Dynamic light scattering

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thAsReaders who have only a qualitative interest in light scatter-ing can skip directly to Sec. II. The remainder of the paper isintended for readers who are interested in setting up theirown apparatus to study Brownian motion or fluid dynamicsquantitatively. The same equipment can be used to makemeasurements in fluids near the critical point.

Suppose that the experimental components have been as-sembled and arranged on a table as indicated in Fig. 1. Thebeam of the low-power HeNe laser is directed at a smallvial or test tube filled with water in which small particleshave been suspended. For the particle suspension, a drop ofskim milk in water will do, but it is best to use particles, allof which have the same diameter d in the 0.01 micron to 50micron range.2 With the proper particle concentration, thesample will be pale blue when viewed in room light. A laser

mum produced by a slit of width L . For studies of Brownianmotion, one might want to choose L , R , and h so that B isless than 10 or so ~for studies of fluid motion discussed inSec. IV, B should be less than unity!.

The photodetector should be operated as a pulse countingdevice rather than in an analog mode, that is, it should sensethe individual photoelectron pulses. The width of thesepulses is not relevant to the experiment; only their spacing intime is important. It is preferable if all the photopulses are ofthe same width and amplitude, which is accomplished bypassing the photodetector output through a discriminator.The individual pulses coming out of the discriminator shouldbe less than 1 ms in width and ideally less than 0.1 ms. Manyphotodetectors come with a built-in discriminator with aTTL output.

1152 1152Am. J. Phys. 67 ~12!, December 1999 1999 American Association of Physics TeachersW. I. GoldburgDepartment of Physics and Astronomy, University of Pittsb~Received 27 May 1999; accepted 27 July 1999!

By scattering light from small particles, their geometrmeasured. An experiment is described for measuringBrownian motion using the technique called photoscattering. The necessary experimental apparatus acorrelation spectroscopy is a powerful tool for studcritical points, and a discussion is given of this phenomcan be used to study laminar or turbulent flows, andsuch experiments to be interpreted. 1999 American

I. INTRODUCTION

Fractals, continuous phase transitions, spinodal decompo-sition, Brownian motion, the WienerKhinchine theorem,the BrownTwiss effect, photon correlation spectroscopy,and turbulence. Many phenomena and ideas of great impor-tance in statistical and condensed matter physics can be in-troduced by studying dynamic light scattering. The goal ofthis paper is to introduce this technique by describing a fewexperiments that can be performed by undergraduate stu-dents. The experimental possibilities are sufficiently rich thatthey can be stimulating to graduate students as well. Themeasurements require inexpensive optical components suchas a small HeNe laser ~output power of a few mW!, a fewlenses and lens holders, and a photodetector. In addition, amore expensive device is needed, namely a photon cor-relator. Many correlators that may have been discarded byresearchers may serve the purpose perfectly well.1 A newcorrelator can be purchased for several thousand dollars.

Considerable emphasis is placed on the Brownian motionof micron-size particles diffusing in a background fluid suchas water, because an understanding of Brownian motionopens the door to understanding a broad range of subjects,including the dynamical behavior of fluids and fluid mix-tures, and magnetic materials near their respective criticalpoints.

Near the critical point of fluids, we can observe fluctua-tions by eye and also by using a very small laser. Section IIdescribes a desktop experiment for doing so. It turns out thatthese fluctuations produce the same effects as those seen inthe scattering of light by small particles in Brownian motion.rgh, Pittsburgh, Pennsylvania 15260

al structure and their state of motion can behe diffusivity of small particles undergoingcorrelation spectroscopy or dynamic lightthe related theory are discussed. Photon

ing the dynamical behavior of fluids nearnon. The same experimental technique alsoe associated theory is introduced to enable

sociation of Physics Teachers.

beam passing through the sample should appear as a brightline which is well defined throughout the entire scatteringvolume.

The experimental setup in Fig. 1 shows a photodetectorthat collects light at a scattering angle u equal to 90. Tosatisfy the spatial coherence condition discussed below thedistance R between the center of the sample cell and thephotodetector should be large; say, 30100 cm. In addition,a small pinhole of diameter h should be placed close to thephotodetector face. It is helpful to mildly focus the incidentlaser beam on the sample volume, using, say, an f 520 cmlens. Ideally, the scattering particles will produce a sharplydefined bright line in the fluid. Only a segment of the scat-tered beam should fall on the face of the photodetector. Thissegment can be defined by a pair of parallel closely spacedblack masking tape strips placed on the sample tube ~sepa-ration of the order of a mm!. Alternatively, a lens can beused to form the image of the scattering volume on a slit ofadjustable width L . The photodetector then accepts lightfrom scattering particles within the segment. Spatial coher-ence is achieved by choosing h , L , and R to satisfy thecondition

B5hL/Rl&1. ~1!

where l is the wavelength of light in the water sample ofrefractive index n , and l is related to the vacuum wave-length of the laser beam lvac by l5lvac /n . For water n51.3.3 The optical coherence condition in Eq. ~1! is familiarfrom elementary optics. When Eq. ~1! is satisfied, the photo-detector collects light from only the first diffraction maxi-

To be sure that all the optics and electronics are arrangedcorrectly, it is useful to observe the chain of TTL pulses onan oscilloscope before sending them to the correlator. If theTTL pulse width is shorter than 0.1 ms, use very short inter-connecting cables or be sure that all outputs and inputs areterminated in the characteristic impedance of the cable,which is 50 V for RG58U cable. This is to avoid reflection ofpulses from the ends of the cable due to impedance mis-matching problems. Data collection will be tedious if themean photon counting rate ^I& is less than 1000 counts/s.

