HW2Problem_Set2

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Audrey Gabbard BME 6931 Problem Set 2 – Due Friday September 15, 2023 at 5:00 PM EST Total 150 points 1. (50 points) – Set up a Michelson interferometer to measure the wavelength of a light beam, and also to measure the index of refraction (n) of a substance. Refer to the accompanying diagram. The light source (a tunable laser) at left shines light onto a beam splitter. The split beams each hit a mirror (M1, M2), one of which is movable (M2), then recombine at the beam splitter and continue to a detector for light intensity. You will write Matlab code to detect the intensity of the recombined beam as a function of the position of M2. A. Assume that both M1 and M2 are initially set to be 10 cm from the beam splitter. Move M2 away from the beam splitter in 0.025 µm increments over a distance of 2µm. At each position of M2 (including the 10cm initial position), compute the resultant intensity of light at the detecto r. Repeat the calculation for wavelengths of light between 350 and 800 nm, in 50 nm increments. a. Plot your intensity data for each wavelength separately, in a 5 row X 2 column array (10 different wavelengths). Keep the scales for the abscissa and ordinate identical across plots. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 350nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 400nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 450nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 500nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 550nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 600nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 650nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 700nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 750nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 800nm Intensity Plots at each Wavelength on M2"s distance from beam splitter
b. For each wavelength, indicate the distance that the mirror was moved between two maxima in your plots. Then use that value to calculate the wavelength of the light , which should match the wavelength that you introduced into the interferometer and quantified by moving M2. c. In words, indicate what measurement is most fundamental for your calculations, and which also determines the accuracy of the interferometer. The most important measurement for the calculations is the wavelength and amplitude that the Intensity is being sampled at for the interferometer to determine the characteristics of light waves. The amplitude of the waves directly influences the brightness or strength of the light, which is key to determining the intensity of the light at a given point. In an interferometer, the accuracy of the interference patterns relies on the precise measurement and control of the amplitude of the two interfering waves. Any variations or discrepancies in the amplitude can significantly impact the accuracy of the interferometer's measurements. Therefore, ensuring precise and reliable amplitude 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 350nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 400nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 450nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 500nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 550nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 600nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 650nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 700nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 750nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance (um) 0 0.5 1 Intensity (W/m 2 ) Intensity at 800nm Intensity Plots at each Wavelength on M2"s distance from beam splitter X 0.7 Y 1 X 1.4 Y 1 X 0.7 Y 1 X 0.35 Y 1 X 0.9 Y 1 X 0.45 Y 1 X 0.8 Y 1 X 0.4 Y 1 X 1 Y 1 X 0.5 Y 1 X 1.1 Y 1 X 0.55 Y 1 X 1.3 Y 1 X 0.65 Y 1 X 1.5 Y 1 X 0.75 Y 1 X 1.2 Y 1 X 0.6 Y 1 X 1.6 Y 1 X 0.8 Y 1
measurements is essential for both intensity calculations and the overall accuracy of interferometric measurements. B. Some guidance for computing intensity: a) You’ll need to calculate the phase shift between the light paths that was introduced by moving M2. b) Next, calculate the sum of two sine waves with that phase shift. a. Sample the sine waves at 0.1*π increments through one complete cycle (2π) and add the values at each sample point. c) To compute intensity, square each value of the summed sine wave (recall Intensity is proportional to Amplitude 2 ). d) Sum the 21 values and divide by 21 to achieve the average intensity. The average intensity is 0.9524 for the 21 samples. 2. (30 points) Using the interferometer from problem 1, we will measure the index of refraction (n) of a thin piece of glass. A. Introduce into the middle of the light path from the beam splitter to M1 (at the 5 cm location) a piece of glass 1mm thick. For the purpose of this problem, assume that the glass has index of refraction n = 1.4. a. For lambda = 500 nm, plot the intensity profile while moving M2 as in problem 1 without and with the piece of glass introduced into the light path. Plot both profiles on the same graph. M1 is 5cm from beam splitter Thickness of glass, h = 1mm Index of refraction of glass, n = 1.4
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Lambda = 500nm Plot intensity vs distance twice without glass and then with glass in light path on same graph b. If you received the data that you plotted, knowing only the thickness of the piece of glass, how would you use the information to calculate n? Write Matlab code to generate your plots and comment on your thought process for each step. Imagine not knowing n (index of refraction), inference from the two plots with and without the glass, then go backwards to prove the calculation of n 3. (20 points) Using the wavelet approach of Huygens, and the attached diagram, describe how the plane wave incident in air at 60° from vertical to a flat piece of glass is refracted in its direction through the glass. The diagram is from lecture, and the incident plane wave in the diagram is not 60° to the surface of the glass, so you’ll need to draw your own diagram, based on the diagram pictured here. Draw on paper the excitation of each atomic center at each step of time as a radiating circle, with the circle diameters in the glass as 2/3 the diameter of the circles outside of the glass. Draw a straight line across the fronts of each set of wavelets at each point in time. Measure the refraction angle on your drawing with a compass and report the actual value.
