Classical electric dipole radiation

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1 B Classical electric dipole radiation In Chapter a classical model was used to describe the scattering of X-rays by electrons. The equation relating the strength of the radiated to incident X-ray electric fields Eq..5) was stated without proof. Here the derivation of this equation is outlined more fully. We imagine that an electromagnetic plane wave with an electric field E in is incident on a charge distribution, which oscillates in response to this driving field, and hence acts as a source of radiation. The problem then is to evaluate the radiated electric field at some observation point X, as shown in Fig. B.a). This is simplified considerably if it is assumed that r is much greater than the spatial extent of the charge distribution, and also if r is much greater than the wavelength of the radiation λ. The first of these is the dipole approximation, while the second assures that we can interpret the electromagnetic effects at X as radiation. Here it is further assumed that the electrons forming the charge distribution are free. The electric and magnetic fields at X can be derived from the scalar potentialφand the vector potential A: E= Φ A t and B= A B.) The task of evaluating the fields at X is further simplified if it is recalled that electromagnetic waves are transverse, with the fields being perpendicular to the propagation direction n, as shown in Fig. B.b). We then have that n is colinear to E B, and by solving the wave equation it can be shown that E =c B. It is therefore sufficient to derive B from A Eq. B.)), and then E follows immediately. The vector potential is given by Jr, t r r /c) Ar, t)= c 2 r r dr where Jr, t) is the current density of the source. As the fields propagate at a finite velocity, the fields experienced at the observation point X at time t depend on the position of the electron at an earlier time t r r /c. For this reason A given by the above is known as the retarded vector potential. Elements of Modern X-ray Physics, Second Edition. Jens Als-Nielsen and Des McMorrow 20 John Wiley & Sons, Ltd. Published 20 by John Wiley & Sons, Ltd.

2 350 Classical electric dipole radiation The dipole approximation allows us to ignore r in comparison with r, so that Ar, t) Jr, t r/c) dr c 2 r To proceed it is noted that the current density is equal to the product of the charge density ρ and the velocity v, J=ρv. For a distribution of discrete charges q i the integral is replaced by a sum so that J dr = ρv dr = q i v i = d dt q i r i The last term is recognizable as the time derivative of the electric dipole moment which is written as ṗ. We now let the incident beam be linearly polarized along the z axis, so that the dipole moment and hence the vector potential will have a component along this direction only Fig. B.b)). Thus for a single dipole we have ) A z = ṗt ) c 2 r and A x = A y = 0. From Eq. B.) the components of the B field follow as B x = A z y ; i i B y = A z x ; B z = 0 B.2) For the x component of the B field we evaluate the partial derivative of A z with respect to y as ) ṗt A z y = ) ) c 2 y r )[ ṗt ] ) = ṗt ) r c 2 r y r 2 y Since we are interested in the far-field limit of B, we can neglect the second term in the above, while the partial derivative of the first term with respect to y can be evaluated by noting that y = t t y = t y = y c r t ) x c 2 + y 2 + z 2 ) t Hence the x component of the B field in the far field limit is ) y B x c 2 cr pt ) r) and the y component follows by interchanging x and y, and allowing for the minus sign in Eq. B.2). Recalling that pt ) is implicitly along the z axis we can generalize to any direction of pt ) by writing ) B c 2 cr pt ) ˆr

3 Fig. B. a) The coordinate system used to calculate the electromagnetic field radiated from a charge distribution when placed in an incident plane wave. b) An electromagnetic plane wave polarized with its electric field along the z axis forces an electric dipole at the origin to oscillate. In the far-field limit the field radiated from the dipole is approximately a plane wave with the E and B fields perpendicular to the propagation direction as indicated in the figure. 35

4 352 Classical electric dipole radiation where ˆr is the unit vector x/r, y/r, z/r). The numerical value of the vector cross product is p cos ψ where ψ is defined in Fig. B.b). The direction of the electric field is perpendicular to both ˆr and B in such a way that the cross product of E B is along ˆr. In particular we note that for ψ=0 the E field has the opposite direction of p. Its magnitude is given by E = c B so that Et)= c 2 ) r pt ) cos ψ B.3) The next step is to calculate the magnitude of p in terms of the incident driving field E in = E 0 e i ωt r/c). By definition we have p=q z=q Force mass = q q E in m = q2 m E ωt r/c) 0e i which when inserted into Eq. B.3) with q= e, and remembering that ω/c=k, leads to e 2 ) ) e i kr Et)= E mc 2 r in t) cos ψ The prefactor is the Thomson scattering length r 0, so that the ratio of the radiated to incident electric fields is given by ) Et) e i kr E in t) = r 0 cos ψ B.4) r The factor cos ψ in Eq. B.3) is the origin of the polarization factor for X-ray scattering, as pt ) cos ψ may be thought of as the apparent acceleration as seen by the observer. This is clear if we return to the case when E in is along the z axis. If ψ=0 the maximum acceleration is observed, whereas for ψ=90 the apparent acceleration is zero. The polarization factor is discussed further in Chapter. We note that the minus sign means that there is a phase shift of π between the incident and scattered fields, and it follows that the index of refraction is necessarily less than unity see Chapter 3). This result holds in the X-ray region, where most if not all of the atomic electrons may be treated as though they are essentially free. In the visible part of the spectrum, however, we have to allow for the fact that the electrons are bound. This produces resonances in the frequency dependence of the index of refraction, and on the low frequency side of the resonances, corresponding to the visible part of the spectrum, the index of refraction is greater than one. One way to characterize the efficiency with which an electron scatters the incident radiation is to calculate the total scattering cross-section. The power per unit area is proportional to E 2, and by definition the differential cross-section is the power scattered into the solid angle dω, normalized by the incident flux see Appendix A). From Eq. B.4) it follows that the differential cross-section is ) dσ = r0 2 dω cos2 ψ where the factor of r in the denominator of B.4) cancels on taking the square with a factor of r 2 that arises in converting from surface area to solid angle. The total cross-section for Thomson scattering is

5 353 found by integration over the polar angles ϕ and θ: σ T = r0 2 cos 2 ψ sin θdθdϕ=r0 2 = cm 2 sin 2 θ sin θdθdϕ= ) 8π r0 2 3 = barn B.5) The classical cross-section for the scattering of an electromagnetic wave by a free electron is therefore a constant, independent of energy. Further reading Foundations of Electromagnetic Theory, J.R. Reitz, F.J. Milford, and R.E. Christy Addison- Wesley Publishing Company, 992) Classical Electromagnetic Radiation, M.A. Heald and J.B. Marion Saunders College Publishing, 995)

PHYS Introduction to Synchrotron Radiation

PHYS Introduction to Synchrotron Radiation C. Segre (IIT) PHYS 570 - Spring 2018 January 09, 2018 1 / 20 PHYS 570 - Introduction to Synchrotron Radiation Term: Spring 2018 Meetings: Tuesday & Thursday 13:50-15:05 Location: 213 Stuart Building Instructor: