“neutron man” personifies the tice (green), and becomes, · matter. this makes electrons...

“Neutron Man” personifies theneutron’s dual nature, exhibit- .

ties. Here he enters a crystallattice as a plane wave (blue),interacts with the crystal lat- . .,tice (green), and becomes,through interference effects,an outgoing plane wave (red)with a direction dictated by

, . .of flight is measured. .

I

I

I

. .,

$ - -

ow can we determine the rel-ative positions and motionsof atoms in a bulk sample ofsolid or liquid? Somehow we jects separated by about a microme-

need to see inside the material with a t e r ( 1 06 meter), which is more than asuitable magnifying glass, But, seeing thousand times longer than the typicalwith light in an everyday sense will not interatomic distance in a solid (aboutsuffice. First, we can only see inside the 10 -10 meter or so).few materials that are Transparent, and X rays have wavelengths much shortersecond, there is no microscope that willallow us to see individual atoms. Theseare not merely technical hurdles, likethose of sending a man to the moon.but intrinsic limitations. We cannotmake an opaque body transparent norcan we see detail on a scale finer thanthe wavelength of the radiation we areusing to observe it. For observationswith visible light this limits us to ob-

than those of visible light, so we mighttry using them to find atomic positions.For many crystalline materials thistechnique works quite well.x rays are diffracted by thematerial, and one can work outthe relative atomic positions from thepattern of’ spots the diffracted rays makeon a photographic plate. However, notall atoms are equally "visible" to x rays:

Neutron Scattering-A Primer

the light atoms in the soft tissue of ourjowls do not stop x rays as well asthe heavy mercury atoms in the dentalamalgam used to fill teeth. Althoughthis phenomenon is useful to the dentalprofession, it is often an embarrassmentfor scientists measuring atomic posi-tions.

X rays are scattered by the electronssurrounding the nucleus of an atom.As a result, heavy atoms with manyelectrons (such as mercury) scatter xrays more efficiently than light atoms(such as oxygen or, worse, hydrogen).Thus, x rays pass right through lightmaterials without being greatly attenu-ated or deflected. It is for this reasonthat the structure of the much-heraldedhigh-temperature superconductors wasnot determined by x-ray diffraction—in spite of the fact that most universityphysics departments worldwide have anx-ray machine. One of the first high-temperature superconductors discov-ered contained yttrium and copper, bothof which are heavy and scatter a rel-atively large percentage of the x raysincident on a sample. Unfortunately,the superconductors also contained oxy-gen, whose feeble scattering of x raysis swamped by that of its heavy neigh-bors. It was impossible to determine thepositions of the oxygen atoms using x-ray diffraction because the x rays passedthrough the superconductor almost with-out noticing the oxygen.

We might try to find atomic posi-tions by “seeing” with electron beams.After all, quantum mechanics tells usthat particles have wave properties,and the wavelength of electrons caneasily be matched to interatomic dis-tances by changing the electron en-ergy. However, as anyone who hasever rubbed a balloon on the family catknows, the interaction between electricalcharges is strong. Not surprisingly then,a charged particle, such as an electronor a positron, does not travel far throughsolids or liquids before it is attracted or

2

0 20 4 0 6 0 80Atomic Number

NEUTRON, ELECTRON, AND X-RAY PENETRATION DEPTHS

Fig. 1. The plot shows how deeply a beam of electrons, x rays, or thermal neutrons penetrates aparticular element in its solid or liquid form before the beam’s intensity has been reduced by a

having a wavelength of 1.4 angstroms (1.4 x 10-10 meter).

repelled by the electrons already in thematter. This makes electrons unsuitablefor looking inside bulk materials: theysuffer from the same opacity problemas light, and specially prepared, thinsamples are required for electron mi-croscopy.

Neutron Scattering

What about neutrons? They have nocharge, and their electric dipole mo-ment is either zero or too small to bemeasured by the most sensitive of mod-ern techniques. For these reasons, neu-trons can penetrate matter far better thancharged particles. Furthermore, neutronsinteract with atoms via nuclear rather

than electrical forces, and nuclear forcesare very short range-of the order ofa few fermis (1 fermi = 10-15 meter).Thus, as far as the neutron is concerned,solid matter is not very dense becausethe size of a scattering center (nucleus)is typically 100,000 times smaller thanthe distance between such centers. As aconsequence, neutrons can travel largedistances through most materials with-out being scattered or absorbed (see theopening illustration to “Putting Neu-trons in Perspective”). The attenuation,or decrease in intensity, of a beam oflow-energy neutrons by aluminum, forexample, is about 1 percent per millime-ter compared with 99 percent or moreper millimeter for x rays. Figure 1 illus-

Los Alamos Science Summer 1990

Neutron Scattering—A Primer

Neutron

Neutron

X Ray

SCATTERING INTERACTIONS

Fig. 2. Beams of neutrons, x rays, and electrons interact with material by different mechanisms.X rays (blue) and electron beams (yellow) both interact with electrons in the material; with x raysthe interaction is electromagnetic, whereas with an electron beam it is electrostatic. Both ofthese interactions are strong, and neither type of beam penetrates matter very deeply. Neutrons(red) interact with atomic nuclei via the very short-range strong nuclear force and thus penetratematter much more deeply than x rays or electrons. If there are unpaired electrons in the material,neutrons may also interact by a second mechanism: a dipole-dipole interaction between themagnetic moment of the neutron and the magnetic moment of the unpaired electron.

trates this point for other atoms and forelectrons as well as x rays and neutrons.

Like so many things in life, the neu-tron’s penetrating power is a two-edgedsword. On the plus side, the neutroncan penetrate deep within a sample evenif it first has to pass through a container(necessary, for example, if the sampleis a fluid or has to be kept at low tem-peratures or high pressures). The corol-lary is that neutrons are only weakly

scattered once they do penetrate. Also,detection of a subatomic particle in-volves the observation of that particle’sinteraction with some other particle, soneutron detection requires a certain in-genuity (in practice, detectors make useof one of the few atoms, such as boron,helium-3, or lithium, that absorb neu-trons strongly to produce ionizing radia-tion). To make matters worse. availableneutron beams inherently have low in-

tensities. X-ray instruments at synchro-tron-radiation facilities can providefluxes of 1018 photons per second persquare millimeter compared with 104

neutrons per second per square millime-ter in the same energy bandwidth forpowerful neutron-scattering instruments.

The combination of a weak interac-tion and low fluxes makes neutron scat-tering a signal-limited technique, whichis practiced only because it provides in-formation on the structure of materialsthat cannot be obtained by other means.This constraint means that no genericinstrument can be designed to examineall aspects of neutron scattering. In-stead, a veritable zoo of instruments hasarisen with each species specializing ina particular aspect of the scattering phe-nomenon.

In spite of its unique advantages, neu-tron scattering is only one of a batteryof techniques for probing the struc-tures of materials. All of the techniques,such as x-ray scattering and electron mi-croscopy, are needed if scientists are tounderstand the full range of structuralproperties of matter. In most cases, thedifferent methods used to probe materialstructure give complementary informa-tion because the nature of the interactionbetween the radiation and the sampleare different. For example, neutrons in-teract with nuclei, whereas x rays andelectrons “see” only the electrons inmatter (Fig. 2). To a certain extent themethod of choice depends on the lengthscale of the structure to be investigated(Fig. 3). When two techniques addressthe same scale. additional information,such as the size and chemical composi-tion of the sample, is required to choosethe optimal technique.

Scattering by aSingle Fixed Nucleus

The scattering of neutrons by nucleiis a quantum-mechanical process. For-mally, the process has to be described in

3


Crystallography

Atomic Structures

Neutron Diffraction

X-Ray Diffraction

ElectronDiffraction

Microstructure Structure

Viruses

Neutron Small-Angle Scattering

X-Ray Small-Angle Scattering

Bacteria

Grain Structures

Optical Microscopy

Transmission Electron Microscopy



STRUCTURE PROBES

used to probe structure, but one of the

resin determining factors in the choice of a

technique is the length scale of the structure

being examined. Techniques range from

neutron diffraction, which can be used to

study atomic structure with length scales of

10 -11 to 10-9 meter, to optical microscopy,

which can be used to study bacteria and

crystalline grain structures at much greater

length scales.

terms of the wave functions of the neu-tron and the nucleus. The wave functionof the neutron, as its name suggests, hasthe form of a wave—that is, a functionthat oscillates sinusoidally in space andtime. The square of the amplitude ofthis wave at any point gives the proba-bility that the neutron will be found atthat point. It does not matter whetherwe talk about the wave that representsthe neutron or the probability that a par-ticle called the neutron is at a givenlocation. Both descriptions will giverise to the same mathematics and are,therefore, equivalent. Sometimes it isconvenient to refer to the neutron as awave because the picture thus conjuredis easier to understand. At other times itis more useful to think of the neutron asa particle. We can switch from one de-scription to the other at will, and if wedo the mathematics correctly, we willalways get the same answer.

The neutrons used for scattering ex-periments usually have energies simi-lar to those of atoms in a gas such asair. Not surprisingly, the velocities atwhich they move are also comparablewith those of gas molecules—a fewkilometers per second. Quantum me-chanics tells us that the wavelength ofthe neutron wave is inversely propor-tional to the magnitude of the neutron

we will use a bold variable to representa vector quantity and a nonbold ver-

LOS Alamos Science Summer 1990

sion of the same variable to representthe corresponding magnitude). For theneutrons used in scattering experiments,

angstroms (1 angstrom = 10-10 meter).It is often useful to work in terms of theso-called neutron wave vector, k, which

points along the neutron’s trajectory.The vectors k and v are collinear andrelated by the equation

(1)

where h is Planck’s constant, m is themass of the neutron (1.67495 x 10-27

kilogram), and mv is the momentum ofthe neutron.

The scattering of a neutron by a sin-gle nucleus can be described in terms of

(1 barn = 10-28 square meter), thatis equivalent to the effective area pre-sented by the nucleus to the passingneutron. If the neutron hits this area, itis scattered isotropically, that is, withequal probability in any direction. Whyisotropically? The range of the nuclearpotential is tiny compared to the wave-length of the neutron, and so the nu-cleus is effectively a point scatterer.(X rays, on the other hand, do not scat-ter isotropically because the electronclouds around the atom scattering the xrays are comparable in size to the wave-length of the x rays.)

