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From Emil Wolf, History and Solution of the Phase Problem in the Theory of StructureDetermination of Crystals from X-Ray Diffraction Measurements.

In: Peter W. Hawkes, editor, Advances in Imaging and Electron Physics, Vol 165.San Diego: Academic Press, 2011, pp. 283–325.

ISBN: 978-0-12-385861-0c© Copyright 2011 Elsevier Inc.

Academic Press.

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Chapter7

History and Solution of thePhase Problem in the Theoryof Structure Determination ofCrystals from X-Ray DiffractionMeasurements

Emil Wolf

Contents 1. Introduction 2842. Approximate Methods of Solution of the

Phase Problem 2913. Review of Elements of Coherence Theory 305

3.1. Coherence Theory in the Space-Time Domain 3053.2. Coherence Theory in the Space-Frequency

Domain 3103.3. Spatially Coherent Radiation 3123.4. Some Properties of Spatially Completely Coherent

Radiation 3154. Solution of the Phase Problem 316Appendix A. A Method for Determining the Modulus and

the Phase of the Spectral Degree of Coherence fromExperiment 320

Appendix B. Nobel Prizes 322Acknowledgments 323References 324

Department of Physics and Astronomy and the Institute of Optics, University of Rochester, Rochester,NY 14627, USA

Advances in Imaging and Electron Physics, Volume 165, ISSN 1076-5670, DOI: 10.1016/B978-0-12-385861-0.00007-5.Copyright c© 2011 Elsevier Inc. All rights reserved.

283

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284 Emil Wolf

1. INTRODUCTION

The subject reviewed in this article concerns a rather important old prob-lem, first formulated about 100 years ago. It is of considerable interest inphysics, chemistry, biology, and medicine. Its importance can be appreci-ated from the fact that about eleven Nobel Prizes were awarded, either fora partial solution of the problem or for the use of its approximate solu-tion in specific applications. After a brief review of the history of researchin this field, we present a solution of the problem obtained very recently(Wolf, 2009, 2010a).

In 1895 W. C. Roentgen (Figure 1) discovered certain rays, originallycalled Roentgen rays, now more commonly known as X-rays. Their dis-covery was followed by a controversy regarding their nature. Later, C. G.Barkla (Figure 2) conducted experiments that provided strong evidencethat they were a new kind of electromagnetic radiation. Soon afterward,Arnold Sommerfeld (Figure 3) estimated from the analysis of certainexperiments on blackening of photographic plates, that the wavelengthsof the X-rays were about a third of the Angstrom unit. However, there wasno direct way to verify this estimate or to confirm that the rays were a newkind of electromagnetic radiation and that they, therefore, should havewavelike properties. Max Laue (Figure 4), a junior lecturer in Munich anda colleague of Sommerfeld, sought a way to verify their electromagnetic

FIGURE 1 W. C. Roentgen.


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286 Emil Wolf

FIGURE 4 M. T. F. Laue.

nature. Laue realized that, to prove this, one could let a beam of X-raysimpinge on some structure consisting of periodic arrangements of holesor slits that would act as pinholes or as diffraction gratings; but becauseof the exceedingly short wavelengths of X-rays predicted by Sommerfeld,it was not clear how to produce such a periodic structure.

Much earlier, Bravais, in 1850, suggested that solids have periodiccrystalline structure. An estimate of the separation between neighboringatoms in a crystal lattice could be deduced from the knowledge of theAvogadro number and from the density and the molecular weight of thecrystal. The estimate obtained in this way indicated that the separation ofneighboring atoms in a crystal lattice was about an Angstrom. This smalldistance is of the order of magnitude of the wavelength of X-ray radiation,as estimated by Sommerfeld. Laue realized that if Sommerfeld’s estimatewas correct, conditions would be satisfied for producing interference anddiffraction of an X-ray beam that passes through a crystal, which wouldact as a three-dimensional diffraction grating (Laue, 1912). Two youngcolleagues, W. Friedrich and P. Knipping, performed experiments to testthis prediction. They found that the X-rays transmitted through the crys-tal indeed produced a diffraction pattern (Friedrich and Knipping, 1912).The first crystal that they irradiated was copper sulfate. Some of the pat-terns they obtained are reproduced as Figure 5. In 1914, Laue received the


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History and Solution of the Phase Problem 287

FIGURE 5 Diffraction pattern formed by X-rays diffracted by copper sulphate. (AfterW. Friedrich and P. Knipping, 1912.)

Physics Nobel Prize for his discovery of the diffraction of Roentgen raysby crystals.

Soon after the discovery of interference and diffraction of X-rays bycrystals, William Henry Bragg (Figure 6) and his son, William LawrenceBragg (Figure 7), considered what may be called the inverse problem; specif-ically, they estimated the structure of some crystalline media from analysisof diffraction patterns produced by X-rays that passed through the crystal.An example of a model of a crystalline medium obtained by them is shownin Figure 8. The immediate success of the method can be judged from thefact that in 1915, only three years after the publication of Laue’s paper, thetwo Braggs were jointly awarded the Physics Nobel Prize for their servicesin the analysis of crystal structure by means of X-rays.

The work of Laue and the Braggs was a starting point of an impor-tant technique for determining the structure of solids and of other media.Later, similar investigations were carried out—and continue to be carriedout—also with neutrons and with electrons; and these investigations led tothe development of a large and flourishing technique. Delightful accountsof the history of this subject are presented in Ewald (1962).


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FIGURE 8 Model of arrangement of atoms in Fluorspan (CaF2). The black ballsrepresent calcium; the white balls represent fluorine (Reprinted from 1922 Nobel lectureby W. L. Bragg).

