Diffraction from a Ruler

Transcription

1 Lab #27 Diffraction page 1 Diffraction from a Ruler Reading: Giambatista, Richardson, and Richardson Chapter 25 (25.1, 25.6, 25.7, 25.9). Summary: In this lab you will learn the diffraction analysis scientists use to determine the crystal and molecular structure of proteins, minerals, semiconductors, polymers, and metals. However, these materials are very complicated so, as an introduction, we will simulate a single line of atoms or molecules using the engravings on a metal ruler. Then you will shine visible-light photons (from a helium-neon laser) on this single atomic/molecular line, the laser light will bounce off the atoms/molecules, and the interfering light beams will produce a pattern of diffracted spots. From the distance between these diffracted spots, you will deduce the atomic/molecular spacing. Note: Answers to all questions are to be given in full sentences, such that by looking at the answer, the reader knows what question is being answered without having to read the question. Any answers not written this way, even if they are correct, will receive 0 credit. Pre-Lab Analysis Three-dimensional crystal or molecular structures are of utmost importance in determining (1) a material s properties and (2) how the atoms and molecules behave within in a material. The technique most frequently employed in scientific disciplines to analyze the three-dimensional structures is x-ray diffraction. In biology and chemistry, NMR is also used for structure determinations, but NMR only provides information about which atoms are connected to which and it is frequently difficult to determine the three-dimensional arrangement of these atoms. For those of you who have taken organic chemistry, you have learned about stereo-chemistry where the three-dimensional nature of the molecule is important. To determine this three-dimensional structure, about the only definitive technique is x-ray diffraction. Hence the study of this technique here. For diffraction to occur, the wavelength of the incident photons must be smaller and within three to four orders of magnitude of the distance between the things (slits as in your textbook, or atoms/molecules in real materials) off which the photons diffract. If the photon wavelength is too large, no photons will pass through the slits and if the wavelength is too small, the photons will pass through undisturbed by diffraction. For example in your textbook, multiple slit diffraction (e.g. from a diffraction grating) is described where the slits are about 1 µm apart. Your textbook says visible light will diffract from these slits. According to the rule above, the photon wavelengths need to be on the order of µm (1 nm) to 1 µm (1000 nm) to cause diffraction. Since visible light consists of wavelengths from ~400 nm to ~800 nm, visible light will diffract from these diffraction gratings. For crystalline materials, in order for diffraction to occur the photon wavelength therefore must be within the range from roughly the same size and within three to four orders of magnitude smaller than the spacing between the atoms or molecules inside the crystal. Only electron and x-ray wavelengths are small enough and hence electrons and x- rays are used to probe the crystal structure of materials. Once the atoms or molecules diffract the electrons or x-rays, beautiful and intricate spot patterns form and are captured on film. These patterns are unique to each different crystal structure (of which there are only 14 basic ones in all of nature). From these diffraction patterns, x-ray crystallographers work

2 Lab #27 Diffraction page 2 backward to determine the crystal structure that caused the pattern and with that information they find the exact position of atoms and molecules within the crystal. However, such analysis is very complex, especially for three-dimensional crystals. For your introduction to diffraction in this lab, we will tackle a slightly less complicated problem than the three-dimensional structure of biological materials. In superconducting materials and some semiconductors, 1-dimensional structures (like an ultrathin line) offer unique properties that can only be explained by quantum mechanics. To understand this quantum mechanics and to use it to predict the electrical and optical properties of these materials, the inter-atomic distance must be known. Conveniently, the distance between the atoms in the line is relatively easy to determine using x-ray diffraction. In this lab you will use the engraving on a piece of metal to simulate the atoms in a one-dimensional structure, but the engravings will be spaced much farther apart than the atoms in the semiconductors or superconductors. Thus you will need to use photons of longer wavelengths than electrons or x rays. In fact since the engravings will be on the order of 1mm, three to four orders of magnitude smaller than this is between 0.1µm and 1µm. the wavelength of red light is within in the range and in this lab you will diffract red photons (from a helium-neon laser) from the simulated atoms (ruler engravings). Once the atoms diffract the photons, a spot diffraction pattern will form on a screen about 1 m to 2 m away. From the distance between the diffracted spots and the diffraction formula, you will calculate very precisely (as real crystallographers do) how far apart the atoms must have been in order to cause the observed diffraction pattern. You will then check this inter-atomic spacing against the markings on the ruler. The equation that relates the inter-atomic spacing d between the ridges (engravings) on the ruler, to the angle q at which the diffracted spot is observed and the wavelength l of the photons being diffracted (from your textbook) is: sinj = m l, m = 0, ± 1, ± 2,... d where m represents the order of the diffracted spot. Figure 1 illustrates the experimental setup. The beam from a He-Ne laser strikes a horizontal steel ruler at a grazing incident angle Ji. Diffraction spots are observed at heights y n (n =..., -2, -1, 0, 1, 2,...) on a vertical screen or wall. Each maximum corresponds to a diffracted angle Jd,n, which will vary depending on the value of n. According to the geometry shown in the figure tan (Jd,n) = y n /L. To observe diffracted spots at y n the difference in the path lengths between Ray 1 and Ray 2 must be an integral number of wavelengths nl. Writing DSR for the distance from the photon source (laser) to the ruler at point 1 and DSR for the distance to point 2, and similarly DRD for the distance from the ruler to the photon detector (screen) for Ray 1 while DRD is for Ray 2, we find:

