Elastic light scattering

Elastic light scattering 1. Introduction Elastic light scattering in quantum mechanics Elastic scattering is described in quantum mechanics by the Kramers Heisenberg formula for the differential cross section dddd = rr ddω 0 2 mm 2 ωω 4 (εε ssdd 1ii )(εε DD ii1 ) + (εε DD 1ii )(εε ss DD ii1 ) ii 2, where rr ωω ii ωω ωω ii +ωω 0 = ee2 = mmmm 2 2.8 10 13 cccc is the electron radius (with no meaning other than squared in a cross section), mm the electron mass, εε ss and εε are the polarizations of the scattered and incident light, ωω, ωω ii are the energies of the incident light and intermediate states, DD 1ii = 1 rr ii is the dipole transition term, and 1 is the same initial and final state [Fig. 1(a)]. It is assumed that the dipole approximation is good and that ωω is not very close to an atomic resonance (near resonances every system behaves differently and other details must be included). Unlike for absorption (a one-step process), derivation of this expression for scattering in quantum mechanics requires a quantized field and a composite ket nn, αα, ββ where nn is the state of the atom, αα the incident light mode and ββ the scattered light mode. Exercise: check that the units of dddd are correct (ħ = 1 convention is applied). ddω Note: scattering in the K H formula is one transition in second order perturbation theory. The intermediate states ii are the eigenkets of the unperturbed Hamiltonian HH 0 and their energy is a variable (sum over ii) over all eigenvalues of HH 0. Sometimes these are called virtual states (this is not recommended use the term intermediate instead). The measured scattering rate of one scatterer in a solid angle ΔΩ (given by the detector aperture and its distance from the scatterers) is related to the differential cross section II = iiiiii TTTTTTTT AAAAAAAA dddd II ΔΩ, where iiiiii is the incident light flux and ΔΩ is assumed ddω TTTTTTTT AAAAAAAA as II ssssssssss (θθ,φφ) TTTTTTTT sufficiently small so that dddd ddω cccccccccc. If we consider a linearly-polarized incident light, the induced dipole is oriented along the incident light polarization εε. Then, (εε ssdd 1ii )(εε DD ii1 ) (εε ssεε ) (the same for the second term in the K H formula), which gives the angular dependence for scattered light intensity as II ssssssssss (cos θθ) 2 (the dipolar distribution). In particular, there is no scattering along the direction of the induced dipole. Note: the dependence of scattered intensity on ωω at fixed detector angles is more complicated in general. Two limits are well-known: dddd ddω = rr 0 2 (εε εε ss) 2 = rr 0 2 (cos θθ) 2 for ωω ωω RR (the Thomson limit) [exercise: show this with the dipole (TRK) sum rule] and dddd is an atomic resonance, for ωω ωω ii (the Rayleigh limit). ddω ωω ωω RR 4 rr 0 2 (εε ssεε ) 2 where ωω RR Page 1

ωω iiii 1, αα, ββ 1, αα 1, ββ + 1 ii ii, αα 1, ββ Energy ωω ωω iii ωω iii 1 Time (a) (b) Fig.1 (a) A traditional sketch of elastic scattering orders states by energy. These sketches work well for 1 st order transitions, but are less effective in illustrating higher-order processes. (b) The recommended sketch is called a Feynman diagram. Unlike the first sketch, no assumption is made on the energy of the intermediate state ii. This is better as the energy ωω ii is a variable. Another advantage of this method is that each term in the K H formula has its own diagram. 2. Experimental setup We will measure the dependence of scattered intensity on detector angle for a fixed incident light energy ωω, given by the green laser. The incident light beam points vertically along the tube axis (the cover slip on top prevents evaporation) and the detector rotates in the horizontal plane The laser diode output at 808 nm pumps a laser transition at 1064 nm in a NNNN: YYYYOO 4 crystal, which is then doubled to 532 nm (green light) in a non-linear crystal by secondharmonic generation (the opposite process to parametric down-conversion). The polarization of this green laser is not very stable, and we have to first place a polarizer in its beam. The sample is a diluted ( 1% by volume) solution of skim milk in water. The dilution minimizes multiple scattering. This is necessary because each scattering process gives a well-defined dipolar intensity distribution with respect to the direction of the incident beam. If the incident beam direction is variable, as is the case for the second, third, etc. scattering event, the well-defined dipolar intensity-distribution will change to isotropic. In order of decreasing size, milk contains lipids (fat) particles, proteins (casein), sugars (lactose) and minerals (Na, Cl, Ca, K ). The fat particles of regular milk have been removed in our skim milk sample. This is required because these particles have diameters > λλ, for which the dipole approximation cannot be applied. Page 2