Let us mention some highlights resulting from the use of this tech-nique. In 1962, F. H. C. Crick (Figure 9), J. D. Watson (Figure 10), andM. H. Wilkins (Figure 11) determined the molecular structure of nucleicacids and its significance for information transfer in living materials. Morespecifically, they determined the structure of DNA molecules (Figure 12),which carry information about heredity. This achievement was honoredby the award of the 1962 Nobel Prizes in Physiology and Medicine. In1982, A. Klug (Figure 13) received the Nobel Prize in Chemistry for devel-opment of crystallographic electronmicroscopy and his structure elucida-tion of biologically important nucleic-acid protein complexes. In 1988, H.Michel (Figure 14), J. Deisenhofer (Figure 15), and R. Huber (Figure 16)received the Nobel Prize in Chemistry for the determination of the three-dimensional structure of a photosynthetic reaction center. Several otherNobel Prizes were awarded for research in this general area: to P. D. Boyer(Figure 17) and J. E. Walker (Figure 18) for the elucidation of the enzymaticmechanism underlying the synthesis of adenosine triphosphate (ATP),


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FIGURE 11 M. H. F. Wilkins.

and to J. C. Skou (Figure 19) for the first discovery of an ion-transportingenzyme, NA+, K+ −ATPase.

In 2003, the Nobel Prize in Chemistry was awarded to P. Agre(Figure 20) and R. MacKinnon (Figure 21) for structural and mechanis-tic studies of ion channels; and the 2006 Nobel Prize in Chemistry wasawarded to R. Kornberg (Figure 22) for his studies of the molecular basisof eukaryotic transcription. Very recently, V. Ramakrishnan (Figure 23),T. A. Steitz (Figure 24), and A. E. Yonath (Figure 25) were awarded the2009 Nobel Prize in Chemistry for studies of the structure and function ofribosome, by the use of X-ray diffraction techniques.

2. APPROXIMATE METHODS OF SOLUTIONOF THE PHASE PROBLEM

As successful as the reconstructions leading to these discoveries havebeen, they suffered a serious limitation: In mathematical language, thereconstructions were based on the Fourier transform relation betweenthe distribution of the electron density ρ(r), say, throughout the crystal(r being a position vector of a point in the crystal) and the scattered X-ray


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292 Emil Wolf

FIGURE 12 Double-helix structure of DNA molecules (Reproduced from M. Wilkins,1962.)

field in the far zone. For a crystalline medium, ρ(r) is a periodic func-tion of r. The basic relation needed for the reconstruction follows from thefollowing considerations:

Suppose that a plane monochromatic wave

U(i)(r, t) = ei(ks0·r−ωt), (k = ω/c), (2.1)


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300 Emil Wolf

Here, f (s, s0) is the scattering amplitude, given by the formula

f (s, s0) =

∫D

ρ(r′)e−ik(s−s0)·r′d3r′, (2.3)

where D is the volume occupied by the crystal. This formula shows thatthe scattering amplitude f (s, s0) is the Fourier transform of the electrondensity distribution ρ(r) throughout the crystal. Hence, if one measuredthe scattering amplitude for all directions of incidence s0 and of scatter-ing s and then took the Fourier inverse of Eq. (2.3), one would obtainthe basic quantity that represents the structure of the crystal—namely, theelectron density distribution ρ(r) throughout the crystal.1 However, whatone can measure is not the (generally complex) scattered field but ratherthe intensity I, which is proportional to the squared modulus of expres-sion (2.3). Consequently, one can determine only the amplitudes of thescattered fields, not their phases. However, to calculate the electron den-sity distribution via the Fourier transform of the relation (2.3), one needsto know not only the amplitudes but also the phases. Until very recently,no method for measuring the phases has been found and, consequently,full reconstruction of the crystal structure by use of this technique has notproved to be possible.

Many publications have been devoted to estimating the phases, someof them based on the following fact: Since the electron density ρ(r) isnecessarily nonnegative, there is some constraint on its Fourier trans-form. A constraint, well known to mathematicians, is expressed by theso-called Bochner’s theorem (Bochner, 1932, 1937; Goldberg, 1965), which,in one-dimensional form, may be stated as follows.

If g(x) is a nonnegative function (that is, if g(x) ≥ 0, for all values of x),then the Fourier transform

g(u) =

∞∫−∞

g(x)e−iuxdx (2.4)

is necessarily nonnegative definite—that is, for any positive integer N andfor any sets of arbitrary numbers (a1, a2, . . . , aN), real or complex, and anyset of real numbers (u1, u2, . . . , uN):

N∑n=1

N∑m=1

a∗mang(um − un) ≥ 0. (2.5)

1 Strictly speaking, one could determine only the “low-spatial frequency part” of ρ(r)—namely, the “filtered”version of ρ(r) associated with spatial frequency components ρ(K) of ρ(r) for which |K| < 2k = 4π/λ, λ beingthe wavelength of the X-rays. The endpoints of these K-vectors are confined to the interior of the Ewaldlimiting sphere (Born and Wolf, 1999, §13.1.2). These K-components carry information about details of thestructure which, roughly speaking, exceed several wavelengths of the X-rays.


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FIGURE 27 J. Karle.

The condition (2.5) implies a set of inequalities involving determinantsof higher and higher orders of N (Korn and Korn, 1968, §13.5–6). Suchinequalities were used by two mathematicians, J. Karle (Figure 27) andH. A. Hauptman (Figure 28), to obtain constraints on the phases, althoughthey did not explicitly use Bochner’s theorem in their analysis. An exam-ple of such an inequality involving the structure factors Fklh—essentiallythe Fourier coefficients of the electron density (see Bacon, 1966, p. 45)—is reproduced in Table 1, from the basic paper by Karle and Hauptman(1950). The method introduced by Karle and Hauptman, called the directmethod, was followed by numerous other publications about this tech-nique. (For comprehensive accounts of this method see, for example,Woolfson, 1961, and Giacovazzo, 1980.) The method has made a majorimpact on the field, and numerous approximate determinations of crys-tal structures have been based on it. Its importance was recognized by theaward of a Nobel Prize in Chemistry to J. Karle and H. A. Hauptman in1985 for their outstanding achievement in the development of the directmethod for determination of crystal structures.

Another approximate technique that makes it possible to estimatestructures of crystalline media is the so-called heavy atom method, intro-duced by M. F. Perutz (Figure 29) and developed further by J. C. Kendrew(Figure 30). The essential feature of the technique consists of placing


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302 Emil Wolf

FIGURE 28 H. A. Hauptman.

TABLE 1 Example of an inequality satisfied by the structurefactors Fklh, which is a consequence of the nonnegativity of theelectron density.∣∣∣∣∣∣∣∣∣∣

F000 F001 F010 F011 F100 F101 F110 F111F001 F000 F011 F010 . . . . . . . . . . . .

