DOCSLIB.ORG

A.1 DERIVATION of the X-RAY FORM FACTOR Ρ Ρ Dr

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A.1 DERIVATION of the X-RAY FORM FACTOR Ρ Ρ Dr Appendix 1 A.1 DERIVATION OF THE X-RAY FORM FACTOR An X-ray beam is incident in the direction defined by the unit vector s0 . We shall consider radiation scattered in the direction defined by the unit vector s [7] The path difference between the rays is: − = − OR QP S.r S.r 0 … (A.1-1) π Phase difference = 2 × path difference λ Where λ is the wavelength of the incident radiation Then 2π v v v Phase difference = s.(r − s ) … (A.1-2) λ 0 Hence the scattered amplitude from an element dr of electron density ρ(r) is given by: 2πi vvv − )ss.(r Scattered amplitude = ρ e)r( λ 0 dr … (A.1-3) v v = ρ e)r( r.dik dr … (A.1-4) 2π v v v Where k = , d = s − s λ 0 Thus, the total scattered amplitude is obtained by integration over the whole electron density distribution as: v v Total scattered amplitude ∝ ∫ ρ e)r( r.dik dr … (A.1-5) 60 Appendix 1 Let us write the atomic scattering factor )s(f from an electron distribution of density ρ( r )as s = kd For closed shell atom ρ )r( is spherically symmetrical. The angular integrations can then be performed by spherical polar coordinates θ ,,r φ . Then the volume element becomes r 2 dr sin θdθdφ , and since s.r = sr cos θ we can integrate over φ from 0 to 2π to obtain. ∞ π )s(f = 2π ∫ r 2 ρ dr)r( ∫ e cosisr θ sin θdθ … (A.1-6) 0 0 The integration over θ can be carried out, with the result ∞ sin sr )s(f = ∫ ρ )r( 4πr 2 dr … (A.1-7) 0 sr But ρ )r( 4πr2 = D )r( … (A.1-8) Then ∞ sin sr )s(f = ∫ )r(D dr … (A.1-9) 0 sr Q P ρ dr)r( r S0 S O R Fig.(A.1-1) x-ray scattering 61 Appendix 1 The solution of equation (A.1-9) by mathematical analysis is: For K-shell − ζ +ζ + = ( i j ).r ( ni n j ) D( r ) ∑∑ N i N j e r … (A.1-10) i j ( sr )3 ( sr )5 … (A.1-11) sin( sr ) = sr − + − ...... 3! 5! By substituting equation (A.1-11) and (A.1-10) in equation (A.1-9) 3 5 − ( sr ) + ( sr ) − ∞ sr ...... − ζ+ ζ + = ( i j ).r ( ni n j ) 3! 5! )s(f ∫ ∑∑ ci c j N i N j e r dr 0 i j sr … (A.1-12) = x ∑∑ci c j N i N j … (A.1-13) i j = ζ + ζ m i j … (A.1-14) = + … (A.1-15) p ni n j ∞ 2 4 − ( sr ) ( sr ) )s(f = x∫ r p e m.r (1 − + − ......) dr 0 3! 5! ∞ 2 4 p −m.r s p+2 −m.r s p+4 −m.r = x∫ r e − r e + r e dr 0 3! 5! 2 + 4 + = !p − s ( p 2 )! + s ( p 4 )! x p+1 p+3 p+5 m 3! m 5! m 62 Appendix 1 ( p + 2 k( ))! s 2( k ) = − k )s(f x∑( 1) p+2 ( k )+1 … (A.1-16) k =0 ( 2( k ) + 1)! m (n + n + 2(k)) s! 2()k = ()− k i j )s(f ∑∑c c N N ∑ 1 + + ()+ i j i j ni n j 2 k 1 i j k =0 + ()ζ + ζ ( 2( k ) 1)! i j … (A.1-17) The total scattering form factor is given by: For Li-like ions ∞ sin( sr ) = 1 … (A.1-18) )s(f total 3∫ DT ( r1 ) dr 1 0 sr 1 = 1 [ + + ] D r( ) D α β r( ) D α α r( ) D β α r( ) T 1 3 K K 1 