SMALL-ANGLE NEUTRON SCATTERING AND BIOMOLECULES

SMALL-ANGLE NEUTRON SCATTERINGAND BIOMOLECULES

by J. Katsaras, T.A. Harroun, J. Pencer, T. Abraham, N. Kučerka and M.-P. Nieh

LA PHYSIQUE ET L’ÉDUCATION ( SMALL-ANGLE NEUTRON ... )

John Katsaras <[email protected]>a,b,c,Thad A. Harrounc, Jeremy Pencera, Thomas Abrahama,Norbert Kučerkaa, and Mu-Ping Nieha; aCanadian NeutronBeam Centre, National Research Council Canada, ChalkRiver Laboratories, Chalk River, ON, K0J 1J0;bBiophysics Interdepartmental Group, Guelph-WaterlooInstitute for Physics, Guelph, ON, N1G 2W1; cDepartmentof Physics, Brock University, St. Catharines, ON, L2S 3A1

Soft materials, both polymeric and biologically relevant, arerich in hydrogen. By coincidence, neutrons have the uniquecapability of scattering differently from hydrogen (coherentscattering length of hydrogen, bH = -0.37 × 10-12 cm) com-pared to its isotope deuterium (bD = 0.67 × 10-12 cm). As aresult of this marked difference in scattering power (contrast)between native hydrogenated materi-als and their counterparts synthesizedfrom deuterated monomer units, neu-tron scattering techniques have provento be powerful tools for the study ofsoft condensed matter systems. Here,we will discuss the small-angle neu-tron scattering (SANS) technique,which is presently playing a pivotalrole in extracting unique structuralinformation from intrinsically disor-dered systems.

NEUTRONS

Neutrons are electrically neutral, sub-atomic, elementary particles, found inall atomic nuclei, except hydrogen(1H). They are approximately 1,840times more massive than an electronand have a nuclear spin of 1/2. Neutrons are only stablewhen bound by an atomic nucleus, while unstable free neu-trons have a mean lifetime of approximately 900 s, decayinginto a proton, an electron, and an antineutrino [1,2].

Because neutrons interact with atomic nuclei the scattering“power” (cross-section) of an atom is not strongly related toits atomic number. Neighbouring elements in the periodictable can therefore, have substantially different scatteringcross sections [3]. More importantly, the interaction of a neu-tron with the nucleus of an atom allows neutrons to interactdifferentially with an element’s isotopes. The classic exampleis the isotopic substitution of 1H for deuterium (2H) in poly-meric materials [4,5]. As a result of their intrinsic properties,neutrons are used as follows:

(i) Since they interact weakly with atomic nuclei, neutronsare highly penetrating. This feature allows neutrons toprobe samples in complex sample environments, withoutthe need to engineer neutron “windows” or ports into thesample enclosure. This enables the measurement of bulkprocesses under realistic conditions [6-8].

(ii) Because the scattering ability of an atom is not stronglyrelated to its atomic number, neutrons are used exten-sively to locate “light”, low atomic number atoms among

“heavy” atoms. In the case of polymeric materials, neu-trons are used to precisely locate hydrogen atoms [9,10].

(iii) 1H has a negative scattering length giving it “contrast”when surrounded by other, positive scattering lengthatoms. For biological samples intrinsically rich in hydro-

gen, judicious substitution of 2H for 1Hprovides a powerful method for select-ively tuning the contrast of a givenmacromolecule. By doing so, one canaccentuate, or nullify, the scatteringfrom particular parts of a macromolecu-lar complex. This powerful technique iscommonly referred-to as “contrast vari-ation” [11-13].

(iv) Neutron energies are similar to theenergies of atomic and electronicprocesses, i.e. meV to eV range.This allows for the study of thevarious dynamic properties (i.e.,translations, rotations, vibrationsand lattice modes) exhibited bymolecules and eV transitions with-in the electronic structure of mate-rials [14-16].

(v) Because they possess a magnetic moment (spin 1/2 par-ticles), neutrons are ideally suited to the study of magnet-ic structures (short- and long-range) and short wave-length magnetic fluctuations. It is important to note thatthe cross-sections for magnetic scattering are of the samemagnitude to those for nuclear scattering [17-18].