Suppose that at an instant in time werepresent neutrons incident on a fixednucleus by a wave function ei kr, whichis a plane wave of unit amplitude ex-pressed in terms of the position vector

cross section

The effective area presented by a nucleus

to an incident neutron. One unit for cross

section is the barn, as in “can’t hit the side of

a barn!”

r. Note that the square modulus of this vwave function is unity, which meansthe neutron has the same probability of point scattererbeing found anywhere in space but has

An object that scatters incident radiationnodes of the wave—that is, the points isotropically by virtue of being very small

compared with the wavelength of the radia-where n is an integer—are the straight tion.

5


NEUTRON ScatteringFROM A FIXED POINT

Fig. 4. A neutron beam incident on a singlescattering center and traveling in the xdirection can be represented as a plane wavee with unit amplitude. Because the neutronsees the scattering center (a nucleus) as apoint, the scattering will be isotropic. As aresult, the scattered neutron beam spreadsout in spherical wavefronts (here drawn ascircles) of amplitude b/r. The 1/r part ofthis amplitude factor, when squared to getintensity, accounts for the 1/r2 decrease inintensity with distance that occurs as thescattered wavefront grows in size. Becausewe have here taken the scattering center to berigidly fixed, the interaction is elastic, there isno exchange of energy, and the incident andscattered wave vectors both have magnitudek. (To be rigorous, we should have includedthe time dependence eiwt. But since thescattering is elastic, this factor is the same forthe incident and scattered waves and cancelsout of relative expressions, such as the onefor the cross section.)

scattering length

A measure of the strength of the neutron-nucleus interaction, denoted by b and related

isotopic labeling

A technique that takes advantage of theconsiderable variation in scattering crosssection among isotopes. By substituting oneisotope for another (of either the same or adifferent element), the scattering from thoseconstituents containing the substitute maybe varied to reveal their positions relative toother constituents.

k

Y

wavefronts shown in Fig. 4 (for a wave wave function, decreases as the inversetraveling in the x direction). In light square of the distance from the source.of our earlier discussion, we ought to In this case, the source is the scatter-choose the amplitude of the neutron ing nucleus. The constant b, referred towave function (the constant multiplying as the scattering length of the nucleus,the exponential) so that the amplitude measures the strength of the interactionsquared gives a probability of finding between the neutron and the scatteringa neutron at a position r that is con- nucleus. The minus sign in the wavesistent with the number of neutrons inthe beam we are using. However, sincewe shall be interested only in the ratioof the amplitudes of the incident andscattered neutron waves, we can set theamplitude of the incident wave to unityfor the moment.

What is the amplitude of the neutronwave scattered by the nucleus? That de-pends on the strength of the interactionbetween the neutron and the nucleus.Because the scattered neutron waveis isotropic, its wave function can bewritten as (–b/r)e ikr if the scatteringnucleus is at the origin of our coordi-nate system. The spherical wavefrontsof the scattered neutron are representedby the circles spreading out from thenucleus in Fig. 4. The factor (1/r) inthe wave function of the scattered neu-tron takes care of the inverse squarelaw that applies to all wave motions:the intensity of the neutron beam, givenby the square of the amplitude of the

function means that b is a positive num-ber for a repulsive interaction betweenneutron and nucleus.

For the type of collision being imag-ined here, the energy of the neutron istoo small to change the internal energyof the scattering nucleus, and becausewe imagine the nucleus to be fixed, theneutron cannot impart kinetic energy.Thus, the scattering occurs without anychange of the neutron’s energy and issaid to be elastic. Because the neu-tron energy is unchanged by a nuclearcollision, the magnitude of its velocityand thus of its wave vector is also un-changed, and the same k appears in thewave function of the incident and thescattered neutrons.

What is the relationship between scat-tering length, b, and the cross section,

strength of the scattering interaction?

to b, a length, by the simple relation

6 Los Alamos Science Summer 1989


elastic scattering

Scattering with no change in the energy ofthe incident neutron; or, in terms of the wavevector of the neutron, scattering in which thedirection of the vector changes but not itsmagnitude.

were half the radius of the nucleus asseen by the neutron.

For a few nuclei the scattering length,b, varies with the energy of the inci-dent neutrons because compound nu-clei with energies close to those of ex-cited nuclear states are formed duringthe scattering process. This resonancephenomenon gives rise to complex val-ues of b: the real part corresponds toscattering of the neutrons, whereas theimaginary part corresponds to absorp-tion of the neutron by a nucleus. Usu-ally, such resonant effects occur at neu-tron energies greater than those used toprobe the structure of matter. In the ma-jority of cases of interest to scientistsdoing neutron scattering, b is a real andenergy-independent quantity. However,b has to be determined experimentallyfor each nuclear isotope because, unlikethe equivalent quantity for x rays, thescattering length for neutrons cannot becalculated reliably in terms of funda-mental constants.

Also unlike x rays, neutrons interactwith atoms of an element in a mannerthat does not seem correlated with theatomic number of the element (as isevident in Fig. 1). In fact, the neutron’sinteraction with a nucleus of an atomvaries from one isotope to another. For

example, hydrogen and deuterium, bothof which interact weakly with x rays,have neutron scattering lengths that arerelatively large and quite different. Thedifferences in scattering lengths fromone isotope to another can be used invarious isotopic-labeling techniquesto increase the amount of informationavailable from a particular neutron-scattering experiment. We shall discussisotopic labeling in more detail in thesection on small-angle scattering.

Scattering of Neutrons by Matter

To work out how neutrons are scat-tered by matter, we need to add up thescattering from each of the individualnuclei. This is a lengthy and not partic-ularly instructive quantum-mechanicalcalculation. Fortunately, the details ofthe calculation are not very important.The result is, however, both simple andappealing.

When neutrons are scattered by mat-ter, the process can alter both the mo-mentum and the energy of the neutronsand the matter. The scattering is notnecessarily elastic as it is for a single,rigidly fixed nucleus because atoms inmatter are free to move. to some ex-tent. They can therefore recoil during acollision with a neutron, or if they aremoving when the neutron arrives, theycan impart or absorb energy just as abaseball bat does.

AS is usual in a collision, the totalmomentum and energy are conserved:when a neutron is scattered by mat-

is gained by the sample. From Eq. 1 itis easy to see that the amount of mo-mentum given up by the neutron duringits collision, the momentum transfer, is

vector of the incident neutrons and k'is that of the scattered neutrons. Thequantity Q = k – k’ is known as thescattering vector, and the vector rela-tionship between Q, k, and k' can be

inelastic scattering

Scattering in which exchange of energy andmomentum between the incident neutron andthe sample causes both the direction and themagnitude of the neutron’s wave vector tochange. .

Los Alamos Science Summer 1989 7


SCATTERING TRIANGLES

Fig. 5. Scattering triangles are depicted herefor both (a) an elastic scattering event inwhich the neutron is deflected but the neutrondoes not lose or gain energy (so that k’ = k)and (b) inelastic scattering events in whichthe neutron either loses energy (k’ < k) orgains energy (k’ > k) during the interaction.In both elastic and inelastic scattering events,the neutron is scattered through the angle

vector relationship Q = k — k’. For elasticscattering, a little trigonometry shows (lower

neutron-scattering law

The intensity of scattered neutrons as a func-tion of the momentum and energy transferredto the sample during the scattering. The

displayed pictorially in the so-calledscattering triangle (Fig. 5). This trian-

ergy transfer (see Eq. 3 in ‘cThe Mathematical gle also emphasizes that the magnitudeFoundations of Neutron Scattering”). and direction of Q is determined by the

the magnitudes of the wave vectors forthe incident and scattered neutrons andthe angle 2% through which a neutronis deflected during the scattering pro-cess. Generally, 29 is referred to as thescattering angle. For elastic scattering

trigonometry applied to the scattering

In all neutron-scattering experiments,scientists measure the intensity of neu-

8

trons scattered by matter (per incidentneutron) as a function of the variables Q

as the neutron-scattering law for thesample.

In a complete and elegant analysis,Van Hove showed in 1954 that the scat-tering law can be written exactly interms of time-dependent correlationsbetween the positions of pairs of atomsin the sample (see “The MathematicalFoundations of Neutron Scattering” fora more detailed discussion). Van Hove’s



portional to the Fourier transform of afunction that gives the probability offinding two atoms a certain distanceapart. It is the simplicity of this resultthat is responsible for the power of neu-tron scattering. If nature had been un-kind and included correlations betweentriplets or quadruplets of atoms in theexpression for the scattering law, neu-tron scattering could never have beenused to probe directly the structure ofmaterials.

Of course, we have not yet explainedhow one may measure the intensityof scattered neutrons as a function of

Hove’s result provides a way of relatingthe intensity of the scattered neutronsto the relative positions and the relativemotions of atoms in matter. In fact, VanHove’s formalism can be manipulated(see “The Mathematical Foundationsof Neutron Scattering”) to reveal scat-tering effects of two types. The first iscoherent scattering in which the neutronwave interacts with the whole sampleas a unit so that the scattered wavesfrom different nuclei interfere with eachother. This type of scattering dependson the relative distances between theconstituent atoms and thus gives infor-mation about the structure of materials.Elastic coherent scattering tells us aboutthe equilibrium structure, whereas in-

provides information about the collec-tive motions of the atoms, such as thosethat produce vibrational waves in a crys-talline lattice. In the second type ofscattering, incoherent scattering, theneutron wave interacts independentlywith each nucleus in the sample so thatthe scattered waves from different nucleidon’t interfere. Rather the intensitiesfrom each nucleus just add up. Inco-herent scattering may, for example, bedue to the interaction of a neutron wavewith the same atom but at different po-sitions and different times, thus provid-ing information about atomic diffusion.


Diffraction, or Bragg Scattering

The simplest type of coherent neu-tron scattering to understand is diffrac-tion. Suppose that atoms are arranged atfixed positions on a lattice (such as thetwo-dimensional portion of the latticeshown in Fig. 6) and a beam of neu-trons is fired at that lattice. We imaginethat all of the neutrons move on paral-lel paths and have the same velocity, sothat there is only one value for the inci-dent wave vector, k. Because the atomsand their associated nuclei are imaginedto be fixed, there is no change in theneutron’s energy during the scatteringprocess; that is, the scattering is elasticand k' = k.