KL 1 KL 1 (n + n + 2(p)) s! 2()p = ()− p i j )s(f 2∑∑c c N N ∑ 1 + + ()+ i j i j ni n j 2 p 1 i j p=0 + ()ζ + ζ ( 2( k ) 1)! i j (n + n + 2( p )) !s 2( p ) + ()− p i j ∑∑d d N N ∑ 1 + + + i j i j ni n j 2( p ) 1 i j p=0 + ()ζ + ζ ( 2( k ) 1)! i j … (A.1-19) For Be-like ions ∞ sin( sr ) = 1 )s(f total 4∫ DT r( 1 ) dr 1 … (A.1-20) 0 sr 1 63 Appendix 1 1 D )r( = [D r( )+ D )r( + D )r( + D r( )+ D )r( + D r( )] T 1 6 KKα β 1 KLα α 1 KLβ α 1 KLα β 1 KLβ β 1 LLα β 1 (n + n + 2(p)) s! 2()p = ()− p i j )s(f 3∑∑c c N N ∑ 1 + + ()+ i j i j ni n j 2 p 1 i j p=0 + ()ζ + ζ ( 2( k ) 1)! i j ()n + n + 2()p s! 2()p + ()− p i j 3∑∑d d N N ∑ 1 + + ()+ i j i j ni n j 2 p 1 i j p=0 + ()ζ + ζ ( 2( k ) 1)! i j … (A.1-21) 64 Appendix 2 A.2 Γ DERIVATIONS THE TWO-PARTICLE DENSITY HF (,1 2 ) FOR KαKβ , K αLα , K βLα , K αLβ , K βLβ and, L αLβ Γ The two-particle density function HF (,1 2 )for the individual shells for Li- Like ions can be written as [51]: 3 Γ = 1 ∗ ij (1,2 ) ∑ Aij Aij 2 i< j And for Be-like ions can be written as: 6 Γ = 1 ∗ ij (1,2 ) ∑ Aij Aij 2 i< j Γ i j ij (,1 2 ) Shell Γ 1 2 12 (,1 2 ) K α K β Γ 1 3 13 (,1 2 ) Kα Lα Γ 2 3 23 (,1 2 ) K β Lα Γ 1 4 14 (,)1 2 K α Lβ Γ 2 4 24 (,)1 2 K β Lβ Γ 3 4 34 (,)1 2 Lα Lβ = φ φ − φ … φ … (A.2.1) Aij i (1) j ( 2 ) j (1) i ( 2 ) • ≡ In this work Aij Aij due to the S-state symmetry. 65 Appendix 2 (1) Kα Kβ Shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) −φ (1)φ ( 2 ) …(A.2.2) 12 2 1 2 2 1 1 2 Γ = [ϕ α ϕ β −φ β φ α ] …(A.2.4) 12 1s (1) (1) 1s ( 2 ) ( 2 ) 1s (1) (1) 1s ( 2 ) ( 2 ) 2 Since : ϕ ( 1 ) = R ( 1 )Y ( 1 ) s1 s1 s1 ϕ = …(A.2.5) s1 ( 2 ) R s1 ( 2 Y) s1 ( 2 ) After integrated over all spins in equation (A.2.4) and substituted equation (A.2.5) into (A.2.4) we get: Γ ′ = [ ] 2 ….(A.2.6) 12 ( spinless ) R1s (1 Y) 1s (1)R1s ( 2 Y) 1s ( 2 ) now we can integrate over all angular part to get: Γ = 2 2 '12 (1,2 ) R1s (1 R) 1s ( 2 ) …(A.2.7) (2) Kα Lα Shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) 13 2 1 3 3 1 …(A.2.8) Γ = 1 [ϕ α ϕ α − ϕ α ϕ α ]2 13 (1,2 ) 1s (1) (1) 2s ( 2 ) ( 2 ) 2s (1) (1) 1s ( 2 ) ( 2 ) 2 ...(A.2.9) 66 Appendix 2 ϕ (1)ϕ ( 2 )−ϕ (1)ϕ ( 2 ) 2 Γ = []α α 2 1s 2s 2s 1s 13 ,( 21 ) (1) ( 2 ) …(A.2.10) 2 since ϕ = 1 s ( 1 ) R 1 s ( 1 Y) 1 s ( 1 ) ϕ ( 2 ) = R ( 2 Y) ( 2 ) 1 s 1 s 1 s …(A.2.11) ϕ (1) = R (1 Y) (1) 2 s 2 s 2 s ϕ = 2s ( 2 ) R2s ( 2 Y) 2s ( 2 ) And ∫α(1)β(1 d) σ = 0 ∫α(1)α(1 d) σ = 1 ∫ β(1)α(1 d) σ = 0 …(A.2.12) ∫ β(1)β(1 d) σ = 1 by substituted the equation (A.2.11) into (A.2.10) and integrated over all spins using equation (A.2.12) we get: 1 2 Γ ′( spinless ) = [R (1 Y) (1 R) ( 2 Y) ( 2 ) − R (1 Y) (1 R) ( 2 Y) ( 2 )] 13 2 1s 1 s 2s 2s 2s 2s 1s 1s ….