F010 F011 F000 F001 . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

F111 F110 F101 F100 . . . . . . . . . F000

∣∣∣∣∣∣∣∣∣∣≥ 0.

After Karle and Hauptman, 1950.

heavy atoms into certain positions in the crystal. The procedure alters thediffraction pattern, and the changes so introduced can be used to estimatethe structure of the crystal. In 1962, Perutz and Kendrew were awardedthe Nobel Prize in Chemistry for their studies of the structure of globularprotein.

Despite of the great success of such reconstruction techniques, theyhave serious limitations. The inability to measure the phases of thediffracted beams does not make it possible to determine the structureof a crystal with certainty. Here is an account from a book by Ridley(2006, pp. 37–38), about work on this subject by Francis Crick, one of the


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304 Emil Wolf

discoverers of the double-helix structure of DNA molecules, by use of thistechnique:

But the problem Crick was to attack, essentially to choose a protein and discoverits structure—has defeated Perutz for more than a decade for the seeminglyinsuperable reason that an X-ray diffraction pattern records only the intensity ofthe waves, not the relative timing when such wave arrives at the plane of thepicture. This so-called “phase problem” could be circumvented in the case of smallmolecules by trial and error with model-building as Lawrence Bragg has shownmany years before. Crick put it thus: “If the structure could be guessed, it was onlya problem in computation to derive the X-ray pattern it should give. This puts ahigh price on a successful guess.”

Not all the guesses have been successful. This is clear, for example,from the following: Two different structures were predicted for the min-eral bixbyite, one by L. Pauling, the other by W. H. Zachariasen, It is notknown which, if either, is correct.2

Recently, a solution of the phase problem was found (Wolf, 2009,2010a). Before outlining, it seems appropriate to point out the follow-ing: Previous attempts to determine the phases of the diffracted beamsassumed that the X-rays used for the reconstruction are monochromatic.This is an idealization because monochromatic beams are not realiz-able. Any field that can be generated in a laboratory is, at best, quasi-monochromatic; that is, its spectral width 1ω is much smaller than itsmean frequency ω. The amplitudes and the phases of the oscillation ofthe field are random variables. There are several causes of the random-ness, for example, temperature fluctuations of the sources of the radiationand mechanical vibrations of the apparatus used in the measurements.Even if such causes could be eliminated, there is one cause of random-ness that is always present, as Einstein showed many years ago—namely,spontaneous emission of radiation. Consequently, even the output of awell-stabilized laser, for example, which is frequently (but incorrectly)considered to be monochromatic, undergoes random phase fluctuations.

Quantities that are physically meaningful and can be measured arevarious correlation functions of the field well known in coherence theoryof light (Born and Wolf, 1999; Mandel and Wolf, 1995; and Wolf, 2007).We will show that the correlation functions contain information aboutboth the amplitudes and the phases of the diffracted beams; that is, theinformation needed for determining the crystalline structures, providedthat the beams are spatially coherent, a property that is different frommonochromaticity—a distinction which is generally not appreciated.

Before outlining solution of the phase problem, we will briefly sum-marize the basic concepts and results of coherence theory.

2 For discussion of this question, see Kleebe and Lauterbach (2008). I am obliged to Prof. Alberto Grunbaumfor having drawn my attention to this paper.


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3. REVIEW OF ELEMENTS OF COHERENCE THEORY

3.1. Coherence Theory in the Space-Time Domain

Let us consider the light vibrations represented by real functions U1(t)and U2(t), say, of the field at two points P1(r1) and P2(r2). For simplicity,we treat U1(t) and U2(t) as scalar quantities.

Suppose that the field is narrow-band; that is, its bandwidth, 1ω, issmall compared to its mean frequency ω. The oscillations at the two pointswill have the form

U1(t) = a cos [φ1(t)− ωt] , (3.1a)

U2(t) = a cos [φ2(t)− ωt] , (3.1b)

where, for simplicity we assumed that the amplitude a is constant. For anyrealizable beam, φ1(t) and φ2(t) will vary randomly in time. Typical suchoscillations are shown in Figure 31.

On superposing U1(t) and U2(t), after a phase delay, δ say, has beenintroduced between them, the average intensity I(P) of the superposedvibrations is given by the expression

〈I(P)〉 = 〈I1〉 + 〈I2〉 + 〈I12〉, (3.2)

where

I1 = a2 cos2 [φ1(t)− ωt] , (3.3a)

I2 = a2 cos2 [φ2(t)− ωt] , (3.3b)

FIGURE 31 Examples of oscillations of narrow-band field vibrations at two points inspace.


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306 Emil Wolf

and, on using elementary trigonometric identities, one finds that

I12 = a2 cos [φ1(t)+ φ2(t)− 2ωt+ δ]+ a2 cos2 [φ1(t)− φ2(t)− δ] . (3.4)

On taking the average (denoted by angular brackets) over a time intervalthat is large compared to the reciprocal bandwidth of the radiation, oneobtains at once the expressions

〈I1(t)〉 = 12 a2, 〈I2(t)〉 = 1

2 a2, (3.5a)

〈I12〉 = a2〈cos [φ1(t)− φ2(t)+ δ]〉. (3.5b)

The term 〈I12〉 represents an interference term. It may, in general, be presenteven if φ1(t) and φ2(t) fluctuate randomly—for example, when

φ1(t)− φ2(t) = constant. (3.6)

This simple example shows that to obtain sharp interference fringes, thefield may fluctuate randomly, provided only that the vibrations U1(t) andU2(t) undergo essentially the same kind of fluctuations. Such a situationwas referred to by the great French optical scientist E. Verdet in a paperpublished in 1894, as vibrations in unison. In recent years, this concept hasbeen made more precise and called statistical similarity between vibrationsat the points P1 and P2 (Wolf, 2010b). Thus, we may say that in order toobtain interference on superposition of vibrations U1(t) and U2(t), the vibrationsneed not be monochromatic; they may fluctuate randomly, provided that theypossess statistical similarity.