Figure 2~a! shows a chain of photodetector pulses pro-duced by incoherent light falling on the photodetector. Thepulses occur at random times, with their mean spacing beinginversely proportional to ^I&. Figure 2~b! shows the detectoroutput from light that is scattered by diffusing particles in thesample. In both experiments the photon pulses have passedthrough a discriminator, assuring that all the pulses have thesame height and width. In Fig. 2~b! the pulses are correlatedin time, that is, they are bunched together within a correla-tion time tc which is determined by the diffusion coefficientof the particles and the scattering angle. The photon cor-relator identifies this clustering effect by measuring the cor-relation function

Fig. 1. Experimental setup for studying the Brownian motion of particles inwater. The sample is labeled S . The laser L directs the incident beam ver-tically upward. The photodetector PD converts the scattered light into apulse string which is fed to the correlator COR.

Fig. 2. ~a! The output of the PD for immobile scatterers, as might be thecase if the sample has been frozen or if the sample is a piece of milk glassor even a piece of paper. The arrival time of the photopulses is purelyrandom in this case. ~b! The PD output of moving particles which are Dop-pler shifting the frequency of the incident light. Note that the pulses tend tocluster in time. This clustering time or correlation time tc is inversely pro-portional to the diffusivity of the randomly moving particles that scatter thelight.

1153 Am. J. Phys., Vol. 67, No. 12, December 1999K~t!5^I~ t !I~ t1t!& ~2!

or its dimensionless counterpart

g~t!5^I~ t !I~ t1t!&/^I~ t !&2. ~3!

Experimentally, the average designated by the brackets inEq. ~3! is a time average over the data collection interval T:

K~t!51T E0

TI~ t !I~ t1t!dt , ~4!

where T typically ranges from seconds to many minutes,with the choice dictated by the desired signal-to-noise ratio.The mean intensity is the time average, ^I&5(1/T)*0TI(t)dt .

In the experiments discussed here, the system is in thermalequilibrium, or as in Sec. V, in a stationary state. In eithercase ^I(t)&5^I(t1t)&, a fact that will be invoked repeat-edly. Of course, K(t) is not independent of t; itst-dependence is what we are interested in measuring andcalculating. Its characteristic decay time will be denoted astc . Typically tc is in the range of hundreds of microsecondsto milliseconds, depending on the particle size used.

The signal-to-noise ratio, S/N , will always improve if thedata collection time T is increased. Suppose that we are con-tent with data for which g(t) is measured within 1%. Firstassume that the counting rate ^I& is so large that the qualityof the measurements is not determined by the low countingrate, but is limited by the number Nc5T/tc , of fluctua-tions collected in the time T . According to Poissonstatistics,4 S/N51/ANc , and hence, one percent accuracy isachieved by setting T/tc5104.

Next consider the opposite limit where the scattered inten-sity ^I& is very low. In this limit, the fluctuations in g(t) willbe governed by so-called shot noise and will vary as theinverse square root of the measuring time T . In this lowcounting rate limit, where Poisson statistics is again relevant,T/tc is no longer the controlling parameter; to make a mea-surement to 1% accuracy, choose T^I&5104.

Figure 3 shows a measured g(t) produced by small par-ticles in Brownian motion. The solution in which they are

Fig. 3. Plot of g(t) versus t measured in a soap film which was only a fewmm thick. The soap solution was a 1% dish washing detergent in water. Theseed particles were polystyrene spheres of diameter 1 m. Note that g(0).2.

1153W. I. Goldburg

All photodetectors generate output pulses even when thereis no light shining on them. This dark count rate can be a fewmoving is a soap film only a few mm thick. For sphericalBrownian particles diffusing in the background solvent, itcan be shown4 that the function

G~t![g~t!21 ~5!decays exponentially, with the decay rate given by

tc2152 Dq2. ~6!

Here D is the diffusion coefficient of the particles and q isthe scattering wavevector whose magnitude q is given by

q5uqu54pl

sinu

2 , ~7!

where u is the scattering angle and l is lvac /n .In a time t a particle will diffuse a mean square distance

^r2& that is proportional to Dt . The diffusion produces alarge fluctuation in the intensity in a time t8 such that t8.1/(Dq2). This characteristic time t8 is of the order tc . It isthese intensity fluctuations that a photon correlator measures.Helpful discussions of diffusion can be found in Refs. 4 and5.

The diffusion coefficient D depends on the particle diam-eter d . For spherical particles, D5kBT/3phd ,6 where T isthe temperature, kB is Boltzmanns constant, and h is theviscosity of the background fluid. From this relation, wehave h50.01 g cm21 s21 and D54.431029 cm2/s for a dif-fusing particle 1 mm in diameter in water at room tempera-ture. From a measurement of g(t), we can obtain tc21 andfind the radius of the diffusing particles, even if they are toosmall to be observed with a microscope. ~See Fig. 4.! Simi-larly, a measurement of g(t) enables us to measure the ra-dius of gyration of polymer molecules diffusing in solutionand to determine the correlation length of fluctuations in den-sity or concentration in a fluid or mixture near its criticalpoint.7,8 Measurements of g(t) can be a powerful tool forstudying phenomena that have nothing to do with diffusion,for example, the determination of the width of a narrowspectral line9 or the diameter of a distant double star.10 Thelatter experiment gave birth to the technique of photon cor-relation spectroscopy.

Fig. 4. Sketch of diffusing particle with an incident beam traveling from leftto right. The particle is located at the origin of the coordinate system at timet50 and at time t its position is r(t). The photodetector is labeled PD. Thedirections of the incident and scattered beams are shown.