(A.2.13) Since θ ϕ = θ ϕ …(A.2.14) Y1s ( , ) Y2s ( , ) 1 2 2 Γ ′(1,2 ) = [][]Y (1 Y) ( 2 ) R (1 R) ( 2 ) − R (1 R) ( 2 ) 13 2 1s 2s 1s 2s 2s 1s …(A.2.15) Now we can integrate over all the angular part so that we can get: 67 Appendix 2 R (1 R) (2)− R (1 R) (2) 2 Γ = 1s 2s 2s 1s ' 13 ( 1 , 2 ) …(A.2.16) 2 1 Γ )2,1(' = [R2 )1( R2 )2( + R2 )1( R2 )2( 13 2 1s 2s 2s 1s − ] 2R1s )1( R2s )1( R2s )2( R1s )2( ….(A.2.17) (3) Kβ Lα Shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) …(A.2.18) 23 2 2 3 3 2 1 2 Γ (1,2 ) = [ϕ (1)β(1)ϕ ( 2 )α( 2 ) − ϕ (1)α(1)ϕ ( 2 )β( 2 )] 23 2 1s 2s 2s 1s ..(A.2.19) By substituted equation (A.2.11) into (A.2.19) and integrated over all spins using equation (A.2.12) we get: 1 Γ ′ ( spinless ) = [R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 ) + R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 )] 23 2 1 s 1 s 2 s 2 s 2 s 2s 1s 1s …(A.2.20) Using eq.(A.2.14) and integrate over all angular part to get: Γ / = 1 []2 2 + 2 2 23 ( 1 , 2 ) R 1 s ( 1 R) 2 s ( 2 ) R 2 s ( 1 R) 1 s ( 2 ) …(A.2.21) 2 (4) Kα Lβ shell 68 Appendix 2 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) 14 2 1 4 4 1 ...(A.2.22) 1 2 Γ (1,2 ) = [ϕ (1)α(1)ϕ ( 2 )β( 2 ) − ϕ (1)β(1)ϕ ( 2 )α( 2 )] 14 2 1s 2s 2s 1s ..(A.2.23) By substituting equation (A.2.11) into (A.2.23) and integrated over all spins using equation (A.2.12) we get: 1 Γ ′ ( spinless ) = [R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 ) + R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 )] 14 2 1s 1s 2s 2s 2s 2s 1s 1s ...(A.2.24) using equation (A.2.14) and integrate over all angular part to get: 1 Γ ' (1,2 ) = []R 2 (1 R) 2 ( 2 ) + R 2 (1 R) 2 ( 2 ) ...(A.2.25) 14 2 1s 2s 2s 1s (5) K β Lβ Shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) ...(A.2.26) 24 2 2 4 4 2 1 2 Γ (1,2 ) = [ϕ (1)β(1)ϕ ( 2 )β( 2 ) − ϕ (1)β(1)ϕ ( 2 )β( 2 )] 24 2 1s 2s 2s 1s ...(A.2.27) 69 Appendix 2 By substituting equation (A.2.11) into (A.2.27) and integrated over all spins using equation (A.2.12) we get: 1 Γ ′ ( spinless ) = [R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 ) + R 2 (1 Y) 2 (1 R) 2 ( 2 Y) 2 ( 2 )] 24 2 1s 1s 2s 2s 2s 2s 1s 1s ...(A.2.28) using equation (A.2.14) and integrate over all angular part to get: 1 Γ ,(' 21 ) = [R 2 (1 R) 2 ( 2 ) + R 2 (1 R) 2 ( 2 ) 24 2 1s 2s 2s 1s − ] ...(A.2.29) 2R1s (1 R) 2s (1 R) 2s ( 2 R) 1s ( 2 ) (6) Lα Lβ shell : 1 2 Γ (1,2 ) = []φ (1)φ ( 2 ) − φ (1)φ ( 2 ) ...(A.2.30) 34 2 3 4 4 3 1 2 Γ (1,2 ) = [ϕ (1)α(1)ϕ ( 2 )β( 2 ) − φ (1)β(1)φ ( 2 )α( 2 )] 34 2 2 s 2 s 2s 2s ...(A.2.31) Since : ϕ (1)=R (1 Y) (1) 2s 2s 2s …(A.2.32) ϕ = 2s (2) R2s (2 Y) 2s (2) 70 Appendix 2 After integrated over all spins in equation (A.2.31) and substituted equation (A.2.32) into (A.2.31) we get: Γ ′ = [ ] 2 ….(A.2.33) 34 ( spinless ) R2s (1 Y) 2s (1)R2s ( 2 Y) 2s ( 2 ) now we can integrate over all angular part to get: Γ = 2 2 '34 (1,2 ) R2s (1 R) 2s ( 2 ) …(A.2.34) 71 References References 1.