The scatterers are protein aggregates of typical diameter 10 100 nnnn λλ = 532 nnnn, so that the dipole approximation can be applied. Their total ZZ is large, so we expect broad resonances, covering the entire visible spectrum. The sugars and minerals are small (< 1 nnnn) and scatter very weakly in the visible, as can be seen when dissolving regular sugar and salt in water. Their ZZ is relatively small, so that from the dipole sum rule ( σσ aaaaaa dddd = 2ππ 2 rr 0 cccc) relatively narrow and widely spaced resonances are expected. They do not scatter unless the light energy is tuned exactly to their resonances (not our case). Mirror Laser Aperture Lens Polarizer εε 1 xx zz θθ φφ εε 2 yy II iiiiiiiiiiiiiiii Marker Wheel Sample tube II ssssssssss θθ θθ Photodiode Fig.2 Experimental geometry 3. Experiments Angular dependence of scattering The intensity of the scattered light is II εε ssεε 2 where εε ss, εε are the polarization vectors of the scattered and incident beams The polarization of the incident beam depends on how we orient the polarizer before the sample. If its transmission axis is along the xx and yy axes we have εε 1 = (1,0,0) and εε 2 = (0,1,0), respectively. The polarization of the scattered beam depends on how we orient the polarizer before the photodiode. For a vertical and horizontal transmission axis the results are εε ss1 = ( cos θθ cos φφ, cos θθ sin φφ, sin θθ) and εε ss2 = ( sin φφ, cos φφ, 0), respectively, for the range 0 θθ ππ and 0 φφ ππ. The intensity in other octants of the total solid 2 2 angle can be found by symmetry. This gives two possibilities II 1aa = (cos θθ sin φφ) 2, II 1bb = (cos φφ) 2 for incident polarization εε 1 and two possibilities II 2aa = (cos θθ cos φφ) 2, II 1bb = (sin φφ) 2 for incident polarization εε 2 The intensities are plotted in Fig. 3, and illustrate that in the dipole approximation no radiation is scattered in the direction of the incident polarization. Page 3

3.1 No polarization analysis When no polarizer is placed before the detector the intensities are II 1 = II 1aa + II 1bb = 1 (sin φφ sin θθ) 2 and II 2 = II 2aa + II 2bb = 1 (cos φφ sin θθ) 2, respectively. As expected, II 2 = II 1 φφ φφ + ππ 2. Orient the polarization of incident beam along the 0 degree mark on the large white wheel (the polarizer should be rotated to approximately 145 degrees) Observe the difference in the scattered intensity by eye (with no polarization sensitivity) as you scan over the total solid angle in the horizontal and vertical planes Orient the polarization of the incident beam along the 90 degree mark on the large white wheel (the polarizer should be rotated by 90 degrees from the previous orientation) and observe by eye the differences from the previous case For a quantitative result open the Power meter icon on the desktop and enter 532 nm in the Wave box. Measure and the dependence of the scattered intensity on the angle θθ in the horizontal plane for the two cases above. You can achieve an accuracy of 1 oo when rotating the white wheel with the photodiode arm fixed to it and looking at the clamp marker to obtain the angle. 3.2 Polarization analysis Place the large polarizer in front of the photodiode and measure the angular distributions with its transmission axis vertical and horizontal for the two cases above. Plot the results in polar coordinates and compare to the predicted dipolar distribution εε 11 (a) (b) nn (c) Fig.3 Angular dependence of dddd ddω (θθ, φφ) for elastic scattering in the dipole approximation for II 1aa, II 1bb, and II 1 = II 1aa + II 1bb. The case with εε 2 is obtained by rotating φφ by 90 degrees. Page 4

