DOCSLIB.ORG

X-Ray Diffraction • X-Ray Tube (Sealed) • Pure Metal Target (Cu) • Electrons Remove Inner-Shell Electrons Mineral Identification from Target

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
X-Ray Diffraction • X-Ray Tube (Sealed) • Pure Metal Target (Cu) • Electrons Remove Inner-Shell Electrons Mineral Identification from Target X-ray Generation X-ray Diffraction • X-ray tube (sealed) • Pure metal target (Cu) • Electrons remove inner-shell electrons Mineral identification from target. Mode analysis • Other electrons “fall” Structure Studies into hole. X-ray Spectrum from Tube X-ray Generation • The incoming electron must have enough energy to remove inner 1s electrons from the copper atoms. • This energy corresponds to the Cu absorption edge • The 2s and 2p electrons fall back into the 1s shell and emit the Kα1 Kα2 lines. Energy Calculations Energy Calculations • What is the minimum potential in KV that is required to excite Cu K-series • Planck’s constant (h) = 6.6 * 10-34 joule-sec radiation from a Cu-target X-ray tube? • Absorption edge of Cu = 1.380Å • 1 electron-volt = 1.6016 * 10-19 λ joule • E = hc/ = (6.60 10-34)(3*108)/(1.380*10-10) -15 • Speed of light (c) = 3.0 * 108 m/s • E = 1.435*10 joule -15 -19 • Photon frequency ν = c/λ • E = 1.435*10 /1.6016*10 = 8958 ev • Photon Energy E = hν = hc/λ • The potential on the tube must exceed 8.958 KV 1 Diffraction X-Ray Diffraction • Diffraction is the coherent scattering of waves from a periodic array of scatterers. • The wavelength of light is about half a micron • Atoms separated by distance d will scatter • Light is diffracted by the tracks in a in phase when the path length difference is CD. an integral number of wavelengths. • The wavelengths of X-rays is about • Path length difference B-C-D = nλ the same as the interatomic distances • nλ = 2d sin θ in crystals. X-ray Diffraction Experiment • We use the ‘monochromatic’ Kα1-2 lines for our diffraction experiment. • The wavelength is 1.5405Å • We use a diffracted beam monochro- mator to clean up the X-rays entering the detector. • We use a powdered sample so that all orientations are present in the sample. • We move the detector through angle 2θ. 2 Miller Indices • The real use of Miller indices is to describe diffraction planes. • For a lattice plane with Miller indices h k l in an orthorhombic lattice a b c, • d = 1 / [(h/a)2+(k/b)2+(l/c)2]1/2 • For cubic: • d = a/[h2+k2+l2]1/2 Diffraction Calculations XPOW • XPOW uses the unit cell and atom position data to • For forsterite a = 4.75; b = 10.20; c = calculate the diffraction pattern. α=β=γ 5.98Å orthorhombic = 90º • Intensities can be calculated knowing the position • Calculate 2θ for the (201) lattice and scattering characteristics of each atom. spacing for Cukα (λ = 1.5405Å) • Fhkl = square root of integrated intensity. • d = 1 / [(h/a)2+(k/b)2+(l/c)2]1/2 • fj = scattering of atom j at angle 2θ • d = 1/ [(2/4.75)2+(1/5.98)2]1/2 •Atomj located at fractional coordinates xj, yj, zj. • d = 1/0.4530 = 2.207Å • 2θ = 2 sin-1 λ/2d = 2* sin-1 (1.5405/4.414) • 2θ = 2 * 20.43 = 40.86º Uses of X-ray Powder Diffraction • Mineral identification • Determination of Unit Cell Parameters • Modal (phase percentage) Analysis • Crystal Structure Determination 3 X-ray Fluorescence Electron Microprobe X-ray Fluorescence • Chemical analysis • Major and minor element • Uses Ag kα to excite secondary X-rays from sample. • Powdered or flux-fused glass sample. Electron Microprobe Electron Microprobe • Quantitative Chemical analysis • Major and minor element • Uses electrons to excite secondary X- rays from sample. • Electrons can be focussed onto a 10μm spot • Sample is polished thin section 4 Mineral Spectroscopy Electromagnetic Spectrum • Gamma ray (Mössbauer Spectroscopy) • X-ray Fluorescence • Raman spectroscopy • Optical absorbance spectroscopy • Infrared Spectroscopy • NMR (Nuclear Magnetic Resonance) Visible Light: 7700 - 3900Å Mössbauer Spectroscopy Mössbauer spectroscopy • Resonant Gamma Ray spectroscopy Looks at γ-ray interaction at • Uses 57Fe gamma decay at 14.4 KeV nucleus of 57Fe (2%) • Source is 57Co (291 days) • Source is accelerated mechanically to produce ultra-fine doppler energy shifts • Absorption as a function of source velocity • Looks at electric field effects at nucleus due to d-orbital occupancy and perturbations from local coordination effects Mössbauer spectroscopy Mössbauer spectroscopy 5 FTIR Spectrometer Infrared spectroscopy • Near IR 5000 - 13000cm-1 – orbital transitions • Mid-IR 2500 - 5000cm-1 – N-H and O-H bond vibrations • Far IR 500 - 2500 cm-1 – Cation-Oxygen bond vibrations – Structural phonons. Mid IR spectroscopy Mid IR spectroscopy Raman Spectroscopy Raman Spectroscopy • Looks at wavelength shifts in scattered light. • Shifts are in atomic vibrational part of spectrum • 0 - 5000cm-1. (same as mid to far IR) • Excitation is usually by a monochromatic source in the visible region (commonly a laser). 6 Electromagnetic Spectrum Visible Light: 7700 - 3900Å 7.