As the incident neutron wave arrivesat each atom, the atomic site becomesthe center of a scattered spherical wavethat has a definite phase relative to allother scattered waves. In two dimen-sions, it is as if a handful of pebbleshave been thrown into a calm pond.At the point where each pebble strikesthe pond (the atomic site), a circularwave spreads outwards. Because thewaves from each site overlap there willbe places where the disturbances fromdifferent waves reinforce one another

and other places where they cancel out.This is the phenomenon of interference.

As the waves spread out from a reg-ular array of sites, the individual distur-bances will reinforce each other only inparticular directions. In other words, ifwe observe the wave motion at somedistance from the lattice, we will seewaves (scattered neutrons) traveling inwell-defined directions (Fig. 6). Thesedirections are closely related to the sym-metry and spacing (or lattice) of thescattering sites—a hexagonal grid willgenerate a different set of wavefrontsthan a square grid. Consequently, onemay use a knowledge of the directionsin which various incident waves arescattered to deduce both the symmetryof the lattice and the distances between

coherent scattering

Scattering in which an incident neutron wave

interacts with all the nuclei in a sample in

a coordinated fashion; that is, the scattered

waves from all the nuclei have definite relative

phases and can thus interfere with each

other.

incoherent scattering

Scattering in which an incident neutron wave

interacts independently with each nucleus in

the sample; that is, the scattered waves from

different nuclei have random, or indeterminate,relative phases and thus cannot interfere with

each other.

diffraction

A type of scattering in which coherently

scattered waves interfere.

9


DIFFRACTION FROM A LATTICE

interfere with each other. In those directionsin which the interference is constructive,scattered neutrons may be measured. Thefigure depicts such constructive interferencein two dimensions with planar wavefrontsrepresented as lines, spherical wavefrontsas colored circles, and the scattering centersas small circles. To simplify the diagram,the scattering is shown only for four centers(solid black) in each of the two rows ofscattering planes. Also, color is used torelate each incident wavefront to the scatteredwavefronts that have so far been generatedby it. Thus, the incident red wavefront Scatteringhas passed over and scattered from fourscattering centers in Scattering Plane 1;the orange wavefront has passed over andscattered from these scattering centers plusthe leftmost scattering center in ScatteringPlane 2; the yellow wavefront has passed overall eight scattering centers in both planes. For Scattering

constructive interference to take place, Q mustbe perpendicular to the two scattering planes,

distance between the two scattering planesand n is an integer. Combining this conditionwith Q =



where rj and rk represent the positionsof atoms labeled j and k in the latticeand bcoh is the coherent scattering lengthof those atoms.

Equation 2 is the scattered intensitythat would be measured in a neutron-diffraction experiment with a real crys-tal, and is often called the structure fac-tor, S(Q). As we count through theatoms of a lattice performing the sumin Eq. 2, the real and imaginary partsof the exponential function both takevalues that are distributed essentially atrandom between plus and minus one.Because many atoms are involved, thesum usually averages to zero, except atcertain unique values of Q.

Obviously, the values of Q for whichthe structure factor, S(Q), is nonzero arerather special, and it is easy to imaginethat not many values of Q satisfy thiscondition. Further, those values are in-timately related to the structure of thecrystal because the vectors rj – rk. inEq. 2 represent the set of distances be-tween different atoms in the crystal.

We can determine the values of Qat which S(Q) is nonzero and at whichdiffraction occurs by consulting Fig. 6.Suppose Q is perpendicular to a planeof atoms such as Scattering Plane 1 inthis figure. If the value of Q is any in-

distance between parallel, neighboringplanes of atoms (Scattering Planes 1and 2 in Fig. 6), then Q (rj – rk) is

because each exponential term in thesum in Eq. 2 is unity. Thus, Q must beperpendicular to planes of atoms in thelattice and its value must be an integral

do not satisfy this condition, S(Q) = 0,and there is no scattering.

The values of Q at which neutrondiffraction occurs are governed by thesame law that was discovered for xrays in 1912 by William and LawrenceBragg, father and son. To see this, weapply the condition described above


to the scattering triangle for elastic scat-tering. Then using the relationship be-

condition can be rewritten as

(3)

This equation. called Bragg’s law, re-

terplanar spacing in a crystalline sample.Bragg’s law can also be understood in

terms of the path-length difference be-tween waves scattered from neighboringplanes of atoms (Fig. 7). For construc-tive interference to occur between wavesscattered from adjacent planes, the path-length difference must be a multiple

condition to Fig. 7 immediately yieldsBragg’s law in the form given in Eq. 3.Many of the results described in the ar-ticles in this issue will fall back on thispoint of view.

Diffraction, or Bragg scattering, asit is sometimes called, may occur forany set of atomic planes that we canimagine in a crystal, provided the wave-

incident neutron beam and the planesare chosen to satisfy Eq. 3. Bragg scat-tering from a particular set of atomicplanes resembles reflection from a mir-

re la t ive phase

the expression A eik r describing a plane wave

of amplitude A. For a plane wave traveling in

wave vector k and equal amplitude A are in

phase, their phases at any point in space are

the same and the waves add constructively to

yield an intensity of 4 A2. When the relative

waves will interfere with each other so that

their resulting intensity fluctuates in space and

is always less than 4A2. Incoherent scattering

produces random changes in the phase of

the incident wave so that the relative phases

of the scattered waves are indeterminate, the

waves do not interfere with each other, and

the intensity of each wave is added separately

to yield the total intensity.

11

THE PATH-DIFFERENCEAPPROACH TO BRAGG’S LAW

Fig. 7. Constructive interference occurs whenthe waves reflected from adjacent scatteringplanes remain in phase. This happens whenthe difference in distance traveled by wavesreflected from adjacent planes is an integralmultiple of the wavelength. The figure showsthat the extra distance (shown in red) traveledby the wave reflected from Scattering Plane

when n = 1, but higher-order Bragg peaks arealso observed for other values of r?.

Scattering Plane 1

Scattering Plane 2

Reflected Beam

ror parallel to those planes: the anglebetween the incident beam and the planeof atoms equals the angle between thescattered beam and the plane (Fig. 7). Ifa beam of neutrons of a particular wave-length is incident on a single crystal,there will, in general, be no diffraction.To obtain diffraction for a set of planes.the crystal must be rotated to the cor-rect orientation so that Bragg’s law issatisfied-much as a mirror is adjusted

to reflect the sun at someone’s face.The signal thus observed by a neutrondetector at a particular scattering angleis called a Bragg peak because as werotate the crystal to obtain diffractionwe observe a peak in the signal beingrecorded.

According to Eq. 2, the intensity ofthe scattered neutrons is proportionalto the square of the density of atomsin the atomic planes responsible for the



I

I

scattering. We can see this by notingthat as the summation is carried outfor each atom j in one plane, unit ex-ponential factors are added for all theatoms k in another plane. And the moreclosely the atoms are spaced within areflecting plane, the more unit factorswill be summed per unit area. Thus,an observation of Bragg peaks allowsus to deduce both the spacing of planes(from Bragg’s law) and the density ofthe atoms in the planes. To measureBragg peaks corresponding to many dif-ferent atomic planes with neutrons of aparticular wavelength, we have to vary

crystal orientation. First we choose thedetector position so that the scatteringangle satisfies Bragg’s law, then we ro-tate the crystal until a Bragg diffractedbeam falls on the detector.

To this point we have been discussinga simple type of crystal that can be builtfrom unit cells, or building blocks, thateach contain only one atom. In thiscase, each of the exponential factorsthat contribute to S(Q) in Eq. 2 is unity,and the structure is easily deduced fromthe intensities of the Bragg peaks andthe scattering angles at which Bragg dif-fraction occurs. However, the unit cellsof materials of interest to chemists orbiologists almost invariably have morecomplicated shapes and contain manydifferent types of atoms distributedthroughout their volumes. Those atoms,of course, are not positioned randomlyin the unit cell but are arranged in a ge-ometric pattern determined by the waythey bond together. Nevertheless, it maynot be trivial to deduce the atomic po-sitions from an observation of Braggscattering because some of the expo-nential factors that contribute to S(Q)are now complex and the phases ofthese quantities cannot be obtained di-rectly from a measurement of Braggdiffraction. Deducing the structure ofa complex material may take severalmonths and a great deal of ingenuity.


crystals, the sample must be correctlyoriented with respect to the neutronbeam to obtain Bragg scattering. Fur-thermore, if neutrons of a single wave-length are used, the detector must alsobe positioned at the appropriate scatter-ing angle for the atomic planes causingthe scattering. On the other hand, poly-crystalline powders, which consist ofmany randomly oriented single-crystalgrains, will diffract neutrons whateverthe orientation of the sample relative tothe incident beam of neutrons. Therewill always be grains in the powderthat are correctly oriented to diffract.Thus, whenever the scattering angle,

isfy the Bragg equation (Eq. 3) for aset of planes, a reflection will be de-tected, independent of the sample orien-tation. This observation is the basis ofa widely used technique known as pow-der diffracfion, which is implementedin slightly different ways depending onthe nature of the neutron source. Beforedescribing powder diffraction in greaterdetail, we digress to consider the dif-ferent techniques that may be used toproduce neutrons for scattering experi-ments.

Neutron Production

Neutron-scattering facilities through-out the world generate neutrons ei-ther with nuclear reactors or with high-energy particle accelerators. The neu-trons produced have energies up to tensor even hundreds of mega-electron volts(MeV), and the corresponding neu-tron wavelengths are far too short forinvestigating condensed matter. Fur-thermore, neutrons whose energies areabove a few electron volts tend to dam-age solids by knocking atoms out oftheir official positions, producing vacan-cies and interstitial. For this reason,neutrons must be “cooled down” beforebeing used for scattering experiments.

unit cell

The repeating unit of a crystal.