SMALL ANGLE NEUTRON SCATTERING (SANS)

Small angle neutron scattering (SANS) probes structurein materials of length scales ranging from tens of angstroms(10-9 m) to hundreds of nanometers (10-7 m) [19]. The lengthscale, d, is determined by the neutron wavelength, λ, and thescattering angle, θ, through the relationship

λ = 2d sin θ/2,

As a result of a marked dif-ference in scattering power(contrast) between nativehydrogenated materials andtheir counterparts synthe-sized from deuteratedmonomer units, neutronscattering techniques haveproven to be powerful toolsfor the study of soft con-densed matter systems.

LA PHYSIQUE AU CANADA septembre / octobre 2006 233

Francine Ford

Note

This is an official electronic offprint of the article entitled "Small-Angle Neutron Scattering and Biomolecules" by J. Katsaras et al., published in Physics in Canada, Vol. 62 No. 5 (Sept/Oct 2006), pp. 233-240. Copyright 2006, CAP/ACP All rights reserved/ Tous drots de reproduction réservés. F.M. Ford Managing Editor, PiC

PHYSICS AND EDUCATION ( SMALL-ANGLE NEUTRON ... )

234 PHYSICS IN CANADA September / October 2006

trons are chosen by a mechanical velocity selector, basically, ahigh-speed rotor. The helically twisted rotor blades are coat-ed with 10B, a neutron absorbing material, and reasonablymonochromatized neutrons (bandpass, Δλ/λ of ~ 10%) areobtained by varying the rotor speed (revolutions/minute,rpm). Those neutrons whose velocities are not synchronizedto the rotor speed are absorbed by the 10B coated blades.Monochromatic neutrons are then transported over metersand are collimated through a series of nickel-coated guides,which take advantage of the wave-like properties of neutrons.

The propagation characteristics of neutrons involves therefractive index (n) of the medium. Since the critical angle (θc)depends on the refractive indices of the media that the neu-trons traverse, when nmedium < nair neutrons are transportedalong the length of the guide by a mechanism known astotal external reflection (θincident < θc). For neutronsn = 1 – (λ2p/2π) and the scattering length density, ρ is equalto Σbi/V, where bi is the coherent scattering length and V isthe sample volume. Neutron guides are made of optically flatglass whose interior is generally coated with nickel or its iso-tope 58Ni (larger θc and increased Δλ/λ). Since, for neutrons,the index of refraction of 58Ni is slightly less than one, thenall neutrons with an angle < θc (i.e., < 0.5° for λ = 5 Å neu-trons) are transported.

Recently developed supermirrors made up, for example,of Ni/Ti multilayers can increase the effective θc by up toa factor of 3, compared to pure Ni [23]. They do so not onlyby utilizing the total external reflection component, but alsothe superimposed constructive interference (Bragg reflection)from the successive layers of Ni, effectively extendingthe plateau of total external reflection. The desired energyneutrons impinge on the sample, which when scattered, areusually detected by a 3He-filled two-dimensional (2D) detec-tor.

SANS INFORMATION AT A GLANCE: FRACTALDIMENSIONALITY

For objects with a radius of gyration, RG , and Q << 1/RGwhere Q = 4π/λ sin θ/2, plotting ln[I(Q)] vs Q2 results in astraight line of slope –RG

2/3, commonly referred to as aGuinier plot. However, when Q >> 1/RG I(Q) decays as Q-α,where α is the fractal dimension of the scattering object. Inthis case, fractal refers to a complex structure made up of geo-metrical objects (self-similarity). The magnitude of α permitsfor the geometry (i.e. morphology) of the scattering object tobe determined. In the case where the Q-range of the scatter-ing data is sufficiently large (over one decade in Q) [24], onecan estimate α by simply determining the slope of the linefrom a log-log plot of I(Q) vs Q. Table I shows the fractaldimensions corresponding to various morphologies adoptedby biomolecules and polymeric systems.

SANS can also be used to characterize the stability of biologi-cal membranes interacting with additive molecules. Of spe-cial interest are pharmacologically important molecules that,in appropriate concentrations help to either stabilize the lipidbilayer or cause it to undergo structural change (e.g., lamellarto hexagonal transition). For example, non-ionic surfactantmolecules such as, N-dodecyl-N,N-dimethylamine (DDAO)

commonly referred-to as Bragg’s Law. Through the use ofcold (i.e. long wavelength) neutrons and the appropriatebeam collimation, length scales approaching tens of microm-eters are possible [20,21].