1154 Am. J. Phys., Vol. 67, No. 12, December 1999Hz to several kHz. Except at very small t, these dark countpulses occur roughly randomly in time and hence give a flatbackground to g(t), reducing the relative contribution ofG(t), the quantity in which we are really interested. In roomlight the photodetector will have the same effect. It is there-fore necessary to employ a laser-generated counting rate thatis many times larger than the dark count rate and room lightcount rate, in addition to satisfying the condition ^I&tc@1.The contribution to ^I& from room light can be reduced byplacing an inexpensive interference filter11 directly in frontof the photodetector and interposing a tube between it andthe sample. A cardboard tube works well. For the mostnoise-free results, turn off the room lights when the cor-relator is making measurements.

If a flowing fluid is seeded with small particles, we canlearn a lot about the flow by measuring g(t). The fluid flowproblem is treated in Sec. V. It will be seen that g(t) givesinformation about velocity differences between pairs ofpoints in the flow. Laminar and turbulent flows can be stud-ied in the laboratory1214 using the same apparatus as thatemployed in the Brownian motion measurements.

In Sec. III the intensity correlation function is calculatedfor a single Brownian particle, with the assumption that thephotodetector is so small that the scattering wave vector q issharply defined. The calculation will be valid even whenmany particles are present in the sample, as long as they aresufficiently far apart that they do not interact. We will seethat when B is no longer very small, G(t) is diminished bya factor f (A) which takes into account the fact that photonsscattered from the most widely separated diffusing particlesin the sample will interfere with each other to give

G~t!5g~t!215 f ~A !e22Dq2t. ~8!The argument A in f (A) is the area of the photodetector, animportant parameter in the subsequent calculation of f (A) inSec. IV.

The single-particle calculation of g(t) is presented in Sec.III. To find f (A) in Eq. ~8!, we must consider the motion ofmany Brownian particles, because its attenuating effectarises from the interference of the electric fields scattered bythese particles. The calculation of f (A) appears in Sec. IV.

The basic theory developed in Secs. II and III can be ap-plied with little modification to the study of laminar andturbulent flows, which is the subject of Sec. V. Some con-cluding remarks are given in Sec. VI.

II. CRITICAL PHENOMENA, LIGHT SCATTERING,AND BROWNIAN MOTION

Light scattering is a very powerful tool for studying criti-cal phenomena. Some of the central features of continuousphase transitions are easily observed and can be presented ina lecture demonstration or observed on a desktop.

Here we briefly review the subject of critical phenomenaand describe an experiment for observing a striking manifes-tation of it, namely critical opalescence. You can observethis effect with a very small laser of the type used as apointer in lecture halls. In addition, you will need a smallglass container to hold the sample, which might be a mixtureof 2,6-lutidine and water ~LW!, available from a chemicalsupply house. A convenient sample cell is a vial of diameter

1154W. I. Goldburg

1 cm and a height of several centimeters. The vial should15

globally in one phase. These local fluctuations in density orcomposition produce local variations in the refractive indexhave a good screw top, because 2,6-lutidine has a very

unpleasant smell. The trick is to prepare the LW mixture sothat the lutidine concentration c is close to the critical con-centration, cc , which is 28% by weight. In a binary mixturesuch as LW, the concentration variable plays the same roleas the density in a simple fluid such as water or CO2.16

Sample preparation should be carried out under a hood orin a very well ventilated space. After the water and lutidinehave been placed in the container with its composition withina few percent of cc and shaken up, it will be in one phase ifthe temperature is less than the critical temperature, Tc533.4 C. Place the cell in a beaker of water, which willserve as a temperature bath. The bath is then heated by add-ing hot tap water. When the sample is held for a few minutesat a high temperature well above Tc , it will separate into twowell-defined phases, with the heavier water-rich phase on thebottom.

If you were successful in preparing a sample close to thecritical concentration, the volumes of the two phases will bealmost equal. If not, insert a syringe into the cell and drawout the excess phase ~or add a drop or two of the appropriateminority component!, so that the two phases have equal vol-ume. After this single iteration, you have probably preparedthe mixture adequately close to Tc .

To observe critical opalescence, return the sample cell tothe water bath and heat it slowly, shaking the sample cellnow and then to bring the mixture to the same temperature asthat of the surrounding bath. When T is increased through Tcfrom below, droplets will form, and after several minutes,the more dense water-rich phase will settle to the bottom andtwo sharply defined phases will be seen. The most instructiveobservations are made when the system is still in one phase,so increase the bath temperature slowly as Tc is approached.

When the temperature becomes close to Tc , the samplewill develop a brownish appearance because it is stronglyscattering the room light. The scattering may become solarge that the laser beam, which you should now be sendingthrough the sample, may hardly go through it. This phenom-enon is called critical opalescence. Place a sheet of paper sothat you can view the transmitted beam. You should see thatthe scattered light consists of speckles that flash on and off ina random fashion.

How are we to understand this strong scattering phenom-enon when the system is in one phase but close to Tc? Nearthe critical point, small regions of water rich and lutidine richphases momentarily form and then diffuse away. This samephenomenon occurs far from the critical point, but the size ofthe regions is much smaller than the wavelength of light, andthe regions come and go so quickly that most measuringinstruments will not be able to record them. Similarly, aglass of water near room temperature undergoes fluctuationsin density, but these fluctuations are so small in spatial extentand in amplitude that they cannot be easily observed. More-over, the fluctuation rate G is very fast far above or belowTc , because small-scale density variations diffuse awayfaster than large ones, making them unobservable by eye andeven by very fast electronic equipment.