13


I

!

cold neutrons

Neutrons whose energies have been reducedbelow about 5 meV by inelastic scatteringin a cold material such as liquid hydrogenor deuterium. Researchers use such longer-wavelength neutrons to conduct experimentsat larger length scales.

Such cooling is done by bringing theneutrons into thermal equilibrium with a“moderating” material—a material witha large scattering cross section, suchas water or liquid hydrogen. The mod-erator, whose volume may vary froma deciliter to several tens of liters, isplaced close to the neutron source. Neu-trons enter the moderator and, in a se-ries of collisions in the material, loseenergy to recoiling moderator atoms.After a few tens of collisions, the ener-gies of the neutrons are similar to thoseof the atoms of the moderator. Thus,thermal neutrons are emitted from themoderator surface with a spectrum ofenergies around an average value de-termined by the moderator temperature.The average energy of neutrons froma water moderator at ambient tempera-ture is about 25 thousandths of an elec-tron volt (25 meV), and the averageenergy from a liquid-hydrogen modera-tor at 20 kelvins is around 5 meV. Thewavelength of a 25-meV neutron is 1.8angstroms (1.8 x 10-10 meter), which isof the same order as typical interatomicdistances and, therefore, is quite suitablefor diffraction experiments.

Reactor Sources. Neutrons are pro-duced in a nuclear reactor by the fis-sioning of atoms in the reactor fuel,which, for research reactors, is invari-ably uranium. The neutrons are moder-ated in the manner described above andallowed to emerge from the reactor in acontinuous stream with an energy spec-trum similar to the curves of Fig. 8a.

For most scattering experiments at re-actors, the neutrons emerging from themoderator must be reduced to a mon-ochromatic beam: that is, only thoseneutrons in a single, narrow energy bandare selected from the spectrum. Thisselection is usually accomplished byBragg reflection from a large singlecrystal of a highly reflective material,such as pyrolytic graphite, germanium,or copper. A crystal monochromator

(a) Reactor Neutrons

(b) Spallation Neutrons

Energy (meV)

REACTOR ANDSPALLATION NEUTRONS

Fig. 8. (a) The relative flux of neutrons as afunction of energy for the high-flux reactorat the Institut Laue-Langevin in Grenoble,France. The curves show the distributionof neutrons emerging from moderators attemperatures of 20, 300, and 2000 kelvins.(b) Similar distribution curves for neutronsgenerated at the Manuel Lujan, Jr. NeutronScattering Center at Los Alamos (LANSCE)by moderators al temperatures of 20 and 290kelvins.

works because, even though the inci-dent beam contains neutrons of manywavelengths, the spacing of the reflect-ing planes of atoms, d, and the scatter-

those neutrons with a wavelength sat-isfying the Bragg equation are trans-mitted in the direction of the exper-iment. The wavelength of the neu-trons used for experiments can thenbe controlled by changing the scat-tering angle at the monochrornator.



spallation neutrons

Neutrons generated at an accelerator bydriving a highly energetic beam of particles,typically protons, into a target of heavy atoms,such as tungsten. The incident protons knockneutrons loose from the nuclei of the target,creating a pulse of highly energetic spallationneutrons.

Spallation Sources. Other neutron fa-cilities, such as the one at the ManuelLujan, Jr. Neutron Scattering Center atLos Alamos (LANSCE), use acceler-ators to produce spallation neutrons.This is done by allowing high-energyprotons (or, less effectively, electrons)to collide with a heavy-metal target,such as tungsten or uranium, drivingneutrons from the nuclei of the tar-get. The protons are produced by theaccelerators—in this case, LAMPF (theLos Alamos Meson Physics Facility)coupled with a proton storage ring—inbursts that last for less than a microsec-ond. At LANSCE there are 20 suchbursts of 800-MeV protons per second.Each proton in the burst then generatesabout 20 neutrons.

One of the advantages of a spallationsource is that only a small amount ofenergy—about 27 MeV per neutron—is deposited in the spallation target bythe protons. Nuclear fission producesabout four or five times as much energyin generating each of its neutrons. How-ever, the cost of producing the high-energy protons—the electricity bill ofthe accelerators—is not cheap.

The moderated neutrons that finallyemerge into the experimental area froma spallation source have a spectrum re-sembling the curves of Fig. 8b. Clearly,this spectrum is quite different from


that produced by a reactor (Fig. 8a) be-cause there is a greater percentage ofhigh-energy neutrons. However, thespectrum is not the only difference be-tween the two types of neutron sources.Neutrons from a spallation source ar-rive in pulses rather than continuouslyas they do at a reactor. This fact meansthat the monochromator crystal neededat reactors can here be avoided and allthe neutrons can be used (rather thanonly those in a narrow energy band).

The trick that allows the use of allneutrons from a spallation source relieson the measurement of the time it takesfor each detected neutron to traverse the

neutron velocity can be determined, andEq. 1 gives its wavelength. Generat-ing a monochromatic beam is thereforeunnecessary.

A thermal neutron with an energy of25 meV travels at a speed of about 2.2kilometers per second, or about Mach 7.A typical neutron spectrometer is about10 meters long, so the neutron travelsfrom the moderator to the detector in time of flight

about 5 milliseconds. Because the du-ration of the neutron pulse emerging The time it takes a neutron to travel from a

from the moderator of a pulsed source pulsed source to a detector, which is thus a

is typically a few tens of microseconds, measure of the neutron’s velocity and kinetic

the time of flight of the neutron can be energy.

determined with high relative precision.

15


Powder Diffraction

Now let’s return to powder diffrac-tion. In a powder-diffraction instru-ment at a reactor source (Fig. 9), amonochromatic beam of neutrons im-pinges on a powdered sample, and theneutrons scattered from the sample are

Each Bragg peak in a typical scatteringpattern (Fig. 10) corresponds to diffrac-tion from atomic planes with a differ-ent interplanar spacing, or value of d.Many peaks can be recorded simultane-ously by placing detectors at a varietyof scattering angles (such as the sixty-four helium-3 detectors in Fig. 9).

In a powder diffraction instrument ata spallation source (Fig. 11), the sam-ple is irradiated with a pulsed beam ofneutrons having a wide spectrum of en-ergies. Scattered neutrons are recordedin banks of detectors located at differ-ent scattering angles, and the time atwhich each scattered neutron arrivesat the detector is also recorded. At aparticular scattering angle, the resultis a diffraction pattern very similar tothat measured at a reactor, but now theindependent variable is the neutron’stime of flight rather than the scatteringangle. Because the neutron’s time offlight is proportional to its wavelengthand, for constant scattering angle, wave-length is proportional to the spacingbetween atomic planes (Eq. 3), the mea-sured neutron scattering can be plotted

spacing (Fig. 12). (The resemblance be-tween Figs. 10 and 12 is obvious. Thepatterns are equivalent ways of prob-ing Bragg’s law, and in fact, diffractiondata obtained at reactors and spallationsources can be plotted on the same scale

independent variable. )As in the reactor case, detectors at a

spallation source can be placed at dif-ferent scattering angles, allowing manypatterns to be measured simultaneously,

POWDER DIFFRACTION AT A REACTOR SOURCE

Fig. 9. An essential component of a powder diffractometer at a high-flux reactor

ModeratedNeutronBeam

is a very largecrystal whose reflecting surface may be as large as 200 square centimeters. The crystal acts asa monochromator by scattering neutrons of a given energy toward the sample. To help focusthe beam of neutrons, the crystal may also be curved, effectively acting as a concave mirror. Asecond scattering occurs at the powder sample, which scatters the monoenergetic focused beamtoward a set of detectors (here, 64 helium-3 neutron detectors). These detectors are here shownpositioned along an arc on one side of the sample, but the whole array can be moved to otherpositions along the circular support track. The distance between the monochromator and thesample is typically about 2 meters.

120 130 140 150

A POWDER DIFFRACTION PATTERN RECORDED AT A REACTOR

Fig. 10. A typical powder diffraction pattern obtained at a reactor source gives intensity, ornumbers of neutrons, as a function of the scattering angle 219. Each peak represents neutronsthat have been scattered from a particular set of atomic planes in the crystalline Iattice.


Neutron Scattering—A PrimerSample

POWDER DIFFRACTION AT A SPALLATION SOURCE

Fig. 11. The Neutron Powder Diffractometer (NPD) at LANSCE (see photograph on page 54). Theincident beam of neutrons, having been moderated with water chilled to 10” C, is directed ontothe target in a large evacuated chamber. Surrounding this chamber are eight banks of detectorspositioned at fixed scattering angles. Each bank consists of sixteen helium-3 detectors, and thed-spacing that can be measured ranges from about 1.2 to 33.6 angstroms at the 20° detectorbank to about 0.25 to 5.2 angstroms at the 148° detector bank. The distance between the sampleand the detectors at the 90° scattering angle is about 2 meters, so the whole spectrometer isvery much larger than the equivalent instrument at a reactor.

I I I I

0 . 5 0 , 6 0 . 7 0 . 8

A POWDER DIFFRACTION PATTERN RECORDED AT A SPALLATION SOURCE

Fig. 12. A typical powder diffraction pattern obtained at a spallation source (“fat garnet” measuredat one of the 148° bank of detectors in the diffractometer of Fig. 11). As in Fig. 10, the verticalcoordinate is the intensity, or number of neutrons, but the horizontal coordinate is the d-spacing

neutron time of flight (via Eqs. 1 and 3).


Detectors at small scattering angles pro-vide information about widely spacedatomic planes, whereas those at largerangles record data relevant to smallspacings. There is usually some overlapof information provided by the differentdetectors.

Using patterns like those of Figs. 10and 12, the atomic structure of a poly -crystalline sample may be deduced fromEq. 2. In practice, however, one guessesthe atomic positions, evaluates Eq. 2,and from a comparison of the calcu-lated and measured diffraction patterns,refines the atomic coordinates. Thistype of procedure is described in de-tail in the article “X-Ray and NeutronCrystallography—A Powerful Combina-tion” by Robert Von Dreele.