Huygen’s interpretation of light refraction. Draw your own diagram, with equally space radiating centers at the surface of the glass, to explain the angle of refraction. Does your measured angle (theta) and refractive index (n) values approximately follow Snell’s Law? Show your work. θ = 60 ° n = 1 hypotenusewavelength 1 .47 →n = 2.123 2.123 = sin ( 60 ) sin ( ¿ R ) sin ( R ) = sin ( 60 ) 2.123 sin ( R ) =− 0.1437 ¿ sin 1 ( 0.1437 ) = R→ 8.2614 ° = R
4. (5 points) From the ppt of Week 3, Lecture 2, slide 8, perform the last algebraic steps [Substitute (13) into (10)…] and arrive at equation 14. Show clearly each step of your work. Is equation 14 correct? o Yes! One important check in a physics problem is to determine if the units cancel and are correct. Equation 14 should have the units of coulomb (C) for charge (Q). Show clearly each step of your work. Q = 18 π d g V η 3 v f 3 2 ρg ∗( 1 + v r v f ) Q ( Coulombs )= 18 π d g V η 3 v f 3 2 ρg ∗( 1 + v r v f ) Coulomb = Joules Voltage Q is charge o Units: Coulomb (C) or Joules/Volts (J/V) d is distance o Units: meters g is gravity, 9.81m/s^2 o Units: m/s^2 V is voltage o Units: Volts η is viscosity (of air) o Units: Pa*s or kg/(m*s) ρ is rho (of oil) o Units: kg/m^3 v f is falling velocity o Units: m/s v r is rising velocity o Units: m/s
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5. (20 points) Calculate the velocity of electrons moving through the column of an electron microscope . a. From the example equation in the lecture relating 100,000 eV of kinetic energy to particle velocity (100k eV = (1/2)mv 2 , and the deBroglie relation lambda = h/mv , generate an expression relating v to lambda. Show your work a. Symbols are: i. e = electron charge, ii. m = electron mass iii. c = speed of light iv. h = Planck’s constant Momentum p = mv (I think) Lambda = h/mv = h/p E electron = 100,000 eV = 1 2 m v 2 200,00 eV = m v 2 200,000 eV m = v deBrogli e ' s : λ = h mv →λ = h m ( 200,000 eV m ) b. Write a Matlab script, to calculate velocities and wavelengths when electrons are accelerated by the high voltage transmission electron microscope (TEM) (1MeV), intermediate voltage TEMs (400keV, 200keV), standard voltage TEM (100keV), 30keV typical of an SEM, an SEM adjusted for a landing voltage of 2.5keV and 1keV, and 200V more typical of a cathode ray tube. TEM = 1E6 eV, 400E3 eV, 200E3 eV, 100E3 eV, 30E3 eV, 2.5E3 eV, 1E3 eV, 200 eV c. Plot wavelength(lambda) and velocity vs these electron energies, using the left y- axis for wavelength and a second y-axis plotted at the right side of the graph for velocity.