Load more

Recommended publications
  • Glossary Physics (I-Introduction)
    1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay.
    [Show full text]
  • Classical and Modern Diffraction Theory
    Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Classical and Modern Diffraction Theory Edited by Kamill Klem-Musatov Henning C. Hoeber Tijmen Jan Moser Michael A. Pelissier SEG Geophysics Reprint Series No. 29 Sergey Fomel, managing editor Evgeny Landa, volume editor Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Society of Exploration Geophysicists 8801 S. Yale, Ste. 500 Tulsa, OK 74137-3575 U.S.A. # 2016 by Society of Exploration Geophysicists All rights reserved. This book or parts hereof may not be reproduced in any form without permission in writing from the publisher. Published 2016 Printed in the United States of America ISBN 978-1-931830-00-6 (Series) ISBN 978-1-56080-322-5 (Volume) Library of Congress Control Number: 2015951229 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Dedication We dedicate this volume to the memory Dr. Kamill Klem-Musatov. In reading this volume, you will find that the history of diffraction We worked with Kamill over a period of several years to compile theory was filled with many controversies and feuds as new theories this volume. This volume was virtually ready for publication when came to displace or revise previous ones. Kamill Klem-Musatov’s Kamill passed away. He is greatly missed. new theory also met opposition; he paid a great personal price in Kamill’s role in Classical and Modern Diffraction Theory goes putting forth his theory for the seismic diffraction forward problem.
    [Show full text]
  • Intensity of Double-Slit Pattern • Three Or More Slits
    • Intensity of double-slit pattern • Three or more slits Practice: Chapter 37, problems 13, 14, 17, 19, 21 Screen at ∞ θ d Path difference Δr = d sin θ zero intensity at d sinθ = (m ± ½ ) λ, m = 0, ± 1, ± 2, … max. intensity at d sinθ = m λ, m = 0, ± 1, ± 2, … 1 Questions: -what is the intensity of the double-slit pattern at an arbitrary position? -What about three slits? four? five? Find the intensity of the double-slit interference pattern as a function of position on the screen. Two waves: E1 = E0 sin(ωt) E2 = E0 sin(ωt + φ) where φ =2π (d sin θ )/λ , and θ gives the position. Steps: Find resultant amplitude, ER ; then intensities obey where I0 is the intensity of each individual wave. 2 Trigonometry: sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2] E1 + E2 = E0 [sin(ωt) + sin(ωt + φ)] = 2 E0 cos (φ / 2) cos (ωt + φ / 2)] Resultant amplitude ER = 2 E0 cos (φ / 2) Resultant intensity, (a function of position θ on the screen) I position - fringes are wide, with fuzzy edges - equally spaced (in sin θ) - equal brightness (but we have ignored diffraction) 3 With only one slit open, the intensity in the centre of the screen is 100 W/m 2. With both (identical) slits open together, the intensity at locations of constructive interference will be a) zero b) 100 W/m2 c) 200 W/m2 d) 400 W/m2 How does this compare with shining two laser pointers on the same spot? θ θ sin d = d Δr1 d θ sin d = 2d θ Δr2 sin = 3d Δr3 Total, 4 The total field is where φ = 2π (d sinθ )/λ •When d sinθ = 0, λ , 2λ, ..
