallel to the dipole axis \(ox\)). The very same scattering pattern, and moreover for any orientation of the primary beam and the object, will be obtained for a symmetric object with \(A(q)=+\sqrt{Q(q)}\) (i.e. for \(\alpha(q)=0\)).

Fig. 1.

The structure of this symmetric object, giving the same diffraction pattern as the true asymmetric object, is determined by the formula:

\[ D_s(\mathbf{x})=\frac{2\varepsilon}{(2\pi)^3}\int e^{-\frac{q^2\Delta^2}{4}}\left|\sin(q_xl)\right|\times \]

\[ \times e^{-i\mathbf{q}\mathbf{x}}\,dq. \tag{7′} \]

To compute this integral we put

\[ \left|\sin(q_xl)\right|=\sin(q_xl)\frac{2}{\pi}\sum_{s=0}\frac{1}{2s+1}\sin\left[(2s+1)q_xl\right] \tag{9} \]

and, carrying out the integration, we find:

\[ D_s(\mathbf{x})= \frac{\varepsilon}{\pi^{5/2}\Delta^3} \sum_{s=0}^{\infty} \frac{e^{-\frac{\rho^2}{\Delta^2}}}{(2s+1)} \left\{ e^{-\frac{(x-2ls)^2}{\Delta^2}} + e^{-\frac{(x+2ls)^2}{\Delta^2}} - \right. \]

\[ \left. - e^{-\frac{(x-2sl-2l)^2}{\Delta^2}} - e^{-\frac{(x+2sl+2l)^2}{\Delta^2}} \right\}. \tag{10} \]

The distribution \(D_s(\mathbf{x})\) for this structure is shown in Fig. 1 by the dotted curve.

Let us give another example. Let the deviations from symmetry be small, so that the phase \(\alpha(\mathbf q)\) is small in the essential range of values of \(q\). Let us expand this phase in powers of \(q_1=q_x,\ q_2=q_y,\ q_3=q_z\):

\[ \alpha=\sum_{i,k,l}\alpha^{ikl}q_iq_kq_l+\cdots . \tag{11} \]

In doing so we begin the expansion with the third term, since the first can be eliminated by a choice of the origin, while even powers are altogether absent in \(\alpha\).

Fig. 2.

Putting \(A(\mathbf q)=\sqrt{Q(\mathbf q)}\{1+i\alpha(\mathbf q)+\cdots\}\), we find (from formula (2))

\[ D(\mathbf x)=D_s(\mathbf x)-\sum_{i,k,l}\alpha^{ikl}\, \frac{\partial^3 D_s(\mathbf x)}{\partial x_i\partial x_k\partial x_l}. \tag{12} \]

All these structures, for small \(\alpha^{ikl}\), will give one and the same diffraction pattern.

Thus we see that, without resorting to theoretical conceptions of the structure of the object, it is impossible, from observation of its diffraction image alone, to draw unambiguous conclusions about its structure.

This circumstance may also prove to be significant in the study of the structure of small particles by ultramicroscopic methods.