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0 1 2 3 4 5 6 7 8 9 10 Electron Energy (eV) 10 5 0 0.5 1 1.5 2 2.5 3 3.5 Wavelengths (m) 10 -20 0 5 10 15 Velocity (m/s) 10 17 Wavelength (left) and Velocity (right) at Electron Energies (eV) d. Comment on the values you plotted, and indicate if you think there are problems with any of the values. No problems. e. Research the fastest serve recorded in men’s tennis. World’s Fastest Men’s & Women’s Tennis Serves Ever Recorded (tenniscompanion.org) a. Using typical metrics for a tennis ball, calculate the associated wavelength of the tennis ball during the serve. Wavelength,lambdaduring serve = 1.604E-34 m λ = 1.604E-25 nm b. Calculate the wavelength for a baseball traveling at the same speed. Wavelength,lambdabaseball = 6.2712E-35 m λ = 6.2712E-26 nm c. Comment on the different values. i. The difference in the tennis ball and baseball is that the wavelength for the tennis ball is larger than the baseball. The only difference in the calculation of the wavelengths was the mass of the different balls. The baseball has a greater mass than the tennis ball, thus by deBroglie’s formula, this means that with greater mass, the wavelength will decrease.
6. (10 points) Compare a TEM to a SEM. a. Draw the elements of the electron optical train for transmission (TEM) and scanning (SEM) electron microscopes.
- Transmission Electron Microscope (TEM): o Electron Source (Tungsten filament or field emission gun) o Condenser Lens System o Sample (ultrathin section of the specimen) o Objective Lens o Intermediate Lens System o Projector Lens System o Image Plane o Fluorescent Screen or Camera for imaging - Scanning Electron Microscope (SEM): o Electron Source (Tungsten filament or field emission gun) o Condenser Lens System (usually not used in SEM) o Sample o Objective Lens (usually not used in SEM) o Scanning Coils (for scanning the electron beam) o Detector (various types, such as Everhart-Thornley Detector or secondary electron detector) b. List the elements of the TEM that could be used in construction of a SEM without modification. Electron source (e.g., tungsten filament or field emission gun) c. List and indicate the functions of the elements that are found only in the SEM. - Scanning Coils: These coils are used to scan the electron beam across the surface of the sample in a raster pattern. This scanning motion is essential for producing images in SEM. - Detector: SEMs have specialized detectors, such as Everhart-Thornley Detectors or secondary electron detectors, which are used to collect signals generated by interactions between the electron beam and the sample surface. These detectors are not present in TEMs. d. For these SEM elements, indicate why they are placed where they are placed. - Scanning Coils: These coils are placed near the sample chamber to control the position of the focused electron beam and scan it across the sample surface. The precise control of the beam position allows for the creation of high-resolution images. - Detector: The detector is placed near the sample chamber to collect signals efficiently from the interactions between the electron beam and the sample. Its position allows it to capture electrons and signals emitted from the sample's surface during scanning.
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e. What electron detector types are used in SEM? Research and report on the energy range of electrons captured using each detector. - Everhart-Thornley Detector (ETD): This is a common secondary electron detector in SEMs. It is used to detect low-energy secondary electrons emitted from the sample's surface due to electron beam interactions. The energy range for secondary electrons typically ranges from a few eV to a few keV. - Backscattered Electron Detector (BSD): This detector is used to detect backscattered electrons, which are higher-energy electrons that are scattered backward when they collide with the sample's atomic nuclei. The energy range for backscattered electrons typically spans from a few keV to several tens of keV. - In-Lens Detector: This is an advanced type of secondary electron detector integrated into the objective lens of some SEMs. It can detect both low-energy and high-energy secondary electrons, covering a wide energy range. - Energy-Dispersive X-ray Spectroscopy (EDS) Detector: While not an electron detector, EDS detectors are often used in conjunction with SEMs to analyze X- rays generated by the interaction of the electron beam with the sample. EDS can measure X-rays in a wide range of energies, depending on the specific detector and application, from a few keV to tens of keV. 7. (20 points) Re-visit the layer 2 pyramidal cell Narration, and inspect carefully the spines on the dendrites. If you were permitted to generate up to 6 classes of spines from qualitative expression, what would they be? Feel free to choose fewer than 6 categories. Draw your examples, and in no more than a few bullet points, define their features. Draw examples of spines that are most difficult to categorize. Trace along the main apical dendrite (straightest route to the brain surface). Approximately divide the length into 4 equal segments. Record the number of each of your types of spines in each of the 4 segments. Segments Small (Figure 1) Medium (Figure 2) Large (Figure 3) 1 4 9 2 2 6 8 2 3 6 10 1 4 9 9 2
SMALL MEDIUM
LARGE
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