    [Show full text]
  • Light and Matter Diffraction from the Unified Viewpoint of Feynman's
    European J of Physics Education Volume 8 Issue 2 1309-7202 Arlego & Fanaro Light and Matter Diffraction from the Unified Viewpoint of Feynman’s Sum of All Paths Marcelo Arlego* Maria de los Angeles Fanaro** Universidad Nacional del Centro de la Provincia de Buenos Aires CONICET, Argentine *[email protected] **[email protected] (Received: 05.04.2018, Accepted: 22.05.2017) Abstract In this work, we present a pedagogical strategy to describe the diffraction phenomenon based on a didactic adaptation of the Feynman’s path integrals method, which uses only high school mathematics. The advantage of our approach is that it allows to describe the diffraction in a fully quantum context, where superposition and probabilistic aspects emerge naturally. Our method is based on a time-independent formulation, which allows modelling the phenomenon in geometric terms and trajectories in real space, which is an advantage from the didactic point of view. A distinctive aspect of our work is the description of the series of transformations and didactic transpositions of the fundamental equations that give rise to a common quantum framework for light and matter. This is something that is usually masked by the common use, and that to our knowledge has not been emphasized enough in a unified way. Finally, the role of the superposition of non-classical paths and their didactic potential are briefly mentioned. Keywords: quantum mechanics, light and matter diffraction, Feynman’s Sum of all Paths, high education INTRODUCTION This work promotes the teaching of quantum mechanics at the basic level of secondary school, where the students have not the necessary mathematics to deal with canonical models that uses Schrodinger equation.
    [Show full text]
  • Estimating Fire Properties by Remote Sensing
    Estimating Fire Properties by Remote Sensing1. Philip J. Riggan USDA Forest Service Pacific Southwest Research Station 4955 Canyon Crest Drive Riverside, CA 92507 909 680 1534 [email protected] James W. Hoffman Space Instruments, Inc. 4403 Manchester Avenue, Suite 203 Encinitas, CA 92024 760 944 7001 [email protected] James A. Brass NASA Ames Research Center Moffett Federal Airfield, CA 94035 650 604 5232 [email protected] Abstract---Contemporary knowledge of the role of fire in the TABLE OF CONTENTS global environment is limited by inadequate measurements of the extent and impact of individual fires. Observations by 1. INTRODUCTION operational polar-orbiting and geostationary satellites provide an 2. ESTIMATING FIRE PROPERTIES indication of fire occurrence but are ill-suited for estimating the 3. ESTIMATES FROM TWO CHANNELS temperature, area, or radiant emissions of active wildland and 4. MULTI-SPECTRAL FIRE IMAGING agricultural fires. Simulations here of synthetic remote sensing 5. APPLICATIONS FOR FIRE MONITORING pixels comprised of observed high resolution fire data together with ash or vegetation background demonstrate that fire properties including flame temperature, fractional area, and INTRODUCTION radiant-energy flux can best be estimated from concurrent radiance measurements at wavelengths near 1.6, 3.9, and 12 µm, More than 30,000 fire observations were recorded over central Successful observations at night may be made at scales to at Brazil during August 1999 by Advanced Very High Resolution least I km for the cluster of fire data simulated here. During the Radiometers operating aboard polarorbiting satellites of the U.S. daytime, uncertainty in the composition of the background and National Oceanic and Atmospheric Administration.
    [Show full text]
  • 25 Geometric Optics
    CHAPTER 25 | GEOMETRIC OPTICS 887 25 GEOMETRIC OPTICS Figure 25.1 Image seen as a result of reflection of light on a plane smooth surface. (credit: NASA Goddard Photo and Video, via Flickr) Learning Objectives 25.1. The Ray Aspect of Light • List the ways by which light travels from a source to another location. 25.2. The Law of Reflection • Explain reflection of light from polished and rough surfaces. 25.3. The Law of Refraction • Determine the index of refraction, given the speed of light in a medium. 25.4. Total Internal Reflection • Explain the phenomenon of total internal reflection. • Describe the workings and uses of fiber optics. • Analyze the reason for the sparkle of diamonds. 25.5. Dispersion: The Rainbow and Prisms • Explain the phenomenon of dispersion and discuss its advantages and disadvantages. 25.6. Image Formation by Lenses • List the rules for ray tracking for thin lenses. • Illustrate the formation of images using the technique of ray tracking. • Determine power of a lens given the focal length. 25.7. Image Formation by Mirrors • Illustrate image formation in a flat mirror. • Explain with ray diagrams the formation of an image using spherical mirrors. • Determine focal length and magnification given radius of curvature, distance of object and image. Introduction to Geometric Optics Geometric Optics Light from this page or screen is formed into an image by the lens of your eye, much as the lens of the camera that made this photograph. Mirrors, like lenses, can also form images that in turn are captured by your eye. 888 CHAPTER 25 | GEOMETRIC OPTICS Our lives are filled with light.
