History and Solution of the Phase Problem in the Theory of Structure Determination of Crystals from X-Ray Diffraction Measurements

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From Emil Wolf, History and Solution of the Phase Problem in the Theory of Structure Determination 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: 978-0-12-385861-0 c Copyright 2011 Elsevier Inc.

Academic Press. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 283 #1 Author's personal copy

Chapter 7

History and Solution of the Phase Problem in the Theory of Structure Determination of Crystals from X-Ray Diffraction Measurements

Emil Wolf

Contents 1. Introduction 284 2. Approximate Methods of Solution of the Phase Problem 291 3. Review of Elements of Coherence Theory 305 3.1. Coherence Theory in the Space-Time Domain 305 3.2. Coherence Theory in the Space-Frequency Domain 310 3.3. Spatially Coherent Radiation 312 3.4. Some Properties of Spatially Completely Coherent Radiation 315 4. Solution of the Phase Problem 316 Appendix A. A Method for Determining the Modulus and the Phase of the Spectral Degree of Coherence from Experiment 320 Appendix B. Nobel Prizes 322 Acknowledgments 323 References 324

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

283 AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 284 #2 Author's personal copy 284 Emil Wolf

1. INTRODUCTION

The subject reviewed in this article concerns a rather important old problem, first formulated about 100 years ago. It is of considerable interest in physics, chemistry, biology, and medicine. Its importance can be appreciated from the fact that about eleven Nobel Prizes were awarded, either for a partial solution of the problem or for the use of its approximate solution in specific applications. After a brief review of the history of research in 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, originally called Roentgen rays, now more commonly known as X-rays. Their discovery was followed by a controversy regarding their nature. Later, C. G. Barkla (Figure 2) conducted experiments that provided strong evidence that they were a new kind of electromagnetic radiation. Soon afterward, Arnold Sommerfeld (Figure 3) estimated from the analysis of certain experiments on blackening of photographic plates, that the wavelengths of the X-rays were about a third of the Angstrom˚ unit. However, there was no direct way to verify this estimate or to confirm that the rays were a new kind of electromagnetic radiation and that they, therefore, should have wavelike properties. Max Laue (Figure 4), a junior lecturer in Munich and a colleague of Sommerfeld, sought a way to verify their electromagnetic

FIGURE 1 W. C. Roentgen. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 286 #4 Author's personal copy 286 Emil Wolf

FIGURE 4 M. T. F. Laue. nature. Laue realized that, to prove this, one could let a beam of X-rays impinge on some structure consisting of periodic arrangements of holes or slits that would act as pinholes or as diffraction gratings; but because of 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 periodic crystalline structure. An estimate of the separation between neighboring atoms in a crystal lattice could be deduced from the knowledge of the Avogadro number and from the density and the molecular weight of the crystal. The estimate obtained in this way indicated that the separation of neighboring atoms in a crystal lattice was about an Angstrom.˚ This small distance is of the order of magnitude of the wavelength of X-ray radiation, as estimated by Sommerfeld. Laue realized that if Sommerfeld’s estimate was correct, conditions would be satisﬁed for producing interference and diffraction of an X-ray beam that passes through a crystal, which would act as a three-dimensional diffraction grating (Laue, 1912). Two young colleagues, W. Friedrich and P. Knipping, performed experiments to test this prediction. They found that the X-rays transmitted through the crystal indeed produced a diffraction pattern (Friedrich and Knipping, 1912). The ﬁrst crystal that they irradiated was copper sulfate. Some of the patterns they obtained are reproduced as Figure 5. In 1914, Laue received the AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 287 #5 Author's personal copy History and Solution of the Phase Problem 287

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

Physics Nobel Prize for his discovery of the diffraction of Roentgen rays by crystals. Soon after the discovery of interference and diffraction of X-rays by crystals, William Henry Bragg (Figure 6) and his son, William Lawrence Bragg (Figure 7), considered what may be called the inverse problem; specif- ically, they estimated the structure of some crystalline media from analysis of diffraction patterns produced by X-rays that passed through the crystal. An example of a model of a crystalline medium obtained by them is shown in Figure 8. The immediate success of the method can be judged from the fact that in 1915, only three years after the publication of Laue’s paper, the two Braggs were jointly awarded the Physics Nobel Prize for their services in the analysis of crystal structure by means of X-rays. The work of Laue and the Braggs was a starting point of an important technique for determining the structure of solids and of other media. Later, similar investigations were carried out—and continue to be carried out—also with neutrons and with electrons; and these investigations led to the development of a large and ﬂourishing technique. Delightful accounts of the history of this subject are presented in Ewald( 1962). AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 289 #7 Author's personal copy History and Solution of the Phase Problem 289