In general, the vibrations at two points in a wavefield will not be sta-tistically similar but rather will be a mixture of statistically similar anddissimilar vibrations. A measure of the two contributions of these twocomponents is the so-called degree of coherence, which may be expressedin terms of measurable quantities. To see this, let us imagine that weperform Young’s interference experiment, with light emerging from twopinholes at points Q1(ρ1) and Q2(ρ2) in an opaque screen A and that onemeasures the average intensities at some point P(r) in a plane B, parallelto A (Figure 32). The average intensity at the point P(r) may readily beshown to be given by the expression (Born and Wolf, 1999, §10.3, Eq. (11))

I(P) = I(1)(P)+ I(2)(P)+ 2√

I(1)(P)√

I(1)(P)R [γ (ρ1, ρ2, t2 − t1)] , (3.7)

where I(1)(P) is the average intensity at P of the radiation that reached Pfrom the first pinhole only (that is, when the pinhole at Q2 is closed), withI(2)(P) having a similar meaning. Further, t1 and t2 are the times needed


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FIGURE 32 Young’s interference experiments.

for the radiation to reach P from the pinholes at Q1 and Q2, respectively,and R denotes the real part.

The factor γ on the right-hand side of Eq. (3.7) is a certain normalizedcorrelation coefficient, viz.

γ (ρ1, ρ2, τ) =0(ρ1, ρ2, τ)√

0(ρ1, ρ1, 0)√0(ρ2, ρ2, 0)

, (3.8)

where3

0(ρ1, ρ2, τ) = 〈V∗(ρ1, τ)V(ρ2, t+ τ)〉. (3.9)

We have now written γ (ρ1, ρ2, τ) in place of γ (Q1, Q2, τ). The function 0is called the mutual coherence function of the vibrations at the points Q1(ρ1)

and Q2(ρ2). It is the central quantity in the theory of coherence. In mathe-matical language, it is the cross-correlation function of the vibrations at thepoints Q1(ρ1) and Q2(ρ2). The quantities that appear in the denominatorof Eq. (3.8) are just the average intensities at the two pinholes.

The formula (3.7) is the so-called interference law of stationary fields.It shows that in order to determine the (average) intensity at the point P inthe observation plane B, one must know not only the average intensitiesof the two beams at P, but also the real part of the correlation coefficient γcalled the complex degree of coherence of the radiation at the pinholes. It maybe shown that for all values of its arguments (Born and Wolf, 1999, §10.3,Eq. (17))

0 ≤ |γ (ρ1, ρ2, τ)| ≤ 1.(3.10)

3 Usually the field fluctuations are statistically stationary and ergodic. Consequently, the angular brackets maybe regarded as representing either the time average or the ensemble average.


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308 Emil Wolf

It is convenient to rewrite the intensity law (3.7) in a somewhat dif-ferent form. We express the (generally complex) degree of coherence γ interms of its modulus |γ | and its phase φ; that is,

γ (ρ1, ρ2, τ) = |γ (ρ1, ρ2, τ)|eiφ(ρ1,ρ2,τ). (3.11)

On substituting from Eq. (3.11) into Eq. (3.7), one obtains the intensity lawin the form

I(P) = I(1)(P)+ I(2)(P)+ 2√

I(1)(P)√

I(2)(P)|γ (Q1, Q2, τ)| cos [φ(Q1, Q2, τ)] ,

(3.12)

where

τ = t2 − t1. (3.13)

We will consider only the case of narrow-band radiation; that is, radi-ation whose bandwidth 1ω is small compared to the main frequency ω.Such radiation is said to be quasi-monochromatic. One can show that if theargument (phase), φ of γ is expressed in the form (Born and Wolf, 1999,§10.3, Eq. (19))

φ(Q1, Q2, τ) = α(Q1, Q2, τ)− ωτ , (3.14)

the function α(Q1, Q2, τ) varies slowly over time intervals of duration1τ . c/1ω, known as the coherence time of the light (Born and Wolf, 1999,§10.3, Eq. (19)).

Using Eq. (3.14), the intensity law may be expressed in the form

I(P)= I(1)(P)+ I(2)(P)+ 2√

I(1)(P)√

I(2)(P)|γ (Q1, Q2, τ)|cos[α(Q1, Q2,τ)−δ] ,

(3.15)

where

δ = ωτ ≡ ω(t2 − t1) =2πλ(R2 − R1), (3.16)

with λ denoting the mean wavelength of the radiation and where R1 andR2 are the distances Q1P and Q2P, respectively (see Figure 32).

The form (3.15) of the interference law for radiation of any state ofcoherence may readily be seen to imply that both the modulus and thephase of the degree of coherence γ (ρ1, ρ2, τ) may be determined fromintensity measurements in the plane of the interference pattern. One wayof seeing it is to note that if the average intensities I(1)(P), I(2)(P), and I(P)are measured for several values of the phase delay δ, defined by Eq. (3.16),


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one can infer from the data, by use of Eq. (3.15) both the modulus |γ | andthe phase φ [which is trivially related to the phase α via the expression(3.14)].

There are other ways of determining the modulus and the phase of thecomplex degree of coherence. Assuming for simplicity that the averageintensities I(1)(P) and I(2)(P) of the radiation reaching the point P fromeach pinhole are equal to each other, as is frequently the case, one canreadily show (Born and Wolf, 1999, §10.4, Eq. (4)) that

|γ (r1, r2, τ)| = V (P), (3.17)

where

V (P) =Imax(P)− Imin(P)Imax(P)+ Imin(P)

. (3.18)

In this formula, Imax(P) is the maximum and Imin(P) is the minimumof the average intensities in the immediate neighborhood of the pointP. The quantity V (P), defined by Eq. (3.18), is a well-known measureof the “sharpness” of interference fringes, called the visibility of the fringes(Figure 33).

The phase of the complex degree of coherence may be deduced frommeasurements of the positions of the intensity maxima and minima inthe immediate neighborhood of the point P in the interference pattern, asdiscussed, for example, in Born and Wolf (1999), §10.4.1.

Radiation with high degree of spatial coherence, that is, radiationfor which |γ | ≈ 1 is routinely generated at optical wavelengths, has beengenerated in recent years with X-rays (See, for example Figures 34 and 35).

