Lecture: Scattering theory
|
|
- Marshall Bryan
- 6 years ago
- Views:
Transcription
1 Lecture: Scattering theory SS2012: Introduction to Nuclear and Particle Physics, Part 2 2 1
2 Part I: Scattering theory: Classical trajectoriest and cross-sections Quantum Scattering 2
3 I. Scattering experiments Scattering experiment: A beam of incident scatterers with a given flux or intensity (number of particles per unit area da per unit time dt ) impinges on the target (described by a scattering potential); the flux can be written as The number of particles per unit time which are detected in a small region of the solid angle, dω, located at a given angular deflection specified by (θ, φ), can be counted as 3
4 Scattering cross section The differential cross-section for scattering is defined as the number of particles scattered into an element of solid angle dω in the direction (θ,φ) per unit time : [dimensions of an area] dσ ( ϑ, φ ) dω = 1 J inc dn sc dω J inc - incident flux (1.1) The total cross-section corresponds to scatterings through any scattering angle: (1.2) Most scattering experiments are carried out in the laboratory (Lab) frame in which the target is initially at rest while the projectiles are moving. Calculations of the cross sections are generally easier to perform within the center-of-mass (CM) frame in which the center of mass of the projectiles target system is at rest (before and after collision) one has to know how to transform the cross sections from one frame into the other. Note: the total cross section σ is the same in both frames, since the total number of collisions that take place does not depend on the frame in which the observation is carried out. However, the differential cross sections dσ/ σ/dω are not the same in both frames, since the scattering angles (θ,φ) are frame dependent. 4
5 Connecting the angles in the Lab and CM frames Elastic scattering of two structureless particles in the Lab and CM frames: Note: consider for simplicity NON-relativistic kinematics! r V r CM V 1 L To find the connection between the Lab and CM cross sections, we need first to find how the scattering angles in one frame are related to their counterparts in the other. If and if we have denote the position of m 1 in the Lab and CM frames, respectively, denotes the position of the center of mass with respect to the Lab frame, A time derivative of this relation leads to (1.3) where are the velocities of m 1 in the Lab and CM frames before collision and is the velocity of the CM with respect to the Lab frame. Similarly, the velocity of m 1 after collision is (1.4) 5
6 Connecting the angles in the Lab and CM frames Since r V r CM V 1 L (1.5) (1.6) Dividing (1.6) by (1.5), we end up with (1.7) Use that CM momenta r p CM ( m 1 = + r p m 1 L 2 + r )V r p CM 2 L = m 1 r V 1 L + m 2 r V (1.8) 2 L Since from (1.8) (1.9) or from (1.9) (1.10) (1.11) 6
7 Connecting the angles in the Lab and CM frames Since the center of mass is at rest in the CM frame, the total momenta before and after r r r r r collisions are separately zero: v pc = p1c + p 2 C = p C = p 1C + p 2 C = (1.12) (1.13) Since the kinetic energy is conserved: (1.14) (1.15) 0 In the case of elastic collisions, the speeds of the particles in the CM frame are the same before and after the collision; From (1.11) (1.16) Dividing (1.9) by (1.16) (1.17) 7
8 Connecting the angles in the Lab and CM frames Finally, a substitution of (1.17) into (1.7) yields (1.17) and using we obtain (1.18) Note: In a similar way we can establish a connection between θ 2 and θ. From (1.4) we have + using the x and y components of this relation are (1.19) (1.20) 8
9 Connecting the Lab and CM Cross Sections The connection between the differential cross sections in the Lab and CM frames can be obtained from the fact that the number of scattered particles passing through an infinitesimal cross section is the same in both frames: What differs is the solid angle dω : in the Lab frame: in the CM frame: (1.21) (1.22) Since there is a cylindrical symmetry around the direction of the incident beam (1.23) From (1.18) (1.24) 9
10 Connecting the Lab and CM Cross Sections Thus : (1.25) From (1.23) and (1.20) (1.26) Limiting cases: 1) the Lab and CM results are the same, since (1.17) leads to θ 1 =θ (1.27) 2) then from (1.20) from (1.25) (1.28) 10
11 From classical to quantum scattering theory 1. Classical theory: - scattering of hard spheres - individual well-defined trajectories 2. Quantum theory: - scattering of wave pakeges wave particle duality - probabilistic origin of scattering process 11
12 II. Classical trajectoriest and cross-sections Scattering trajectories, corresponding to different impact parameters, b give different scattering angles θ. All of the particles in the beam in the hatched region of area dσ = 2π bdb are scattered into the angular region (θ, θ + dθ) The equations of motion for incident particle: (Newton s law) + initial conditions: since initial kinetic energy defines the trajectory (2.1) (2.2) (2.3) For a particle obeying classical mechanics: the trajectory for the unbound motion, corresponding to a scattering event, is deterministically predictable, given by the interaction potential and the initial conditions; the path of any scatterer in the incident beam can be followed, and its angular deflection is determined as precisely as required 12
13 Classical trajectoriest and cross-sections The number of particles scattered per unit time into the angular region (θ, θ+dθ) with any value of φ can be written as (2.4) (2.5) The knowledge of b(θ), obtained directly from Newton s laws or other methods, is then sufficient to calculate the scattering cross-section. For the scattering from nontrivial central forces, the trajectory can be obtained from the equations of motion by using energy- and angular momentum- conservation methods E.g., one can rewrite L - angular momentum in the form (2.6) 13
14 Classical trajectoriest and cross-sections The angular momentum can be written via the angular velocity from (2.6) (2.7) the angle through which the particle moves between two radial distances r 1 and r 2 : (2.8) (2.9) Scattering trajectories in a central potential : r min - the distance of closest approach, Θ deflection angle: Use that the initial angular momentum and from (2.8), (2.9) (2.10) 14
15 Classical Coulomb scattering The potential corresponding to the scattering of charged particles via Coulomb s law can be written in the general form (2.11) where the constant point-like charges Z 1 e and Z 2 e corresponds to the electrostatic interaction of from (2.10) (2.12) Note that for large energies (E ) or vanishing charges (A 0), one has r min b Use that from (2.12) (2.13) (2.14) 15
16 Classical Coulomb scattering Use that (2.15) Rutherford formula - characteristic for Coulomb scattering from a point-like charge: (2.16) Note: If one attempts to evaluate the total cross-section using Eq. (2.16), one obtains an infinite result. This divergence is due to the infinite range of the 1/r potential, and the infinity in the integral comes from scatterings as θ 0 which corresponds to arbitrarily large impact parameters where the Coulomb potential causes a scatter through an arbitrarily small angle, which still contributes to the total cross-section. 16
17 III. Quantum Scattering The scattering amplitude of spinless, nonrelativistic particlesp articles: Consider the elastic scattering between two spinless, nonrelativistic particles of masses m 1 and m 2. During the scattering process, the particles interact with each other. If the interaction is time independent, we can describe the two-particle system with stationary states (3.1) where E T is the total energy and Schrödinger equation: is a solution of the time-independent (3.2) is the potential representing the interaction between the two particles. In the case where the interaction between m 1 and m 2 depends only on their relative distance one can reduce the eigenvalue problem (3.2) to two decoupled eigenvalue problems: one for the center of mass (CM), which moves like a free particle of mass M= m 1 +m 2 (which is of no concern to us here) and another for a fictitious particle with a reduced mass moves in the potential which (3.3) 17
18 Scattering of spinless particles In quantum mechanics the incident particle is described by means of a wave packet that interacts with the target. After scattering, the wave function consists of an unscattered part propagating in the forward direction and a scattered part that propagates along some direction (θ,ϕ) 18
19 Scattering of spinless particles One can consider (3.3) as representing the scattering of a particle of mass µ from a fixed scattering center described by V(r), where r is the distance from the particle µ to the center of V(r): (3.3) assume that V(r) has a finite range a: Vˆ ( r ) = = 0, if 0, if r r > a a Outside the range, r > a, the potential vanishes, V(r)= 0; the eigenvalue problem (3.3) then becomes (3.4) In this case µ behaves as a free particle before the collision and can be described by a plane wave (3.5) where is the wave vector associated with the incident particle; A is a normalization factor. Thus, prior to the interaction with the target, the particles of the incident beam are independent of each other; they move like free particles, each with a momentum 19
20 Scattering of spinless particles When the incident wave (3.5) collides or interacts with the target, an outgoing wave is scattered out. In the case of an isotropic scattering, the scattered wave is spherically symmetric, having the form In general, however, the scattered wave is not spherically symmetric; its amplitude depends on the direction (θ,ϕ) along which it is detected and hence (3.6) where f (θ,ϕ) is called the scattering amplitude, is the wave vector associated with the scattered particle, and θ is the angle between After the scattering has taken place, the total wave consists of a superposition of the incident plane wave (3.5) and the scattered wave (3.6): (3.7) where A is a normalization factor; since A has no effect on the cross section, as will be shown later, we will take it equal to one. 20
21 Scattering amplitude and differential cross section Introduce the incident and scattered flux densities: (3.8) (3.9) Inserting (3.5) into (3.8) and (3.6) into (3.9) and taking the magnitudes of the expressions thus obtained, we end up with (3.10) We recall that the number dn(θ,ϕ) of particles scattered into an element of solid angle dω in the direction (θ,ϕ) and passing through a surface element da= r 2 dω per unit time is given as follows: (3.11) Using (3.10) (3.12) 21
22 Scattering amplitude and differential cross section By inserting (3.12) and in (1.1), we obtain: (3.13) Since the normalization factor A does not contribute to the differential cross section, it will be taken to be equal to one. For elastic scattering k 0 is equal to k; so (3.13) reduces to (3.14) The problem of determining the differential cross section dσ/dω therefore reduces to that of obtaining the scattering amplitude f (θ,ϕ). 22
23 Part II: Scattering theory: Born Approximation 23
24 Scattering amplitude We are going to show here that we can obtain the differential cross section in the CM frame from an asymptotic form of the solution of the Schrödinger equation: (1.1) Let us first focus on the determination of the scattering amplitude f (θ, φ), it can be obtained from the solutions of (1.1), which in turn can be rewritten as where 2 2µE k = 2 h The general solution of the equation (1.2) consists of a sum of two components: (1.2) 1) a general solution to the homogeneous equation: (1.3) In (1.3) is the incident plane wave 2) and a particular solution of (1.2) with the interaction potential 24
25 General solution of Schrödinger eq. in terms of Green s function The general solution of (1.2) can be expressed in terms of Green s function. (1.4) is the Green s function corresponding to the operator on the left side of eq.(1.3) The Green s function is obtained by solving the point source equation: (1.5) (1.6) (1.7) 25
26 Green s function A substitution of (1.6) and (1.7) into (1.5) leads to (1.8) The expression for can be obtained by inserting (1.8) into (1.6) (1.9) (1.10) To integrate over angle in (1.10) we need to make the variable change x=cosθ (1.11) 26
27 Method of residues Thus, (1.9) becomes (1.12) (1.13) The integral in (1.13) can be evaluated by the method of residues by closing the contour in the upper half of the q-plane: The integral is equal to 2π i times the residue of the integrand at the poles. 27
28 Green s functions Since there are two poles, q =+k, the integral has two possible values: the value corresponding to the pole at q =k, which lies inside the contour of integration in Figure 1a, is given by (1.14) the value corresponding to the pole at q =-k, Figure 1b, is (1.15) Green s function represents an outgoing spherical wave emitted from r and the function corresponds to an incoming wave that converges onto r. Since the scattered waves are outgoing waves, only is of interest to us. 28
29 Born series Inserting (1.14) into (1.4) we obtain for the total scattered wave function: (1.16) This is an integral equation. All we have done is to rewrite the Schrödinger (differential) equation (1.1) in an integral form (1.16), which is more suitable for scattering theory. Note that (1.16) can be solved approximately by means of a series of successive or iterative approximations, known as the Born series. the zero-order solution is given by the first-order solution is obtained by inserting into the integral of (1.16): (1.17) 29
30 Born series the second order solution is obtained by inserting into (1.16): (1.18) the n th order approximation for the wave function is a series which can be obtained by analogy to (1.18). continuing in this way, we can obtain to any desired order; 30
31 Asymptotic limit of the wave function In a scattering experiment, since the detector is located at distances (away from the target) that are much larger than the size of the target, we have r>>r, where r represents the distance from the target to the detector and r the size of the target. If r >> r we may approximate: (1.19) 31
32 Asymptotic limit of the wave function Substitute (1.19) to (1.16): for r>>r ikr ikr ' r µ e e r r 3 = φ inc ( ) V ( ) ( )d r 2 h ψ π r rr (1.16) From the previous two approximations (1.19), we may write the asymptotic form of (1.16) as follows: (1.20) (1.21) where ia a plane wave and k is the wave vector of scattered wave; the integration variable r extends over the spacial degrees of freedom of the target. The differential cross section is given by (1.22) 32
33 The first Born approximation If the potential V(r) is weak enough, it will distort only slightly the incident plane wave. The first Born approximation consists then of approximating the scattered wave function Ψ(r ) by a plane wave. This approximation corresponds to the first iteration in the Born series of (1.16): (1.16) that is, Ψ(r ) is given by (1.17): (1.23) Thus, using (1.21) we can write the scattering amplitude in the first Born approximation as follows: (1.24) 33
34 The first Born approximation Using (1.23), we can write the differential cross section in the first Born approximation as follows: (1.25) where is the momentum transfer; are the linear momenta of the incident and scattered particles, respectively. In elastic scattering, the magnitudes of are equal (Figure 1); hence (1.26) Figure 1: Momentum transfer for elastic scattering: 34
35 The first Born approximation If the potential is spherically symmetric, and choosing the z-axis along q (Figure 1), then and therefore (1.27) Inserting (1.27) into (1.24) and (1.25) we obtain (1.28) (1.29) In summary, we have shown that by solving the Schrödinger equation (1.1) in first-order Born approximation (where the potential V(r ) is weak enough that the scattered wave function is only slightly different from the incident plane wave), the differential cross section is given by equation (1.29) for a spherically symmetric potential. 35
36 Validity of the firts Born approximation The first Born approximation is valid whenever the wave function Ψ(r ) is only slightly different from the incident plane wave; that is, whenever the second term in (1.23) is very small compared to the first: (1.23) (1.30) Since we have (1.31) 36
37 Validity of the firts Born approximation In elastic scattering k 0 = k and assuming that the scattering potential is largest near r=0, we have (1.32) (1.33) Since the energy of the incident particle is proportional to k (it is purely kinetic, ) we infer from (1.33) that the Born approximation is valid for large incident energies and weak scattering potentials. That is, when the average interaction energy between the incident particle and the scattering potential is much smaller than the particle s incident kinetic energy, the scattered wave can be considered to be a plane wave. 37
38 Born approximation for Coulomb potential Let s calculate the differential cross section in the first Born approximation for a Coulomb potential (1.34) where Z1e and Z1e are the charges of the projectile and target particles, respectively. In a case of Coulomb potential, eq.(1.29): (1.35) becomes (1.36) (1.37) 38
39 Born approximation for Coulomb potential Now, since an insertion of (1.37) into (1.36) leads to (1.38) where is the kinetic energy of the incident particle. Eq. (1.38) is Rutherford formula Note: (1.38) is identical to the purely classical case! cf. Part I, eq.(2.16) 39
Lecture 5 Scattering theory, Born Approximation. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 5 Scattering theory, Born Approximation SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Scattering amplitude We are going to show here that we can obtain the differential cross
More informationLecture 6 Scattering theory Partial Wave Analysis. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 6 Scattering theory Partial Wave Analysis SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 The Born approximation for the differential cross section is valid if the interaction
More informationLet b be the distance of closest approach between the trajectory of the center of the moving ball and the center of the stationary one.