Probing Larger Structures

Another way of thinking about coher-ent elastic neutron scattering is shown inFig. 13. One can imagine the incidentand scattered neutron waves setting upa “probe wave” in the sample—much astwo misaligned picket fences generate aset of moire fringes. One can alter the

bychanging the angle between the ingoingand outgoing waves (that is, the scatter-ing angle) or by increasing or decreas-ing the wavelength of the neutrons used.To obtain information about structures

must be chosen to be approximately thesame as the size of the structure. Forcrystallography this means that &~&~needs to be of the same order as inter-atomic spacings. We already know thisfrom Bragg’s law. A little trigonom-etry applied to Fig. 13 will show that

adjacent scattering planes, Bragg’s lawis satisfied.

The probe-wave idea shows us howwe can measure structures that are largerthan typical interatomic distances. We

17


THE PROBE-WAVE VIEWOF NEUTRON SCATTERING

Fig. 13. Another way to view neutron scat-tering is to imagine that the incident neutronwave (In) and the scattered neutron wave (Out)form a secondary “probe wave” (here seen asa moire pattern in both examples) that mustmatch the average periodicity of the structurein the scattering sample. Because the averageperiodicity of the top sample is larger thanthat of the lower one, the wavelength of the

must be smaller (here 30). Another way to

In



either by decreasing the scattering an-gle or by increasing the neutron wave-length. In practice, to examine someof the larger structures displayed inFig. 3—polymers, colloids, or viruses,for example—we need to use neutronwavelengths greater than 5 angstromsand scattering angles less than 1 de-gree. Because of the latter constraint,this technique is known as small-angleneutron scattering, or SANS.

The Van Hove formulation for neu-tron scattering may be manipulated (see“The Mathematical Foundations of Neu-tron Scattering”) to provide the follow-ing equation for the intensity of neu-trons scattered at small angles (that is,for small values of Q):

where the integral extends over theentire scattering sample and b(r), thescattering-length density, is calculatedby summing the coherent scatteringlengths of all the atoms over a smallvolume and dividing by that volume.

In many cases, samples measured bySANS consist of particles with a uni-form scattering-length density bp thatare dispersed in a uniform matrix witha scattering-length density bm. Exam-ples include pores in rock, colloidal dis-persions, biological macromolecules in

water, and many more. The integralin Eq. 4 can, in this case, be separatedinto a uniform integral over the wholesample and a term that depends on thedifference, bp – bm, between the scat-tering length’ of the particles and that ofthe matrix. This difference is called thecontrast factor. If all the particles areidentical and their positions are uncorre-lated, Eq. 4 becomes

where the integral is now over the vol-ume Vp of one of the particles and NP

is the number of such particles in thesample.

The integral above of the phase factore iQ r over a particle is called the form

factor for that particle. For many sim-ple particle shapes, the form factor canbe evaluated without difficulty: the ex-pression for spherical objects was firstderived by Lord Rayleigh in 1911.

Equation 5 allows us to understand animportant technique used in small-anglescattering known as contrast match-ing. The total scattering is proportionalto the square of the scattering contrastbetween a particle and the matrix inwhich it is embedded. If we embed theparticle in a medium whose scatteringlength is equal to that of the particle,

small-angle neutron scattering

A technique for studying structural details withdimensions between 10 and 1000 angstromsby measuring the intensity of neutronsscattered through small angles, usually lessthan 1 degree.

contrast matching

An isotopic-labeling technique based on thedramatic difference between the scatteringlengths of hydrogen and deuterium, whichis particularly useful in neutron-scatteringstudies of complex biological molecules inaqueous solution. The technique involvesmatching the scattering from the solvent withthat from one component of the biologicalmolecules by replacing the hydrogen atomsin the solvent or the component or both withdeuterium. The observed scattering is thendue to only the unmatched components.



A SMALL-ANGLENEUTRON-SCATTERINGSPECTROMETER

Fig. 14. (a) The spectrometer illustrated here,the Low-Q Diffractometer (LQD) at LANSCE,measures neutron scattering at small angles.The neutrons are first moderated in liquidhydrogen to increase the percentage of verycool, long-wavelength neutrons in the beamthat hits the sample. The moderated beamthen passes through a collimating system thatis more than 7 meters long before impingingon the sample. To increase the accuracywith which the small scattering angles canbe measured, the large position-sensitivedetector is placed far from the sample (about4 meters). (b) Neutrons from a spallationsource have a range of speeds and are thusunder the influence of gravity for differentamounts of time, an effect that smears thesignal at the detector. However, the beam canbe “focused” by placing a fixed aperture at thebeginning of the collimator and a moveableaperture at the end of the collimator andaccelerating the latter aperture upward duringthe pulse of neutrons. Such an arrangementselects only those neutrons with parabolictrajectories that end at the center, or focus, ofthe detector. Small-angle scattering is suitablefor studying structures with dimensions in therange of 10 to 1000 angstroms.

Neutron

the latter will be invisible. (This tech-nique is used by the manufacturers ofgel toothpaste—there really is gritty ma-terial in there to clean your teeth. butyou can’t see it because the grit and thegel have similar refractive indices!)

Suppose the particles we are inter-ested in are spherical eggs rather thanuniform spheres: they have a core (theyolk) with one scattering length and acovering (the white) of a different scat-tering length. If such particles are im-mersed in a medium whose scatteringlength is equal to that of the egg white,then a neutron-scattering experimentwill only “see” the yolk. The form fac-tor will be evaluated by integrating overthis central region only. On the otherhand, if our particles are suspended in amedium whose scattering length is thesame as that of the yolk, only the eggwhite will be visible; the form factorwill correspond to that of a thick, hol-low shell. The scattering pattern will bedifferent in the two cases, and from twoexperiments, we will discover the struc-tures of both the covering and the coreof the particle.

Variation of the scattering-length den-sity of the matrix is often achieved bychoosing a matrix that contains hy-drogen (such as water). By replac-

ing different fractions of the hydrogenatoms with deuterium atoms, a largerange of scattering-length densitiescan be achieved for the matrix. Thiscontrast-matching technique works, aswe pointed out earlier, because of thesignificantly different scattering-lengthdensities of hydrogen and deuterium,and it is one of the main reasons for thesuccessful application of neutron scatter-ing to problems in biology. Both DNAand protein can be contrast matched bywater containing different fractions ofdeuterium. Several problems in struc-tural biology that have been studied bycontrast matching are described in “Bi-ology on the Scale of Neglected Dimen-sions” by Jill Trewhella.

Small-angle scattering is perhaps theeasiest neutron-scattering technique torealize in practice. Like diffraction ex-periments, SANS experiments at a re-actor source require a monochromator,whereas at a spallation source measure-ment of times of flight determine thewavelengths of the incident and scat-tered neutrons.

The Low-Q Diffractometer at theLANSCE facility (Fig. 14a) is an exam-ple of a SANS spectrometer at a spal-Iation source. One essential componentof the instrument is a large position-



sensitive neutron detector located behindthe sample directly in line with the inci-dent beam. Another important compo-nent (invented by Phil Seeger at LAN-SCE) is the gravity focuser (Fig. 14b),which accounts for the fact that neutronsfall under the influence of gravity. Ifthe aperture at the exit of the collimatorthat defines the trajectory of the incidentneutron beam was fixed, neutrons of dif-ferent velocities could only pass throughthat slit if they were following parabolicpaths that fell on the detector at differ-ent heights, blurring the image producedthere. To avoid this blurring, the exitaperture of the collimator is moved up-ward during each neutron pulse. Slowerneutrons then have to go through anopening that is higher relative to thecenter of the detector. The position ofthe aperture at each instant is chosen sothat all neutrons, independent of theirspeed, arrive at the center of the de-tector, if they are not scattered by thesample. The whole thing is a little likea stone-throwing contest: weak throwershave to throw stones on a higher trajec-tory to hit the target.

Inelastic Scattering

In reality, atoms are not frozen infixed positions in a crystal. Thermal en-ergy causes them to oscillate about theirlattice sites and to move around insidea small volume with the lattice site atits center. Since an atom can fully con-tribute to the constructive interference ofBragg scattering only when it is locatedexactly at its official position in the lat-tice, this scattering becomes weaker themore the atoms vibrate and the less timethey spend at their official positions.

When a crystal structure is deter-mined from single-crystal or powder dif-fraction, the extent of the thermal mo-tion of the atoms is found at the sametime as the atomic positions. Often, thethermal motions are anisotropic, indicat-ing that it is easier for an atom to move


in particular directions away from itsequilibrium position. Sometimes thisinformation can be related to other prop-erties, such as structural changes thatoccur at a phase transition or elasticanisotropy.

Although such weakening of thescattering signal is the only effect ofthe thermal motion of atoms on elas-tic Bragg scattering, it is not the onlyway to use neutrons to observe atomicmotion. In fact, one of the great ad-vantages of neutrons as a probe of con-densed matter is that they can be usedto measure the details of atomic andmolecular motions by measuring inelas-tic scattering. In other words, when theneutron bounces off a molecular frame-work that is not totally rigid, we canhave an inelastic interaction with an ex-change of energy between neutrons andthe lattice.

To explain this, we begin with an-other simple analogy. If one end ofa rope is tied to a fixed point and theother end is jerked up and down, awave can be observed traveling alongthe rope. A discontinuous version ofthis effect can be obtained with a chorusline (for this analogy I am indebted toa colleague who once choreographed itfor a midwestern television station). Ifeach member of the line swings a legbut starts the swing slightly after hisor her nearest neighbor to one side, thenet effect is the appearance of a wavetraveling along the line. The thermalmotion of atoms in a crystal can be de-scribed in terms of a superposition ofwaves of this sort. One may imaginethe atoms to be the feet of the membersof the chorus line.

The analogy, if not the image, canbe improved by replacing the swinginglegs with rigid pendulums with weightsat their extremities. Rather than watch-ing for a neighbor to swing a leg, weachieve coupling by attaching identicalsprings between each pendulum and itstwo nearest neighbors. Now, if we dis-

place one pendulum, the springs tendto cause the neighboring pendulums tomove as well, and a wave starts pass-ing up and down the line, just as it didfor the chorus. The frequency of motiondepends on the mass of the pendulumsand the stiffness of the springs that con-nect them.

Waves similar to those in the chainof pendulums pass through a lattice ofatoms connected by the binding forcesthat are responsible for the cohesionof matter. The whole effect is muchmore difficult to visualize in this case,however, because it happens in three di-mensions. Nevertheless, it is possibleto prove that any atomic motion in acrystal can be described by a superpo-sition of waves of different frequenciesand wavelengths traveling in differentdirections. In other words, the thermalmotion of the atoms about their latticesites can be described as a superpositionof waves moving through the lattice,and these waves are known as phonons.Their energies are quantized so thateach phonon has an energy hv, wherev is the frequency of atomic motion as-sociated with that phonon. Just as in thependulum analogy, the frequency of aphonon depends on the wavelength ofthe distortion, the masses of the atoms,and the stiffness of the “springs,” orbinding forces, that connect them.

When a neutron is scattered by acrystalline solid, it can absorb or emitan amount of energy equal to a quantumof phonon energy, hv. This gives rise toinelastic coherent scattering of neutronsin which the neutron energy before andafter the scattering event differ by an

In most solids v is a few times 1012

hertz, and the corresponding phononenergy is a few meV (1012 hertz cor-responds to an energy of 4.18 meV).Because the thermal neutrons used forscattering experiments also have ener-gies in the meV range, scattering by aphonon causes an appreciable fractional

21


phonons

Fundamental vibrational waves in a crystal inwhich nuclei oscillate in a coordinated mannerabout their “official” positions.

change in the neutron energy. This al-lows an accurate measurement of theenergy change and makes neutrons anideal tool for measuring phonon fre-quencies and hence for obtaining infor-mation about the forces that hold mattertogether.

For inelastic scattering—from pho-nons, for example—a neutron has differ-ent velocities, and thus different wavevectors, before and after it interactswith the sample; so the correspondingsides of the scattering triangle (k andk’ in Fig. 5b) are of unequal lengths.To determine the phonon energy andthe scattering vector, Q, we need to de-termine the neutron wave vector be-fore and after the scattering event. Ata reactor we may resort to the methodalready discussed—Bragg scatteringfrom single crystals. A first crystal, themonochromator, directs neutrons of agiven energy at the sample (as was donefor the powder diffractometer shownin Fig. 9). After the sample scattersthese neutrons in various directions, asecond crystal—positioned at a well-defined scattering angle and called theanalyzer -Bragg reflects only thoseneutrons that have a particular energyinto a suitably placed detector. Thistype of instrument is called a three-axisspectrometer (Fig. 15) because there arethree centers (monochromator, sample,and analyzer) at which the scattering an-gles can be altered. Such instrumentsare the workhorses for the measurementof phonons at reactors.

Three-axis spectrometers have con-tributed prolifically to the various sci-entific problems studied by neutronscattering, probably because they areso inefficient. At each setting of the

22

spectrometer-corresponding to partic-ular scattering angles at the monochro-mator, sample, and analyzer—a mea-surement is made for a single scatter-

Each measurement usually takes sev-eral minutes; a complete scan at a series

days. This inefficiency has advantages,though—it allows the experimenter toconcentrate on measuring particular ex-citations at particular values of Q and

each new measurement in light of thedata already accumulated.

The success of three-axis spectrome-ters leads to an interesting philosophicaldilemma. Does materials science byits very nature require for its study aninstrument such as a three-axis spec-trometer? That is, is there some reasonto believe that a majority of interest-ing and important effects occur, likeBragg scattering, only in a restricted

understanding of materials actually beenhampered because three-axis spectrom-eters have been so popular and prolific?Have we seen only a part of the truthbecause three-axis spectrometers canonly probe a single scattering vector andenergy transfer at one time? Would welearn more if we could make measure-ments for a wide range of values of Q

the extensive use of alternative types ofspectrometers can answer this question.Many of the instruments that are bestsuited to surveys of neutron scatteringfor large ranges of scattering vector andenergy transfer are located at spallationsources such as the one at LANSCE.

There is no real equivalent of the



THREE-AXISNEUTRON SPECTROMETER

Fig. 15. A three-axis spectrometer built bythe author at the Institut Laue Langevin inGrenoble, France. The scattering angles atthe monochromator, sample, and analyzercan be varied by moving these connectedunits on the air pads seen in the photograph.This spectrometer is equipped for polarizationanalysis. The hollow box-like object on thesample table has current-carrying wires alongeach edge that can produce a field of about100 oersteds at any direction on a sampleplaced at the center of the box. Various spinflippers, diaphragms, and filters are mountedon the optical benches before and after thesample position.

three-axis spectrometer that can be builtat a spallation source. Inelastic scat-tering can, however, be measured in avariety of ways. Perhaps the simplestis to place an analyzing crystal in thescattered neutron beam just as one doeswith the three-axis machine. This crys-tal determines the final energy of theneutrons scattered by the sample. Oncethis energy and the total time of flightfrom the moderator to the detector areknown, the incident energy can also bededuced.

Another method of measuring in-elastic scattering at a pulsed spallationsource has been used to obtain someof the data discussed by Juergen Eck-ert and Phil Vergamini (see “Neutronsand Catalysis”). This method uses a fil-ter rather than an analyzing crystal inthe scattered neutron beam. The filterallows only neutrons whose energy isless than a certain cutoff value to passthrough to a detector behind the filter.Filters of this type can be made, forexample, from a block of cooled poly-crystalline beryllium that is several cen-timeters thick, When neutrons impingeon the block, they are scattered just as


they would be from any polycrystallinematerial. But there is a maximum valueof the neutron wavelength beyond whichBragg scattering cannot occur becausethere are no atomic planes spaced farenough apart to diffract these long-wavelength neutrons. Neutrons withwavelengths greater than the cut-offtherefore pass through, the filter withoutbeing scattered out of’ the beam. In thecase of beryllium, neutrons with wave-lengths greater than about 4 angstroms(energies less than about 5 meV) aretransmitted. In the Filter-DifferenceSpectrometer at LANSCE, two filtersare used, beryllium and beryllium ox-ide. The latter material transmits neu-trons with energies below 3.7 meV. Bysubtracting data obtained with the BeOfilter from that obtained with the Be fil-ter, we obtain a result that includes onlythose neutrons with final energies in thenarrow window between 3.7 meV and5 meV, the two filtering energies. Thistechnique allows the energy of the scat-tered neutrons to be determined accu-rately. As usual, the total time of flightlets us deduce the incident energy of theneutrons.

The Filter Difference Spectrometeris not well suited for measurements ofphonons because the geometry of theinstrument makes it inherently difficultto determine the scattering vector, Q, toa high degree of accuracy. This is anadvantage when one is measuring in-coherent inelastic scattering, however,

independent of Q, and one may sumscattered intensities for many values ofQ, thereby increasing the statistical ac-curacy of the data obtained. This sum-mation is accomplished automaticallywith the Filter Difference Spectrometerat LANSCE.

The final method of measuring inelas-tic scattering at a spallation source—amethod that does determine the scat-tering vector accurately—makes use ofa so-called chopper spectrometer. Thechopper, which can be thought of as ashort (20-centimeter) pipe rotating aboutan axis perpendicular to its length, isplaced in the neutron beam ahead ofthe scattering sample. If the pipe is ro-tating at a frequency that is an integralmultiple of that of the pulsed neutronsource, it briefly becomes aligned with

23


the neutron beam at the same time dur-ing each neutron pulse from the mod-erator. Because the chopper is usuallyseveral meters from the neutron mod-erator, the fast neutrons in each pulsearrive at the chopper ahead of theirslower brethren. Only those neutronsthat arrive at the chopper when it isopen—that is, aligned with the beam—get through. Thus, the chopper selectsneutrons in a small band of velocitiesand allows them to impinge on the sam-ple. Neutrons outside this band willarrive either too late or too early at thechopper and will be stopped. The chop-per thus determines the wave vector ofthe neutrons incident on the sample,whereas a measurement of the total timeof flight allows the wave vector of thescattered neutrons to be calculated aswell.

A great advantage of chopper spec-trometers is that neutron detectors canbe placed at many different scatteringangles simultaneously, allowing scatter-ing to be recorded at many values of Q

perimenter is inundated with data andmust rely heavily on computers to re-duce the massive array of numbers tosomething comprehensible.

In fact, massive amounts of data arethe norm for spectrometers at spallationsources. At each detector, a series of Q

spond to a full range of differing flighttimes of the detected neutrons. One au-tomatically obtains these values whetherone wants the flood of data or not. Inshort, a three-axis spectrometer at a re-actor source is a rifle, whereas its equiv-alent at a spallation source is a shot-gun. Which source is more efficient fora given experiment really depends onwhat type of information one wants—a single bull’s-eye or a barn door fullof interesting holes! More seriously,we can obtain a detailed knowledge ofthe scattering law for a few values of

24

extended picture covering a wide rangeof these variables at a spallation source.

Magnetic Scattering

So far we have discussed only theinteraction between neutrons and atomicnuclei. But there is another interactionbetween neutrons and matter—one thatresults from the fact that a neutron has amagnetic moment (Fig. 2). Just as twobar magnets either attract or repel oneanother, the neutron experiences a forceof magnetic origin whenever it passesclose to another magnetic particle, suchas an electron in matter.

Most electrons in atoms or in matterare paired so that the magnetic momentof one electron cancels that of its part-ner. Occasionally, however, not all theouter, or binding, electrons are paired ina particular compound, and neutrons arescattered by the resulting magnetic mo-ments. Diffraction experiments, similarto those described earlier, can be usedto measure the density of such unpairedelectrons between the atoms of a solid.

Ferromagnetic materials, such as iron,are magnetic because the moments oftheir unpaired electrons tend to alignspontaneously. For many purposes, suchmaterials behave as if a small magneticmoment were located at each atomicsite with all the moments pointed in thesame direction. These moments giverise to Bragg diffraction of neutrons inthe same manner as the nuclear inter-action. Because nuclear and magneticinteractions experienced by the neu-tron are of similar magnitude, the cor-responding Bragg reflections are also ofcomparable intensity.

One difference between the two typesof scattering, however, is that magneticscattering, unlike its nuclear counter-part, is not isotropic. The magnetic in-teraction has a dipolar nature, whichcan easily be observed by bringing twobar magnets close to one another. Sup-pose the two magnets are parallel with

their north poles pointing upward. Ifone magnet is above the other, unlikepoles will be close, and the magnetswill attract; if they are side by side,like poles will be close, and the mag-nets will repel. For neutrons, the dipolarnature of magnetic interaction meansthat only the component of the sample’smagnetization that is perpendicular tothe scattering vector, Q, is effective inscattering neutrons. Neutron scatteringis therefore sensitive to the direction ofmagnetization in a material as well as toits spatial distribution.

The anisotropic nature of the mag-netic interaction can be used to separatenuclear and magnetic Bragg peaks inferromagnets, for which both types ofBragg peaks occur at the same valuesof Q. If the electronic moments can bealigned by an applied magnetic field,magnetic Bragg peaks for which Q isparallel to the induced magnetizationvanish, leaving only the nuclear com-ponent. On the other hand, an equiva-lent Bragg peak for which the scatteringvector is perpendicular to the field willmanifest both nuclear and magnetic con-tributions.

In an antiferromagnet (a material withunpaired electrons that have an alter-nating, or antiparallel, arrangement),the repeat distance between planes ofmagnetic moments is twice that of thespacing between corresponding planesof atoms. As a result, Bragg’s law issatisfied at scattering angles whose sinesare half those for normal Bragg scat-tering, as well as at the normal angles.Half the magnetic Bragg peaks fall be-tween their nuclear counterparts, andthe problem of separating magnetic andnuclear contributions does not arise.Nevertheless, the dipolar character ofthe magnetic interaction again allowsthe electronic spin directions to be es-tablished. A recent example of thisis to be found in the superconductingcuprates—the so-called high-temperaturesuperconductors—some of which are an-



(a)

Neutron

(b)

CoolingFan

A FLAT-COIL NEUTRON-SPIN FLIPPER

Fig. 16. (a) Schematic diagram of one type of neutron flipper. A direct current in the horizontalcoil of aluminum wires (blue) produces a field HI inside the device that is equal but oppositeto the neutron guide field Hguide, effectively canceling that component. The vertical coil (red)produces a field Hz that is at right angles to the guide field and thus to the moment of theneutron, causing it to precess. The strength of this field and the thickneas d of the flipper arechosen so that the neutron precesses exactly 160 degrees during its passage. (b) Photographof disassembled neutron flipper. The penetrating power of neutrons is apparent in the fact thatthere is no “window” in the two coils of wire; the neutrons pass on through the aluminum wireunimpeded. The component on the right produces a vertical guide fieid of about 40 oersteds.

tiferromagnetic when oxygen deficient.

Polarized Neutrons. Usually, a neu-tron beam contains neutrons with mag-netic moments pointing in all directions.If we could measure the number of neu-trons with moments parallel and antipar-allel to a particular direction—say anapplied magnetic field-we would findequal populations. However, various

special techniques can generate a po-larized beam, that is, one with a largefraction of its neutron moments in thesame direction. The polarization of sucha beam can be maintained by applyinga modest magnetic field (a few tens ofoersteds) all along the beam. Such afield is called the guide field.

There are several ways to polarizeneutron beams: Bragg diffraction from

suitable magnetized crystals, reflec-tion from magnetized mirrors made ofcobalt and iron (CoFe), and transmis-sion through polarized helium-3, for ex-ample. Each of these methods alignsthe neutron moments parallel or an-tiparallel to an applied magnetic field.If the neutron moments are parallelto the field, they are said to be’ up; ifthe moments are antiparallel, they aresaid to be down. An ‘up’ polarizer willnot transmit ‘down’ neutrons, just as a‘down’ polarizer blocks ‘up’ neutrons.Thus, by placing an cup’ polarizer be-fore and after a scattering sample, theneutron scattering law can be measuredfor those scattering processes in whichthe direction of the neutron momentis not changed. To measure the othercombinations—such as ‘up’ neutrons be-ing flipped to ‘down’ neutrons—requireseither a variety of different ‘up’ and‘down’ polarizers or a device calleda flipper. Because polarizers tend tobe expensive, flippers are the practicalchoice.

A flipper is a device that can changethe direction of a neutron moment fromup to down or vice versa. This can bedone in one of two ways. Either theguide-field direction can be invertedwithout changing the direction of theneutron moment in space, or the neutronmoment can be inverted without alter-ing the direction of the guide field. Ineither case, the direction of the neutronmoment with respect to the field (whichis all that counts) has been changed.

An example of the second type offlipper is shown in Fig. 16. It consistsof two flat coils of wire wrapped one ontop of the other. One of the coils pro-duces a field inside the flipper that isequal and opposite to the guide field, ef-fectively canceling that component, andthe other coil produces a field perpen-dicular to the guide field. Thus, whena neutron enters the flipper, it suddenlyexperiences a magnetic field that is atright angles to the direction of its mag-



netic moment. In this situation the clas-sical equations that describe the motionof the neutron moment are similar tothose of a rotating top that has beenpushed by a force from the side andso begins processing about its originalaxis of rotation. The neutron does thesame thing—its moment starts to pre-cess about the local field direction at arate known as the Larmor frequency,which depends on the magnitude of thefield inside the flipper. By choosing thethickness of the flipper and the strengthof the field in the second coil appropri-ately, one can arrange for the neutronmoment to rotate precisely 180 degreesduring its passage through the flipper.Clearly, if a neutron’s moment was upbefore the flipper, it will be down afterthe flipper, and vice versa.

Now suppose we have a spectrome-ter with polarizers before and after thescattering sample. If flippers are in-serted on either side of the sample, wecan measure all of the neutron scatter-ing laws—up to down. up to up, andso forth-simply by turning the appro-priate flipper on or off. This technique,known as polarization analysis, is usefulbecause some scattering processes flipthe neutron’s moment whereas others donot.

Scattering from a sample that is mag-netized provides a good example. Mag-netic scattering will flip the neutron’smoment if the magnetization responsi-ble for the scattering is perpendicular tothe guide field used to maintain the neu-tron polarization. If the magnetizationis parallel to the guide field, no flippingoccurs. Thus, like the dipolar interactiondescribed earlier, polarization analysis isa technique that helps determine the di-rection of electronic moments in matter.

Incoherent scattering that arises fromthe random distribution of nuclear spinstates in materials provides another ex-ample of the use of polarization anal-ysis. Most isotopes have several spinstates, and the scattering cross section

26

for a nucleus varies with spin state. Therandom distribution of nuclear spins inthe sample gives rise to incoherent scat-tering of neutrons. It turns out that two-thirds of the neutrons scattered by thisincoherent process have their momentsflipped, whereas the moments of the re-maining third are unaffected. This resultis independent of the isotope that is re-sponsible for the scattering and of thedirection of the guide field. Althoughincoherent scattering can also arise if asample contains a mixture of isotopes ofa particular element, neither this secondtype of incoherent scattering nor coher-ent nuclear scattering flip the neutron’smoment. Polarization analysis thus be-comes an essential tool for sorting outthese different types of scattering, al-lowing nuclear coherent scattering to bedistinguished from magnetic scatteringand spin-incoherent scattering.

Polarization analysis has been partic-ularly useful in the study of magneticphenomena because it has helped to de-termine the directions of the magneticfluctuations responsible for scattering.Without this technique, many of the el-egant experiments that have providedconfirmation for ideas about nonlinearphysics (see “Nonlinear Science—FromParadigms to Practicalities” by David K.Campbell, Los Alamos Science No. 15,1987) could not have been performed.The three-axis spectrometer of Fig. 15,for example, is equipped for polarizationanalysis.

Magnons. Another important aspectof magnetized materials is the fact thatthe directions of the atomic momentsin a material such as iron can oscillatelike the pendulums considered earlierfor lattice vibrations. Here again, thereis a coupling between magnetization atdifferent atomic sites, and a wave ofmagnetic oscillations can pass throughthe material. These magnetic excita-tions, or magnons, are the magneticanalogue of the phonon displacement

waves described earlier. Not surpris-ingly, magnon frequencies can be mea-sured by inelastic neutron scattering inthe same way as phonon frequencies.Since the magnetic oscillations thatmake up the magnons are perpendicu-lar to the equilibrium direction of theatomic moments, the scattering causesthe magnetic moment of the neutrons tobe flipped, provided the neutron guidefield is parallel to the equilibrium di-rection of the atomic moments. This,of course, allows one to distinguish be-tween phonons and magnons.

Surface Structure

So far we have described only exper-iments in which the structure of bulkmatter is probed. One may ask whetherneutrons can provide any informationabout the structure of the surfaces ofmaterials. At first sight, one mightexpect the answer to be a resounding"No!” After all, one of the advantagesof neutrons is that they can penetratedeeply into matter without being af-fected by the surface. Furthermore,because neutrons interact only weaklywith matter, large samples are generallyrequired. Because there are far feweratoms on the surface of a sample thanin its interior, it seems unreasonable toexpect neutron scattering to be sensitiveto surface structure.

In spite of these objections, it turnsout that neutrons are sensitive to sur-face structure when they impinge onthe surface at sufficiently low angles.In fact, for smooth surfaces, perfect re-flection of neutrons occurs for almostall materials at angles of incidence (theangle between the incident beam andthe surface) less than a critical angle,

to the coherent scattering-length densityof the material and the neutron wave-length. For a good reflector, such asnickel, the critical angle measured in de-grees is about one-tenth of the neutron

Los Alamos .Science Summer 1990


0.05 0.1 0.15 0.2 0.25

wavelength measured in angstroms—it is well under a degree for thermalneutrons. As the angle of incidence in-creases above the critical angle, lessand less of the incident neutrons are re-flected by the surface. In fact, reflectiv-ity, which measures the fraction of neu-trons reflected from the surface, obeysthe same law, discovered by Fresnel,that applies to the reflection of light: re-flectivity decreases as the fourth powerof the angle of incidence at sufficientlylarge grazing angles.

However, Fresnel’s law applies to re-flection of radiation from the smooth,flat surface of a homogeneous material.If the material is inhomogeneous andthere is a variation of the scattering-length density perpendicular to the sur-face, the neutron reflectivity, measuredas a function of the angle of incidence,shows a more complicated behavior. Bykeeping the reflection angle, 8, small,neutron reflectometry can be used toprobe density variations in the surfaceto depths of a few thousand angstromswith a resolution of a few angstroms.