Near the critical point, the free energies of the one-phasestates and the two-phase states are almost the same. As aresult, the system makes brief excursions or fluctuationsfrom the one-phase state to the two phase state, that is, itlocally tries to phase separate, even though the system is

1155 Am. J. Phys., Vol. 67, No. 12, December 1999n . It is these local variations in n that scatter the light, just asa small lens would. As Tc is approached, these compositionfluctuations grow in magnitude and in spatial extent j andcan become even larger than the wavelength of light. Thefluctuations of size j cause G(t)5g(t)21 to vary ase22Dq

2t, just as for Brownian particles ~provided qj,1).

The diffusivity D in G(t) is the same as that of a Brownianparticle, except that the radius d/2 of the diffusing particle isreplaced by j.

The reader interested in learning more about critical phe-nomena and the role of light scattering has many sources ofinformation available, such as those collected in Refs. 8, 16,and 17.

III. INTENSITY CORRELATION FUNCTION FORBROWNIAN PARTICLES

We first deduce the limiting value of g(t) for t@tc . Inthis limit the intensities at t and t1t are unrelated, soK(t)5^I(t)&^I(t1t)&^I(t)&25^I(t1t)&2 in steady orequilibrium states. Therefore g(t)51 in Eq. ~3!.

In the opposite limit t!tc , the scattered intensities at thetimes t and t1t are almost equal, so that g(t)^I2&/^I&2,which is in general greater than unity. For the Brownianmotion problem, we will see that g(t0)52. If the samplecontains large specks of dust that produce occasional brightflashes of light when floating through the laser beam, we areno longer dealing with the Gaussian statistics of Brownianmotion, and g(0) can greatly exceed two.

We are interested in g(t) generated by non-interactingBrownian particles such as polystyrene latex spheres of di-ameter d51 mm in a small container of water. As a startingpoint, consider a particular particle located at r(t) at time tthat was at r(0) at t50. A laser beam traveling along the yaxis illuminates the particle, which then scatters the incidentbeam. The photodetector, which is located a large distance Rfrom the origin, senses the scattered field, producing photo-pulses at the ~fluctuating! counting rate I(t). The intensityI(t) is proportional to the square of the incident electric fieldEi , which is assumed to be of complex form:

Ei5E 0e2ikir(t0)1ivt0, ~9!where v is an optical frequency of the order of 1015 Hz andt0 is the instant when the scattered wave leaves the particle.Here ki has the direction of the incident beam and its mag-nitude is k052p/l . Because we are ultimately interested inthe ratio of scattered intensities, nothing is lost by taking thisfield to be expressed in dimensionless units. The scatteredfield Es is a spherical wave whose amplitude is proportionalto the incident field amplitude:

Es~R,t !}Ei~ t0!e2ik0uR2r(t0)u

uR2r~ t0!ueiv(t2t0). ~10!

The intensity is

I~ t !5Const Es~R,t !Es*~R,t !. ~11!The constant of proportionality depends on many param-eters, including E0 and R . The factor eivt in Eq. ~10! willsometimes be suppressed because it will always be canceled

1155W. I. Goldburg

out for any measurable quantity. The units of the electricfield will be chosen such that E (t)E*(t)5I(t). In the next

number of independent random variables dri , has a Gauss-

s s

section, quantities like K and Q ~the analog of Q ) will havedifferent units, but the quantities of interest, namely g(t)and f (A), will remain dimensionless.

Equation ~10! can be simplified by taking advantage of thefact that uRu is much greater than ur(t)u in all cases of experi-mental interest. Then the exponent in Eq. ~10! can be ex-panded to give the phase factor

Df~ t !5k0uR2r~ t0!u

5k0AR222Rr1r~ t0!2.k0R2k0Rr~ t0!/R .We define the scattered light wave vector ks5k0R/R , sothat, to a good approximation, Df5k0R2ksr(t0). Defin-ing

q5ks2ki , ~12!

we can write the total scattered field at time t produced bythe Brownian particle as

Es~ t !}e2iqr(t0)1ivt

R ,

where we have dropped the r-independent factor eik0R. Inaddition, the denominator is approximated by R , because inall experiments R@r . The vectors appearing here are shownin Fig. 4.

Because the flight time of the photon, t2t0 is very short~much shorter than the time for a particle to diffuse an opti-cal wavelength!, t0 can be replaced by t , so that the aboveequation, combined with Eq. ~9!, can be written

Es~ t !}e2iqr(t)1ivt

R . ~13!

The scattering of photons by the small particles is almostperfectly elastic, so that the magnitude of the incident andscattered photons is equal, uksu5ukiu52p/l[k0 . If thescattering angle is u, then q52k0 sin(u/2). This result isquickly obtained from a sketch of the two vectors ks and kidrawn so that their tails touch each other, with the angle ubetween them. As expected, when u is zero and p, q is zeroand 2k0 , respectively.

In this discussion, the vector nature of the electric field hasbeen ignored, so polarization effects were not considered.Such effects are not important if the refractive index of thescattering particle is close to that of the surrounding fluid.18If this condition is fulfilled, the scattered light will have thesame polarization as that of the incident beam, because thescattering particles behave almost like point dipoles, theirdipole moment being induced by the incident electric field.19

The above derivation is for an ensemble of non-interactingBrownian particles, a condition that is well fulfilled if theparticle density is low enough that the beam traces a sharpline through the fluid. In the limit of extreme multiple scat-tering, ks would no longer be well-defined, and the scatteringwave vector q5ks2ki would have lost its meaning.