Scattering Classical model As a model for the classical approach to collision, consider the case of a billiard ball colliding with a stationary one. The scattering direction quite clearly depends rather
More informationI. Elastic collisions of 2 particles II. Relate ψ and θ III. Relate ψ and ζ IV. Kinematics of elastic collisions
I. Elastic collisions of particles II. Relate ψ and θ III. Relate ψ and ζ IV. Kinematics of elastic collisions 49 I. Elastic collisions of particles "Elastic": KE is conserved (as well as E tot and momentum
More informationElastic Scattering. R = m 1r 1 + m 2 r 2 m 1 + m 2. is the center of mass which is known to move with a constant velocity (see previous lectures):
Elastic Scattering In this section we will consider a problem of scattering of two particles obeying Newtonian mechanics. The problem of scattering can be viewed as a truncated version of dynamic problem
More informationClassical Scattering
Classical Scattering Daniele Colosi Mathematical Physics Seminar Daniele Colosi (IMATE) Classical Scattering 27.03.09 1 / 38 Contents 1 Generalities 2 Classical particle scattering Scattering cross sections
More informationQuantum Physics III (8.06) Spring 2008 Assignment 10
May 5, 2008 Quantum Physics III (8.06) Spring 2008 Assignment 10 You do not need to hand this pset in. The solutions will be provided after Friday May 9th. Your FINAL EXAM is MONDAY MAY 19, 1:30PM-4:30PM,
More informationQuantum Physics III (8.06) Spring 2005 Assignment 9
Quantum Physics III (8.06) Spring 2005 Assignment 9 April 21, 2005 Due FRIDAY April 29, 2005 Readings Your reading assignment on scattering, which is the subject of this Problem Set and much of Problem
More informationDecays and Scattering. Decay Rates Cross Sections Calculating Decays Scattering Lifetime of Particles
Decays and Scattering Decay Rates Cross Sections Calculating Decays Scattering Lifetime of Particles 1 Decay Rates There are THREE experimental probes of Elementary Particle Interactions - bound states
More informationElastic Collisions. Chapter Center of Mass Frame
Chapter 11 Elastic Collisions 11.1 Center of Mass Frame A collision or scattering event is said to be elastic if it results in no change in the internal state of any of the particles involved. Thus, no
More informationSummary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization:
The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory:
More informationScattering is perhaps the most important experimental technique for exploring the structure of matter.
.2. SCATTERING February 4, 205 Lecture VII.2 Scattering Scattering is perhaps the most important experimental technique for exploring the structure of matter. From Rutherford s measurement that informed
More informationIntroduction to Elementary Particle Physics I
Physics 56400 Introduction to Elementary Particle Physics I Lecture 2 Fall 2018 Semester Prof. Matthew Jones Cross Sections Reaction rate: R = L σ The cross section is proportional to the probability of
More information5.62 Physical Chemistry II Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 5.6 Physical Chemistry II Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.6 Spring 008 Lecture #30
More informationNon-relativistic scattering
Non-relativistic scattering Contents Scattering theory 2. Scattering amplitudes......................... 3.2 The Born approximation........................ 5 2 Virtual Particles 5 3 The Yukawa Potential
More informationQuantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie
Quantum Mechanics II Lecture (www.sp.phy.cam.ac.u/~dar/pdf) David Ritchie Michaelmas. So far we have found solutions to Section 4:Transitions Ĥ ψ Eψ Solutions stationary states time dependence with time
More informationScattering theory I: single channel differential forms
TALENT: theory for exploring nuclear reaction experiments Scattering theory I: single channel differential forms Filomena Nunes Michigan State University 1 equations of motion laboratory Center of mass
More informationScattering Cross Sections, Classical and QM Methods
Scattering Cross Sections, Classical and QM Methods Jean-Sébastian Tempel Department of Physics and Technology University of Bergen 9. Juni 2007 phys264 - Environmental Optics and Transport of Light and
More informationScattering amplitudes and the Feynman rules
Scattering amplitudes and the Feynman rules based on S-10 We have found Z( J ) for the phi-cubed theory and now we can calculate vacuum expectation values of the time ordered products of any number of
More informationPHY492: Nuclear & Particle Physics. Lecture 3 Homework 1 Nuclear Phenomenology
PHY49: Nuclear & Particle Physics Lecture 3 Homework 1 Nuclear Phenomenology Measuring cross sections in thin targets beam particles/s n beam m T = ρts mass of target n moles = m T A n nuclei = n moles
More information16. Elastic Scattering Michael Fowler
6 Elastic Scattering Michael Fowler Billiard Balls Elastic means no internal energy modes of the scatterer or of the scatteree are excited so total kinetic energy is conserved As a simple first exercise,
More informationLecture 10. Central potential
Lecture 10 Central potential 89 90 LECTURE 10. CENTRAL POTENTIAL 10.1 Introduction We are now ready to study a generic class of three-dimensional physical systems. They are the systems that have a central