In general, SANS can provide information regarding a parti-cle’s size and shape, distribution of scattering inhomo-geneities, conformational changes and molecular associa-tions in solution. More importantly, because of the proper-ties of neutrons individual components within a macromole-cule can be systematically manipulated either through iso-topic labelling or the judicious use of solvents. Below we willdiscuss SANS instrumentation and provide a few examplesof SANS data obtained from lipid/water andsurfactant/water systems.

SANS INSTRUMENTATION

Figure 1 shows a schematic of a typical SANS instrumentlocated at a neutron source capable of producing long wave-length (λ ~ 5 – 20 Å) or commonly referred-to, “cold neu-trons” [22]. Velocities (i.e., wavelengths) of the incoming neu-

Fig. 1 Schematic of a typical SANS instrument utilizinglong wavelength or commonly referred-to cold neu-trons. Velocities (i.e., wavelengths) of incoming neu-trons are chosen by a mechanical velocity selector (a).For a given cylinder length, L, and a spiral pitch, p, ifthe cylinder spins on its axis at an angular velocity ωωonly neutrons of velocity, V (= pωω/2ππ) are transmit-ted. Reasonably monoenergetic cold neutrons aretransported over meters and collimated through aseries of nickel-coated or supermirror (e.g., Ni/Timultilayers) guides (b). The evacuated guides, whichtransport cold neutrons via total external reflection,are made of optically flat glass and their interiors arecoated with either nickel, or its isotope 58Ni (largercritical angle, θθc, increased ΔΔλλ/λλ), or multilayers ofNi/Ti, which offer an even greater θθc. Neutrons theninteract with the sample (c), which scatters neutronsusually detected by a 3He-filled 2D detector (neutron+ 3He 66 3H + 1H + 0.76 MeV) (d). The flight path inwhich the 2D detector is housed is evacuated, result-ing in a reduced background.



destabilize dioleoyl phosphatidylcholine(DOPC) bilayers forming mixed micelles whoseshape changes, as a function of increasingDDAO concentration, result in rod-like particles(e.g., tubular or cylindrical micelles) and hardsphere objects (e.g., globular micelles) [25].

MORPHOLOGIES OF “BICELLE” MIX-TURE LIPIDS DETERMINED BY SANS

Amphipathic phospholipids are one of the maincomponents of biological membranes. They arecomposed of hydrophobic fatty acid chains andhydrophilic headgroups (Fig. 2), and along withcholesterol are the primary constituents of cellmembranes. In purified forms, lipid/water sys-tems form a variety of interesting structures(e.g., lamellar, cubic and hexagonal phases,micelles, etc.) (Fig. 3) which for a number of rea-sons have been the focus of both experimen-tal [26-32] and theoretical interest [33-38]. Many of these struc-tures exhibit features on length scales ranging from nanome-ters to microns.

In the recent past there has been a great deal of scientificactivity in a system forming bilayered micelles, or commonlyreferred-to ‘‘bicelles’’ [39-40]. As we shall show below, neu-tron scattering has proven extremely useful in characterizingthese systems. Although bicelles were commonly formed inaqueous solutions of ionic surfactants and alcohols [41-43], forbiologists a more pertinent system is where the detergent molecules have been substituted by a short chain phospho-

lipid, such as dihexanoylp h o s p h a t i d y l c h o l i n e(DHPC) [44,45]. In this sys-tem, a typical saturatedacyl chain lipid, such asdimyristoyl phosphatidyl-choline (DMPC, di-14:0hydrocarbon chains),forms a disk-shaped bilay-er whose edges are stabi-lized by a curved monolay-er of detergent [46]. Sincetheir discovery, bilayeredmicelles have been used ina number of studiesattempting to elucidate thestructure of proteins underphysiologically relevantc o n d i t i o n s [ 3 9 , 4 5 , 4 7 , 4 8 ] .

However, as we will show, the bilayered micelle morpholo-gy is just one of many structures that these lipid mixtures arecapable of adopting.