In a macroscopic time interval t , the distance r(t) tra-versed by a Brownian particle is a Gaussian random variable;its position in a liquid is the sum of many independent stepsof atomic size. According to the central limit theorem,5 arandom variable r5(dri which is the sum of a very large

1156 Am. J. Phys., Vol. 67, No. 12, December 1999ian distribution, regardless of the functional form of the dis-tribution of the individual terms. In three dimensions20

P~r !5~4pDt !23/2e2r2/4Dt

. ~14!This probability density function describes the position dis-tribution of a large assembly ~or ensemble! of particles ini-tially located at r50 at t50. The integral of P over all spaceis unity at t50 and at all subsequent times. According to Eq.~14!, P is an infinitely sharp spike or delta function at t50. As time progresses, P broadens, but its maximum doesnot shift. This broadening of P(t) describes the diffusion ofan ink drop that is initially infinitely small and subsequentlyspreads out as a spherically symmetric cloud, fading at itsedges. Of course, ink particles are Brownian particles them-selves, rendered observable by their color.

We now turn our attention to a calculation of the electricfield correlation function g1(t), a complex quantity definedas

g1~t![^Es*~ t !Es~ t1t!&/^I&

5^eiqr(t)&eivt5E P~r !eiqrd3reivt, ~15!so that

g1~t!5~4pDt!23/2eivtE2

e2r2/2Dte2iqrd3r

5e2Dq2teivt. ~16!

Written out in full, K(t) in Eq. ~2! isK~t!5^Es*~ t !Es~ t !Es*~ t1t!Es~ t1t!& . ~17!

The motion of a single particle will Doppler shift the fre-quency of plane waves falling upon it, but a photodetectorwill not record a fluctuation in the total scattered intensityEs(t)Es*(t), since the time-dependent phase factors in Es(t)and Es*(t) cancel. To observe intensity fluctuations in a pho-todetector, one must have at least two particles present, sothat their relative motion will generate a beating of the scat-tered electric fields. It turns out, as one will see, that the onlyterms contributing to g(t) are such pairwise terms.

We are interested in g(t)5^I(t)I(t1t)&/^I&2, when thesample consists of a very large number N of particles ex-posed to the incident plane wave. Writing out the numeratorand denominator of g(t) in full and dropping the subscripts ,

g~t!5K~t!/Q 2

5 (i , j ,k ,l

^Ei~ t !E j*~ t !Ek~ t1t!El*~ t1t!&/Q 2, ~18!

where

^Q ~ t !&5^I~ t !&5(i , j

E02^ei(qri(t)e2i(qrj(t)&.

There are N4 terms in K and in Q 2, but in the ensembleaveraging operation, most of them are zero because they cor-respond to fluctuations at optical frequencies or because theparticles are, by assumption, uncorrelated in their motion.~See the next section for more details.! The only surviving

1156W. I. Goldburg

terms are those for which i5l and j5k . The absence of acorrelation between the particles enables one to factor func-

tions like ^Ei(t)El*(t1t)E j*(t)Ek(t1t)& into the product ofpairwise correlation functions with i5l and j5k , so that thisproduct may be written ^Ei(t)Ei*(t1t)&^E j*(t)E j(t1t)&with i j . Because all particles scatter identically, this termhas the same value as u^Ei(t)Ei*(t1t)&u2, even though itrepresents a beating term between different particles i and j .There will be N22N.N2 such electric field correlationfunction terms in K(t).

Now consider those terms for which i5 j and k5l ,namely ( i , j^Ei(t)Ei*(t)E j(t1t)E j*(t1t)&. It is equal toN2E0

25^I&2, or equivalently N2u^E(t)&u2, where E is thefield scattered by a single particle.

Collecting these results and putting them in Eq. ~15! per-mits one to write

g~t!511u^~E~ t !E*~ t1t!&u2/u^E~ t !u2&u2. ~19!Or equivalently,

g~t!511ug1~t!u2. ~20!The correlation function is written in the above way be-

cause the result holds more generally than suggested by thisderivation; Eq. ~20! is valid for any N-particle system, aslong as the the total scattered field, Es5( jE j is a ~two-dimensional! Gaussian random variable. This importantequation is called the BlochSiegert theorem. Its full deriva-tion can be found in Ref. 21.

Because ^uE*(t)E(t1t)&/^uE(t)u2&25e22Dq2t, we have,g~t!511e22Dq

2t, ~21!

which is our central result.Equation ~21! gives the correct t dependence of G(t)

5g(t)21, even when B is large and f (A) in Eq. ~8! iscorrespondingly small. On the other hand, it fails when ap-plied to flowing fluids, where the particles are no longermoving independently. In that case, the factorization of thefourfold correlation function in K(t) is no longer permis-sible, and one must turn to the formalism presented in Sec.V.

For a sample containing a spread of particle diameters,such as dilute milk, g(t) will consist of a sum or integral ofexponentially decaying contributions of the forme22Di(d)q

2t, with each term appropriately weighted by a con-

centration factor ci(d). There exists a large body of literaturedealing with this weighting problem, as it is of great practicalimportance.22 Further complications arise if the Brownianparticles are not spherical in shape.7

So far we have considered only the limit B!1, so that thephotodetector is receiving light from a single scattering vec-tor, ks . That restriction is lifted in the next section. In manyexperiments, the finite area of the photodetector changesonly the amplitude of G(t), but for fluid flow, the shape ofthis function can be appreciably changed if B is not less thanunity.12,13

IV. THE EFFECT OF SPATIAL INCOHERENCE

The goal of this section is to calculate the ~de!coherencefactor f (A) in Eq. ~8!. This function will depend on theprecise shape and size of the source as seen by the photode-

1157 Am. J. Phys., Vol. 67, No. 12, December 1999tector and on the same parameters for the photodetector sur-face with area A . Consider the special geometrical arrange-ment shown in Fig. 5. The wave number ki of the incidentlaser beam has been chosen to lie in the plane of the paper,and the photodetector receives light from a range of ks val-ues. The mean value of the scattered wavevector ^ks& pointsalong the z axis in Fig. 5, so the average scattering vector qis given by q5^ks&2ki . Its direction and magnitude areshown in Fig. 5.