    [Show full text]
  • The Electromagnetic Spectrum
    The Electromagnetic Spectrum Wavelength/frequency/energy MAP TAP 2003-2004 The Electromagnetic Spectrum 1 Teacher Page • Content: Physical Science—The Electromagnetic Spectrum • Grade Level: High School • Creator: Dorothy Walk • Curriculum Objectives: SC 1; Intro Phys/Chem IV.A (waves) MAP TAP 2003-2004 The Electromagnetic Spectrum 2 MAP TAP 2003-2004 The Electromagnetic Spectrum 3 What is it? • The electromagnetic spectrum is the complete spectrum or continuum of light including radio waves, infrared, visible light, ultraviolet light, X- rays and gamma rays • An electromagnetic wave consists of electric and magnetic fields which vibrates thus making waves. MAP TAP 2003-2004 The Electromagnetic Spectrum 4 Waves • Properties of waves include speed, frequency and wavelength • Speed (s), frequency (f) and wavelength (l) are related in the formula l x f = s • All light travels at a speed of 3 s 108 m/s in a vacuum MAP TAP 2003-2004 The Electromagnetic Spectrum 5 Wavelength, Frequency and Energy • Since all light travels at the same speed, wavelength and frequency have an indirect relationship. • Light with a short wavelength will have a high frequency and light with a long wavelength will have a low frequency. • Light with short wavelengths has high energy and long wavelength has low energy MAP TAP 2003-2004 The Electromagnetic Spectrum 6 MAP TAP 2003-2004 The Electromagnetic Spectrum 7 Radio waves • Low energy waves with long wavelengths • Includes FM, AM, radar and TV waves • Wavelengths of 10-1m and longer • Low frequency • Used in many
    [Show full text]
  • Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
    Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: [email protected] e-mail: [email protected] tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions.
    [Show full text]
  • 9.2 Refraction and Total Internal Reflection
    9.2 refraction and total internal reflection When a light wave strikes a transparent material such as glass or water, some of the light is reflected from the surface (as described in Section 9.1). The rest of the light passes through (transmits) the material. Figure 1 shows a ray that has entered a glass block that has two parallel sides. The part of the original ray that travels into the glass is called the refracted ray, and the part of the original ray that is reflected is called the reflected ray. normal incident ray reflected ray ␪ ␪i r ␪r ϭ ␪i air glass ␪2 refracted ray Figure 1 A light ray that strikes a glass surface is both reflected and refracted. Refracted and reflected rays of light account for many things that we encounter in our everyday lives. For example, the water in a pool can look shallower than it really is. A stick can look as if it bends at the point where it enters the water. On a hot day, the road ahead can appear to have a puddle of water, which turns out to be a mirage. These effects are all caused by the refraction and reflection of light. refraction The direction of the refracted ray is different from the direction of the incident refraction the bending of light as it ray, an effect called refraction. As with reflection, you can measure the direction of travels at an angle from one medium the refracted ray using the angle that it makes with the normal. In Figure 1, this to another angle is labelled θ2.
    [Show full text]
  • Hydraulics Manual Glossary G - 3
    Glossary G - 1 GLOSSARY OF HIGHWAY-RELATED DRAINAGE TERMS (Reprinted from the 1999 edition of the American Association of State Highway and Transportation Officials Model Drainage Manual) G.1 Introduction This Glossary is divided into three parts: · Introduction, · Glossary, and · References. It is not intended that all the terms in this Glossary be rigorously accurate or complete. Realistically, this is impossible. Depending on the circumstance, a particular term may have several meanings; this can never change. The primary purpose of this Glossary is to define the terms found in the Highway Drainage Guidelines and Model Drainage Manual in a manner that makes them easier to interpret and understand. A lesser purpose is to provide a compendium of terms that will be useful for both the novice as well as the more experienced hydraulics engineer. This Glossary may also help those who are unfamiliar with highway drainage design to become more understanding and appreciative of this complex science as well as facilitate communication between the highway hydraulics engineer and others. Where readily available, the source of a definition has been referenced. For clarity or format purposes, cited definitions may have some additional verbiage contained in double brackets [ ]. Conversely, three “dots” (...) are used to indicate where some parts of a cited definition were eliminated. Also, as might be expected, different sources were found to use different hyphenation and terminology practices for the same words. Insignificant changes in this regard were made to some cited references and elsewhere to gain uniformity for the terms contained in this Glossary: as an example, “groundwater” vice “ground-water” or “ground water,” and “cross section area” vice “cross-sectional area.” Cited definitions were taken primarily from two sources: W.B.