Load more

Recommended publications
  • Atomic Form Factor Calculations of S-States of Helium
    American Journal of Modern Physics 2019; 8(4): 66-71 http://www.sciencepublishinggroup.com/j/ajmp doi: 10.11648/j.ajmp.20190804.12 ISSN: 2326-8867 (Print); ISSN: 2326-8891 (Online) Atomic Form Factor Calculations of S-states of Helium Saïdou Diallo *, Ibrahima Gueye Faye, Louis Gomis, Moustapha Sadibou Tall, Ismaïla Diédhiou Department of Physics, Faculty of Sciences, University Cheikh Anta Diop, Dakar, Senegal Email address: *Corresponding author To cite this article: Saïdou Diallo, Ibrahima Gueye Faye, Louis Gomis, Moustapha Sadibou Tall, Ismaïla Diédhiou. Atomic Form Factor Calculations of S-states of Helium. American Journal of Modern Physics . Vol. 8, No. 4, 2019, pp. 66-71. doi: 10.11648/j.ajmp.20190804.12 Received : September 12, 2019; Accepted : October 4, 2019; Published : October 15, 2019 Abstract: Variational calculations of the helium atom states are performed using highly compact 26-parameter correlated Hylleraas-type wave functions. These correlated wave functions used here yield an accurate expectation energy values for helium ground and two first excited states. A correlated wave function consists of a generalized exponential expansion in order to take care of the correlation effects due to N-corps interactions. The parameters introduced in our model are determined numerically by minimization of the total atomic energy of each electronic configuration. We have calculated all integrals analytically before dealing with numerical evaluation. The 1S2 11S and 1 S2S 21, 3 S states energies, charge distributions and scattering atomic form factors are reported. The present work shows high degree of accuracy even with relative number terms in the trial Hylleraas wave functions definition.
    [Show full text]
  • With Synchrotron Radiation Small-Angle X-Ray Scattering (SAXS)
    Small-angleSmall-angle X-rayX-ray scatteringscattering (SAXS)(SAXS) withwith synchrotronsynchrotron radiationradiation Martin Müller Institut für Experimentelle und Angewandte Physik der Christian-Albrechts-Universität zu Kiel • Introduction to small-angle scattering • Instrumentation • Examples of research with SAXS Small-angleSmall-angle X-rayX-ray scatteringscattering (SAXS)(SAXS) withwith synchrotronsynchrotron radiationradiation • Introduction to small-angle scattering • Instrumentation • Examples of research with SAXS WhatWhat isis small-anglesmall-angle scattering?scattering? elastic scattering in the vicinity of the primary beam (angles 2θ < 2°) at inhomogeneities (= density fluctuations) typical dimensions in the sample: 0.5 nm (unit cell, X-ray diffraction) to 1 µm (light scattering!) WhatWhat isis small-anglesmall-angle scattering?scattering? pores fibres colloids proteins polymer morphology X-ray scattering (SAXS): electron density neutron scattering (SANS): contrast scattering length OnOn thethe importanceimportance ofof contrastcontrast …… ScatteringScattering contrastcontrast isis relativerelative Babinet‘s principle two different structures may give the same scattering: 2 I(Q) ∝ (ρ1 − ρ2 ) r 4π scattering vector Q = sinθ λ 2θ DiffractionDiffraction andand small-anglesmall-angle scatteringscattering cellulose fibre crystal structure scattering contrast crystals - matrix M. Müller, C. Czihak, M. Burghammer, C. Riekel. J. Appl. Cryst. 33, 817-819 (2000) DiffractionDiffraction andand small-anglesmall-angle scatteringscattering
    [Show full text]
  • Data on the Atomic Form Factor: Computation and Survey Ann T
    • Journal of Research of the National Bureau of Standards Vol. 55, No. 1, July 1955 Research Paper 2604 Data on the Atomic Form Factor: Computation and Survey Ann T. Nelms and Irwin Oppenheim This paper presents the results of ca lculations of atomic form factors, based on tables of electron charge distributions compu ted from Hartree wave functions, for a wide range of atomic numbers. Compu tations of t he form factors for fi ve elements- carbon, oxygen, iron, arsenic, and mercury- are presented and a method of interpolation for other atoms is indi­ cated. A survey of previous results is given and the relativistic theory of Rayleigh scattering is reviewed. Comparisons of the present results with previous computations and with some sparse experimental data are made. 1. Introduction The form factor for an atom of atomic number Z is defined as the matrix: element: The atomic form factor is of interest in the calcu­ lation of Rayleigh scattering of r adiation and coher­ en t scattering of charged par ticles from atoms in the region wherc relativist.ic effects can be n eglected. The coheren t scattering of radiation from an atom where 0 denotes the ground state of the atom , and consists of Rayleigh scattering from the electrons, re onant electron scattering, nuclear scattering, and ~ is the vector distance of the jth clectron from the Delbrucl- scattering. When the frequency of the nucleus. For a spherically symmetric atom incident photon approaches a resonant frequency of the atom, large regions of anomalous dispersion occur fC0 = p (r)s i ~rkr dr, (4) in which the form factor calculations are not sufficient 1'" to describe th e coherent scattering.
    [Show full text]
  • Solution Small Angle X-Ray Scattering : Fundementals and Applications in Structural Biology
    The First NIH Workshop on Small Angle X-ray Scattering and Application in Biomolecular Studies Open Remarks: Ad Bax (NIDDK, NIH) Introduction: Yun-Xing Wang (NCI-Frederick, NIH) Lectures: Xiaobing Zuo, Ph.D. (NCI-Frederick, NIH) Alexander Grishaev, Ph.D. (NIDDK, NIH) Jinbu Wang, Ph.D. (NCI-Frederick, NIH) Organizer: Yun-Xing Wang (NCI-Frederick, NIH) Place: NCI-Frederick campus Time and Date: 8:30am-5:00pm, Oct. 22, 2009 Suggested reading Books: Glatter, O., Kratky, O. (1982) Small angle X-ray Scattering. Academic Press. Feigin, L., Svergun, D. (1987) Structure Analysis by Small-angle X-ray and Neutron Scattering. Plenum Press. Review Articles: Svergun, D., Koch, M. (2003) Small-angle scattering studies of biological macromolecules in solution. Rep. Prog. Phys. 66, 1735- 1782. Koch, M., et al. (2003) Small-angle scattering : a view on the properties, structures and structural changes of biological macromolecules in solution. Quart. Rev. Biophys. 36, 147-227. Putnam, D., et al. (2007) X-ray solution scattering (SAXS) combined with crystallography and computation: defining accurate macromolecular structures, conformations and assemblies in solution. Quart. Rev. Biophys. 40, 191-285. Software Primus: 1D SAS data processing Gnom: Fourier transform of the I(q) data to the P(r) profiles, desmearing Crysol, Cryson: fits of the SAXS and SANS data to atomic coordinates EOM: fit of the ensemble of structural models to SAXS data for disordered/flexible proteins Dammin, Gasbor: Ab initio low-resolution structure reconstruction for SAXS/SANS data All can be obtained from http://www.embl-hamburg.de/ExternalInfo/Research/Sax/software.html MarDetector: 2D image processing Igor: 1D scattering data processing and manipulation SolX: scattering profile calculation from atomic coordinates Xplor/CNS: high-resolution structure refinement GASR: http://ccr.cancer.gov/staff/links.asp?profileid=5546 Part One Solution Small Angle X-ray Scattering: Basic Principles and Experimental Aspects Xiaobing Zuo (NCI-Frederick) Alex Grishaev (NIDDK) 1.
    [Show full text]
  • Problems for Condensed Matter Physics (3Rd Year Course B.VI) A
    1 Problems for Condensed Matter Physics (3rd year course B.VI) A. Ardavan and T. Hesjedal These problems are based substantially on those prepared anddistributedbyProfS.H.Simonin Hilary Term 2015. Suggested schedule: Problem Set 1: Michaelmas Term Week 3 • Problem Set 2: Michaelmas Term Week 5 • Problem Set 3: Michaelmas Term Week 7 • Problem Set 4: Michaelmas Term Week 8 • Problem Set 5: Christmas Vacation • Denotes crucial problems that you need to be able to do in your sleep. *Denotesproblemsthatareslightlyharder.‡ 2 Problem Set 1 Einstein, Debye, Drude, and Free Electron Models 1.1. Einstein Solid (a) Classical Einstein Solid (or “Boltzmann” Solid) Consider a single harmonic oscillator in three dimensions with Hamiltonian p2 k H = + x2 2m 2 ◃ Calculate the classical partition function dp Z = dx e−βH(p,x) (2π!)3 ! ! Note: in this problem p and x are three dimensional vectors (they should appear bold to indicate this unless your printer is defective). ◃ Using the partition function, calculate the heat capacity 3kB. ◃ Conclude that if you can consider a solid to consist of N atoms all in harmonic wells, then the heat capacity should be 3NkB =3R,inagreementwiththelawofDulongandPetit. (b) Quantum Einstein Solid Now consider the same Hamiltonian quantum mechanically. ◃ Calculate the quantum partition function Z = e−βEj j " where the sum over j is a sum over all Eigenstates. ◃ Explain the relationship with Bose statistics. ◃ Find an expression for the heat capacity. ◃ Show that the high temperature limit agrees with the law of Dulong of Petit. ◃ Sketch the heat capacity as a function of temperature. 1.2. Debye Theory (a) State the assumptions of the Debye model of heat capacity of a solid.
    [Show full text]
  • Magnetic Form Factor Measurements by Polarised Neutron Scattering: Application to Heavy Fermion Superconductors
    o (1993 ACTA PHYSICA POLONICA A o 1 oceeigs o e I Ieaioa Scoo o Mageism iałowiea 9 MAGEIC OM ACO MEASUEMES Y OAISE EUO SCAEIG AICAIO O EAY EMIO SUECOUCOS K.-U. NEUMANN AND K.R.A. ZIEBECK eame o ysics ougooug Uiesiy o ecoogy ougooug E11 3U Gea iai e eeimea ecique o si oaise euo scaeig as use i mageic om aco measuemes is esee A ioucio o e ieeaio a e cacuaio o mageic om acos a magei- aio esiies is gie e eeimea ecique o euo scaeig eoy as aie o easic si oaise scaeig eeimes is iey iouce e cacuaio o e mageic om aco a e mageia- io esiies ae cosiee o sime moe sysems suc as a coecio o ocaise mageic momes o a iiea eeco sysem e iscussio is iusae y a eeimea iesigaio o e mageic om aco i e eay emio suecoucos Ue13 a U3 Mageiaio esiy mas a mageic om acos ae esee a ei imicaios o oe ysica quaiies ae iey iscusse ACS umes 755+ 71+ 1 Ioucio euo scaeig a i aicua si oaise euo scaeig is a oweu oo o e caaceiaio a iesigaio o ysica oeies o a aomic scae e cage euaiy a mageic ioe mome o e euo make i a iea oe o e iesigaio o mageic egees o eeom I aiio o usig euo scaeig o iesigaig eomea coece wi mageism (y eg esaisig e aue o mageicay oee saes eeimes may e eise o e caaceiaio o eais o eecoic wae ucios Suc a eeime is eaise we si oaise euos ae use o e eemiaio o e mageic om aco a o e mageiaio esiy Coay o secoscoic eciques wic ie eais o e wae ucio y a iesigaio o e eeimeay eemie ee sceme (eg e cysa ie ees o ocaise momes mageic om aco measuemes eee (3 44 K.-U.
    [Show full text]
  • Detailed New Tabulation of Atomic Form Factors and Attenuation Coefficients
    AU0019046 DETAILED NEW TABULATION OF ATOMIC FORM FACTORS AND ATTENUATION COEFFICIENTS IN THE NEAR-EDGE SOFT X-RAY REGIME (Z = 30-36, Z = 60-89, E = 0.1 keV - 8 keV), ADDRESSING CONVERGENCE ISSUES OF EARLIER WORK C. T. Chantler. School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia ABSTRACT Reliable knowledge of the complex X-ray form factor (Re(f) and f') and the photoelectric attenuation coefficient aPE is required for crystallography, medical diagnosis, radiation safety and XAFS studies. Discrepancies between currently used theoretical approaches of 200% exist for numerous elements from 1 keV to 3 keV X-ray energies. The key discrepancies are due to the smoothing of edge structure, the use of non-relativistic wavefunctions, and the lack of appropriate convergence of wavefunctions. This paper addresses these key discrepancies and derives new theoretical results of substantially higher accuracy in near-edge soft X-ray regions. The high-energy limitations of the current approach are also illustrated. The associated figures and tabulation demonstrate the current comparison with alternate theory and with available experimental data. In general experimental data is not sufficiently accurate to establish the errors and inadequacies of theory at this level. However, the best experimental data and the observed experimental structure as a function of energy are strong indicators of the validity of the current approach. New developments in experimental measurement hold great promise in making critical comparisions with theory in the near future. Key Words: anomalous dispersion; attenuation; E = 0.1 keV to 8 keV; form factors; photoabsorption tabulation; Z = 30 - 89. 31-05 Contents 1.
