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5 X-Ray Crystallography Introductory biophysics A. Y. 2016-17 5. X-ray crystallography and its applications to the structural problems of biology Edoardo Milotti Dipartimento di Fisica, Università di Trieste The interatomic distance in a metallic crystal can be roughly estimated as follows. Take, e.g., iron • density: 7.874 g/cm3 • atomic weight: 56 3 • molar volume: VM = 7.1 cm /mole then the interatomic distance is roughly VM d ≈ 3 ≈ 2.2nm N A Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 The atomic lattice can be used a sort of diffraction grating for short-wavelength radiation, about 100 times shorter than visible light which is in the range 400-750 nm. Since hc 2·10−25 J m 1.24 eV µm E = ≈ ≈ γ λ λ λ 1 nm radiation corresponds to about 1 keV photon energy. Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 !"#$%&'$(")* S#%/J&T&U2*#<.%&CKET3&VG$GG./"#%G3&W.%-$/; +(."J&AN&>,%()&CTDB3&S.%)(/3&X.1*&W.%-$/; Y#<.)&V%(Z.&(/&V=;1(21&(/&CTCL&[G#%&=(1&"(12#5.%;&#G&*=.& "(GG%$2*(#/&#G&\8%$;1&<;&2%;1*$)1] 9/(*($));&=.&1*:"(."&H(*=&^_/F*./3&$/"&*=./&H(*=&'$`&V)$/2I&(/& S.%)(/3&H=.%.&=.&=$<()(*$*."&(/&CTBD&H(*=&$&*=.1(1&[a<.% "(.& 9/*.%G.%./Z.%12=.(/:/F./ $/&,)$/,$%$)).)./ V)$**./[?& 7=./&=.&H#%I."&$*&*=.&9/1*(*:*.&#G&7=.#%.*(2$)&V=;1(213&=.$"."& <;&>%/#)"&Q#--.%G.)"3&:/*()&=.&H$1&$,,#(/*."&G:))&,%#G.11#%&$*& *=.&4/(5.%1(*;&#G&0%$/IG:%*&(/&CTCL3&H=./&=.&$)1#&%.2.(5."&=(1& Y#<.)&V%(Z.?& !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE Arnold Sommerfeld (1868-1951) ... Four of Sommerfeld's doctoral students, Werner Heisenberg, Wolfgang Pauli, Peter Debye, and Hans Bethe went on to win Nobel Prizes, while others, most notably, Walter Heitler, Rudolf Peierls, Karl Bechert, Hermann Brück, Paul Peter Ewald, Eugene Feenberg, Herbert Fröhlich, Erwin Fues, Ernst Guillemin, Helmut Hönl, Ludwig Hopf, Adolf KratZer, Otto Laporte, Wilhelm LenZ, Karl Meissner, Rudolf Seeliger, Ernst C. Stückelberg, Heinrich Welker, Gregor WentZel, Alfred Landé, and Léon Brillouin became famous in their own right. Three of Sommerfeld's postgraduate students, Linus Pauling, Isidor I. Rabi and Max von Laue, won Nobel Prizes, and ten others, William Allis, Edward Condon, Carl Eckart, Edwin C. Kemble, William V. Houston, Karl HerZfeld, Walther Kossel, Philip M. Morse, Howard Robertson, and Wojciech RubinowicZ went on to become famous in their own right. Walter Rogowski, an undergraduate student of Sommerfeld at RWTH Aachen, also went on to become famous in his own right. Max Born believed Sommerfeld's abilities included the "discovery and development of talents." Albert Einstein told Sommerfeld: "What I especially admire about you is that you have, as it were, pounded out of the soil such a large number of young talents." Sommerfeld's style as a professor and institute director did not put distance between him and his colleagues and students. He invited collaboration from them, and their ideas often influenced his own views in physics. He entertained them in his home and met with them in cafes before and after seminars and colloquia. Sommerfeld owned an alpine ski hut to which students were often invited for discussions of physics as demanding as the sport. ... from