The rectangular coordinate system is centered at a pointnear the center of the sample. Another coordinate system x8,y8, z8 is also employed. The origin of this coordinate systemis at the geometrical center of the photodetector; its face isassumed to be square, with dimensions h . If the photosensi-tive area is circular, its radius is also taken to be h .

The detector collects light from a range of scattered vec-tors ks , and the photon momentum transfer is designated as

Dk5ks2ki5q1dk. ~22!

For the j th particle the scattered electric field isE j5E0e2iDkrj(t)1ivt. ~23!

The subscript s in the scattered field has now beendropped. It will be assumed, as before, that all scatteringparticles have the same diameter, so that the scattering am-plitude E0 is common to all of them. Therefore it does notappear in g(t), and we set it equal to unity.

The correlation function K(t) in Eq. ~8! is the product offour electric field factors, each of which is now a sum overall N particles in the sample. The scattered photons will go toall points on the face of the photodetector, and for eachpoint, the momentum transfer Dk will be different. We needto sum over all such points or rather integrate over the pho-tosensitive area A .

At time t , the product Ei(t)E j*(t) will contain terms suchas ei(q1dk)(rj(t)2ri(t)), where we have replaced Dk by itsequivalent, q1dk. In writing this expression, it has beenrecognized that to a very good approximation q1dk is inde-

Fig. 5. Schematic diagram showing the scattering geometry and the coordi-nate directions discussed in Sec. IV. The incident beam ki , like all the otherwave vectors shown, lies in the plane of the paper. The scattering wavevector ~or the momentum transfer! for the photon striking the photodetectorsurface at the point (x850,y8) is Dk5ks2ki5q1dk. Here q is the aver-age value of Dk. In the detailed calculations presented here, the incidentbeam is travelling in the 2y direction, and the photodetector is a square ofarea h3h or a circle of diameter h . A second coordinate system x8, y8, z8is at the geometrical center of the photodetector.

1157W. I. Goldburg

pendent of the particle positions rn(t), so this momentum Because N is typically very large, we can neglect the N

change is independent of particle position in the sample.

The notation is simplified by letting ri j5rj(t)2ri(t). Inthe short time interval t, particles i and j change their rela-tive positions by a distance dri j(t), so at time t1t , thedifference in positions of this particle pair is ri j(t)1dri j .Integrating the scattered intensity over the area of the photo-detector for both I(t) and I(t1t), we obtain

g~t!5K~t!/Q25K EA

I~ t !dA8EA

I~ t1t!dA8L Y Q2, ~24!where

Q~ t !5K EA

I~ t !dA8L 5K (i , j

EA

ei(q1dk)ri j(t) dA8L5(

i , j K EAei(q1dk)ri j(t) dA8L .~25!

The summations and integrations are freely interchangedhere.

Writing out K(t) in full, we have

K~t!5(i , j K EAei[(q1dk)ri j(t)] dA83(

l ,mE

Ae2i(q1dk)[rlm(t)1(drlm(t)] dA8L . ~26!

Remember that each little area element dA on the face of thephotodetector has a different dk associated with it. This factwill be used when we come to evaluate this integral in twospatial cases: a square and a circular photodetector.

Because the particles have random positions at all times,the ensemble average implied by the brackets, will give zerocontribution to Q unless i5 j and in that case, ri j(t)50.Thus Q5NA and

Q25N2A2. ~27!For K(t), we have no such simplification. Every term in Eq.~26! not involving dk can be taken out of the integral, so thatthis equation can be written in full as

K~t!5 (i , j ,l ,m

^eiq(drl(t)2drm(t))e2iq(rj2ri1rl2rm)BlmCi j&,

~28!where, for any i and j

Ci j5EA

eidkri j dA85C ji* ~29!

and

Blm5EA

eidk(rlm(t)1drlm(t)) dA8. ~30!

For every particle pair, the relative change in the particlepositions drlm(t) is small compared to their separationrlm(t), so we can set Clm5Blm . Also, the randomness of theparticle positions again allows us to simplify the expressions;the only terms in Eq. ~28! that will not average to zero arethose for which i5 j and l5m or i5l and j5m . For theseterms (rj2ri1rl2rm) in Eq. ~28! is zero.

1158 Am. J. Phys., Vol. 67, No. 12, December 1999terms for which i5 j5l5m , there being only N of theseterms compared to the N(N21).N2 particle pairs that donot satisfy this fourfold equality. Taking advantage of thisfact, we write

K~t!5N2A21 (l ,ml

^eiq[drl(t)2drm(t)]ClmClm* & . ~31!

The first term on the right comes from setting i5 j and l5m . Particles undergoing Brownian motion are uncorre-lated, so we can write

K~t!5N2A21N2^eiqdrlm(t)ClmClm* &

5N2A21N2^ClmClm* &^eiqdrml(t)&. ~32!We are justified in averaging the product ClmClm* 5uClmu2separately from the remaining phase factor term, because thechange in separation dri j(t) in the interval t has nothing todo with the initial ~random! separation ri j(t) of the Brownianparticles. This major simplification is lost when the particlesare moving coherently, as discussed in Sec. V for fluid flow.

In the present notation, for any l

^e6iqdrl(t)&5e2Dq2t

, ~33!and the independence of the motion of the particles permitsus to write

^eiqdrml(t)&5^eiqdrl(t)&^e2iqdrm(t)&. ~34!Thus

K~t!5N2A21N2^uClmu2&e22Dq2t

. ~35!Dividing by Q25N2A2 from Eq. ~28! gives the result weseek for the case of ~uncorrelated! Brownian motion of par-ticles in the sample:

g~t!511 f ~A !e22Dq2t, ~36!where f (A)5^uClmu2&/A2.