3 Lab #27 Diffraction page 3 path difference between Ray 1 and Ray 2 = D 1 SR 42 + D 4 RD 3 - D SR' +D RD 43 ' Ray 1 ( ) Ray 2 Ray 2 laser Ray Ray 1 Ray 2 J d,n J i 2 1 d yn L Figure 1: Experimental arrangement. = ( D SR - D SR' )+ ( D RD - D RD' ) =(distance between Point 1 and Point 2) + (distance between Point 1 and Point 2 ) = d cosjd,n d cosji The last expression above was obtained using trigonometry of the two right triangles 1 12 and Hence, the criterion for a diffraction maximum of order n is nl = d cosjd,n d cosji (1)

4 Lab #27 Diffraction page 4 This equation can be rewritten in terms of the measurable distances yn, y0 (the height of the zero th order spot, which is when Jd,0 = Ji), and the distance d (that you are trying to find): where z = n ( y 2 n - y 2 ) 0 L 2. nl = d y n - y 0 L 2 = d z 2 n fi z n = 2 l d n (2) Equation (2) is the working equation for this experiment. Thus, if you measure y n for several n, calculate the corresponding z n, and fit a line to the equation involving z n and n, you will able to experimentally determine d (if you know l), the distance between the markings on the ruler! yn (cm) n 1.) Photons of wavelength 682 nm are incident on a single line of engravings and the photons produce a diffraction pattern on a screen 1.3 m away with the diffracted spots at the distances given in Table a.) Calculate each zn. [10 pts] b.) Make a plot in Excel and fit a Trendline to it so that from the Table 1: Diffraction data. slope of the Trendline you can find the engraving spacing that produced the observed diffraction pattern. Print your plot with the fit and equation of the fit showing. [7 pts] 2 c.) Calculate the engraving spacing. Don t forget to label your answer, include units, and use the correct number of significant figures. [6 pts] 3 d.) What is the engraving spacing in inches? To what fraction does this decimal correspond? [7 pts] 4 2.) Outline the lab following the format of Outline Format posted on the Electronic Reserves web page. [20 pts] 5 Equipment to be used in this lab: He-Ne gas laser 1 engraved steel ruler (with 1 /64 or 1 /32 marked on it) 1 meter stick 1.) Equipment Set-up: Laser, ruler, and white board r Turn on the laser and make sure it illuminates about an inch of the metal ruler. r In your notebook record which engravings on the metal ruler are illuminated by the laser (i.e., note the value of d). [3 pts] 6

5 Lab #27 Diffraction page 5 r The diffraction pattern produced by the laser light diffracting from the metal ruler should be projected onto the white board on the wall. Take a few minutes to familiarize yourself with the apparatus and the pattern on the wall. Adjust the laser beam and ruler to optimize the sharpness and intensity of the diffraction pattern. 2.) Measuring the diffraction pattern A. Finding the position of the zero th order spot r There will be many spots up to perhaps 20 on the wall. Most of them will correspond to positive values of n but a few may correspond to negative values of n. To analyze your data correctly, you must know which spot corresponds to n = 0. r The zero th -order spot appears when the path difference is zero, i.e., when Jd,0 = Ji, or when the angle of incidence equals the angle of reflection. How will you determine which spot is the zero th -order spot? [3 pts] 7 r Once you have identified a procedure, check with your the instructor to make sure it will work. B. Determining origin from which all yn will be measured r The way angles were introduced in the Pre-lab, the yn s are measured from a specific origin. This origin is the extrapolation of the ruler to the white board. You must figure out a way to extrapolate the ruler to the wall. Describe your method in your notebook with a sketch. [3 pts] 8 To get you started: note where the laser beam strikes the wall when the ruler is moved completely out of its path. Then compare this position with the position of the zero th order spot. Where is the origin relative to these two points? Why? [5 pts] 9 r Mark your origin on the white board. C. Identifying the point within the diffraction spot to which to measure for yn r Each bright spot will have a width (in other words the diffraction spots will not be tiny pin dots of light). This complicates the yn measurements. r There are two methods you can choose for determining the point to which you will measure yn and you will need to think carefully about which one will produce a more accurate determination for d. Choose one method, describe it in your notebook, and explain why you think it is the better method. [5 pts] 10 One method is to choose the absolute brightest part of each diffraction spot. The second method is to measure the width of each diffracted spot and locate its true center. D. Measuring yn r Mark the location of each diffraction maximum on the white board. [3 pts] 11 r Carefully measure y n for as many diffracted spots as you can. r Record the value of n you have determined for each spot. (A table may be the easiest way of organizing your data.) [5 pts] 12

6 Lab #27 Diffraction page 6 r Do not erase your data from the white board and do not bump or move your experimental setup, until you have found d (Part 3 below) and it is within 5% of the numbers marked on the metal ruler. E. Measuring L r There is some ambiguity about the distance between the ruler and the wall (L), since a large portion of the metal ruler is illuminated. From where on the ruler should L be measured? Why? [4 pts] 13 r Measure L and record it in your notebook. [4 pts] 14 3.) Determining d r Choose 8 yn, find the corresponding zn, and record them in your notebook. [9 pts] 15 r Plot your data in Excel and fit a line to it (see Prelab Question 1). Print your results with the equation of the fitted line showing. [5 pts] 16 r Calculate d (don t forget units) and label/circle your answer. [10 pts] 17 r Compare your calculated d to the value given on the ruler. [4 pts] 18 r List any possible sources of error. [4 pts] 19 4.) Cleaning Up r When you are finished with your experiments and calculations, make sure the laser illuminates a section of the ruler and that the diffraction pattern is easily readable on the wall, as in Part 1. r Call your TA over and show him or her that everything is lined up correctly. r Once your TA has checked everything, shut off the laser. [5 pts] 20 Note: the laser wavelength will be given in lab.