COMPLETE UNBINDING OF LAMELLAE:FORMATION OF UNILAMELLAR VESICLES

Figure 4 shows SANS profiles of varying wt% DMPC/DHPClipid mixtures doped with the negatively charged lipid,dimyristoyl phosphatidylglycerol (DMPG) [49]. A 25 wt%sample was diluted in single steps, at 45oC, to final wt% con-centrations of 18.0, 12.5, 9.0, 5.0, 2.5, 1.25, 0.5, and 0.1. Theprofiles for lipid concentrations, clp $ 2.5 wt% exhibit quasi-Bragg maxima, characteristic of equidistant lamellae (i.e.,multibilayers). As a function of increasing amounts of water,the lamellar repeat spacing (d-spacing) varied linearly withchanges in clp

-1, from 104 to 1348 Å with the lamellar reflec-

Fig. 2 Chemical composition and space-fillingmodel of 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine (18:0-18:1 PC). This lipid iscomposed of a hydrophilic phosphoryl-choline headgroup, a glycerol backbone, andtwo hydrophobic hydrocarbon chains.

Fig. 3 Example morphologies adopted by lipids: (a) Prolate micelle;(b) Inverse prolate micelle; (c) Hexagonal; (d) Inverted hexago-nal (e) Micelle; (f) Inverted micelle; (g) Unilamellar vesicle; (h)Bilayered micelle; (e) Bilayer; (j) Cubic.

TABLE 1FRACTAL EXPONENTS FOR VARIOUS

MORPHOLOGIES



range, representative of isolated bilayers. The datawere fit using a model of noninteracting polydis-perse ULV.

So what happens if we take some of these mor-phologies, cool them down to 10oC and reheat backto 45oC? Figure 5 includes SANS data of 1.25 and2.5 wt% samples. At 10oC the data do not showany sharp peaks and are well described by thebicelle morphology. The data can be best fit to abilayered disk morphology using a combination ofthe core-shell-discoidal (CSD) model and theHayter-Penfold structure factor, SHP(Q), resultingin a disk core radius, R, of 590 and 220 Å for the1.25 and 2.5 wt% samples, respectively. Not sur-prisingly, both samples have the same bilayerthickness (42 Å). On reheating to 45oC, the lamel-lar morphology is recovered in the case of the2.5 wt% sample. However, of greater interest isthat on reheating the 1.25 wt% sample to 45oC, thescattering pattern shows an oscillatory behavior asa function of Q, the fingerprint of monodisperseULV, instead of the monotonic decay seen initiallyat 45oC. The data were fit to a ULV model with aSHP(Q) structure factor and a Schulz size distribu-tionyielding an average core radius<Ri>of ~300 Å,a bilayer thickness of 33 Å, and a polydispersity of0.14. Whereas the ULV were initially large andhighly polydisperse (Fig. 5), after temperaturecycling they became smaller and more monodis-perse. The formation of polydisperse ULV fromlamellae is not surprising, since the unbinding ofthe bilayers does not select any particular lengthscale. However, the situation is very differentwhen ULV are formed from bicelles, whereby thebilayered micelle morphology dictates the size ofULV formed. Figure 6 pictorially summarizes thevarious morphologies observed by Nieh et al. [49].

SANS AND CONTRAST VARIATION

For polymeric materials rich in hydrogen, the useof contrast variation and SANS makes for a power-ful combination. By judiciously exchanging themolecule’s hydrogen atoms for deuteriums, or bychanging the solvent’s scattering length density(ρ), one can enhance the “visibility” of a molecule’smoieties. For example, the optimum contrast con-ditions for studying the overall bilayer structureare a fully hydrogenated lipid in 100% D2O sol-vent. On the other hand a solvent composed of50:50 D2O:H2O provides the best contrast for lipidswith perdeuterated chains while the same lipid ina pure D2O provides information mainly about thelipid’s headgroup. The data obtained from theseexperiments can then be analyzed using eithermodel dependent or model independent methods.

A model independent method based on the Guinier approxi-mation (i.e., low Q region) provides a reasonably straightfor-ward procedure for extracting the bilayer’s structural param-eters [13]. By analyzing the SANS data obtained at several dif-ferent contrast conditions, the average bilayer scattering

tions moving systematically to lower values of Q. However,at clp # 1.25 wt% the lamellar reflections disappear, theresult of a complete unbinding transition whereby, theextended lamellar stacks have disintegrated forming variableradii unilamellar vesicles (ULV). The scattered intensity forclp # 1.25 wt% follows a Q-2 dependence over an extended Q

Fig. 5 SANS profiles of 2.5 and 1.25 wt% samples prepared at 45°C (topcurve) on cooling to 10°C (middle), and on reheating to 45°C (bot-tom). The arrow represents the sequence of temperatures. Thelamellar phase is recovered in the 2.5 wt % sample, whereas forthe 1.25 wt % sample, initially polydisperse ULV becomemonodisperse on reheating (profile exhibits an oscillatory behav-iour as a function of Q). The solid lines are fits to the data.