FIGURE 33 Intensity distribution in the interference pattern produced by twoquasi-monochromatic beams of equal intensity I(1) and with degree of coherence |γ |:(a) coherent superposition (|γ | = 1); (b) partially coherent superposition (0 < |γ | < 1);(c) incoherent superposition (γ = 0). (Reproduced from Born and Wolf, 1999, p. 569.)


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310 Emil Wolf

(a)

(b)

FIGURE 34 The layout of a Young’s interference experiment with soft X-rays (a) andthe average intensity distribution across the interference pattern (b). (Adapted from Liuet al., 2001.)

3.2. Coherence Theory in the Space-Frequency Domain

The theory of coherence that we just briefly outlined is known as coher-ence theory in the space-time domain. It provides a basis for a rigoroustreatment of the intuitive concepts of coherence, based on the notion ofstatistical similarity, which we mentioned earlier. There is an alternativeformulation of the theory, known as coherence theory in the space-frequencydomain. It is particularly useful for treatments of problems involving quasi-monochromatic radiation and in connection with propagation of radiationin dispersive and absorbing media. In this section, we outline this alterna-tive formulation of coherence theory which, as we will see later, has madeit possible to solve the phase problem of X-ray crystallography.


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FIGURE 35 Interference pattern obtained in a Young’s interference experiment withX-rays of energy 1.1 keV. (After Paterson et al., 2001.)

The basic quantity of coherence theory in the space-frequency domainis the so-called cross-spectral density function, W(r1, r2,ω), which is theFourier transform of the mutual coherence function 0(r1, r2, τ):

W(r1, r2,ω) =1

2π

∞∫−∞

0(r1, r2, τ)eiωτdτ . (3.19)

It may be shown that the cross-spectral density function is also acorrelation function (Mandel and Wolf, 1995, §4.7.2; Wolf, 2007, §4.1).More specifically, one can construct a statistical ensemble of frequency-dependent fields U(r,ω) such that

W(r1, r2,ω) = 〈U∗(r1,ω)U(r2,ω)〉ω, (3.20)

where the angular bracket on the right, with the subscript ω, indicatesthe ensemble average, taken over an ensemble of frequency-dependentrealizations U(r,ω). The function

S(r,ω) ≡W(r, r,ω) = 〈U∗(r,ω)U(r,ω)〉ω (3.21)

represents the spectral density (intensity at frequency ω) of the field at thepoint P(r).


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312 Emil Wolf

In terms of the cross-spectral density, one may introduce the quantity

µ(r1, r2,ω) ≡ |µ(r1, r2,ω)| exp [iβ(r1, r2,ω)] =W(r1, r2,ω)

√W(r1, r1,ω)

√W(r2, r2,ω)

,

(3.22)

known as the spectral degree of coherence of the field at the points P1(r1) andP2(r2), at frequency ω. It can be shown that it is bounded by zero and unityin absolute value; that is, that

0 ≤ |µ(r1, r2,ω)| ≤ 1. (3.23)

The extreme values, |µ| = 1 and 0, are said to represent complete spectralcoherence and complete spectral incoherence, respectively, at frequency ω,at the points P1(r1) and P2(r2). The intermediate values (0 < |µ| < 1) aresaid to represent partial coherence at frequency ω at the two points.

In analogy with the interference intensity law (3.15) in the space-timeformulation, there is a spectral interference law (see, for example, Mandeland Wolf, 1995, p. 173; Wolf, 2007, §4.3)

S(P,ω) = S(1)(P,ω)+ S(2)(P,ω)

+ 2√

S(1)(P,ω)√

S(2)(P,ω)|µ(Q1, Q2,ω)| cos [β(Q1, Q2,ω)− δ] ,

(3.24)

where the quantities on the right have meanings analogous to those thatappear in the corresponding “space-time” intensity law (3.15). The spec-tral interference law may be used to determine both the modulus and thephase of the spectral degree of coherenceµ(Q1, Q2,ω) from measurementsof the spectral densities S(P,ω), S(1)(P,ω), and S(2)(P,ω). A procedure fordoing so is discussed in detail in Appendix A.

Several determinations of the modulus and the phase of the spectraldegree of coherence based on that procedure have been carried out (see,for example, Titus et al., 2000, and Kumar and Rao, 2001); some of theresults are shown in Figure 36.

3.3. Spatially Coherent Radiation

As we already pointed out, an assumption that the beam incident ona crystalline medium is monochromatic is not realistic. Instead we willassume that it is spatially coherent, an assumption that is not equivalent tomonochromaticity, as is frequently incorrectly assumed (in this correction,see Roychowdhury and Wolf, 2005).


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FIGURE 36 Measured cosine and sine of the spectral degree of coherence of a partiallycoherent light beam. (After Titus et al., 2000.)

Coherent beams are routinely produced and used in the optical rangeof the electromagnetic spectrum and can also be generated with X-rays(see, for example, Figures 34 and 35). The very low values of the inten-sity minima in the interference patterns shown in these figures imply thatthe modulus of the spectral degree of coherence has values close to unity;that is, that the radiation in almost completely spatially coherent, havingproduced almost complete cancellation of intensity by interference.

Radiation of a high degree of spatial coherence may be generatedover large regions of space, even when the source is incoherent, just bythe process of propagation. An example is illustrated in Figure 37, whichshows the following: Light from a distant star enters a telescope on thesurface of the earth. The light originates in millions of atoms in thestars, which radiate independently of each other by the process of spon-taneous emission. Consequently, the radiation is spatially incoherent inthe vicinity of the stellar surface. Yet when it reaches the Earth’s sur-face, it is essentially spatially coherent over large regions as is evidentfrom the fact that it produces diffraction patterns with zero minima in thefocal plane of a telescope. This example indicates that spatial coherencefrom a spatially incoherent source has been generated by the process ofpropagation.

Figure 38 is another example illustrating the generation of spatialcoherence on propagation of waves. It shows the surface of water on apond into which several ducks jumped at slightly different times and indifferent places. Initially, the surface of the water disturbed by the ducks


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314 Emil Wolf

FIGURE 37 Illustrating the generation of spatial coherence in starlight.

FIGURE 38 Generation of spatially coherent water waves from randomly distributedwave disturbances, produced by ducks jumping into a pool of water. (After Knox et al.,2010.)

exhibits rather irregular oscillations, showing an incoherent pattern; butwith increasing distance and time, the pattern evolves into a more regularone, i.e., becoming more coherent as shown in the progression of the figure.