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

Let us mention some highlights resulting from the use of this technique. In 1962, F. H. C. Crick (Figure 9), J. D. Watson (Figure 10), and M. H. Wilkins (Figure 11) determined the molecular structure of nucleic acids and its signiﬁcance for information transfer in living materials. More speciﬁcally, they determined the structure of DNA molecules (Figure 12), which carry information about heredity. This achievement was honored by the award of the 1962 Nobel Prizes in Physiology and Medicine. In 1982, A. Klug (Figure 13) received the Nobel Prize in Chemistry for development of crystallographic electronmicroscopy and his structure elucidation 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 other Nobel 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 enzymatic mechanism underlying the synthesis of adenosine triphosphate (ATP), AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 291 #9 Author's personal copy History and Solution of the Phase Problem 291

FIGURE 11 M. H. F. Wilkins. and to J. C. Skou (Figure 19) for the ﬁrst discovery of an ion-transporting enzyme, 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 mechanistic studies of ion channels; and the 2006 Nobel Prize in Chemistry was awarded to R. Kornberg¨ (Figure 22) for his studies of the molecular basis of eukaryotic transcription. Very recently, V. Ramakrishnan (Figure 23), T. A. Steitz (Figure 24), and A. E. Yonath (Figure 25) were awarded the 2009 Nobel Prize in Chemistry for studies of the structure and function of ribosome, by the use of X-ray diffraction techniques.

2. APPROXIMATE METHODS OF SOLUTION OF THE PHASE PROBLEM

As successful as the reconstructions leading to these discoveries have been, they suffered a serious limitation: In mathematical language, the reconstructions were based on the Fourier transform relation between the 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 AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 292 #10 Author's personal copy 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 function of r. The basic relation needed for the reconstruction follows from the following considerations: Suppose that a plane monochromatic wave

(i) i(ks r ωt) U (r, t) e 0· − , (k ω/c), (2.1) = = AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 300 #18 Author's personal copy 300 Emil Wolf

Here, f (s, s0) is the scattering amplitude, given by the formula Z ik(s s ) r 3 f (s, s ) ρ(r0)e− − 0 · 0 d r0, (2.3) 0 = D where D is the volume occupied by the crystal. This formula shows that the scattering amplitude f (s, s0) is the Fourier transform of the electron density distribution ρ(r) throughout the crystal. Hence, if one measured the scattering amplitude for all directions of incidence s0 and of scattering s and then took the Fourier inverse of Eq. (2.3), one would obtain the basic quantity that represents the structure of the crystal—namely, the electron density distribution ρ(r) throughout the crystal.1 However, what one can measure is not the (generally complex) scattered field but rather the intensity I, which is proportional to the squared modulus of expression (2.3). Consequently, one can determine only the amplitudes of the scattered fields, not their phases. However, to calculate the electron density distribution via the Fourier transform of the relation (2.3), one needs to 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 not proved to be possible. Many publications have been devoted to estimating the phases, some of them based on the following fact: Since the electron density ρ(r) is necessarily nonnegative, there is some constraint on its Fourier transform. A constraint, well known to mathematicians, is expressed by the so-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 Z∞ iux g(u) g(x)e− dx ˜ = (2.4) −∞ is necessarily nonnegative definite—that is, for any positive integer N and for any sets of arbitrary numbers (a1, a2, ... , aN), real or complex, and any set of real numbers (u1, u2, ... , uN): N N X X a∗ a g(u u ) 0. m n ˜ m − n ≥ (2.5) n 1 m 1 = =

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 being the wavelength of the X-rays. The endpoints of these K-vectors˜ are conﬁned to the| | interior= of the Ewald limiting sphere (Born and Wolf, 1999, §13.1.2). These K-components carry information about details of the structure which, roughly speaking, exceed several wavelengths of the X-rays. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 301 #19 Author's personal copy History and Solution of the Phase Problem 301

FIGURE 27 J. Karle.

The condition (2.5) implies a set of inequalities involving determinants of higher and higher orders of N (Korn and Korn, 1968, §13.5–6). Such inequalities were used by two mathematicians, J. Karle (Figure 27) and H. A. Hauptman (Figure 28), to obtain constraints on the phases, although they did not explicitly use Bochner’s theorem in their analysis. An example of such an inequality involving the structure factors Fklh—essentially the 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 direct method, was followed by numerous other publications about this technique. (For comprehensive accounts of this method see, for example, Woolfson, 1961, and Giacovazzo, 1980.) The method has made a major impact on the ﬁeld, and numerous approximate determinations of crystal structures have been based on it. Its importance was recognized by the award of a Nobel Prize in Chemistry to J. Karle and H. A. Hauptman in 1985 for their outstanding achievement in the development of the direct method for determination of crystal structures. Another approximate technique that makes it possible to estimate structures of crystalline media is the so-called heavy atom method, introduced by M. F. Perutz (Figure 29) and developed further by J. C. Kendrew (Figure 30). The essential feature of the technique consists of placing AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 302 #20 Author's personal copy 302 Emil Wolf

FIGURE 28 H. A. Hauptman.

TABLE 1 Example of an inequality satisﬁed by the structure factors Fklh, which is a consequence of the nonnegativity of the electron density.

F000 F F F F F F F 001 010 011 100 101 110 111 F001 F000 F F ...... 011 010 F F F F ...... 0. 010 011 000 001 ≥ ......

