Interaction of X-Rays, Neutrons and Electrons with Matter

Interaction of X-rays, Neutrons and Electrons with Matter David P. DiVincenzo This document has been published in Manuel Angst, Thomas Brückel, Dieter Richter, Reiner Zorn (Eds.): Scattering Methods for Condensed Matter Research: Towards Novel Applications at Future Sources Lecture Notes of the 43rd IFF Spring School 2012 Schriften des Forschungszentrums Jülich / Reihe Schlüsseltechnologien / Key Tech- nologies, Vol. 33 JCNS, PGI, ICS, IAS Forschungszentrum Jülich GmbH, JCNS, PGI, ICS, IAS, 2012 ISBN: 978-3-89336-759-7 All rights reserved. A 4 Interaction of X-rays, Neutrons and Electrons with Matter David P. DiVincenzo PGI-2 Forschungszentrum Julich¨ GmbH Contents 1 The Basics of Scattering Theory 2 2 The Scattering of Electromagnetic Radiation from Atoms 4 3 Atomic Form Factor for X-rays 7 4 X-ray Absorption and Dispersion 9 5 Electron Scattering 12 6 Neutron Scattering 13 7 Magnetic Neutron Scattering 15 A4.2 David P. DiVincenzo 1 The Basics of Scattering Theory Each of the scattering probes that we discuss here, be they particles or waves, permit, according to the tenets of quantum mechanics, a description of either sort. In fact, the wave theory is the best adapted as the unified framework that we will set up here. The incident beam will be treated as monoenergetic and unidirectional – and thus as a plane wave, with incident wave field ik·r Ψinc(r) = Ae (1) k’ → → k Fig. 1: A schematic of the scattering process from an atomic target. The incident plane wave (wavy line) has wavevector k; its constant-phase fronts are shown as straight lines. The scattered wave is an outgoing spherical wave (circles) going out in all directions, in- cluding in the wavevector direction k0. The energy of the incident scatterer is a function of the magnitude of the wavevector jkj; for nonrelativistic electrons or neutrons, E = ~2k2=2m, and for light, E = ~ck. The direction of propagation is of course the direction of the vector k, which will be conveniently described in spherical polar coordinates using angles (θ; ') = Ω. We assume that there is a small scattering target fixed at the origin. In the relevant wave equation, this scatterer will be described by a potential energy function V (~r). The “interaction region“ jrj < r0 is assumed to be the only region in which V (~r) 6= 0. Outside this interaction region the wave field also contains an outgoing spherical wave of the form eikr Ψ = Af(Ω) (2) scat r We have specialized to elastic scattering (appropriate for most of the scattering experiments considered in this chapter), so that the magnitude of the incident and scattered wavevectors k are the same. The quantity f(Ω) is the central focus of our attention, describing the amount of scattering in the direction of the solid angle Ω. Note that the complex quantity f has units of length – it is a Scattering Interactions A4.3 detector v∆t ∆Ω ←→ → ∆S →← v∆t → → → r ← Fig. 2: Geometry of the scattering process. ”scattering length”. It in fact directly indicates the normalized scattering flux ∆σ(Ω) in a cone of solid angle ∆Ω (see Fig. 2), in the direction Ω, for unit incident wave flux density: ∆σ = jf(Ω)j2dΩ (3) The total scattering cross section is Z σ = jf(Ω)j2dΩ (4) We see that the phase of the complex scattering length f(Ω) does not appear in any of our expressions; however it is very important in the interference that occurs in scattering from two different scattering centers. This effect is beyond the scope of the present chapter. For completeness, we note the other important quantity, the differential scattering cross section, which is simply the integrand of the quantity above: dσ = jf(Ω)j2 (5) dΩ We end this section with a simple physical picture of the scattering cross section. Naturally, the above discussion implies that the full wave field is given by the sum of the incident and scattered waves, which is correct in the Born approximation: eikr Ψ (r) = A eik·r + f(Ω) (6) tot r This Born approximation expression does not take account of the fact that the flux of the incident beam is affected (and depleted) by scattering. The amount by which it is depleted is exactly the flux density through the area σ. One can have a simple picture of this result: the depleting effect of the scatterer is exactly the same as that of a fully absorbing screen with area σ (Figure 2). The common unit for σ in scattering physics is the barn, which, at 10−28m2, is actually a large unit of area in many areas of particle and nuclear physics. The term originates from the 20th century American taunt to a poor thrower, “You couldn’t hit the broad side of a barn.” A4.4 David P. DiVincenzo Fig. 3: A schematic of the square of ther real part of the polarizability (χ0(!))2 versus frequency !. We see the low frequency, Rayleigh part (frequency independent), a complex intermediate frequency range in which anomalous dispersion and absorption occur, and then a high-frequency, Thompson part (going line 1=!4). 