Most of today’s technical gadgets


are either painted or coated in somefashion to prevent corrosion or wear.Reflectometry can often provide usefulinformation about such protective lay-ers. Figure 17, for example, shows thereflectivity, measured on the LANSCESurface Profile Analysis Reflectome-ter (SPEAR), from a 1500-angstromlayer of diblock copolymer (polystyrene-polymethylmethacrylate) multilayerdeposited on a silicon substrate. Thespacing of the undulations in this resultprovides a direct measure of the aver-age thickness of the polymer layers inthe film. When the detailed shape ofthe reflectivity profile is compared withtheoretical predictions, the density andthickness of the polymer layers, as wellas the thickness of the interface betweenlayers, can be deduced.

Neutron reflectometry is a relativelynew technique. It is also one ideallysuited to spallation sources. In the nextfew years I expect the method to pro-vide new information on subjects as di-verse as the recycling of polymers, mag-netic recording media, and the cleanupof oil spills. For someone like me who

SURFACE REFLECTIVITYMEASUREMENTS

Fig. 17. Neutron reflectivity as a function

thick diblock copolymer (polystyrene-poly-methylmethacrylate) multilayer deposited ona silicon substrate. The solid line representscalculated reflectivity for the data shown. Thecalculation was performed by Tom Russell,IBM Almaden Research Labs.

has been associated with neutron scat-tering for more than twenty years, thebirth of this new technique is a happyevent. It means that there are still qual-itatively new ways in which neutronscan help unravel the complex struc-tures of the materials on which we de-pend. ■

27


The Mathematical Foundationsof Neutron Scattering

I trons scattered by any assembly of nuclei. His result makes use of Fermi ob-servation that the actual interaction between a neutron and a nucleus may be

replaced by an effective potential that is much weaker than the actual interaction.This pseudo-potential causes the same scattering as the actual interaction but is weakenough to be used in the perturbation expansion derived by Max Born. The Born ap-proximation says that the probability of an incident plane wave of wave vector k be-ing scattered by a weak potential V (r) to become an outgoing plane wave with wavevector k’ is proportional to

(1)

where the integration is over the volume of the scattering sample. (We should notethat even though individual nuclei scatter spherically, V (r) represents the potentialdue to the entire sample, and the resulting disturbance for the assembly of’ atoms is aplane wave.)

The potential to be used in Eq. 1 is Fermi’s pseudo-potential. which, for a single

vector r coincides with rj. Thus, for an assembly of nuclei, such as a crystal, thepotential V (r) is the sum of individual neutron-nuclei interactions:

(2)

where the summation is over all the nuclear sites in the crystal.Using Eqs. 1 and 2, Van Hove was able to show that the scattering law—that is,

the number of neutrons scattered per incident neutron-can be written as

(3)

Note that the sum here is over pairs of nuclei j and k and that the nucleus labeled jis at position r;(f) at time t, whereas the nucleus labeled k is at position rk(0) at timet = 0. The angular brackets (. .) denote an average over all possible starting timesfor observations of the system, which is equivalent to an average over all the possiblethermodynamic states of the sample.

The position vectors rj in Eq. 3 are quantum-mechanical operators that haveto be manipulated carefully. Nevertheless, it is instructive to ignore this subtletyand treat the equation as if it described a system obeying classical mechanics be-cause such an approach clarifies the physical meaning of the equation. The sum overatomic sites in Eq. 3 can then be rewritten as



in which the Dirac delta function appears again, this time in terms of r and a differ-ence vector between the position of nucleus j at time t and that of nucleus k at timezero.

Let us suppose for the moment that the scattering lengths of all the atoms in oursample are the same (b j = bk = b). In this case, the scattering lengths in Eq. 4 can beremoved from the summation, and the right side becomes

(5a)

and N is the number of atoms in the sample. The delta function in the definition ofG(r, t) is zero except when the position of an atom k at time zero and the positionof atom j at time t are separated by the vector r. Because the delta functions aresummed over all possible pairs of atoms to obtain G (r, t), this function is equal tothe probability of an atom being at the origin of a coordinate system at time zeroand an atom being at position r at time t. G (r, t) is generally referred to as the time-dependent pair-correlation function because it describes how the correlation betweentwo particles evolves with time.

Van Hove’s neutron-scattering law (Eq. 3) can now be written as

(6)

forms of the time-dependent pair-correlation function. This general result gave a uni-fied description for all neutron-scattering experiments and thus provided the frame-work for defining neutron scattering as a field.

Fourier transform of a function that gives the probability of finding two atoms a cer-tain distance apart-is responsible for the power of neutron scattering. By invertingEq. 6, information about both structure and dynamics of condensed matter may beobtained from the scattering law.

Coherent and Incoherent Scattering

Even for a sample made up of a single isotope, all of the scattering lengths thatappear in Eq. 3 will not be equal. This is because the scattering length of a nucleusdepends on its spin state, and most isotopes have several spin states. Generally, how-ever, there is no correlation between the spin of a nucleus and its position in a sam-ple of matter. For this reason, the scattering lengths that appear in Eq. 3 can be av-eraged over the nuclear spin states without affecting the thermodynamic average (de-noted by the angular brackets).

value of b2 (b2). In terms of these quantities, the sum in Eq. 3 canthe nuclear spins to give

and the averagebe averaged over

where Ajk is shorthand for the integral in Eq. 3. The first term on the right side ofEq. 7 represents the so-called coherent scattering, whereas the second represents


(7)

29

Neutron Scattering—A Primerthe incoherent scattering. Thus, we can define the coherent and. incoherent scatteringlengths as

(8)

The expression for the coherent scattering law is a sum over both j and k andthus involves correlations between the position of an atom j at time zero and theposition of a second atom k at time t. Although j and k are occasionally the sameatom, in general they are not the same because the number N of nuclei in the sampleis large. We can thus say that coherent scattering essentially describes interferencebetween waves produced by the scattering of a single neutron from all the nuclei ina sample. The intensity for this type of scattering varies strongly with the scatteringangle.

Incoherent scattering, on the other hand, involves correlations between the posi-tion of an atom j at time zero and the position of the same atom at time t. Thus, inincoherent scattering, the scattered waves from different nuclei do not interfere witheach other. For this reason, incoherent scattering provides a good method of exam-ining processes in which atoms diffuse. In most situations, the incoherent scatteringintensity is isotropic; that is, it is the same for any scattering angle. This effect of-ten allows incoherent scattering to be ignored when observing coherent scatteringbecause the incoherent effects just add intensity to a structureless background.

The values of the coherent and incoherent scattering lengths for different ele-ments and isotopes do not vary in any obviously systematic way throughout the peri-odic table. For example, hydrogen has a large incoherent scattering length (25.18 fer-mis) and a small coherent scattering length (–3.74 fermis). Deuterium, on the otherhand, has a small incoherent scattering length (3.99 fermis) and a relatively large co-herent scattering length (6.67 fermis). As mentioned in the main article, the differ-ence between the coherent scattering lengths of hydrogen and deuterium is the basisof an isotopic-labeling technique, called contrast matching, that is especially impor-tant in applications of neutron scattering to structural biology and polymer science.

DiffractionOne of the important applications of Van Hove’s equation (Eq. 3) is the scatter-

ing law for diffraction, which we develop here for a crystal containing a single iso-

neutron diffractometers actually integrate over the energies of scattered neutrons.

to be evaluated at t = O for diffraction. The result, for a crystal containing a singleisotope, is

(9)j,k

where the atomic positions rj and rk are evaluated at the same instant.If the atoms in a sample were truly stationary, the thermodynamic averaging

brackets could be removed from Eq. 9 because rj and rk would be constant. In re-ality the atoms oscillate about their equilibrium positions and only spend a fractionof their time at these positions, When this is taken into account, the thermodynamicaverage introduces another factor, called the Debye-Wailer factor, and Eq. 9 then be-comes



,1’

librium position and diffracted intensity is now also called S (Q), the structure factor.This equation is the basis of any crystallographic analysis of neutron-diffraction data.

Small-Angle Scattering.

An important simplification of Eq. 3 occurs when the scattering angle is small.This approximation leads to the formula for one of the most popular neutron-scatter-ing techniques—SANS, or small-angle neutron scattering.

Although Eq. 3 correctly describes neutron scattering at any scattering angle,when the magnitude of Q is very small compared to a typical interatomic distance,the exponential factors in Eq. 3 do not vary much from atom to atom, and the sumover the atomic sites may be replaced by an integral. As a result, the small-anglescattering law for coherent, elastic scattering from an assembly of “objects” (such asthose depicted in Fig. 13 in the main text) can be written

where b (r) is the scattering-length density and the integral extends over the entiresample. To calculate b (r) for a large molecule, for example, we simply add up thecoherent scattering lengths of the atoms in the molecule and divide by the molecularvolume. Equation 11 is essentially a coarse-grained version of the “truth” given byEq. 3 and is valid only when Q is small. However, it is the basic analytic tool ofsmall-angle scattering. ■

Roger Pynn was born and educated in England.He received his M.A. from the Unlversity ofCambridge in 1966 and his Ph.D. in neutronscatterlng, also from the University of Cam-bridge, in 1969. He was a Royal Society Eu-ropean Fellow to Sweden in 1970; he did twoyears of postdoctoral research in Norway; andthen he was an associate physicist for two yearsat Brookhaven National Laboratory. After spend-ing eleven years at the world's leading center forneutron scattering, the Institut Laue Langevin inGrenoble, France, he was appointed as the Direc-tor of the Manuel Lujan, Jr. Neutron ScatteringCenter at Los Alamos.

Acknowledgments

I would like to thank the staff of Los AlamosScience for encouraging me to write this arti-cle in a coherent fashion and for being patientthroughout its many iterations.

Further Reading

G. L. Squires. 1978. Introduction to ThermalNeutron Scattering. Cambridgc: Cambridge Uni-versity Press.

C. G. Windsor. 1981. Pulsed Neutron Scatter-ing. London: Taylor and Francis.

Physics Today, January 1985. A special issue onneutron Scattering.


“neutron man” personifies the tice (green), and becomes, · matter. this makes electrons...

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