    [Show full text]
  • Atomic Pair Distribution Function (PDF) Analysis
    Atomic Pair Distribution Function (PDF) Analysis 2018 Neutron and X-ray Scattering School Katharine Page Diffraction Group Neutron Scattering Division Oak Ridge National Laboratory [email protected] ORNL is managed by UT-Battelle for the US Department of Energy Diffraction Group, ORNL SNS & HFIR CNMS Diffraction Group Group Leader: Matthew Tucker About Me (an Instrument Scientist) • 2004: BS in Chemical Engineering University of Maine • 2008: PhD in Materials, UC Santa Barbara • 2009/2010: Director’s Postdoctoral Fellow; 2011- 2014: Scientist, Lujan Neutron Scattering Center, Los Alamos National Laboratory (LANL) • 2014-Present: Scientist, Neutron Scattering Division, Oak Ridge National Laboratory (ORNL) 2 PDF Analysis Dr. Liu, his Dr. Olds, wife (Dr. Luo), AJ, and Max and baby boy on the way Dr. Usher- Ditzian Dr. Page, and Ellie Wriston, and Abbie 3 PDF Analysis What is a local structure? ▪ Disordered materials: The interesting properties are often governed by the defects or local structure ▪ Non crystalline materials: Amorphous solids and polymers ▪ Nanostructures: Well defined local structure, but long-range order limited to few nanometers (poorly defined Bragg peaks) S.J.L. Billinge and I. Levin, The Problem with Determining Atomic Structure at the Nanoscale, Science 316, 561 (2007). D. A. Keen and A. L. Goodwin, The crystallography of correlated disorder, Nature 521, 303–309 (2015). 4 PDF Analysis What is total scattering? Cross section of 50x50x50 unit cell model crystal consisting of 70% blue atoms and 30% vacancies. Properties might depend on vacancy ordering! Courtesy of Thomas Proffen 5 PDF Analysis Bragg peaks are “blind” Bragg scattering: Information about the average structure, e.g.
    [Show full text]
  • The Chemically-Specific Structure of an Amorphous Molybdenum Germanium Alloy by Anomalous X-Ray Scattering∗
    SLAC-R-598 UC-404 (SSRL-M) THE CHEMICALLY-SPECIFIC STRUCTURE OF AN AMORPHOUS MOLYBDENUM GERMANIUM ALLOY BY ANOMALOUS X-RAY SCATTERING∗ Hope Ishii Stanford Synchrotron Radiation Laboratory Stanford Linear Accelerator Center Stanford University, Stanford, California 94309 SLAC-Report-598 May 2002 Prepared for the Department of Energy under contract number DE-AC03-76SF00515 Printed in the United States of America. Available from the National Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Road, Springfield, VA 22161 ∗Ph.D. thesis, Stanford University, Stanford, CA 94309. THE CHEMICALLY-SPECIFIC STRUCTURE OF AN AMORPHOUS MOLYBDENUM GERMANIUM ALLOY BY ANOMALOUS X-RAY SCATTERING a dissertation submitted to the department of materials science and engineering and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy Hope Ishii May 2002 Abstract Since its inception in the late 1970s, anomalous x-ray scattering (AXS) has been employed for chemically-specific structure determination in a wide variety of non- crystalline materials. These studies have successfully produced differential distribu- tion functions (DDFs) which provide information about the compositionally-averaged environment of a specific atomic species in the sample. Despite the wide success in obtaining DDFs, there are very few examples of successful extraction of the fully- chemically-specific partial pair distribution functions (PPDFs), the most detailed description of an amorphous sample possible by x-ray scattering. Extracting the PPDFs is notoriously difficult since the matrix equation involved is ill-conditioned and thus extremely sensitive to errors present in the experimental quantities that enter the equation.