The two examples just outlined illustrate the so-called van CittertZernike theorem (Mandel and Wolf, 1995, §4.4.4; Wolf, 2007, §3.2) ofelementary coherence theory. The theorem explains quantitatively how


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coherence from a spatially incoherent source is generated by the process ofpropagation. Such radiation is evidently not monochromatic and, unlikemonochromatic radiation, it is frequently generated in nature and can beproduced in a laboratory.

3.4. Some Properties of Spatially Completely Coherent Radiation

We now present an important theorem concerning radiation that is com-pletely spatially coherent in some region of space. The theorem turns outto be of basic importance for solution of the phase problem. It may bestated as follows (Mandel and Wolf, 1981, 1995, §4.5.3):

If a field is completely spatially coherent at frequency ω throughout a three-dimensional domain D; that is, if |µ(r1, r2,ω)| = 1 for all r1 ∈ D and r2 ∈

D, then the cross-spectral density function of the field at that frequency hasnecessarily the factorized form

W(r1, r2,ω) = u∗(r1,ω)u(r2,ω). (3.25)

Moreover, throughout the domain D, u(r,ω) satisfies the Helmholtz equation

(∇2+ k2)u(r,ω) = 0. (3.26)

If we set

u(r,ω) = |u(r,ω)|eiφ(r,ω), (3.27)

we readily find from the definition (3.22) of the spectral degree of coher-ence µ and from Eqs. (3.25) and (3.22) that in this case the spectral degreeof coherence has the form

µ(r1, r2,ω) = ei[φ(r2,ω)−φ(r1,ω)]. (3.28)

Because u(r,ω) satisfies the Helmholtz equation (3.26), it may be identi-fied with the space-dependent part of a monochromatic wave of frequencyω. It is to be understood that this wave is not an actual wave but is equiv-alent to it in the sense indicated by the product relation (3.25) for thecross-spectral density function of a spatially coherent field. Loosely speak-ing, it represents a wave function of an associated average field.4 Thatmakes it possible to calculate the cross-spectral density function of theactual spatially coherent field via the product relation (3.25).

4 In this connection see also Wolf (2011).


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316 Emil Wolf

As we will soon see, the “average” wave function u(r,ω), rather thanthe idealized (nonexistent) monochromatic wave function of the usualtreatments, may be used to analyze diffraction of X-ray beams by crystals;and because its phase is associated with the spectral degree of coherenceof the beam by the formula (3.28), it may be measured. The possibilityof such measurements has been pointed out by Wolf (2003) and con-firmed experimentally by Dogariu and Popescu (2002). Another techniquefor determining both the phase and the modulus of the spectral degreeof coherence, whether or not the radiation is spatially fully coherent, isdescribed in Appendix A.

4. SOLUTION OF THE PHASE PROBLEM

We will now show that the properties of spatially coherent radiation justdiscussed may be used to provide a solution to the phase problem of X-raycrystallography.

Suppose that a spatially coherent, quasi-monochromatic beam of unitamplitude and of mean frequency ω, propagating in the direction speci-fied by a unit vector s0, is incident on a crystalline medium. As seen inthe previous section, one may associate with such a beam an “average”monochromatic wave function

u(r,ω) = exp (iks0 · r) (4.1)

of frequency ω, where k = ω/c, c being the speed of light in free space.By analogy with Eqs. (2.2) and (2.3) encountered earlier, the scattered

field in the far zone of the crystal is then given by the formula

u(∞)(rs,ω) = f (s, s0;ω)eikr

r, (4.2)

where the scattering amplitude

f (s, s0;ω) ≡ f (s− s0,ω) =∫D

ρ(r′) exp [−ik(s− s0) · r′]d3r′. (4.3)

Let us set

k(s− s0) = K (4.4)

in Eq. (4.3) and take the Fourier inverse of the resulting expression. Onethen obtains the basic expression for the electron density ρ(r) throughout


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the crystal in terms of the scattering amplitude:

ρ(r′) =1

(2π)3

∫f (K/k,ω)eiK.r′d3K. (4.5)

Since both the vectors s0 and s are unit vectors, Eq. (4.4) implies that|K| ≤ 2k. Hence the K-components that are accessible to measurementsfill a certain finite domain—the interior of the Ewald limiting sphere (Bornand Wolf, 1999, p. 301), of radius

|K| = 2k. (4.6)

Each point within the Ewald sphere over which the integration on theright-hand side of Eq. (4.5) extends is associated with a 3D spatial Fouriercomponent of the electron density ρ(r′) throughout the crystal.

As already noted, the modulus of the scattering amplitude f , whichenters the basic expression (4.5) for the electron density, is just the square-root of the average intensity in the far zone in direction s, when the crystalis illuminated by the coherent plane wave (4.1), in direction s0. Measure-ments of the intensity and, consequently, determination of the amplitudeof the scattering amplitude present no problem. The situation is quite dif-ferent with measurements of the phase of the scattering amplitude, whichup until now has not proved to be possible. We will now show how thephase may be determined, with the help of some of the properties ofcoherent fields that we have discussed.

We return to the situation described at the beginning of this sectionwhen we assumed that the crystal is illuminated by the spatially coherentplane wave of unit amplitude and of mean frequency ω, propagating indirection specified by a unit vector s0 [Eq. (4.1)]. According to Eq. (3.28),the spectral degree of coherence of the associated average wave in thefar zone, at distance r from the scatter and at points Q1(r1), Q2(r2), (r1 =

r1s1, r2 = r2s2, s21 = s2

2 = 1) (Figure 39a), is given by the expression

µ(rs1, rs2,ω) = exp {i[φ(rs2,ω)− φ(rs1,ω)]}. (4.7)

Let us choose s1 to be along the direction of incidence (that is, s1 = s0) ands2 along the direction of scattering (s) (Figure 39b). Then the formula (4.7)becomes

µs0(rs1, rs2,ω) = µs0(rs0, rs,ω) = exp {i[φs0(rs,ω)− φs0(rs0,ω)]}, (4.8)

where we have attached the subscript s0 to µ and to the φs to stress thatthe spectral degree of coherence and the phases pertain to values whenthe incident beam propagates along the direction s0.


