NNSP neutron summer course, 2021 (online)

Description of intro session: Math and Physics for approx. 35 non-proficient students

Kim Lefmann, Pia J. Ray

Original document for 2019 school by KL, September 8, 2019. Edited by PJR, October 12, 2021.

Day 1    Thursday, October 14, 2021

1. Introduction and Motivation (0.5 h) (blackboard + slides) (Kim)

1. Intro to course
2. Plan for intro days
3. Scattering picture: 1 and N atoms

2. Mathematical background (6 h) (short lectures + individual teaching with KhanAcad.) (Kim)

1. Trigonometry  
2. Exponential functions
3. Complex numbers
4. Integrals

Day 2    Friday, October 15, 2021

3. Wave Physics (3 h) (lecture + individual) (Kim)

1. Definition: wave = media + disturbance
2. Wave descriptors, λ, k, ω, ν, c / Demonstration with a string/spring
3. Wave field sine/cosine
4. Superposition, interference
5. Complex wave description
6. Experiment, demonstration (string)
7. Problems (from Mastering Physics and Hecht)

4. Fourier Transforms (2 h) (mostly lecture) (Kim, it was Johan in 2019)

1. Integrals of trigonometric functions
2. Finite system, Periodic function, Fourier Series as a postulate, Finite FT
3. Infinite system, generalization, infinite FT as a postulate, FT of a lattice and a particle
4. MATLAB demo online (did not work last time)

5. Scattering Physics (1 h) (Kim)

1. Neutrons and X-rays as waves
2. 1-atom scattering, the complex scattered wave
3. 2-atom scattering, complex description of interference, motivation to define q = ki – kf
4. N-atom scattering, diffraction cross section as postulate, Fourier transform
5. Examples: lattice, particle
6. Experiment: laser on optical lattice

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1. Motivation

Overview: Teaching plan for course

Overview: Teaching plan for math-phys intro

Reminder on neutrons, light, and X-rays, materials, experimental setup

Discussion: What is similar between these types of radiation? (wave nature, wavelength size)

Discussion: How do we measure the radiation? (particle detectors)

Discussion: How do we determine scattering angle? (collimators, slits)

Neutrons and X-rays

• Particle-wave duality; energy quanta of the two
• Same duality, same scattering Picture: 1 + atoms

Motivation: We will understand the scattering on an atomic level; therefore we need to know about waves and the corresponding math.

Material:

Slide: intro

Slide: Teaching plan

Slide: Teaching plan for M-P intro

Presentation: Kim

2. Mathematical background

Trigonometry

• Unit circle, define sine and cosine
• Rectangular and polar coordinates
• Addition formulae
• Problems about the unit circle, the Pythagorean identity, trigonometric functions:

• https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func 
  parts belonging to quiz 1 and 2

Exponential functions

• Distinction between powers and exponentials
• 2x + 2y = 2x + y;  20 = 1
• e = 2.718...;  e–x = 1/ex;  shape of function
• Problems about powers and exponential growth:

• https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay
  parts belonging to quiz 1
• https://www.khanacademy.org/math/algebra-home/alg-exp-and-log 
  only exponential part

Complex numbers

• i ;  i2 = –1
• Complex plane;  c = a + i b
• Complex conjugate;  c* ;  |c|2 = c c* = a2 + b2
• Complex number as polar coordinate;   c = r (cos φ + i sin φ)
• Definition of complex exponential;   c = r exp(i φ)
• Show that   exp(i φ) exp(i χ) = exp(i (φ + χ))   - the “usual” behaviour of exponentials
• Problems about complex numbers:

• https://www.khanacademy.org/math/algebra-home/alg-complex-numbers 
  all
• Problems 1, 2, and 3 from the Math 307 (The Complex Exponential Function) lecture notes

Integrals

• Definition drawing, finding area, (inverse derivative)
• Example, integrate y = x
• Integrate complex functions = Integrate real part + i integrate imaginary part
• Problems about integration:

• https://www.khanacademy.org/math/ap-calculus-ab/ab-antiderivatives-ftc
  parts belonging to quiz 1, 2, and 3