    [Show full text]
  • Strain Analysis of Hydrogen Absorption in Cr/V Superlattices
    1 Strain analysis of hydrogen absorption in Cr/V superlattices Uppsala university, Institute of Physics and Astronomy, June 2018 Author: Nikolay Palchevskiy Supervisor: Sotirios Droulias Subject reader: Max Wolff Degree work C, 15 credits 2 Abstract Superlattices are special structures, where the composition of a thin film varies periodically [2, 11]. They may consist of two or more elements (for example: n monolayers of element A, m monolayers of element B, n monolayers of element A, m monolayers of element B, etc.) [2, 11]. Superlattices are grown epitaxial on crystalline substrates. Semiconductor superlattices are considered to have a wide range of applications in future electronics [1]. This work is about the analysis of strain that occurs under hydrogen absorption in a superlattice. The superlattice consists of vanadium and chromium layers and is grown epitaxially on MgO and covered by a protective layer of palladium [2, 13]. Hydrogen is absorbed only by vanadium and causes expansion of the superlattice [2]. Hydrogen is loaded in the samples when we put them in a chamber with hydrogen gas in it. The concentration of hydrogen in the superlattice is increasing with hydrogen pressure. The data sets collected with X-ray diffraction for 18 different pressures of hydrogen were analysed by fitting Gaussian curves to the respective peaks. Values the peaks' position and intensity were extracted in order to calculate the lattice plane distance, the bilayer thickness and the strain wave's amplitude. It was discovered that the lattice plane distance, the bilayer thickness and the strain wave's amplitude increase with increasing pressure. It was also discovered that Cr/V, Fe/V and Fe0.5V0.5/V superlattices have similar pressure-concentration isotherms.
    [Show full text]
  • Atomic Responses to General Dark Matter-Electron Interactions
    PHYSICAL REVIEW RESEARCH 2, 033195 (2020) Atomic responses to general dark matter-electron interactions Riccardo Catena ,1,* Timon Emken ,1,† Nicola A. Spaldin ,2,‡ and Walter Tarantino2,§ 1Chalmers University of Technology, Department of Physics, SE-412 96 Göteborg, Sweden 2Department of Materials, ETH Zürich, CH-8093 Zürich, Switzerland (Received 23 January 2020; revised 23 March 2020; accepted 2 July 2020; published 5 August 2020) In the leading paradigm of modern cosmology, about 80% of our Universe’s matter content is in the form of hypothetical, as yet undetected particles. These do not emit or absorb radiation at any observable wavelengths, and therefore constitute the so-called dark matter (DM) component of the Universe. Detecting the particles forming the Milky Way DM component is one of the main challenges for astroparticle physics and basic science in general. One promising way to achieve this goal is to search for rare DM-electron interactions in low-background deep underground detectors. Key to the interpretation of this search is the response of detectors’ materials to elementary DM-electron interactions defined in terms of electron wave functions’ overlap integrals. In this work, we compute the response of atomic argon and xenon targets used in operating DM search experiments to general, so far unexplored DM-electron interactions. We find that the rate at which atoms can be ionized via DM-electron scattering can in general be expressed in terms of four independent atomic responses, three of which we identify here for the first time. We find our new atomic responses to be numerically important in a variety of cases, which we identify and investigate thoroughly using effective theory methods.
    [Show full text]
  • Phys 446: Solid State Physics / Optical Properties
    Phys 446: Solid State Physics / Optical Properties Fall 2015 Lecture 3 Andrei Sirenko, NJIT 1 Solid State Physics Lecture 3 (Ch. 2) Last week: • Crystals, Crystal Lattice, Reciprocal Lattice, Diffraction from crystals • Today: • Scattering factors and selection rules for diffraction • HW2 discussion Lecture 3 Andrei Sirenko, NJIT 2 1 The Bragg Law Conditions for a sharp peak in the intensity of the scattered radiation: 1) the x-rays should be specularly reflected by the atoms in one plane 2) the reflected rays from the successive planes interfere constructively The path difference between the two x-rays: 2d·sinθ the Bragg formula: 2d·sinθ = mλ The model used to get the Bragg law are greatly oversimplified (but it works!). – It says nothing about intensity and width of x-ray diffraction peaks – neglects differences in scattering from different atoms – assumes single atom in every lattice point – neglects distribution of charge around atoms Lecture 3 Andrei Sirenko, NJIT 3 Diffraction condition and reciprocal lattice Von Laue approach: – crystal is composed of identical atoms placed at the lattice sites T – each atom can reradiate the incident radiation in all directions. – Sharp peaks are observed only in the directions for which the x-rays scattered from all lattice points interfere constructively. Consider two scatterers separated by a lattice vector T. Incident x-rays: wavelength λ, wavevector k; |k| = k = 2/; k'kT 2m Assume elastic scattering: scattered x-rays have same energy (same λ) wavevector k' has the same magnitude |k'| =
    [Show full text]