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318 Emil Wolf

(a)

(b)

FIGURE 39 Notation relating to spectral degree of coherence µs0(rs1, rs2,ω) of thediffracted field in the far zone of a crystal.

The second term on the right of Eq. (4.8) is actually independent of thedirection of incidence s0. This fact follows from the expression (4.3) for thescattering amplitude f (s, s0,ω) when one sets s = s0. The expression thenreduces to

f (s0, s0;ω) =∫D

ρ(r′)d3r′, (4.9)

which, evidently, is a real constant. Consequently, its phase (arg)

φs0(rs0,ω) ≡ arg[f (s0, s0,ω)

]= 0. (4.10)

Making use of Eq. (4.10), Eq. (4.8) reduces to

µs0(rs0, rs,ω) = exp {i[φs0(rs,ω)]}. (4.11)

The formula (4.11) provides a solution to the phase problem of the theoryof diffraction of X-rays on crystals. To see this, let us recall that the phaseφs0(rs,ω) is the “average” phase of the diffracted beam at distance r fromthe crystal, in direction s, when the crystal is illuminated by a quasi-monochromatic, spatially coherent beam of X-rays of mean frequency ωalong the s0 direction. Formula (4.11) shows that this phase is equal to the


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(a)

(b)

FIGURE 40 Schematic sketch of the usual arrangements for study of structure ofcrystalline solids by X-ray diffraction experiments (a); and by the new techniquedescribed in this article (b), which makes it possible to determine not only theamplitudes but also the phases of diffracted beams. (After Wolf, 2009, 2010a.)

phase of the spectral degree of coherence of the diffracted beam in the far zone forthe pair of points rs and rs0. As mentioned earlier and, as is discussed indetail in Appendix A, the phase of the spectral degree of coherence canbe determined from intensity measurements in interference experiments.A schematic sketch showing the usual setup of measurements with detec-tors D1 and D2 is shown in Figure 40a, and that pertaining to the presentmethod is indicated in Figure 40b.

In order to determine all the 3D spatial Fourier components of the elec-tron density distribution in the crystals that are represented by pointswithin the Ewald limiting sphere, the phases of the spectral degree ofcoherence would have to be determined from interference experiments forwhich the angle of scattering θ ≡ cos−1 (s0, s) takes on all possible valuesin the range 0 ≤ θ ≤ 2π . For large angles of scattering, such measurementsseem to be feasible at optical wavelengths with the help of mirrors or opti-cal fibers. To do so with X-rays presents a challenge yet to be met. However,the method for determining the phases of the diffracted beams as just out-lined makes it possible to determine at least phases of beams diffractedat not too large angles. Such beams carry information about details ofthe crystal structure of the order of a few mean wavelengths of the X-raybeams.


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320 Emil Wolf

APPENDIX A

A Method for Determining the Modulus and the Phase of theSpectral Degree of Coherence from Experiment5

We begin with the spectral interference law (3.24) viz.

S(P,ω) = S(1)(P,ω)+ S(2)(P,ω)

+ 2√

S(1)(P,ω)√

S(2)(P,ω)|µ(Q1, Q2,ω)| cos [β(Q1, Q2,ω)− δ] ,

(A.1)

which represents the spectral density at a point P in the Young interferencepattern formed by light of any state of spatial coherence. In this formula,S(1)(P,ω) represents the spectral density at a point P when the radiationreaches that point from the pinhole at Q1 only (that is, with the pinhole atQ2 closed), S(2)(P,ω) having a similar meaning. Further, |µ(Q1, Q2,ω)| isthe modulus and β12(ω) the phase of the spectral degree of coherence.

We will assume, for simplicity, that the spectral density of the lightreaching the observation point P in the plane B of the fringes are the same,as is frequently the case—that is, that S(1)(P,ω) = S(2)(P,ω). The formula(A.1) then takes the form

S(P,ω) = 2S(1)(P,ω){1+ |µ(Q1, Q2,ω)| cos [β(Q1, Q2,ω)− ωT]}, (A.2)

where

T =R2 − R1

c, (A.3)

with R1 = Q1P, R2 = Q2P. Evidently T is the time difference between thetimes needed for the radiation to reach the point P in the interferencepattern from the two pinholes.

It is convenient to introduce a function

f (T,ω) =S(P,ω)

2S(1)(P,ω)− 1. (A.4)

Because both S(P,ω) and S(1)(P,ω) can be determined from spectroscopicmeasurements, the function f (T,ω) can be experimentally determined.

5 The analysis in this appendix follows very closely that of James and Wolf (1998).


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We will show that in order to deduce the values of the (generally com-plex) spectral degree of coherence µ(Q1, Q2,ω), one needs only to measuref (T,ω) for several values of the parameter T.

From Eqs. (A.2) and (A.4) one can readily see that the function f (T,ω)can be expressed in terms of the real and the imaginary parts of thespectral degree of coherence µ(Q1, Q2,ω) by the formula

f (T,ω) = C12(ω) cosωT + S12(ω) sinωT, (A.5)

where

C12(ω) =R{µ12(ω)} = |µ12(ω)| cos [β12(ω)] , (A.6a)

S12(ω) = I{µ12(ω)} = |µ12(ω)| sin [β12(ω)] , (A.6b)

where R and I denote the real and the imaginary parts, respectively, andwe have simplified the notation by writing C12 instead of C(Q1, Q2,ω), etc.