More informationComparative study of scattering by hard core and absorptive potential
6 Comparative study of scattering by hard core and absorptive potential Quantum scattering in three dimension by a hard sphere and complex potential are important in collision theory to study the nuclear
More information221B Lecture Notes Scattering Theory II
22B Lecture Notes Scattering Theory II Born Approximation Lippmann Schwinger equation ψ = φ + V ψ, () E H 0 + iɛ is an exact equation for the scattering problem, but it still is an equation to be solved
More informationPhysics Dec The Maxwell Velocity Distribution
Physics 301 7-Dec-2005 29-1 The Maxwell Velocity Distribution The beginning of chapter 14 covers some things we ve already discussed. Way back in lecture 6, we calculated the pressure for an ideal gas
More informationPhysics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010
Physics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010 1. (a) Consider the Born approximation as the first term of the Born series. Show that: (i) the Born approximation for the forward scattering amplitude
More informationScattering Theory. Two ways of stating the scattering problem. Rate vs. cross-section
Scattering Theory Two ways of stating the scattering problem. Rate vs. cross-section The first way to formulate the quantum-mechanical scattering problem is semi-classical: It deals with (i) wave packets
More informationQuestion. Why are oscillations not observed experimentally? ( is the same as but with spin-1 instead of spin-0. )
Phy489 Lecture 11 Question K *0 K *0 Why are oscillations not observed experimentally? K *0 K 0 ( is the same as but with spin-1 instead of spin-0. ) K 0 s d spin 0 M(K 0 ) 498 MeV /c 2 K *0 s d spin 1
More informationPhys 622 Problems Chapter 6
1 Problem 1 Elastic scattering Phys 622 Problems Chapter 6 A heavy scatterer interacts with a fast electron with a potential V (r) = V e r/r. (a) Find the differential cross section dσ dω = f(θ) 2 in the
More informationQFT. Unit 11: Cross Sections and Decay Rates
QFT Unit 11: Cross Sections and Decay Rates Decays and Collisions n When it comes to elementary particles, there are only two things that ever really happen: One particle decays into stuff Two particles
More informationPHY 5246: Theoretical Dynamics, Fall Assignment # 7, Solutions. Θ = π 2ψ, (1)
PHY 546: Theoretical Dynamics, Fall 05 Assignment # 7, Solutions Graded Problems Problem ψ ψ ψ Θ b (.a) The scattering angle satisfies the relation Θ π ψ, () where ψ is the angle between the direction
More information8/31/2018. PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103
PHY 7 Classical Mechanics and Mathematical Methods 0-0:50 AM MWF Olin 03 Plan for Lecture :. Brief comment on quiz. Particle interactions 3. Notion of center of mass reference fame 4. Introduction to scattering
More informationApplied Nuclear Physics (Fall 2006) Lecture 19 (11/22/06) Gamma Interactions: Compton Scattering
.101 Applied Nuclear Physics (Fall 006) Lecture 19 (11//06) Gamma Interactions: Compton Scattering References: R. D. Evans, Atomic Nucleus (McGraw-Hill New York, 1955), Chaps 3 5.. W. E. Meyerhof, Elements
More informationPhysics 221B Spring 2019 Notes 37 The Lippmann-Schwinger Equation and Formal Scattering Theory
Copyright c 208 by Robert G. Littlejohn Physics 22B Spring 209 Notes 37 The Lippmann-Schwinger Equation and Formal Scattering Theory. Introduction In earlier lectures we studied the scattering of spinless
More informationLorentz invariant scattering cross section and phase space
Chapter 3 Lorentz invariant scattering cross section and phase space In particle physics, there are basically two observable quantities : Decay rates, Scattering cross-sections. Decay: p p 2 i a f p n
More informationRutherford Backscattering Spectrometry
Rutherford Backscattering Spectrometry EMSE-515 Fall 2005 F. Ernst 1 Bohr s Model of an Atom existence of central core established by single collision, large-angle scattering of alpha particles ( 4 He
More informationChapter 10: QUANTUM SCATTERING
Chapter : QUANTUM SCATTERING Scattering is an extremely important tool to investigate particle structures and the interaction between the target particle and the scattering particle. For example, Rutherford
More informationDecays, resonances and scattering
Structure of matter and energy scales Subatomic physics deals with objects of the size of the atomic nucleus and smaller. We cannot see subatomic particles directly, but we may obtain knowledge of their
More informationLecture 7 From Dirac equation to Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 7 From Dirac equation to Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Dirac equation* The Dirac equation - the wave-equation for free relativistic fermions
More informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 1 (2/3/04) Overview -- Interactions, Distributions, Cross Sections, Applications
.54 Neutron Interactions and Applications (Spring 004) Chapter 1 (/3/04) Overview -- Interactions, Distributions, Cross Sections, Applications There are many references in the vast literature on nuclear
More informationFebruary 18, In the parallel RLC circuit shown, R = Ω, L = mh and C = µf. The source has V 0. = 20.0 V and f = Hz.