F111 F110 F101 F100 ...... F000

After Karle and Hauptman, 1950. heavy atoms into certain positions in the crystal. The procedure alters the diffraction pattern, and the changes so introduced can be used to estimate the structure of the crystal. In 1962, Perutz and Kendrew were awarded the Nobel Prize in Chemistry for their studies of the structure of globular protein. Despite of the great success of such reconstruction techniques, they have serious limitations. The inability to measure the phases of the diffracted beams does not make it possible to determine the structure of 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 AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 304 #22 Author's personal copy 304 Emil Wolf discoverers of the double-helix structure of DNA molecules, by use of this technique: But the problem Crick was to attack, essentially to choose a protein and discover its structure—has defeated Perutz for more than a decade for the seemingly insuperable reason that an X-ray diffraction pattern records only the intensity of the waves, not the relative timing when such wave arrives at the plane of the picture. This so-called “phase problem” could be circumvented in the case of small molecules by trial and error with model-building as Lawrence Bragg has shown many years before. Crick put it thus: “If the structure could be guessed, it was only a problem in computation to derive the X-ray pattern it should give. This puts a high 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 not known 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 following: Previous attempts to determine the phases of the diffracted beams assumed that the X-rays used for the reconstruction are monochromatic. This is an idealization because monochromatic beams are not realizable. Any field that can be generated in a laboratory is, at best, quasi- monochromatic; that is, its spectral width 1ω is much smaller than its mean frequency ω. The amplitudes and the phases of the oscillation of the field are random variables. There are several causes of the random- ness, for example, temperature fluctuations of the sources of the radiation and 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 a well-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 are various correlation functions of the field well known in coherence theory of light (Born and Wolf, 1999; Mandel and Wolf, 1995; and Wolf, 2007). We will show that the correlation functions contain information about both the amplitudes and the phases of the diffracted beams; that is, the information needed for determining the crystalline structures, provided that the beams are spatially coherent, a property that is different from monochromaticity—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 Grunbaum for having drawn my attention to this paper. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 305 #23 Author's personal copy History and Solution of the Phase Problem 305

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, is small compared to its mean frequency ω. The oscillations at the two points will have the form

U (t) a cos [φ (t) ωt] , (3.1a) 1 = 1 − U (t) a cos [φ (t) ωt] , (3.1b) 2 = 2 − where, for simplicity we assumed that the amplitude a is constant. For any realizable beam, φ1(t) and φ2(t) will vary randomly in time. Typical such oscillations are shown in Figure 31. On superposing U1(t) and U2(t), after a phase delay, δ say, has been introduced between them, the average intensity I(P) of the superposed vibrations is given by the expression

I(P) I I I , (3.2) h i = h 1i + h 2i + h 12i where

I a2 cos2 [φ (t) ωt] , (3.3a) 1 = 1 − I a2 cos2 [φ (t) ωt] , (3.3b) 2 = 2 −

FIGURE 31 Examples of oscillations of narrow-band ﬁeld vibrations at two points in space. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 306 #24 Author's personal copy 306 Emil Wolf and, on using elementary trigonometric identities, one ﬁnds that

I a2 cos [φ (t) φ (t) 2ωt δ] a2 cos2 [φ (t) φ (t) δ] . (3.4) 12 = 1 + 2 − + + 1 − 2 − On taking the average (denoted by angular brackets) over a time interval that is large compared to the reciprocal bandwidth of the radiation, one obtains at once the expressions

I (t) 1 a2, I (t) 1 a2, (3.5a) h 1 i = 2 h 2 i = 2 I a2 cos [φ (t) φ (t) δ] . (3.5b) h 12i = h 1 − 2 + i The term I represents an interference term. It may, in general, be present h 12i even if φ1(t) and φ2(t) ﬂuctuate randomly—for example, when

φ (t) φ (t) constant. (3.6) 1 − 2 = This simple example shows that to obtain sharp interference fringes, the field may fluctuate randomly, provided only that the vibrations U1(t) and U2(t) undergo essentially the same kind of fluctuations. Such a situation was referred to by the great French optical scientist E. Verdet in a paper published in 1894, as vibrations in unison. In recent years, this concept has been made more precise and called statistical similarity between vibrations at the points P1 and P2 (Wolf, 2010b). Thus, we may say that in order to obtain interference on superposition of vibrations U1(t) and U2(t), the vibrations need not be monochromatic; they may fluctuate randomly, provided that they possess statistical similarity. In general, the vibrations at two points in a wavefield will not be statistically similar but rather will be a mixture of statistically similar and dissimilar vibrations. A measure of the two contributions of these two components is the so-called degree of coherence, which may be expressed in terms of measurable quantities. To see this, let us imagine that we perform Young’s interference experiment, with light emerging from two pinholes at points Q1(ρ1) and Q2(ρ2) in an opaque screen A and that one measures the average intensities at some point P(r) in a plane B, parallel to A (Figure 32). The average intensity at the point P(r) may readily be shown to be given by the expression (Born and Wolf, 1999, §10.3, Eq. (11))

q q I(P) I(1)(P) I(2)(P) 2 I(1)(P) I(1)(P)R [γ (ρ , ρ , t t )] , (3.7) = + + 1 2 2 − 1 where I(1)(P) is the average intensity at P of the radiation that reached P from the ﬁrst pinhole only (that is, when the pinhole at Q2 is closed), with (2) I (P) having a similar meaning. Further, t1 and t2 are the times needed AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 307 #25 Author's personal copy History and Solution of the Phase Problem 307

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 normalized correlation coefﬁcient, viz.

0(ρ1, ρ2, τ) γ (ρ1, ρ2, τ) p p , (3.8) = 0(ρ1, ρ1, 0) 0(ρ2, ρ2, 0) where3 0(ρ , ρ , τ) V∗(ρ , τ)V(ρ , t τ) . (3.9) 1 2 = h 1 2 + i

We have now written γ (ρ1, ρ2, τ) in place of γ (Q1, Q2, τ). The function 0 is 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 mathematical language, it is the cross-correlation function of the vibrations at the points Q1(ρ1) and Q2(ρ2). The quantities that appear in the denominator of 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 in the observation plane B, one must know not only the average intensities of the two beams at P, but also the real part of the correlation coefﬁcient γ called the complex degree of coherence of the radiation at the pinholes. It may be shown that for all values of its arguments (Born and Wolf, 1999, §10.3, Eq. (17)) (3.10) 0 γ (ρ , ρ , τ) 1. ≤ | 1 2 | ≤

3 Usually the ﬁeld ﬂuctuations are statistically stationary and ergodic. Consequently, the angular brackets may be regarded as representing either the time average or the ensemble average. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 308 #26 Author's personal copy 308 Emil Wolf

It is convenient to rewrite the intensity law (3.7) in a somewhat different form. We express the (generally complex) degree of coherence γ in terms of its modulus γ and its phase φ; that is, | | γ (ρ , ρ , τ) γ (ρ , ρ , τ) eiφ(ρ1,ρ2,τ). (3.11) 1 2 = | 1 2 | On substituting from Eq. (3.11) into Eq. (3.7), one obtains the intensity law in the form

q p I(P) I(1)(P) I(2)(P) 2 I(1)(P) I(2)(P) γ (Q , Q , τ) cos [φ(Q , Q , τ)] , = + + | 1 2 | 1 2 (3.12) where

τ t t . (3.13) = 2 − 1 We will consider only the case of narrow-band radiation; that is, radiation whose bandwidth 1ω is small compared to the main frequency ω. Such radiation is said to be quasi-monochromatic. One can show that if the argument (phase), φ of γ is expressed in the form (Born and Wolf, 1999, §10.3, Eq. (19))

φ(Q , Q , τ) α(Q , Q , τ) ωτ, (3.14) 1 2 = 1 2 − the function α(Q1, Q2, τ) varies slowly over time intervals of duration 1τ . 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 q p I(P) I(1)(P) I(2)(P) 2 I(1)(P) I(2)(P) γ (Q , Q , τ) cos[α(Q , Q ,τ) δ] , = + + | 1 2 | 1 2 − (3.15) where 2π δ ωτ ω(t2 t1) (R2 R1), (3.16) = ≡ − = λ − with λ denoting the mean wavelength of the radiation and where R1 and R2 are the distances Q1P and Q2P, respectively (see Figure 32). The form (3.15) of the interference law for radiation of any state of coherence may readily be seen to imply that both the modulus and the phase of the degree of coherence γ (ρ1, ρ2, τ) may be determined from intensity measurements in the plane of the interference pattern. One way of 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), AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 309 #27 Author's personal copy History and Solution of the Phase Problem 309 one can infer from the data, by use of Eq. (3.15) both the modulus γ and | | the 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 the complex degree of coherence. Assuming for simplicity that the average intensities I(1)(P) and I(2)(P) of the radiation reaching the point P from each pinhole are equal to each other, as is frequently the case, one can readily show (Born and Wolf, 1999, §10.4, Eq. (4)) that

γ (r , r , τ) V (P), (3.17) | 1 2 | = where

Imax(P) Imin(P) V (P) − . (3.18) = I (P) I (P) max + min

In this formula, Imax(P) is the maximum and Imin(P) is the minimum of the average intensities in the immediate neighborhood of the point P. The quantity V (P), deﬁned by Eq. (3.18), is a well-known measure of the “sharpness” of interference fringes, called the visibility of the fringes (Figure 33). The phase of the complex degree of coherence may be deduced from measurements of the positions of the intensity maxima and minima in the immediate neighborhood of the point P in the interference pattern, as discussed, for example, in Born and Wolf( 1999), §10.4.1. Radiation with high degree of spatial coherence, that is, radiation for which γ 1 is routinely generated at optical wavelengths, has been | | ≈ generated in recent years with X-rays (See, for example Figures 34 and 35).

FIGURE 33 Intensity distribution in the interference pattern produced by two quasi-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.)| = AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 310 #28 Author's personal copy 310 Emil Wolf

(a)

(b)

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

3.2. Coherence Theory in the Space-Frequency Domain The theory of coherence that we just brieﬂy outlined is known as coherence theory in the space-time domain. It provides a basis for a rigorous treatment of the intuitive concepts of coherence, based on the notion of statistical similarity, which we mentioned earlier. There is an alternative formulation of the theory, known as coherence theory in the space-frequency domain. It is particularly useful for treatments of problems involving quasi- monochromatic radiation and in connection with propagation of radiation in dispersive and absorbing media. In this section, we outline this alternative formulation of coherence theory which, as we will see later, has made it possible to solve the phase problem of X-ray crystallography. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 311 #29 Author's personal copy History and Solution of the Phase Problem 311

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

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

Z∞ 1 iωτ W(r , r2, ω) 0(r , r2, τ)e dτ. (3.19) 1 = 2π 1 −∞ It may be shown that the cross-spectral density function is also a correlation 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(r , r , ω) U∗(r , ω)U(r , ω) ω, (3.20) 1 2 = h 1 2 i where the angular bracket on the right, with the subscript ω, indicates the ensemble average, taken over an ensemble of frequency-dependent realizations U(r, ω). The function

S(r, ω) W(r, r, ω) U∗(r, ω)U(r, ω) ω (3.21) ≡ = h i represents the spectral density (intensity at frequency ω) of the ﬁeld at the point P(r). AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 312 #30 Author's personal copy 312 Emil Wolf

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

W(r1, r2, ω) µ(r1, r2, ω) µ(r1, r2, ω) exp [iβ(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) and P2(r2), at frequency ω. It can be shown that it is bounded by zero and unity in absolute value; that is, that

0 µ(r , r , ω) 1. (3.23) ≤ | 1 2 | ≤ The extreme values, µ 1 and 0, are said to represent complete spectral | | = coherence and complete spectral incoherence, respectively, at frequency ω, at the points P (r ) and P (r ). The intermediate values (0 < µ < 1) are 1 1 2 2 | | said to represent partial coherence at frequency ω at the two points. In analogy with the interference intensity law (3.15) in the space-time formulation, there is a spectral interference law (see, for example, Mandel and Wolf, 1995, p. 173; Wolf, 2007, §4.3)

S(P, ω) S(1)(P, ω) S(2)(P, ω) = + q p 2 S(1)(P, ω) S(2)(P, ω) µ(Q , Q , ω) cos [β(Q , Q , ω) δ] , + | 1 2 | 1 2 − (3.24) where the quantities on the right have meanings analogous to those that appear in the corresponding “space-time” intensity law (3.15). The spectral interference law may be used to determine both the modulus and the phase of the spectral degree of coherence µ(Q1, Q2, ω) from measurements of the spectral densities S(P, ω), S(1)(P, ω), and S(2)(P, ω). A procedure for doing so is discussed in detail in Appendix A. Several determinations of the modulus and the phase of the spectral degree of coherence based on that procedure have been carried out (see, for example, Titus et al., 2000, and Kumar and Rao, 2001); some of the results are shown in Figure 36.

3.3. Spatially Coherent Radiation As we already pointed out, an assumption that the beam incident on a crystalline medium is monochromatic is not realistic. Instead we will assume that it is spatially coherent, an assumption that is not equivalent to monochromaticity, as is frequently incorrectly assumed (in this correction, see Roychowdhury and Wolf, 2005). AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 313 #31 Author's personal copy History and Solution of the Phase Problem 313

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

Coherent beams are routinely produced and used in the optical range of the electromagnetic spectrum and can also be generated with X-rays (see, for example, Figures 34 and 35). The very low values of the intensity minima in the interference patterns shown in these ﬁgures imply that the modulus of the spectral degree of coherence has values close to unity; that is, that the radiation in almost completely spatially coherent, having produced almost complete cancellation of intensity by interference. Radiation of a high degree of spatial coherence may be generated over large regions of space, even when the source is incoherent, just by the process of propagation. An example is illustrated in Figure 37, which shows the following: Light from a distant star enters a telescope on the surface of the earth. The light originates in millions of atoms in the stars, which radiate independently of each other by the process of spontaneous emission. Consequently, the radiation is spatially incoherent in the vicinity of the stellar surface. Yet when it reaches the Earth’s surface, it is essentially spatially coherent over large regions as is evident from the fact that it produces diffraction patterns with zero minima in the focal plane of a telescope. This example indicates that spatial coherence from a spatially incoherent source has been generated by the process of propagation. Figure 38 is another example illustrating the generation of spatial coherence on propagation of waves. It shows the surface of water on a pond into which several ducks jumped at slightly different times and in different places. Initially, the surface of the water disturbed by the ducks AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 314 #32 Author's personal copy 314 Emil Wolf

FIGURE 37 Illustrating the generation of spatial coherence in starlight.

FIGURE 38 Generation of spatially coherent water waves from randomly distributed wave disturbances, produced by ducks jumping into a pool of water. (After Knox et al., 2010.) exhibits rather irregular oscillations, showing an incoherent pattern; but with increasing distance and time, the pattern evolves into a more regular one, i.e., becoming more coherent as shown in the progression of the ﬁgure. The two examples just outlined illustrate the so-called van Cittert Zernike theorem (Mandel and Wolf, 1995, §4.4.4; Wolf, 2007, §3.2) of elementary coherence theory. The theorem explains quantitatively how AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 315 #33 Author's personal copy History and Solution of the Phase Problem 315 coherence from a spatially incoherent source is generated by the process of propagation. Such radiation is evidently not monochromatic and, unlike monochromatic radiation, it is frequently generated in nature and can be produced in a laboratory.

3.4. Some Properties of Spatially Completely Coherent Radiation We now present an important theorem concerning radiation that is completely spatially coherent in some region of space. The theorem turns out to be of basic importance for solution of the phase problem. It may be stated 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(r , r , ω) 1 for all r D and r | 1 2 | = 1 ∈ 2 ∈ D, then the cross-spectral density function of the ﬁeld at that frequency has necessarily the factorized form

W(r , r , ω) u∗(r , ω)u(r , ω). (3.25) 1 2 = 1 2 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 coherence µ and from Eqs. (3.25) and (3.22) that in this case the spectral degree of coherence has the form

i[φ(r ,ω) φ(r ,ω)] µ(r , r , ω) e 2 − 1 . (3.28) 1 2 = 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 frequency ω. It is to be understood that this wave is not an actual wave but is equivalent to it in the sense indicated by the product relation (3.25) for the cross-spectral density function of a spatially coherent field. Loosely speaking, it represents a wave function of an associated average field.4 That makes it possible to calculate the cross-spectral density function of the actual spatially coherent field via the product relation (3.25).

4 In this connection see also Wolf (2011). AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 316 #34 Author's personal copy 316 Emil Wolf

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

4. SOLUTION OF THE PHASE PROBLEM

We will now show that the properties of spatially coherent radiation just discussed may be used to provide a solution to the phase problem of X-ray crystallography. Suppose that a spatially coherent, quasi-monochromatic beam of unit amplitude and of mean frequency ω, propagating in the direction speci- ﬁed by a unit vector s0, is incident on a crystalline medium. As seen in the previous section, one may associate with such a beam an “average” monochromatic wave function u(r, ω) exp (iks r) (4.1) = 0 · 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

ikr ( ) e u ∞ (rs, ω) f (s, s ; ω) , (4.2) = 0 r where the scattering amplitude Z 3 f (s, s ; ω) f (s s , ω) ρ(r0) exp [ ik(s s ) r0]d r0. (4.3) 0 ≡ − 0 = − − 0 · D

Let us set

k(s s ) K (4.4) − 0 = in Eq. (4.3) and take the Fourier inverse of the resulting expression. One then obtains the basic expression for the electron density ρ(r) throughout AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 317 #35 Author's personal copy History and Solution of the Phase Problem 317 the crystal in terms of the scattering amplitude: Z 1 iK.r 3 ρ(r0) f (K/k, ω)e 0 d K. (4.5) = (2π)3

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 measurements | | ≤ ﬁll a certain ﬁnite domain—the interior of the Ewald limiting sphere (Born and Wolf, 1999, p. 301), of radius

K 2k. (4.6) | | = Each point within the Ewald sphere over which the integration on the right-hand side of Eq. (4.5) extends is associated with a 3D spatial Fourier component of the electron density ρ(r0) throughout the crystal. As already noted, the modulus of the scattering amplitude f , which enters 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 crystal is illuminated by the coherent plane wave (4.1), in direction s0. Measure- ments of the intensity and, consequently, determination of the amplitude of the scattering amplitude present no problem. The situation is quite different with measurements of the phase of the scattering amplitude, which up until now has not proved to be possible. We will now show how the phase may be determined, with the help of some of the properties of coherent ﬁelds that we have discussed. We return to the situation described at the beginning of this section when we assumed that the crystal is illuminated by the spatially coherent plane wave of unit amplitude and of mean frequency ω, propagating in direction 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 the far zone, at distance r from the scatter and at points Q (r ), Q (r ), (r 1 1 2 2 1 = r s , r r s , s2 s2 1) (Figure 39a), is given by the expression 1 1 2 = 2 2 1 = 2 = µ(rs , rs , ω) exp i[φ(rs , ω) φ(rs , ω)] . (4.7) 1 2 = { 2 − 1 } Let us choose s to be along the direction of incidence (that is, s s ) and 1 1 = 0 s2 along the direction of scattering (s)(Figure 39b). Then the formula (4.7) becomes

µ (rs , rs , ω) µ (rs , rs, ω) exp i[φ (rs, ω) φ (rs , ω)] , (4.8) s0 1 2 = s0 0 = { s0 − s0 0 } where we have attached the subscript s0 to µ and to the φs to stress that the spectral degree of coherence and the phases pertain to values when the incident beam propagates along the direction s0. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 318 #36 Author's personal copy 318 Emil Wolf

(a)

(b)

