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DNA Diffraction - PHYS 2 LAB REPORT

PHYS 2 LAB REPORT
Course

General Physics II (PHYS 1325)

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Wave Optics:

Diffraction Gratings and DNA

Introduction

Figure 1: X-ray diffraction pattern of B-DNA (the hydrated form of the molecule) taken by Rosalind Franklin.

O

ne of the triumphs of 20th-century biology was the determination of the double-helix structure of DNA (deoxyribonucleic acid). James Watson and Fran- cis Crick proposed the structure in 1953 (and received the Nobel Prize in 1962), and the key to their discovery was the X-ray diffraction pattern of DNA produced by the crystallographer Rosalind Franklin (who, sadly, died of ovarian cancer in 1958 and did not share in the No- bel Prize, which is never awarded posthumously). The diffraction pattern is reproduced in Fig. 1. 1

Prior to her experiments it was known that DNA is a long fibrous polymer made up of phosphate groups, de- oxyribose sugar, and the four bases (each bonded to a sugar and a phosphate) thymine (T), adenine (A), cyto- sine (C) and guanine (G). Franklin’s experiments with hydration DNA had shown that the phosphates were on the outside of the molecule. With that information, plus her knowledge of diffraction, Franklin was able to un- derstand the structure of the molecular as revealed by the pattern. In this lab activity, you will learn how to interpret the pattern as she did. 2

Learning Goals

In this lab, you’ll explore some of the quantitative aspects of diffraction gratings. In particular, by the end of this activity, you should be able to:

  • Use a diffraction grating to determine a wavelength
  • Determine the largest order of interference observable for a given wavelength and diffraction grating.
  • Relate key features of a diffraction pattern to aspects (e., pitch of a helix, offset of two helices and a double helix) of the crystalline structure.

Approximate timing

  • Introduction................................ 10 min.
  • Multi-slit interference and gratings.......... 30 min.
  • Diffraction gratings in biology............... 15 min.
  • X-ray diffraction of DNA................... 45 min.
  • Applications and extensions................. 10 min. 1 R. Franklin and R. Gosling, “Molecular configuration in sodium thymonucleate,” Nature 171 , 740-741 (1953). 2 Adapted from the UNC “Physics Activities for Life Sciences” activity set which was in turn inspired by an activity developed by the Institute of Chemical Education at the Univ. of Wisconsin-Madison ice.chem.wisc, and which is described in A. Lucas, Ph. Lambin, R. Mairess, and M. Mathot, “Revealing the backbone structure of B-DNA from laser optical simulations of its X-ray diagram,” J. Chemical Education 76 , 378-383 (1999).

1 Multi-slit interference and gratings

The double helix of the DNA molecule acts like a large number of slits in a regular array that produces a diffraction patterns when illuminated with X-rays (as in Fig. 1). To better understand its properties, we begin by considering the interaction of visible light with an array of a large number of closely-spaced slits, called a “diffraction grating” (which is an unfortunate name since this phenomena is interference, not diffraction). A grating is generally characterized by the number of lines (i. slits) it has per millimeter. When light of wavelength λ passes through a grating, bright spots will appear on a distant screen at angles θ given by the grating equation:

d sin θm = mλ (1)

where m is the order of the diffraction: m = 0 (the central dot), m = ±1 (the two dots to either side of the central dot), m = ±2 (the two dots to either side of these),...

  1. Shine a laser through a diffraction grating; qualitatively, what pattern is produced on the screen?

  2. What is the spacing d between the slits in a grating that has 300 lines/mm? What about one that has 100 lines/mm?

  3. The spacing d is quite small, so this means that the angle θm may not be small enough to apply the small angle approximation (i. sin θ ≈ tan θ), but let’s suppose that we can use this approximation. If L is the distance between the grating and the screen where the interference pattern is recorded, come up with an expression for the distance from the center of the screen to each bright spot (called “fringes”), ym, in terms of d, λ, and L.

  4. Select a diffraction grating from the central table and first predict where you would expect to see the bright fringes if you shine green laser light (λ = 520 nm) through the grating. Then set this up and check that the bright spots appear where you predict they should. (Don’t forget to write down the grating size).

  5. Are the lines that produced pattern b closer together or farther apart than the lines that produced pattern a? Answer the same question for patterns c and d. Support you answer with both empirical evidence and with the theoretical model (i. the grating equation.)

Find the “Mask Patterns” image on Canvas (or in one of the hard-copy manuals); this is a larger version of the transparency we’ll use to simulate DNA. The top row of masks is designated A, B, and C from left to right, and the remaining masks are similarly designated D, E, F (second row) etc. Observe the diffraction patterns produced by masks A, B, and C.