Material:

Link document to Khan Academy problems

Math 307 (The Complex Exponential Function) lecture notes

Presentation: Kim

3. Wave physics

Definition: wave = media + disturbance

• Discussion: examples (water-height, string-displacement, light-EM field, sound-pressure)
• Harmonic waves

Wave descriptors

• Drawing:  y(t), y(x), y(x,t)
• Discussion:  find λ, k, ω, ν, c 
• Demonstration with a string/spring

Wave field sine/cosine

• Harmonic approximation
• Description of wave as cos(kx), cos(ωt)
• Drawing of A(x,t)
• Description as A = A0 cos(kx – ωt), A0 cos(kx – ωt)
• Plane waves

Superposition, interference

• Scattering from two sources, Huygens, add waves
• Standing wave

Complex wave description

• Construction to ease calculations, real part is the classical wave
• In QM, the imaginary part is “real”
• One spherical wave from scattering
• Complex description of interference

Experiment, demonstration (string or spring)

Problems: There are some suggested problems from the two books below, focus on the ones from University Physics first

Material:

Spring experiment

Scanned textbooks: Hecht (chapter 2), University Physics/Mastering Physics (chapter 15)

Selected problems from University Physics (note: not the discussion questions, but the exercises starting on page 501): 15.4, 15.8, 15.11, 15.35, 15.39, 15.76

Selected problems from Hecht: 2.1, 2.2, 2.13

Extra: Describe standing waves with complex description

                still needed: solutions to problems in Hecht

Presentation: Kim

Integrals of trigonometric functions

• ∫02π sin(x), sin(2x), sin(x)sin(2x) etc. all = 0; graphical solution
• ∫02π sin^2(x) = π; graphical solution

Finite system, Periodic function

• Example: f(x) = A1 sin(x) + A2 sin(2x);  tabulated values;  find A1, A2 by integration
• Fourier Series on (0,2π);  on (0, L)
• Postulate. Describe any periodic function
• Finite FT

Infinite system

• Generalization, let L → ∞
• Infinite FT equation
• Example: FT of a lattice and a particle

• MATLAB demo online (not ready)
• FFT webcam demonstration (draw some patterns and play around with it!):

• https://sciencedemos.org.uk/fft_webcam.php 
  

• Demonstrations online / in browser:

• http://demonstrations.wolfram.com/FourierSeriesOfSimpleFunctions/ 
• http://demonstrations.wolfram.com/FourierSynthesisForSelectedWaveforms/ 
• http://demonstrations.wolfram.com/ComparingFourierSeriesAndFourierTransform/ 
• http://demonstrations.wolfram.com/SamplingTheorem/ 
  

• Problems: 

• https://pan-learning.org/wiki/index.php/Problem:Fourier_transform 

• Problem suggestions from Arfken-Weber-Harris: 

• 19.1.3, 19.2.7, 19.2.9, 20.2.1, 20.3.2, 20.3.3

Material:

Lecture notes (hand written by Johan Hellsvik)

Scanned textbook chapter: Arfken, Weber, et al, Fourier Series and Integral Transforms (chapter 19+20, plus a document with exercise solutions for those two chapters)

Handout of slides: Reciprocal Lattice

PDF of Fourier Transform problems from e-neutrons/pan-learning wiki, with hints and solutions

Presentation: Kim (was Jacob)

1-atom scattering, the complex scattered wave

• A = b/|r – rj| exp(i k |r – rj|) exp(–i ωt)
• Example: find scattering intensity

2-atom scattering

• Derive A = b/r ∑j exp(i q·rj ) exp(–i ωt); motivation to define q = ki – kf
• Example: find scattering intensity

N-atom scattering

• Diffraction cross section as generalisation; | ∑j bj exp(i q·rj) |2 
• Compare to Fourier transform of atomic positions
  

Examples:

• Periodic lattice; sharp features
• Spherical particle; broad features
  

Experiments:

• Laser on optical lattice
• Laser on hair

Material:

Slides, same as day 1

Neutron notes chap. 2

Exp: Laser, lattice, hair

Presentation: Kim