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

The second term on the right of Eq. (4.8) is actually independent of the direction of incidence s0. This fact follows from the expression (4.3) for the scattering amplitude f (s, s , ω) when one sets s s . The expression then 0 = 0 reduces to Z 3 f (s , s ; ω) ρ(r0)d r0, (4.9) 0 0 = D which, evidently, is a real constant. Consequently, its phase (arg)

φ (rs , ω) arg f (s , s , ω) 0. (4.10) s0 0 ≡ 0 0 = Making use of Eq. (4.10), Eq. (4.8) reduces to

µ (rs , rs, ω) exp i[φ (rs, ω)] . (4.11) s0 0 = { s0 } The formula (4.11) provides a solution to the phase problem of the theory of diffraction of X-rays on crystals. To see this, let us recall that the phase

φs0 (rs, ω) is the “average” phase of the diffracted beam at distance r from the crystal, in direction s, when the crystal is illuminated by a quasi- monochromatic, spatially coherent beam of X-rays of mean frequency ω along the s0 direction. Formula (4.11) shows that this phase is equal to the AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 319 #37 Author's personal copy History and Solution of the Phase Problem 319

(a)

(b)

FIGURE 40 Schematic sketch of the usual arrangements for study of structure of crystalline solids by X-ray diffraction experiments (a); and by the new technique described in this article (b), which makes it possible to determine not only the amplitudes 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 for the pair of points rs and rs0. As mentioned earlier and, as is discussed in detail in Appendix A, the phase of the spectral degree of coherence can be 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 present method is indicated in Figure 40b. In order to determine all the 3D spatial Fourier components of the electron density distribution in the crystals that are represented by points within the Ewald limiting sphere, the phases of the spectral degree of coherence would have to be determined from interference experiments for which the angle of scattering θ cos 1 (s , s) takes on all possible values ≡ − 0 in the range 0 θ 2π. For large angles of scattering, such measurements ≤ ≤ seem to be feasible at optical wavelengths with the help of mirrors or optical 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 outlined makes it possible to determine at least phases of beams diffracted at not too large angles. Such beams carry information about details of the crystal structure of the order of a few mean wavelengths of the X-ray beams. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 320 #38 Author's personal copy 320 Emil Wolf

APPENDIX A

A Method for Determining the Modulus and the Phase of the Spectral 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) = + q p 2 S(1)(P, ω) S(2)(P, ω) µ(Q , Q , ω) cos [β(Q , Q , ω) δ] , + | 1 2 | 1 2 − (A.1) which represents the spectral density at a point P in the Young interference pattern 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 radiation reaches that point from the pinhole at Q1 only (that is, with the pinhole at Q closed), S(2)(P, ω) having a similar meaning. Further, µ(Q , Q , ω) is 2 | 1 2 | the modulus and β12(ω) the phase of the spectral degree of coherence. We will assume, for simplicity, that the spectral density of the light reaching 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(Q , Q , ω) cos [β(Q , Q , ω) ωT] , (A.2) = { + | 1 2 | 1 2 − } where

R R T 2 − 1 , (A.3) = c with R Q P, R Q P. Evidently T is the time difference between the 1 = 1 2 = 2 times needed for the radiation to reach the point P in the interference pattern from the two pinholes. It is convenient to introduce a function

S(P, ω) f (T, ω) 1. (A.4) = 2S(1)(P, ω) −

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

5 The analysis in this appendix follows very closely that of James and Wolf( 1998). AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 321 #39 Author's personal copy History and Solution of the Phase Problem 321

We will show that in order to deduce the values of the (generally complex) spectral degree of coherence µ(Q1, Q2, ω), one needs only to measure f (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 the spectral degree of coherence µ(Q1, Q2, ω) by the formula

f (T, ω) C (ω) cos ωT S (ω) sin ωT, (A.5) = 12 + 12 where

C (ω) R µ (ω) µ (ω) cos [β (ω)] , (A.6a) 12 = { 12 } = | 12 | 12 S (ω) I µ (ω) µ (ω) sin [β (ω)] , (A.6b) 12 = { 12 } = | 12 | 12 where R and I denote the real and the imaginary parts, respectively, and we 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 bandwidth for a few different values T1 and T2 of the time delay T. Eq. (A.5) then gives

f (T , ω) C (ω) cos(ωT ) S (ω) sin(ωT ), (A.7a) 1 = 12 1 + 12 1 f (T , ω) C (ω) cos(ωT ) S (ω) sin(ωT ). (A.7b) 2 = 12 2 + 12 2

From these two equations one may determine the functions C12(ω) and S12(ω), which, according to Eqs. (A.6), are just the real and the imaginary parts of the (generally complex) spectral degree of coherence µ12(ω). The solution is readily found to be

sin(ωT2)f (T1, ω) sin(ωT1)f (T2, ω) C12(ω) − , (A.8a) = sin[ω(T T )] 2 − 1 cos(ωT2)f (T1, ω) cos(ωT1)f (T2, ω) S12(ω) − , (A.8b) = − sin[ω(T T )] 2 − 1 provided that

sin[ω(T T )] 0. (A.9) 2 − 1 6=

It follows from Eqs. (A.8) that in terms of C12(ω) and S12(ω), the modulus µ (ω) of the spectral degree of coherence is then given by the | 12 | expression q µ (ω) [C (ω)]2 [S (ω)]2 (A.10) | 12 | = 12 + 12 AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 322 #40 Author's personal copy 322 Emil Wolf