Fig. 4 SANS profiles of (DMPC/DHPC/DMPG) samples prepared anddiluted at 45 °C. For all samples, the molar ratios of ([DMPC]+[DMPG])/[DHPC] and [DMPG]/[DMPC] were fixed at 3.2 and0.01, respectively. Bragg maxima are evident for 2.5 wt % ## clp ##25 wt % samples, the result of multibilayers with a precise lamel-lar periodicity, d-spacing. For samples < 2.5 wt %, the multilamel-lar stacks unbind forming variable size ULV. Note that the SANSprofiles decay monotonically and lack the oscillations which arecharacteristic of monodisperse ULV.



length density is evalu-ated from the quadraticdependence of theintensity at the originversus the solvent scat-tering length density.The radius of gyration(RG) is evaluated fromthe slope of the Kratky-Porod plot and thenplotted against theinverse of the differencebetween the solvent andbilayer average scatter-ing length densities. Insuch a graph, one canobtain RG at infinitelylarge contrast corre-sponding to a point atthe graph’s origin.Compared to a singleSANS measurement,this value - obtainedfrom multiple contrastvariation experiments -is a more precise meas-ure of the bilayer’sapparent thickness andcan be used to study therelative changes in abilayer using a model-free approach.

Contrast variation experiments analyzed using amodel-based approach enables one to increasethe number of independent model parametersleading to more realistic models with betterresolved structural features. Scattering curvesobtained at different contrast conditions (Fig. 7)are used to capture the different features of thebilayer. A single molecular model of the bilayeris then used to simultaneously fit the differentcontrast scattering curves. This model is madeup of the probability distributions correspon-ding to the different functional groups (e.g.,choline headgroup, hydrocarbon chains, etc.) ofa bilayer (inset to Fig. 7).

MORPHOLOGY OF GEMINI SURFAC-TANT AGGREGATES

The aggregation behaviour of Gemini surfac-tants is another problem that has been examinedwith SANS. Gemini surfactants are composed oftwo or more pairs of hydrophilic and hydropho-bic groups connected to each other with a spacer(Fig. 8). In order to modify the surface tension ofa solution, only small amounts of Gemini surfac-tants are required as their critical micellar con-centration (cmc) in aqueous solutions is muchlower than the cmc of conventional surfactantshaving the same hydrophilic and hydrophobic

Fig. 6 Schematic summary of the morphological transformations observed by Nieh et al. [49]. Ondiluting below a critical lipid concentration clpu at T > TM (chain melting transition ofDMPC), extended bilayer sheets unbind into a polydisperse ULV dispersion. On coolingbelow TM and clpu $$ clp $$ 1.25 wt %, polydisperse ULV transform into an isotropic bicellarsolution, which on reheating to T > TM, gives rise to monodisperse ULV. For clp ## 0.5 wt %polydisperse ULV are trapped and cannot, at low T, transform into bicelles. MonodisperseULV can also be obtained by diluting the bicellar phase below clpu at T < TM, followed byheating above TM. In the case of very dilute mixtures, i.e., clp ## 0.1 wt % and T < TM, bilay-ered micelles do not reform. Instead, oblate ellipsoids are created. The dashed lines indicateplausible transformations not probed by the experiments carried out by Nieh et al. [49].

Fig. 7 SANS curves obtained at different contrast variation conditions.ULV composed of fully hydrogenated DPPC and DPPC withperdeuterated hydrocarbon chains (d62-DPPC) were prepared inthree different D2O:H2O (100%, 70% and 50%) mixtures. A molecu-lar model of the bilayer is shown in the inset to the figure. Thebilayer profile is represented by probability distribution functionscorresponding to solvent molecules, the PC headgroup, and the CH2and CH3 making up the lipid’s hydrocarbon chains.



groups. One example of what SANS canachieve in studying the structure of such sur-factant systems is the molecule α,α’-[2,4,7,9-tetramethyl-5-decyne-4,7-diyl]-bis-[ω-hydrox-yl-polyoxyethylene] (Fig. 8), which contains10 ethylene oxide (EO) segments. Conclusionsfrom previous studies were that the systemunderwent two possible transitions namely amonomer 6 micelle I and a micelle I 6 micelleII at 0.9 and 2 wt%, respectively [50-53].However, recent SANS data, outlined below,have contradicted these findings [54].