Physics Qualifying Examination Part I 7- Minute Questions February 18, 2012 1. In the parallel RLC circuit shown, R = 800.0 Ω, L = 160.0 mh and C = 0.0600 µf. The source has V 0 = 20.0 V and f = 2400.0
More informationPhysic 492 Lecture 16
Physic 492 Lecture 16 Main points of last lecture: Angular momentum dependence. Structure dependence. Nuclear reactions Q-values Kinematics for two body reactions. Main points of today s lecture: Measured
More informationPhysics 221B Spring 2018 Notes 34 The Photoelectric Effect
Copyright c 2018 by Robert G. Littlejohn Physics 221B Spring 2018 Notes 34 The Photoelectric Effect 1. Introduction In these notes we consider the ejection of an atomic electron by an incident photon,
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationScattering Theory: Born Series
Scattering Theory: Born Series Stefan Blügel This document has been published in Manuel Angst, Thomas Brücel, Dieter Richter, Reiner Zorn (Eds.): Scattering Methods for Condensed Matter Research: Towards
More informationParticles and Fields
Particles and Fields J.W. van Holten NIKHEF Amsterdam NL and Physics Department Leiden University Leiden NL c 1. Atoms Atoms are the smallest electrically neutral units of chemical elements. They consist
More informationElectron Atom Scattering
SOCRATES Intensive Programme Calculating Atomic Data for Astrophysics using New Technologies Electron Atom Scattering Kevin Dunseath Mariko Dunseath-Terao Laboratoire de Physique des Atomes, Lasers, Molécules
More information10. Scattering from Central Force Potential
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 215 1. Scattering from Central Force Potential Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More informationScattering Theory. In quantum mechanics the basic observable is the probability
Scattering Theory In quantum mechanics the basic observable is the probability P = ψ + t ψ t 2, for a transition from and initial state, ψ t, to a final state, ψ + t. Since time evolution is unitary this
More information1.2 Deutsch s Problem
.. DEUTSCH S PROBLEM February, 05 Lecture VI. Deutsch s Problem Do the non-local correlations peculiar to quantum mechanics provide a computational advantage. Consider Deutsch s problem. Suppose we have
More informationWeak interactions. Chapter 7
Chapter 7 Weak interactions As already discussed, weak interactions are responsible for many processes which involve the transformation of particles from one type to another. Weak interactions cause nuclear
More informationNotes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015)
Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Interaction of x-ray with matter: - Photoelectric absorption - Elastic (coherent) scattering (Thomson Scattering) - Inelastic (incoherent) scattering
More informationCompton Scattering I. 1 Introduction
1 Introduction Compton Scattering I Compton scattering is the process whereby photons gain or lose energy from collisions with electrons. It is an important source of radiation at high energies, particularly
More information1. Kinematics, cross-sections etc
1. Kinematics, cross-sections etc A study of kinematics is of great importance to any experiment on particle scattering. It is necessary to interpret your measurements, but at an earlier stage to determine
More informationMaxwell's Equations and Conservation Laws
Maxwell's Equations and Conservation Laws 1 Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law, since identically. Although for magnetostatics, generally Maxwell suggested: Use Gauss's Law to rewrite continuity
More informationSchrödinger equation for the nuclear potential
Schrödinger equation for the nuclear potential Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 January 24, 2011 NUCS 342 (Lecture 4) January 24, 2011 1 / 32 Outline 1 One-dimensional
More informationConservation of Linear Momentum : If a force F is acting on particle of mass m, then according to Newton s second law of motion, we have F = dp /dt =
Conservation of Linear Momentum : If a force F is acting on particle of mass m, then according to Newton s second law of motion, we have F = dp /dt = d (mv) /dt where p =mv is linear momentum of particle
More informationChapter 18: Scattering in one dimension. 1 Scattering in One Dimension Time Delay An Example... 5
Chapter 18: Scattering in one dimension B. Zwiebach April 26, 2016 Contents 1 Scattering in One Dimension 1 1.1 Time Delay.......................................... 4 1.2 An Example..........................................
