Name ___________________________________ Take Home Assignment 1 Due: Mon, Oct. 14th, 2019, start of class.

1

Show your work where appropriate to receive full credit. 100 pts total

Feel free to use any software you want for this. You can work with up to two colleagues. Please provide the names of those colleagues who you worked with. 1. Quantifying the error in a measurement (25 pts). Using the intensity background data posted on Canvas in the “Take Home Assignment Announcement” page (filename = BackgroundIntensityData.xlxs), do the following a) (5 pts) compute the arithmetic mean of the data b) (5 pts) compute the arithmetic standard deviation of the data c) (5 pts) make a figure (with x and y axes labels) that shows the distribution of OD values. d) (10 pts) fit the distribution with a Gaussian function and overlay the fit on the data (like the we

did in class), and report the true population mean and the true standard deviation. The standard deviation is the error in intensity measurement of I, which will be used below.

2. Error propagation (75 pts). The optical density of a single nanoparticle is measured using single particle electro-optical imaging (Evans, R.C. et al. Proc. Natl. Acad. Sci., 2019, 116 (26) 12666-12671). In a typical measurement, we measure optical transmission images (top panels in the Figure below) as a function of time t. We determine the particle dimensions using electron microscopy images (bottom panels). We calculate the particle thickness (d)-corrected change in the optical density at any time t (DOD(t)d) by computing the light transmitted through the nanoparticle at any time t (I(t)) and the background at the same t (I0(t)). The equation is given below. The I values come from the sum of all the integer pixel values in the red ovals. The units of I are counts, which refers to the number of photons that strike the camera pixel, and you can consider counts as an arbitrary unit, and therefore I is unitless.

∆OD$() = OD()$ − OD(0)$ = +,(-)⬚

$ − +,

(/)⬚ $

= 0123456

7(8) 75(8)

9

$ +

123456 7(5) 75(5)

9

$

Given: I (t = 0) = 21505 counts I0 (t = 0) = 21105 counts I (t = 1000) = 20012 counts I0 (t = 1000) = 21035 counts d = 0.105 µm, percent relative error in d is 5%, The error in I was determined above. Calculate: a) DOD(t)d when t = 1000 s. b) Propagate the error through the calculation so that you can report the absolute uncertainty in DOD(1000)d.