Figure 9 shows SANS patterns for various surfactant concen-trations [54]. At low concentration (0.5 wt%) and a Q-regimeof < 0.02 Å–1, I(Q) decays as Q-4 decay indicating the presenceof large particles (> 50 nm) in solution - denoted later on as“clusters”. Between 0.03 and 0.1 Å-1 I(Q) plateaus anddecays as Q-2 for Q > 0.1 Å-1, characteristic of particles with amuch smaller length scale, possibly monomer surfactant mol-ecules. As the surfactant concentration increases to 1 wt%, theslope of the scattered intensity decreases at small Q values,indicative of scattering contributions from larger sized “clus-ters”. Moreover, the intensity plateau starts to decay earlierthan that seen in the 0.5 wt% sample, implying that the small-er aggregates are getting larger at higher concentrations, pre-sumably due to micellation. The SANS data of the 1 wt%sample also shows a slight upturning at very low Q(< 0.005 Å–1), implying either the coexistence of micelles withsmall amounts of clusters, or that the size of the clusters, atthis concentration, are so large that they are beyond theSANS detecting limit. Above 2 wt% this low Q behaviourdisappears completely, indicating that either the clustershave become too large to detect or that they no longer exist.

Analysis of the SANS data can reveal the size and aggrega-tion number of the surfactant. For Q· RG # 1 (correspondingto a Q range of 0.01 < Q < 0.04 Å-1) I(Q) can be related to theradius of gyration, RG , aggregation number, ns and the sec-ond virial coefficient, A2 (an index for interparticle interac-tion), as follows

Fig. 8 The molecular structure of αα,αα’-[2,4,7,9-tetramethyl-5-decyne-4,7-diyl]bis-[ωω-hydroxyl-polyoxyethylene].

Fig. 9 SANS scattering curves of Gemini surfactants at concentrations vary-ing from 0.5 to 5 wt%.

Fig. 10 Zimm plot constructed for SANS data of surfactantswith concentrations between 1.1 to 5 wt%. The Qvalues range from 0.008 to 0.1 Å-1.


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where νs is the volume of one aggregate and Δρ is the differ-ence in scattering length density between the solvent(i.e., D2O) and the surfactant. This equation is the basis of theZimm plot. By plotting the extrapolated φ=0 (justifiesneglecting intermolecular interferences i.e., structure factor)values from φ/I(Q) versus Q2 plots, a straight line is obtained

with slope . On the other hand, by plotting

the extrapolated Q=0 (justifies the treatment of intramolecu-lar interference i.e. form factor) values, a line is obtained with

slope . A Zimm plot (Fig. 10) can therefore be

constructed with φ/I versus (cφ+Q2), where c is an arbitraryconstant allowing for the various φ lines to be separated. Theobtained RG , ns and A2 values for micelles are (14.7 ± 3.5) Å,(17.2 ± 0.2), and (-2 ± 6) x 10-5 mol/cm3, respectively. Fromthe analysis of the SANS data the two morphological transi-tions identified were clusters/monomers 6 clusters/micellesand clusters/micelles 6 micelles.

CONCLUDING REMARKS

With regards to polymeric materials, SANS is arguably thesingle most important neutron scattering technique. It isused routinely to probe the size, shape and conformation ofmacromolecular complexes whose size ranges from ten toone thousand Ångstroms. In the case of biologically relevantmaterials, SANS is employed to study molecules and molec-ular assemblies under physiologically meaningful condi-tions, and also allows for the study of disordered materialsthat are difficult or impossible to crystallize. It should bepointed out that the use of SANS has flourished in the lastdecade from the production of cold neutrons (i.e., 5 – 20 Å)whose long wavelengths have greatly facilitated the interro-gation of materials with large unit cells (e.g. proteins).

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φρ

φI Q Q R v n

AG s s( ) /

=− +( ) + +

⎛

⎝⎜

⎞

⎠⎟

11 3

1 22 2 2 2Δ

∼1 3 1 2

2 2

2 2

+ +( )+ +

⎛

⎝⎜

⎞

⎠⎟

Q Rv n

AG

s s

/Δρ

φ

Rv n

G

s s

3

23 ⋅ Δρ

2 22

AΔρ



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