More informationLecture 3. Experimental Methods & Feynman Diagrams
Lecture 3 Experimental Methods & Feynman Diagrams Natural Units & the Planck Scale Review of Relativistic Kinematics Cross-Sections, Matrix Elements & Phase Space Decay Rates, Lifetimes & Branching Fractions
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler April, 20 Lecture 2. Scattering Theory Reviewed Remember what we have covered so far for scattering theory. We separated the Hamiltonian similar to the way in
More informationLoop corrections in Yukawa theory based on S-51
Loop corrections in Yukawa theory based on S-51 Similarly, the exact Dirac propagator can be written as: Let s consider the theory of a pseudoscalar field and a Dirac field: the only couplings allowed
More informationThoughts concerning on-orbit injection of calibration electrons through thin-target elastic scattering inside the Mu2e solenoid
Thoughts concerning on-orbit injection of calibration electrons through thin-target elastic tering inside the Mue solenoid George Gollin a Department of Physics University of Illinois at Urbana-Champaign
More informationINTERACTIONS. may be introduced as the number of interactions per unit volume at
(Notes 4) INTERACTIONS 1. Basic concepts When two objects achieve proximity they may interact, a process leading to some change of state of the objects. For practical purposes a laboratory frame of reference
More informationUNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II
UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II August 23, 208 9:00 a.m. :00 p.m. Do any four problems. Each problem is worth 25 points.
More informationSolutions to exam : 1FA352 Quantum Mechanics 10 hp 1
Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem (4 p): (a) Define the concept of unitary operator and show that the operator e ipa/ is unitary (p is the momentum operator in one dimension) (b)
More informationUnits. In this lecture, natural units will be used:
Kinematics Reminder: Lorentz-transformations Four-vectors, scalar-products and the metric Phase-space integration Two-body decays Scattering The role of the beam-axis in collider experiments Units In this
More informationLecture Models for heavy-ion collisions (Part III): transport models. SS2016: Dynamical models for relativistic heavy-ion collisions
Lecture Models for heavy-ion collisions (Part III: transport models SS06: Dynamical models for relativistic heavy-ion collisions Quantum mechanical description of the many-body system Dynamics of heavy-ion
More informationWave Packet with a Resonance
Wave Packet with a Resonance I just wanted to tell you how one can study the time evolution of the wave packet around the resonance region quite convincingly. This in my mind is the most difficult problem
More informationScattering. 1 Classical scattering of a charged particle (Rutherford Scattering)
Scattering 1 Classical scattering of a charged particle (Rutherford Scattering) Begin by considering radiation when charged particles collide. The classical scattering equation for this process is called
More informationDynamical (e,2e) Studies of Bio-Molecules
Dynamical (e,2e) Studies of Bio-Molecules Joseph Douglas Builth-Williams Submitted in fulfillment for the requirements of the degree of Masters of Science March 2013 School of Chemical and Physical Sciences
More information2 Feynman rules, decay widths and cross sections
2 Feynman rules, decay widths and cross sections 2.1 Feynman rules Normalization In non-relativistic quantum mechanics, wave functions of free particles are normalized so that there is one particle in
More informationAngular momentum. Quantum mechanics. Orbital angular momentum
Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular
More informationLecture 22 Highlights Phys 402
Lecture 22 Highlights Phys 402 Scattering experiments are one of the most important ways to gain an understanding of the microscopic world that is described by quantum mechanics. The idea is to take a
More informationfrom which follow by application of chain rule relations y = y (4) ˆL z = i h by constructing θ , find also ˆL x ˆL y and
9 Scattering Theory II 9.1 Partial wave analysis Expand ψ in spherical harmonics Y lm (θ, φ), derive 1D differential equations for expansion coefficients. Spherical coordinates: x = r sin θ cos φ (1) y
More informationApplied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation
22.101 Applied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation References -- R. M. Eisberg, Fundamentals of Modern Physics (Wiley & Sons, New York, 1961). R. L. Liboff, Introductory
More informationPhysics 139B Solutions to Homework Set 4 Fall 2009
Physics 139B Solutions to Homework Set 4 Fall 9 1. Liboff, problem 1.16 on page 594 595. Consider an atom whose electrons are L S coupled so that the good quantum numbers are j l s m j and eigenstates
More informationHIGH ENERGY ASTROPHYSICS - Lecture 7. PD Frank Rieger ITA & MPIK Heidelberg Wednesday
HIGH ENERGY ASTROPHYSICS - Lecture 7 PD Frank Rieger ITA & MPIK Heidelberg Wednesday 1 (Inverse) Compton Scattering 1 Overview Compton Scattering, polarised and unpolarised light Di erential cross-section
More informationMolecular energy levels
Molecular energy levels Hierarchy of motions and energies in molecules The different types of motion in a molecule (electronic, vibrational, rotational,: : :) take place on different time scales and are
More informationKern- und Teilchenphysik I Lecture 2: Fermi s golden rule
Kern- und Teilchenphysik I Lecture 2: Fermi s golden rule (adapted from the Handout of Prof. Mark Thomson) Prof. Nico Serra Dr. Patrick Owen, Mr. Davide Lancierini http://www.physik.uzh.ch/de/lehre/phy211/hs2017.html
More informationWhy Does Uranium Alpha Decay?