Suppose that the function f (T,ω) is measured over some narrow band-width for a few different values T1 and T2 of the time delay T. Eq. (A.5)then gives

f (T1,ω) = C12(ω) cos(ωT1)+ S12(ω) sin(ωT1), (A.7a)

f (T2,ω) = C12(ω) cos(ωT2)+ S12(ω) sin(ωT2). (A.7b)

From these two equations one may determine the functions C12(ω) andS12(ω), which, according to Eqs. (A.6), are just the real and the imaginaryparts of the (generally complex) spectral degree of coherence µ12(ω). Thesolution is readily found to be

C12(ω) =sin(ωT2)f (T1,ω)− sin(ωT1)f (T2,ω)

sin[ω(T2 − T1)], (A.8a)

S12(ω) = −cos(ωT2)f (T1,ω)− cos(ωT1)f (T2,ω)

sin[ω(T2 − T1)], (A.8b)

provided that

sin[ω(T2 − T1)] 6= 0. (A.9)

It follows from Eqs. (A.8) that in terms of C12(ω) and S12(ω), the mod-ulus |µ12(ω)| of the spectral degree of coherence is then given by theexpression

|µ12(ω)| =

√[C12(ω)]2

+ [S12(ω)]2 (A.10)


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322 Emil Wolf

and its phase, β12(ω), by the formulas

cos [β12(ω)] =C12(ω)√

[C12(ω)]2+[S12(ω)]2

, (A.11a)

sin [β12(ω)] =S12(ω)√

[C12(ω)]2+[S12(ω)]2

. (A.11b)

For any two values of the time delays T1 and T2, one may expect thatfor certain frequencies, ω0 say, in the spectral band used, the condition(A.9) is violated; that is, for which sin [ω0(T2 − T1)] = 0. For such frequen-cies Eqs. (A.10) and (A.11) do not hold. One may overcome this difficultyby measuring the function f (T,ω) for another value of the time delay, sayfor T3, for which the condition (A.9) holds.

Determination of the modulus and the phase of the spectral degree ofcoherence by use of this method have been carried out by Titus et al. (2000).Some of their results are shown in Figure 36.

APPENDIX B

Nobel Prizes

Awarded for contributions relating to structure determination ofcrystalline media by diffraction techniques

Physics 1914

M. von Laue, for his discovery of the diffraction of X-rays by crystals.

Physics 1915

W. H. Bragg and W. L. Bragg, for their services in the analysis of crystalstructure by means of X-rays.

Chemistry 1962

M. F. Perutz and J. C. Kendrew, for their studies of the structures ofglobular proteins.

Physiology and Medicine 1962

F. H. C. Crick, J. D. Watson, and M. H. F. Wilkins, for their discoveriesconcerning the molecular structure of nucleic acids and its significancefor information transfer in living material.

Chemistry 1982

A. Klug, for his development of crystallographic electron microscopy andhis structural elucidation of biologically important nucleic acid–proteincomplexes.


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Chemistry 1985

H. A. Hauptman and J. Karle, for their outstanding achievements in thedevelopment of direct methods for the determination of crystal structures.

Chemistry 1988

H. Michel, J. Deisenhofer, and R. Huber, for the determination of the three-dimensional structure of a photosynthetic reaction centre.

Chemistry 1997

P. D. Boyer and J. E. Walker, for their elucidation of the enzymatic mech-anism underlying the synthesis of adenosine triphosphate (ATP); andJ. C. Skou, for the first discovery of an ion-transporting enzyme, NA+,K+ −ATPase.

Chemistry 2003

P. Agre and R. MacKinnon, for structural and mechanistic studies of ionchannels.

Chemistry 2006

R. Kornberg, for his studies of the molecular basis of eukaryotic transcrip-tion.

Chemistry 2009

V. Ramakrishnan, T. A. Steitz, and A. E. Yonath, for studies of the structureand function of the ribosome.

Nobel Prizes

Awarded for related investigations

Physics 1901

W. C. Roentgen, in recognition of the extraordinary services he has ren-dered by the discovery of the remarkable rays subsequently named afterhim.

Physics 1917

C. G. Barkla, for his discovery of the characteristic Roentgen radiation ofthe elements.

ACKNOWLEDGMENTS

I acknowledge with thanks permissions of the editors of Physical Review Journals, theAmerican Institute of Physics publications, IUCr Journals, and the American Physical


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324 Emil Wolf

Society journals to reproduce several of the figures which appear in this article. I amobliged to Mr. Mayukh Lahiri for helpful comments and useful suggestions and toDr. Mohamed Salem for assistance with locating pertinent references and preparing many ofthe figures. I am also grateful to Mr. Thomas Kern and Miss Krista Lombardo for assistancewith the typing and checking the text.

Research relating to the solution of the phase problem was supported by the U.S. AirForce Office of Scientific Research (AFOSR) under grant No. FA95500-08-1-0417.

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Wolf, E. (2010b). Statistical similarity as a unifying concept of the theories of coherence andpolarization of light. Optics Communications, 283, 4427–4429.

Woolfson, M. M. (1961). Direct methods in crystallography. Oxford, UK: Clarendon Press.Zernike, F. (1938). The concept of degree of coherence and its application to the optical

problems. Physica, 5, 785–795.


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290 Emil Wolf

FIGURE 9 F. H. C. Crick.

FIGURE 10 J. D. Watson.


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288 Emil Wolf

FIGURE 6 W. H. Bragg.

FIGURE 7 W. L. Bragg.


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FIGURE 2 C. G. Barkla.

FIGURE 3 A. J. W. Sommerfeld.


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FIGURE 13 A. Klug.

FIGURE 14 H. Michel.


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294 Emil Wolf

FIGURE 15 J. Deisenhofer.

FIGURE 16 R. Huber.


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FIGURE 17 P. D. Boyer.

FIGURE 18 J. E. Walker.


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296 Emil Wolf

FIGURE 19 J. C. Skou.

FIGURE 20 P. Agre.


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FIGURE 21 R. MacKinnon.

FIGURE 22 R. Kornberg.


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298 Emil Wolf

FIGURE 23 V. Ramakrishnan.

FIGURE 24 T. A. Steitz.


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FIGURE 25 A. E. Yonath.

FIGURE 26 Notation relating to diffraction of an X-ray beam by a crystalline medium.

where c is the speed of light in a vacuum, propagating in the directionspecified by a unit vector s0 is incident on the crystal (Figure 26). Assum-ing that the beam is unpolarized and making use of elementary scatteringtheory, one has the well-known Fourier relationship between the scatteredfield U(∞)(rs), in the far zone, in the direction specified by a unit vector sand the electron density distribution, ρ(r) throughout the crystal (Papas,1965, p. 20 et seq.):

U(∞)(rs) = f (s, s0)eikr

r. (2.2)


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FIGURE 29 M. F. Perutz.

FIGURE 30 J. C. Kendrew.


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