    [Show full text]
  • X-Ray Production Compton Scattering Pair Production
    Welcome back to PHY 3305 Today’s Lecture: X-ray Production Compton Scattering Pair Production Arthur Compton 1892 - 1962 Physics 3305 - Modern Physics Professor Jodi Cooley Meeting of the Texas Section of the American Physical Society When: Fri-Sat Oct 20-21 Where: UT Dallas ñ Physics majors are expected to attend, and present any research project if applicable. ñ See the email from Dr. Dalley about registration (deadline Sept 27). ñ See Michele Hill (Physics office) well before the meeting to get set up for registration and banquet fee reimbursement. Physics 3305 - Modern Physics Professor Jodi Cooley Anouncements • Reading Assignment: Chapter 3.6; 4.1 - 4.2 • Problem set 5 is due Tuesday, Sept. 26th at 12:30 pm. • Regrade for problem set 4 is due Tuesday, Sept 26th at 12:30 pm. • Midterm exam 1 covering chapters 1-2 and related material will be in class on Thursday, Sept 21st. There will be a seating chart. Physics 3305 - Modern Physics Professor Jodi Cooley Review Question 1 In a photoelectric effect experiment, if the intensity of the incident light is doubled the stopping potential will: a) Double b) Double, but only of if the light’s frequency is above the cut-off Current between the electrodes c) Half stops when the opposing potential energy difference equals the d) Remain unchanged maximum KE of the photoelectrons. KE = hf φ max − Physics 3305 - Modern Physics Professor Jodi Cooley Electrons are accelerated in television tubes through a potential difference of 10.0 kV. Find the lowest wavelength of the electromagnetic waves emitted when these electrons strike the screen.
    [Show full text]
  • The Path Integral Approach to Quantum Mechanics Lecture Notes for Quantum Mechanics IV
    The Path Integral approach to Quantum Mechanics Lecture Notes for Quantum Mechanics IV Riccardo Rattazzi May 25, 2009 2 Contents 1 The path integral formalism 5 1.1 Introducingthepathintegrals. 5 1.1.1 Thedoubleslitexperiment . 5 1.1.2 An intuitive approach to the path integral formalism . .. 6 1.1.3 Thepathintegralformulation. 8 1.1.4 From the Schr¨oedinger approach to the path integral . .. 12 1.2 Thepropertiesofthepathintegrals . 14 1.2.1 Pathintegralsandstateevolution . 14 1.2.2 The path integral computation for a free particle . 17 1.3 Pathintegralsasdeterminants . 19 1.3.1 Gaussianintegrals . 19 1.3.2 GaussianPathIntegrals . 20 1.3.3 O(ℏ) corrections to Gaussian approximation . 22 1.3.4 Quadratic lagrangians and the harmonic oscillator . .. 23 1.4 Operatormatrixelements . 27 1.4.1 Thetime-orderedproductofoperators . 27 2 Functional and Euclidean methods 31 2.1 Functionalmethod .......................... 31 2.2 EuclideanPathIntegral . 32 2.2.1 Statisticalmechanics. 34 2.3 Perturbationtheory . .. .. .. .. .. .. .. .. .. .. 35 2.3.1 Euclidean n-pointcorrelators . 35 2.3.2 Thermal n-pointcorrelators. 36 2.3.3 Euclidean correlators by functional derivatives . ... 38 0 0 2.3.4 Computing KE[J] and Z [J] ................ 39 2.3.5 Free n-pointcorrelators . 41 2.3.6 The anharmonic oscillator and Feynman diagrams . 43 3 The semiclassical approximation 49 3.1 Thesemiclassicalpropagator . 50 3.1.1 VanVleck-Pauli-Morette formula . 54 3.1.2 MathematicalAppendix1 . 56 3.1.3 MathematicalAppendix2 . 56 3.2 Thefixedenergypropagator. 57 3.2.1 General properties of the fixed energy propagator . 57 3.2.2 Semiclassical computation of K(E)............. 61 3.2.3 Two applications: reflection and tunneling through a barrier 64 3 4 CONTENTS 3.2.4 On the phase of the prefactor of K(xf ,tf ; xi,ti) ....
    [Show full text]