Why Does Uranium Alpha Decay? Consider the alpha decay shown below where a uranium nucleus spontaneously breaks apart into a 4 He or alpha particle and 234 90 Th. 238 92U 4 He + 234 90Th E( 4 He) = 4.2
More information11.D.2. Collision Operators
11.D.. Collision Operators (11.94) can be written as + p t m h+ r +p h+ p = C + h + (11.96) where C + is the Boltzmann collision operator defined by [see (11.86a) for convention of notations] C + g(p)
More informationLecture 3: Propagators
Lecture 3: Propagators 0 Introduction to current particle physics 1 The Yukawa potential and transition amplitudes 2 Scattering processes and phase space 3 Feynman diagrams and QED 4 The weak interaction
More informationPhysics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics
Physics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics We now consider the spatial degrees of freedom of a particle moving in 3-dimensional space, which of course is an important
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma sensarma@theory.tifr.res.in Scattering Theory Ref : Sakurai, Modern Quantum Mechanics Taylor, Quantum Theory of Non-Relativistic Collisions Landau and Lifshitz,
More informationPhysics 216 Spring The Optical Theorem
Physics 6 Spring 0 The Optical Theorem. The probability currents In the quantum theory of scattering, the optical theorem is a consequence of the conservation of probability. As usual, we define ρ(x,t)
More informationEikonal method for halo nuclei
Eikonal method for halo nuclei E. C. Pinilla, P. Descouvemont and D. Baye Université Libre de Bruxelles, Brussels, Belgium 1. Motivation 2. Introduction 3. Four-body eikonal method Elastic scattering 9
More informationExperimental Aspects of Deep-Inelastic Scattering. Kinematics, Techniques and Detectors
1 Experimental Aspects of Deep-Inelastic Scattering Kinematics, Techniques and Detectors 2 Outline DIS Structure Function Measurements DIS Kinematics DIS Collider Detectors DIS process description Dirac
More informationParticle collisions and decays
Particle collisions and decays Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 September 21, 2012 A particle decaying into two particles Assume that there is a reaction A B+C in which external
More informationPHY492: Nuclear & Particle Physics. Lecture 4 Nature of the nuclear force. Reminder: Investigate
PHY49: Nuclear & Particle Physics Lecture 4 Nature of the nuclear force Reminder: Investigate www.nndc.bnl.gov Topics to be covered size and shape mass and binding energy charge distribution angular momentum
More informationψ( ) k (r) which take the asymtotic form far away from the scattering center: k (r) = E kψ (±) φ k (r) = e ikr
Scattering Theory Consider scattering of two particles in the center of mass frame, or equivalently scattering of a single particle from a potential V (r), which becomes zero suciently fast as r. The initial
More informationwhich implies that we can take solutions which are simultaneous eigen functions of
Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3-D Quantum mechanics in 3-D For most physical systems, the dynamics is in 3-D. The solutions to the general 3-d problem are quite complicated,
More informationChapter 3-Elastic Scattering Small-Angle, Elastic Scattering from Atoms Head-On Elastic Collision in 1-D
1 Chapter 3-Elastic Scattering Small-Angle, Elastic Scattering from Atoms To understand the basics mechanisms for the scattering of high-energy electrons off of stationary atoms, it is sufficient to picture
More informationPhysics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 20: Rotational Motion. Slide 20-1
Physics 1501 Fall 2008 Mechanics, Thermodynamics, Waves, Fluids Lecture 20: Rotational Motion Slide 20-1 Recap: center of mass, linear momentum A composite system behaves as though its mass is concentrated
More informationUniversity of Illinois at Chicago Department of Physics. Quantum Mechanics Qualifying Examination. January 7, 2013 (Tuesday) 9:00 am - 12:00 noon
University of Illinois at Chicago Department of Physics Quantum Mechanics Qualifying Examination January 7, 13 Tuesday 9: am - 1: noon Full credit can be achieved from completely correct answers to 4 questions
More informationSelected Topics in Mathematical Physics Prof. Balakrishnan Department of Physics Indian Institute of Technology, Madras
Selected Topics in Mathematical Physics Prof. Balakrishnan Department of Physics Indian Institute of Technology, Madras Module - 11 Lecture - 29 Green Function for (Del Squared plus K Squared): Nonrelativistic
More informationPHY-494: Applied Relativity Lecture 5 Relativistic Particle Kinematics
PHY-494: Applied Relativity ecture 5 Relativistic Particle Kinematics Richard J. Jacob February, 003. Relativistic Two-body Decay.. π 0 Decay ets return to the decay of an object into two daughter objects.
More informationLecture 3. Solving the Non-Relativistic Schroedinger Equation for a spherically symmetric potential
Lecture 3 Last lecture we were in the middle of deriving the energies of the bound states of the Λ in the nucleus. We will continue with solving the non-relativistic Schroedinger equation for a spherically
More informationIntro to Nuclear and Particle Physics (5110)
Intro to Nuclear and Particle Physics (5110) March 23, 2009 From Nuclear to Particle Physics 3/23/2009 1 Nuclear Physics Particle Physics Two fields divided by a common set of tools Theory: fundamental
More informationBethe-Block. Stopping power of positive muons in copper vs βγ = p/mc. The slight dependence on M at highest energies through T max
Bethe-Block Stopping power of positive muons in copper vs βγ = p/mc. The slight dependence on M at highest energies through T max can be used for PID but typically de/dx depend only on β (given a particle
More information