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

C12(ω) cos [β12(ω)] , (A.11a) = √[C (ω)]2 [S (ω)]2 12 + 12 S12(ω) sin [β12(ω)] . (A.11b) = √[C (ω)]2 [S (ω)]2 12 + 12 For any two values of the time delays T1 and T2, one may expect that for certain frequencies, ω0 say, in the spectral band used, the condition (A.9) is violated; that is, for which sin [ω (T T )] 0. For such frequen- 0 2 − 1 = cies Eqs. (A.10) and (A.11) do not hold. One may overcome this difﬁculty by measuring the function f (T, ω) for another value of the time delay, say for T3, for which the condition (A.9) holds. Determination of the modulus and the phase of the spectral degree of coherence 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 of crystalline 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 crystal structure by means of X-rays.

Chemistry 1962 M. F. Perutz and J. C. Kendrew, for their studies of the structures of globular proteins.

Physiology and Medicine 1962 F. H. C. Crick, J. D. Watson, and M. H. F. Wilkins, for their discoveries concerning the molecular structure of nucleic acids and its signiﬁcance for information transfer in living material.

Chemistry 1982 A. Klug, for his development of crystallographic electron microscopy and his structural elucidation of biologically important nucleic acid–protein complexes. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 323 #41 Author's personal copy History and Solution of the Phase Problem 323

Chemistry 1985 H. A. Hauptman and J. Karle, for their outstanding achievements in the development 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 mechanism underlying the synthesis of adenosine triphosphate (ATP); and J. 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 ion channels. Chemistry 2006 R. Kornberg,¨ for his studies of the molecular basis of eukaryotic transcription. Chemistry 2009 V. Ramakrishnan, T. A. Steitz, and A. E. Yonath, for studies of the structure and 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 after him. Physics 1917 C. G. Barkla, for his discovery of the characteristic Roentgen radiation of the elements.

ACKNOWLEDGMENTS

I acknowledge with thanks permissions of the editors of Physical Review Journals, the American Institute of Physics publications, IUCr Journals, and the American Physical AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 324 #42 Author's personal copy 324 Emil Wolf

Society journals to reproduce several of the figures which appear in this article. I am obliged to Mr. Mayukh Lahiri for helpful comments and useful suggestions and to Dr. Mohamed Salem for assistance with locating pertinent references and preparing many of the figures. I am also grateful to Mr. Thomas Kern and Miss Krista Lombardo for assistance with the typing and checking the text. Research relating to the solution of the phase problem was supported by the U.S. Air Force Office of Scientific Research (AFOSR) under grant No. FA95500-08-1-0417.

REFERENCES

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FIGURE 9 F. H. C. Crick.

FIGURE 10 J. D. Watson. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 288 #6 Author's personal copy 288 Emil Wolf

FIGURE 6 W. H. Bragg.

FIGURE 7 W. L. Bragg. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 285 #3 Author's personal copy History and Solution of the Phase Problem 285

FIGURE 2 C. G. Barkla.

FIGURE 3 A. J. W. Sommerfeld. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 293 #11 Author's personal copy History and Solution of the Phase Problem 293

FIGURE 13 A. Klug.

FIGURE 14 H. Michel. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 294 #12 Author's personal copy 294 Emil Wolf

FIGURE 15 J. Deisenhofer.

FIGURE 16 R. Huber. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 295 #13 Author's personal copy History and Solution of the Phase Problem 295

FIGURE 17 P. D. Boyer.

FIGURE 18 J. E. Walker. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 296 #14 Author's personal copy 296 Emil Wolf

FIGURE 19 J. C. Skou.

FIGURE 20 P. Agre. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 297 #15 Author's personal copy History and Solution of the Phase Problem 297

FIGURE 21 R. MacKinnon.

FIGURE 22 R.Kornberg.¨ AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 298 #16 Author's personal copy 298 Emil Wolf

FIGURE 23 V. Ramakrishnan.

FIGURE 24 T. A. Steitz. AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 299 #17 Author's personal copy History and Solution of the Phase Problem 299

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 direction specified by a unit vector s0 is incident on the crystal (Figure 26). Assum- ing that the beam is unpolarized and making use of elementary scattering theory, one has the well-known Fourier relationship between the scattered ( ) field U ∞ (rs), in the far zone, in the direction specified by a unit vector s and the electron density distribution, ρ(r) throughout the crystal (Papas, 1965, p. 20 et seq.): ikr ( ) e U ∞ (rs) f (s, s ) . (2.2) = 0 r AIEP 12-ch07-283-326-9780123858610 2011/1/25 19:25 Page 303 #21 Author's personal copy History and Solution of the Phase Problem 303

FIGURE 29 M. F. Perutz.

FIGURE 30 J. C. Kendrew.