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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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    WKB wavefunctions for simple harmonics Masatsugu Sei Suzuki Department of Physics, SUNY at Binmghamton (Date: November 19, 2011) _________________________________________________________________________ Gregor Wentzel (February 17, 1898, in Düsseldorf, Germany – August 12, 1978, in Ascona, Switzerland) was a German physicist known for development of quantum mechanics. Wentzel, Hendrik Kramers, and Léon Brillouin developed the Wentzel– Kramers–Brillouin approximation in 1926. In his early years, he contributed to X-ray spectroscopy, but then broadened out to make contributions to quantum mechanics, quantum electrodynamics, and meson theory. http://en.wikipedia.org/wiki/Gregor_Wentzel _________________________________________________________________________ Hendrik Anthony "Hans" Kramers (Rotterdam, February 2, 1894 – Oegstgeest, April 24, 1952) was a Dutch physicist. http://en.wikipedia.org/wiki/Hendrik_Anthony_Kramers _________________________________________________________________________ Léon Nicolas Brillouin (August 7, 1889 – October 4, 1969) was a French physicist. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory. http://en.wikipedia.org/wiki/L%C3%A9on_Brillouin _______________________________________________________________________ 1. Determination of wave functions using the WKB Approximation 1 In order to determine the eave function of the simple harmonics, we use the connection formula of the WKB approximation. V x E III II I x b O a The potential energy is expressed by 1 V (x) m 2 x 2 . 2 0 The x-coordinates a and b (the classical turning points) are obtained as 2 2 a 2 , b 2 , m0 m0 from the equation 1 V (x) m 2 x2 , 2 0 or 1 1 m 2a 2 m 2b2 , 2 0 2 0 where is the constant total energy. Here we apply the connection formula (I, upward) at x = a.
  • Chapter 6 Free Electron Fermi Gas

    Chapter 6 Free Electron Fermi Gas

    理学院 物理系 沈嵘 Chapter 6 Free Electron Fermi Gas 6.1 Electron Gas Model and its Ground State 6.2 Thermal Properties of Electron Gas 6.3 Free Electrons in Electric Fields 6.4 Hall Effect 6.5 Thermal Conductivity of Metals 6.6 Failures of the free electron gas model 1 6.1 Electron Gas Model and its Ground State 6.1 Electron Gas Model and its Ground State I. Basic Assumptions of Electron Gas Model Metal: valence electrons → conduction electrons (moving freely) ü The simplest metals are the alkali metals—lithium, sodium, 2 potassium, cesium, and rubidium. 6.1 Electron Gas Model and its Ground State density of electrons: Zr n = N m A A where Z is # of conduction electrons per atom, A is relative atomic mass, rm is the density of mass in the metal. The spherical volume of each electron is, 1 3 1 V 4 3 æ 3 ö = = p rs rs = ç ÷ n N 3 è 4p nø Free electron gas model: Suppose, except the confining potential near surfaces of metals, conduction electrons are completely free. The conduction electrons thus behave just like gas atoms in an ideal gas --- free electron gas. 3 6.1 Electron Gas Model and its Ground State Basic Properties： ü Ignore interactions of electron-ion type (free electron approx.) ü And electron-eletron type (independent electron approx). Total energy are of kinetic type, ignore potential energy contribution. ü The classical theory had several conspicuous successes 4 6.1 Electron Gas Model and its Ground State Long Mean Free Path: ü From many types of experiments it is clear that a conduction electron in a metal can move freely in a straight path over many atomic distances.
  • Catalogue 51: Mostly New Acquisitions

    Catalogue 51: Mostly New Acquisitions

    Catalogue 51: Mostly New Acquisitions For the 48th California International Antiquarian Book Fair Oakland Marriott, City Center, Oakland February 6 – 8, 2015 Visit us at Booth 809 HistoryofScience.com Jeremy Norman & Co., Inc. P.O. Box 867 Novato, CA 94948 Voice: (415) 892-3181 Fax: (415) 276-2317 Email: [email protected] Copyright ©2015 by Jeremy Norman & Co., Inc. / Historyofscience.com First Edition of the First English Sex Manual — One of Three Known Complete Copies 1. [Pseudo-Aristotle]. Aristoteles master-piece, or the secrets of generation displayed in all the parts thereof . 12mo. [4, including woodcut frontispiece], 190, [2, blank], [12, including 6 wood- cut illustrations, one a repeat of the frontispiece]pp. Title leaf is a cancel. London: Printed for J. How, 1684. 144 x 85 mm. Early sheep, worn, binding separating from text block; preserved in a cloth box. Occasional worming and staining, edges frayed, woodcuts with holes, tears and chips slightly affecting images, last woodcut with marginal loss affecting a few words of text, a few leaves slightly cropped, but a good, unsophisticated copy of a book that was mostly read out of existence. $65,000 First Edition. Aristotle’s Masterpiece (neither by Aristotle nor a masterpiece) was the first sex manual in English, providing its readers with practical advice on copulation, conception, pregnancy and birth. This anony- mous, inexpensively printed work proved to be enormously popular: At least three editions were issued by J. How in 1684 (see ESTC and below in our description), and it went through well over 100 editions in the following two centuries.Versions even continued to be published into the early twentieth century, with one appearing as late as 1930! Although Aristotle’s Masterpiece was not intended as pornography, its frank discussion of sex and reproduction was seen as unfit for polite society; the book was often issued under false imprints and sold “under the table.” The publication history of the work is discussed in some detail in Roy Porter and Lesley Hall’s The Facts of Life (pp.
  • Otto Stern Annalen 4.11.11

    Otto Stern Annalen 4.11.11

    (To be published by Annalen der Physik in December 2011) Otto Stern (1888-1969): The founding father of experimental atomic physics J. Peter Toennies,1 Horst Schmidt-Böcking,2 Bretislav Friedrich,3 Julian C.A. Lower2 1Max-Planck-Institut für Dynamik und Selbstorganisation Bunsenstrasse 10, 37073 Göttingen 2Institut für Kernphysik, Goethe Universität Frankfurt Max-von-Laue-Strasse 1, 60438 Frankfurt 3Fritz-Haber-Institut der Max-Planck-Gesellschaft Faradayweg 4-6, 14195 Berlin Keywords History of Science, Atomic Physics, Quantum Physics, Stern- Gerlach experiment, molecular beams, space quantization, magnetic dipole moments of nucleons, diffraction of matter waves, Nobel Prizes, University of Zurich, University of Frankfurt, University of Rostock, University of Hamburg, Carnegie Institute. We review the work and life of Otto Stern who developed the molecular beam technique and with its aid laid the foundations of experimental atomic physics. Among the key results of his research are: the experimental test of the Maxwell-Boltzmann distribution of molecular velocities (1920), experimental demonstration of space quantization of angular momentum (1922), diffraction of matter waves comprised of atoms and molecules by crystals (1931) and the determination of the magnetic dipole moments of the proton and deuteron (1933). 1 Introduction Short lists of the pioneers of quantum mechanics featured in textbooks and historical accounts alike typically include the names of Max Planck, Albert Einstein, Arnold Sommerfeld, Niels Bohr, Max von Laue, Werner Heisenberg, Erwin Schrödinger, Paul Dirac, Max Born, and Wolfgang Pauli on the theory side, and of Wilhelm Conrad Röntgen, Ernest Rutherford, Arthur Compton, and James Franck on the experimental side. However, the records in the Archive of the Nobel Foundation as well as scientific correspondence, oral-history accounts and scientometric evidence suggest that at least one more name should be added to the list: that of the “experimenting theorist” Otto Stern.