2 The Scattering of Electromagnetic Radiation from Atoms While we will in this school primarily be concerned with high-energy scattering probes such as X-rays, we begin this discussion at the low-energy (that is, the low-frequency) end of the spectrum. An electromagnetic wave comprises transverse, perpendicular oscillating electric and magnetic fields. We first consider the effect of the electric fields on a target atom. At low frequencies, below that of any atomic resonances, the applied field will polarize the electrons bound to the atom, producing an electric polarization P proportional to the strength of the electric field: P (!) = χ(!)E(!) (7) At low frequencies, the electric polarizability of the atom χ(!) is independent of frequency !. The resulting electric dipole oscillating with angular frequency !, P (!)ei!t, will radiate an outgoing spherical wave – this is the scattered wave of our general scattering theory. From classical electromagnetic theory, the efficiency with which this dipole radiates energy scales like the fourth power of the frequency; the net result for the scattering cross section is the formula for Rayleigh scattering: 4 8π ! 2 σR(!) = 2 2 (χ(0)) (8) 3 (4π0c ) Recall that this !4 dependence gives Rayleigh’s explanation that the sky is blue. Passing over the visible and ultraviolet region of the spectrum where atoms show complex resonant behavior in their scattering cross section (Figure 3), we consider a regime where the frequency is high enough that the binding of the electron to the atom is irrelevant; the electron oscillates as if it were in free space. In this regime a calculation of the oscillating dipole P (!) is again straightforward, since it simply requires the calculation of the periodic displacement of a free particle subject to a sinusoidal force. The result in this regime is Scattering Interactions A4.5 e2 χ(!) = 2 (9) me! The higher the frequency, the smaller the polarizability, because the electron has a shorter time in which to move. The scattering formula Eq. (8) still applies, so we get the simple, frequency- independent result for the cross section contribution per electron: Fig. 4: Polar plot of the Klein-Nishina formula for the differential scattering cross section of X-rays by electrons. At low frequency the scattering goes to that of a classical dipole; at high frequency (the Compton regime) the cross section becomes more and more forward directed, best described as energy- and momentum-conserving photon-electron collisions. From [1]. 8π σ = r2 (10) T 3 e This is the regime of Thompson scattering. Here 2 1 e −13 re = 2 ≈ 2:8 × 10 cm (11) 4π0 mec is the so-called classical electron radius. Even though the binding of the electron to the atomic nucleus is irrelevant in the Thompson scattering regime, it should be understood that, in the regime of low excitation intensity, the nevertheless remains associated with the atom, so long as the distance over which the electron travels under the influence of the time-oscillatory force is much smaller than the atomic radius. In this regime, X-ray scattering is non-destructive. Naturally, if the excitation intensity is raised to the point where this oscillation distance becomes comparable to or greater than the atomic A4.6 David P. DiVincenzo radius, we enter the regime of high-intensity effects, which can very realistically be achieved with strong X-ray sources such as the free-electron laser (FEL). In that case the X-ray probe is destructive, causing ionization and disruption of chemical structure, so the time available for this scattering probe to give useful information about condensed matter is limited. Of course a more complete calculation is possible; the result for the differential cross section is dσ r2 = e 1 + cos2 θ : (12) dΩ 2 The angular dependence appears in Fig. 3, showing the dipolar form that is also characteristic of the Rayleigh scattering regime. This figure shows the result of a much more general calculation due to Klein and Nishina [1], who calculated this scattering taking quantum and relativistic effects into account. The Klein-Nishina formula for the differential cross section is Fig. 5: Scattering geometry for discussion of atomic form factor. dσ r2 = e P (!; θ)2 P (!; θ) + P (!; θ)−1 − 1 + cos2 θ : (13) dΩ 2 Here the factor 1 P (!; θ) = 2 (14) 1 + (~!=mec )(1 − cos θ) has a simple kinematical interpretation when we take the quantum point of view and consider the light to consist of particles (photons): it is the ratio of the photon energy after the scattering event to its original energy before scattering.

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