https://en.wikipedia.org/wiki/Arnold_Sommerfeld Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 ... Such was the state of affairs as, one evening in February 1912, P. P. Ewald came to visit me. (...) he was faced at that time with certain difficulties and came to me with a request for advice. Now it was not, however, possible for me to assist him at that time. But during the conversation I was suddenly struck by the obvious question of the behaviour of waves which are short by comparison with the lattice-constants of the space lattice. And it was at that point that my intuition for optics suddenly gave me the answer: lattice spectra would have to ensue. The fact that the lattice constant in crystals is of an order of 10-8 cm was sufficiently known from the analogy with other interatomic distances in solid and liquid substances, and, in addition, this could easily be argued from the density, molecular weight and the mass of the hydrogen atom which, just at that time, had been particularly well determined. The order of X-ray wavelengths was estimated by Wien and Sommerfeld at 10-9 cm. Thus the ratio of wavelengths and lattice constants was extremely favourable if X-rays were to be transmitted through a crystal. I immediately told Ewald that I anticipated the occurrence of interference phenomena with X-rays. ... (from von Laue’s Nobel Lecture) Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 R#/&g$:.n1&.`,.%(-./*$)&)$;#:* !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE 9-$F.&*$I./&(/&CTCA&#G&$/&\8%$;& (/*.%G.%./2.&#G&$&Z(/2&<)./".&2%;1*$)?& o(/2&<)./".&bo/Q3&1,=$).%(*.c&H$1&#/.& #G&*=.&G(%1*&2%;1*$)1&(/5.1*(F$*."&<;& g$:.3&0%(."%(2=&$/"&e/(,,(/F? bV=#*#F%$,=J&+.:*12=.1 ':1.:-c !"#$#%&"'()#%*))+,#-.&%#/)%0(-1# 23 %.4*)5#-.$-#%4(&5%#$/)#4/5)/)5#$-#-.)#64()70($/#()8)( 93 &-#%)--()5#$((#:0)%-&4"%#4"#-.)#"$-0/)#4;#<=/$>% !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE “Dear Mr. Laue! I cordially salute you on your marvelous success. Your experiment counts among the most glorious that Physics has seen so far.” Albert Einstein (on a postcard to von Laue, dated June 1912) Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 /:,$8:--:"0$9*',?$@,"<< S#%/J&A&p:);&CKDA3&X(F*#/3&4/(*."&e(/F"#- +(."J&CA&'$%2=&CTLA3&g#/"#/3&4/(*."&e(/F"#- Y#<.)&V%(Z.&(/&V=;1(21&(/&CTCO&[G#%&*=.(%&1.%5(2.1&(/&*=.&$/$);1(1& #G&2%;1*$)&1*%:2*:%.&<;&-.$/1&#G&\8%$;1] !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE 8:--:"0$("B,*'5*$@,"<< S#%/J&NC&'$%2=&CKTB3&>".)$(".3&>:1*%$)($ +(."J&C&p:);&CTEC3&9,1H(2=3&4/(*."&e(/F"#- Y#<.)&V%(Z.&(/&V=;1(21&(/&CTCO&[G#%&*=.(%&1.%5(2.1&(/&*=.&$/$);1(1& #G&2%;1*$)&1*%:2*:%.&<;&-.$/1&#G&\8%$;1] !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE Y$h) 2%;1*$)1 !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE Q,=$).%(*.3&+#)#-(*.3&h=$)2#,;%(*.?&g#2$)(*;J&p#,)(/&0(.)"3&7%(8Q*$*.& +(1*%(2*3&p$1,.%&h#:/*;3&'(11#:%(3&4Q>& b=**,JMM./?H(I(,."($?#%FMH(I(MQ,=$).%(*.c !