4 Estimate of the cross section from the scattering intensity The laser outputs 4.5 mmmm before the aperture. Measure the intensity of the incident vertical beam and that of the scattered light in the direction of maximum intensity [should be a few tens of nnnn]. The expression for the scattered intensity for one scatterer in the introduction allows an estimate the scattering cross-section. In practice, we have to account for the many scatterers inside the scattering volume given by the light beam in the solution. Specifically, for relatively weak scattering (no appreciable depletion of the incident beam intensity, as in our case) the Beers attenuation law II tttttttttt = II iiiiii ee σσ ρρ ssssssssssll simplifies to II ssssssssss(tttttttttt) σσρρ II ssssssssss LL, where ρρ ssssssssss is the density of scatterers, LL 3 cccc is iiiiii the length of the light beam in the solution, and II ssssssssss (tttttttttt) is the scattered intensity over the entire solid angle. Take the detector solid angle as ΔΩ 1ccmm2 (dd cccc) 2, where dd is the distance from the detector to the sample, and integrate the measured intensity over the total 4ππ solid angle to estimate II ssssssssss(tttttttttt) II iiiiii We have LL and II ssssssssss(tttttttttt), so to find σσ we need ρρ II ssssssssss. iiiiii The mass of the proteins in milk is ρρ ssssssssss,mmmmmmmm = 8 gg = 2 gggggggggggg 10 3 gg ccmm3. Each protein weights mm pppppppppppppp 25 kkkkkk = 25 10 3 1.6 10 24 gg. Proteins form aggregates with a distribution of sizes, dependent on many factors. Take an aggregate made of 10 proteins with mm aaaaaaaaaaaaaaaaaa 10 mm pppppppppppppp = 4 10 19 gg. Then, the number density of scatterers is ρρ ssccaaaaaa,nnnnnnnnnnnn 0.5 10 16 ccmm 3 which, when diluted to 1%, gives ρρ ssssssssss 0.5 10 14 ccmm 3. Estimate σσ and compare to the Thomson cross section for scattering by one free electron σσ TThoooooooooo,tttttttttt = 4ππ (2.8 10 13 ) 2 ccmm 2 10 24 ccmm 2 = 1 bbbbbbbb (Rayleigh cross section is even smaller because of the ωω ωω RR 4 suppression factor). Give a possible explanation for the difference. 5 Conclusion In the visible range of the spectrum, elastic scattering has often been eclipsed in practical applications by the much weaker inelastic Raman and resonance fluorescence scattering. Raman scattering is useful in the measurements of material compositions, while resonance fluorescence is useful in microscopy. In the X-ray energy range, where light sources are typically not as bright as the visible range lasers, elastic scattering remains a powerful technique in the experimental studies of materials. Page 5

6 Notes 1. A more complicated version of the Kramers Heisenberg formula describes both elastic and inelastic Raman scattering in quantum mechanics: ee 4 3 ωω ωω ωω ff dddd = ωω ff <ωω εε ssdd ffff (εε DD ii1 ) + εε DD ffff (εε ss DD ii1 ) ddω ff 16ππ 2 εε 2 0 ħ 2 cc 4 ii, where ff = 1, ωω ωω ii ωω ωω ii ω f +ωω ff = 0 is the elastic Rayleigh scattering and ff > 1, ωω ff > 0 is the inelastic Raman scattering. In contrast, the resonance fluorescence is made of two transitions, described with two first order perturbation theory expressions. 2 ωω ffffffffffffffffffffffff ωω eeeeeeeeeeeeee Line width Raman ωω SSSSSSSSSSSS, ωω AAAAAAAA SSSSSSSSSSSS Fig. 4: Contributions from three processes to a scattering spectrum. In practice, in scattering spectroscopy, Raman and resonant fluorescence are separated with changes in the incident beam energy: the relatively broad and incoherent fluorescence peak does not move when incident frequency is tuned, while the narrow and coherent Raman lines move together with the elastic peak when incident beam frequency is tuned. 2. Classical Lorentz dipoles can give elastic scattering only because they oscillate and radiate at the forcing field frequency only. The expression above for the QM elastic scattering away from resonances is identical to an expression that can be obtained with the Lorentz classical dipoles. 3. If the dipole approximation is not good, additional terms must be added. The next higher order term is a combination of magnetic dipole electric quadrupole scattering, which in our case is (kkkk) 2 vv cc 2 ffffff % of the electric dipole scattering. Page 6

Name Phys-602 Quantum Mechanics Laboratory Elastic light scattering lab report Dates of measurements: Page 7