The function Clm depends on the particle separations rlm ,and for all applications of interest, we can treat this separa-tion as a continuous variable. Then Clm can be replaced byC(r) where r is the separation of pairs of diffusing particles.We must integrate over all r lying in the sample volume. Inaddition, there is an integration over the face x8, y8 of thephotodetector ~see Fig. 5!. For algebraic simplicity, we con-sider the special case of the incident laser beam ki travelingalong the y axis and illuminating the Brownian particles overthe vertical distance 0

h3h , and its face is normal to the direction of ^ks& , the Again focus attention on a pair of particles, l and m , mov-

scalar product in Eq. ~24! is dkr5k0yy8/R . Then

C~r!5hE2h/2

h/2eik0yy8/R dy85

h2 sin~k0hy !k0hy

5h2 sinc~k0hy !.

~37!

We use Eq. ~37! and the fact that Q25N2h4, because thesquare photodetector has sides h , we have

f ~A !5E0

L~2/L !~12y /L !sinc2~k0yh/R !dy . ~38!

It has been implicitly assumed in the above calculation thatthe incident laser beam has a square profile. In reality, laserbeams usually have radially Gaussian intensity variation.This fact does not alter the above derivation as long as thebeam length L seen by the photodetector is long compared tothe Gaussian beam radius.

Consider next the case where the aperture in front of thephotodetector is circular with diameter a . Again the incidentlaser beam is a thin line of light traveling along the y-axis inFig. 5. We switch to cylindrical coordinates, dksr5k0ryy8/R , where ry is the component of source coordinater in the y direction. Taking a to be the diameter of thedetector, we obtain

C~r !5C~ry!5E0

a/2E0

2pei(k0ryr8/R)cos f df dry

52pE0

a/2J0Fk0ryr8R Gr8 dr8, ~39!

where Jn(u) is a Bessel function of order n . Using*0

uJ0(u8)u8 du85uJ1(u), we find

G~t!54E0

L~2/L !~12ry /L !

3@J1~k0ary/2R !/~k0ary/2R !#2dry e22Dq2t

5 f ~A !e22Dq2t. ~40!Again, f (A) falls below unity when B is no longer small,because the photodetector is accepting light from many in-dependently fluctuating speckles ~intensity maxima!, whichtend to average out the intensity fluctuations.

V. FLUID FLOW

We now ignore the effect of Brownian motion and insteadcalculate G(t) under conditions where the velocity is chang-ing from point to point and is a function of time t . The abovederivation requires surprisingly little modification, eventhough the functional form of G(t) will be entirely differentwhen the particles in the flow are not moving independently.The calculation will show that G(t) decays in a time tcgoverned by the velocity variation measured across L . LetV(L) be the rms velocity difference across the source widthL . We will find that tc.(qV(L))21 if B!1. If, on the otherhand, the photodetector collects light from many speckles,the shape of G(t) will be altered.

1159 Am. J. Phys., Vol. 67, No. 12, December 1999ing at velocities vl(t) and vm(t). We can replace dri j(t), therelative change in position of this particle pair in a time t, by

Vlmt5Et

t1t

@vl~ t8!2vm~ t8!#dt8. ~41!

We have used the fact that in the short time t where G(t) isappreciable, the change in velocity difference Vlm is almostconstant. If the flow is turbulent, Vlm is a random variable asmeasured in an experiment that spans many correlation timesin the measuring time T .

With this change, the above results can be used, except forthe fact that the factorization in Eq. ~32! is no longer permis-sible. Therefore we have

K~t!5N2A21 (i , j ,i j

^eiqVi jtCi jCi j*& . ~42!

Here there is no justification for averaging the factor Ci jCi j*separately. To understand this point, consider a pair of par-ticles that are near each other in the source and another pairthat is widely separated. There will be a coherence factor Bassociated with each of these pairs, because closely spacedparticles will produce a broader speckle than those that arenot, just as a narrow slit produces a broader diffraction maxi-mum than does a wide slit. Thus, the widely spaced pair willreceive a smaller weight in G(t). It is therefore necessary toassure that B5hL/Rl!1 if the measurements are to have areasonably direct interpretation. Reference 13 discusses acalculation and a measurement of the effect on G(t) when Bis not small, so that the photodetector collects light frommany speckles.

As before, the seeding density of the particles is assumedto be sufficiently high that Vi j can be replaced by V(r),where r is the distance between a particle pair. Because V(r)is a random variable that generally will not be Gaussian, theprobability distribution P must be left as an unknown func-tion to be determined by the experiment. Let Vq(r) be thecomponent of the velocity difference V along q. We will seethat the measured G(t) is closely related to the Fouriertransform of P(Vq). @If light is carried from the sample tothe photodetector by single-mode optical fibers, no integralover A is required, and G(t) and P(Vq) are cosine transformpairs.23#.

By repeating the same steps needed to obtain G(t) forBrownian motion, we calculate that for a source in the formof a thin line of length L and for a scattering angle of 90,

G~t!51

A2 EVq(r)52Vq52 E

0

Lw~r !P~Vq ,r !

3cos~qVqt!uC~r !u2 dVq~r !dr . ~43!In Eq. ~43! the exponential factor in Eq. ~42! has been re-placed by the cosine, because K(t) is real.

For a photodetector with a square face of area h3h ,

G~t!5E0

LEVq

~2/L !~12y /L !P~Vq ,r !cos~qVq~r !t!

3@sinc~k0hry/2R#2 dVq~r !. ~44!