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE Crystal structure In order to proceed, and explain the contributions by von Laue and the Braggs, we must describe order in crystals basis lattice Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 a1 a2 a1 and a2 are the primitive lattice vectors, and the translation vectors T = u1a1 + u2a2 (u1,2 integers) generate the whole lattice Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 a1 a2 This pair of a1 and a2 is not primitive because they do not generate the whole lattice Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 a1 The primitive vectors also define the crystal axes a2 a 1 The associated parallelogram is the primitive cell a2 Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 a 1 a3 The primitive vectors also define the crystal axes a2 The associated parallelepiped a 1 a3 is the primitive cell a2 Volume of primitive cell: V = a ⋅a × a 3D 1 2 3 Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 7=.%.&$%.&-$/;&*;,.1&#G&)$**(2.1?&7=.;&$%.&1;1*.-$*(2$));& 2)$11(G(."&<;&"(12%.*.&1,$2.&F%#:,1?& 7=.&2#--#/&/#-./2)$*:%.&(1&*=$*&#G&*=.&S%$5$(1 g$**(2.1?& !`$-,).1 L10'>030' T.@7.(-?1*> %0@E>-$'&70' !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE 7=.&2:<(2&)$**(2.1 Q(-,).&2:<(2&b12c S#";82./*.%."&2:<(2&b<22c&&&0$2.82./*.%."&2:<(2& bG22c !"#$%"#&'()#**(&8 9/*%#":2*#%;&<(#,=;1(21&8 >?@?&ABCD8CE Reciprocal lattice vectors a2 × a3 a3 × a1 a1 × a2 b1 = 2π ; b2 = 2π ; b3 = 2π ; a1 ⋅a2 × a3 a2 ⋅a3 × a1 a3 ⋅a1 × a2 These vectors define the reciprocal lattice and have the property ai ⋅b j = 2πδ ij and they define a reciprocal lattice, by means of the translation vectors G = v1b1 + v2b2 + v3b3 Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 Example: reciprocal lattice to a simple cubic (sc) lattice ⎛ a ⎞ ⎛ 0 ⎞ ⎛ 0 ⎞ a = ⎜ 0 ⎟ a = ⎜ a ⎟ a = ⎜ 0 ⎟ 1 ⎜ ⎟ 2 ⎜ ⎟ 3 ⎜ ⎟ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ a ⎠ ⎛ 1 a ⎞ ⎛ 0 ⎞ ⎛ 0 ⎞ b = ⎜ ⎟ b = ⎜ 1 a ⎟ b = ⎜ 0 ⎟ 1 ⎜ 0 ⎟ 2 ⎜ ⎟ 3 ⎜ ⎟ ⎝⎜ 0 ⎠⎟ ⎝⎜ 0 ⎠⎟ ⎝⎜ 1 a ⎠⎟ The reciprocal lattice is again sc; the sc lattice is self-dual. Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 Example: body-centered cubic lattice (bcc) ed. (Wiley, 2005) th from C. Kittel, “Introduction to Solid State Physics, 8 Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 a a a a1 = (−x + y + z); a2 = (x − y + z); a3 = (x + y − z); 2 2 2 a2 ⌢ ⌢ a2 ⌢ ⌢ a2 ⌢ ⌢ a2 × a3 = (y + z); a3 × a1 = (x + z); a1 × a2 = (x + y); 2 2 2 a3 a1 ⋅(a2 × a3 ) = 2 2π ⌢ ⌢ 2π ⌢ ⌢ 2π ⌢ ⌢ b1 = (y + z); b2 = (x + z); b3 = (x + y); a a a Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 2π 2π 2π b1 = (y + z); b2 = (x + z); b3 = (x + y); a a a The reciprocal lattice vectors of the bcc lattice correspond to the lattice vectors of the fcc lattice. Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 Diffraction of waves by crystals a b θ θ crystal surface d θ Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 path difference between rays a and b: 2d sinθ constructive interference condition: 2d sinθ = nλ a (Bragg law) b θ θ crystal surface d θ Edoardo Milotti - Introductory biophysics - A.Y. 2016-17 Remarks: • the scattering cross-section is small, thus X-rays penetrate the crystal and are scattered by different planes • the scattering cross-section is small, thus the X-ray beam is not significantly attenuated by previous crystal planes • X-rays are scattered by electrons, and they are scattered more where the electron density is higher • the Bragg law implies that nλ 2d = sinθ ≤ 1 ⇒ n ≤ 2d λ Edoardo Milotti - Introductory biophysics - A.Y.
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