For laminar flows Vq is not a random variable and nointegration over dVq is necessary. For turbulent flows it has

1159W. I. Goldburg

not been possible to derive P(Vq ,r), even when the turbu-lence is isotropic. For this reason the subject of turbulenceremains a very active one,24 to which the photon correlationtechnique can be employed to advantage.

For a simple example of a laminar flow that is easily stud-ied, consider the seeded sample to be cylindrical in shapeand rotating about its axis on, say a record turntable. Let itsangular frequency be v, so that the azimuthal velocity isgiven by vf5rv . To avoid having to take into accountBrownian motion of the seed particles, one might want tochoose a solvent having a large viscosity, such as a mixtureof glycerine and water. This choice will make D small, sothat the decay term e22Dq2t can be ignored. If parameters are

1For example, older models of Malvern, Langley Ford, and Brookhavencorrelators should work fine.

2A good source of such particles is PolyScience, 6600 W. Touhy Ave., P.O.Box 48312, Niles, IL 60714.

3D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics ~Wiley,New York, 1997!, 5th ed., Chap. 37.

4C. Kittel, Elementary Statistical Physics ~Wiley, New York, 1958!.5F. Reif, Fundamentals of Statistical and Thermal Physics ~McGrawHill,New York, 1965!.

6A. Einstein, in Investigations on the Theory of Brownian Motion, edited byR. Furth and A. D. Cowper ~Dover, New York, 1956!, Chap. I.

7B. J. Berne and R. Pecora, Dynamic Light Scattering ~Wiley-Interscience,New York, 1976!.

8W. I. Goldburg, The dynamics of phase separation near the critical point:spinodal decomposition and nucleation, in Scattering Techniques Ap-chosen to assure B!1, so that the factors C(r) can be ig-nored as well, a simple integration of Eq. ~43! establishesthat

g~t!511@sinc~x !#2, ~45!where x5qVqt/25k0Lvt/2. The decaying oscillations ofthis function are easily observed.

VI. CONCLUSION

Dynamic light scattering has long been a powerful tool forstudying a wide range of phenomena, only a few of whichhave been discussed here. In these and many other applica-tions, the scattering is generated by small particles that havebeen introduced into a solvent, which is assumed to scatterlight weakly. But many fluid-like and glassy systems, such asgels, strongly scatter light by themselves, so that no probeparticles need be introduced. The experiments suggestedhere could easily provide a launching point for original in-vestigations.

No mention has been made of the use of optical fibers tocouple the light from the source to the photodetector. Withthis new technical development, dynamic light scattering ex-periments become much more convenient. Because single-mode fibers will conduct light having a single wavenumber,we can satisfy the coherence condition B!1 without havingto put the photodetector and sample far from each other. Auseful introduction to this technique can be found in recentwork by Du et al.23 The measurements shown in Fig. 3 weremade with a relatively inexpensive optical fiber that did notpreserve polarization.

ACKNOWLEDGMENTS

I am especially indebted to P. Tong, K. J. Maloy, and J. R.Cressman for their contribution to my understanding of thissubject and to Rob Cressman and Quentin Bailey for theircareful reading of this paper. This work is supported bygrants from the NSF and NASA.

1160 Am. J. Phys., Vol. 67, No. 12, December 1999plied to Supramolecular and Nonequilibrium Systems, edited by S.-H.Chen, B. Chu, and R. Nossal ~Plenum, New York, 1981!, pp. 383409,and other articles in this volume.

9L. Mandel and E. Wolf, Coherence Properties of Optical Fields, Rev.Mod. Phys. 37, 231287 ~1965!.

10R. Hanbury Brown and R. Q. Twiss, Correlation between photons in twocoherent beams of light, Nature ~London! 177, 2729 ~1956!.

11Edmund Scientific in Barrington, NJ is a good source for interferencefilters and other optical components.

12P. Tong et al., Turbulent transition by photon-correlation spectroscopy,Phys. Rev. A 37, 21252133 ~1988!.

13K. J. Maloy et al., Spatial coherence of homodyne light scattering fromparticles in a convective velocity field, Phys. Rev. A 46, 32883291~1992!.

14H. Kellay et al., Experiments with Turbulent Soap Films, Phys. Rev. A74, 39753978 ~1995!.

15Also called 2,6-dimethylpyridine.16H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena

~Oxford, New York, 1971!.17W. I. Goldburg, Light scattering investigations of the critical region in

fluids, in Light Scattering Near Phase Transitions, edited by H. Z. Cum-mins and A. P. Levanyuk ~North-Holland, New York, 1983!, pp. 531581;J. D. Gunton, M. San Miguel, and P. S. Sahni, Phase Transitions andCritical Phenomena, edited by C. Domb and J. L. Lebowitz ~Academic,London, 1983!, Vol. 8.

18H. C. Van de Hulst, Light Scattering by Small Particles ~Wiley, NewYork, 1957!.

19J. D. Jackson, Classical Electrodynamics ~Wiley, New York, 1992!, 2nded.

20N. Clark, J. H. Lunacek, and G. B. Benedek, A study of Brownianmotion using light scattering, Am. J. Phys. 38, 575585 ~1970!.

21C. L. Mehta, Coherence and statistics of radiation, in Lectures in The-oretical Physics, Vol. 7C, edited by Wesley E. Brettin ~University ofColorado, Colorado, 1965!, pp. 345401.

22B. Chu, Laser Light Scattering ~Academic, New York, 1991!, 2nd ed.,Chap. VII.

23Y. Du, B. J. Ackerson, and P. Tong, Velocity difference measurementwith a fiber-optic coupler, J. Opt. Soc. Am. A 15, 24332439 ~1998!.

24U. Frisch, Turbulence, the Legacy of A. N. Kolmogorov ~Cambridge U. P.,New York, 1995!.

1160W. I. Goldburg