  1. Describe how the array of lines in mask A create the corresponding diffraction pattern (e. how does the spacing, positioning, and orientation of the lines correspond to the diffraction pattern?).

  2. Qualitatively describe how the orientation and spacing of the lines in mask B compare to those in mask A.

  3. Qualitatively describe how the orientation and spacing of the lines in mask C compare to those in mask A.

  4. What would the diffraction pattern look like if patterns B and C were superposed? Do you see anything similar in Franklin’s diffraction pattern?

In masks D through H, the arrays of short lines found in masks A-C are connected to form sine waves, as in the figure below. The length A represents the amplitude of the sine wave and l represents the distance between two adjacent parallel lines. The distance l also corresponds to the wavelength of the sine wave (which is not the same as the wavelength of light you are shining through the mask).

  1. Describe the diffraction pattern you would expect such an array of lines to produce. Afterwards, set this up and observe the pattern from mask D. Does it match your prediction?

  2. The distance l in mask E is one half of the distance l of the sine wave in mask D, but the two waves have the same amplitude. Predict how the diffraction pattern from mask E will differ from that of mask D. Record the diffraction pattern from mask E and resolve any discrepancies with your prediction.

  3. The amplitude of the sine wave in mask G is one half of the amplitude of the sine wave in mask D but distance l is the same. Predict how the diffraction pattern from mask G will differ from that of mask D. Record the diffraction pattern from mask G and resolve any discrepancies with your prediction.

  4. Now compare the diffraction pattern from mask H to that of mask D. Based on your answers above, what can you conclude about the ratio A/l for the two masks, i., how does A/l for mask H compare to A/l for mask D?

3 Applications and Extensions

To assess your understanding of some of this lab’s key ideas, your group must answer the following questions together without help from the instructors or other groups:

  1. Summarize the diffraction rules you have formulated in your analysis of these diffraction patterns. Specifically,

(a) What aspect of the structure of the DNA molecule causes the basic “X” shape diffraction pattern? (b) What aspect of the structure of the DNA molecule causes the missing layer lines?

  1. If you want to measure the difference in wavelength between light from two sources that have wave- lengths that are very close together (i. one has wavelength λ 1 and the other has wavelength λ 2 = λ 1 + ∆λ and ∆λ is much smaller than λ 1 ), would you want to use a diffraction grating with a small number of lines/mm or a large number of lines/mm? Explain.

Appendix: The geometry of the grating equation

Light that shines on a diffraction grating will emerge from each of the multiple slits and travel to an image screen (or your eye). For any given point on the screen, light beams can be traced back to each of these slits—but the light beams must travel different distances from all of the different slits! Comparing any two beams, one will have traveled a little bit farther than the other—a small extra distance we call ∆r. If an integer number of wavelengths can fit into this extra distance (∆r = mλ, where m = 0, 1 , 2 ,... ) then the arriving beams of light will be in phase and you will see a bright dot (“constructive interference”). However, if ∆r 6 = mλ then the arriving light will be out of phase and you will see nothing (“destructive interference”). The resulting pattern of dots arrises because only a few points on the screen satisfy the condition of constructive interference (each point represents a different value of m).

Figure 2: Diagrams of the geometry of 2-slit diffraction. (a) a diagram of the whole set-up, (b) a close-up of ∆r, (c) a recreation of the triangles that appear in the setup.

The grating equation is largely an application of geometry to this situation. Notice the triangles in the diagrams; these can guide us to some trigonometric relationships. From digram (b) and (c), we can see that the extra distance traveled, ∆r, forms one leg of a right triangle 3 and follows the relationship d sin(θ) = ∆r. Combining this with the condition of constructive interference, we get

d sin(θm) = mλ. (2)

And from diagram (a) and (c) we can similarly see that

L tan(θm) = ym. (3)

A third relationship that is useful is the small angle approximation: for very small angles (θ < 20 ◦ or so),

sin(θ) ≈ tan(θ) (4)

(try it in a calculator!) These three relationships can all be recombined to find other algebraic relationships, as you are asked to do throughout the worksheet. 3 Sharp-eyed readers will notice that the triangle in diagram (b) isn’t actually a right triangle in diagram (a) and indeed it’s not. But so long as L  ∆r then it’s close enough that we can both apply the trigonometry of right triangles and equate the small angle to the θm that appears in digram (a) (which is a proper right triangle).

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DNA Diffraction - PHYS 2 LAB REPORT

Course: General Physics II (PHYS 1325)

3 Documents
Students shared 3 documents in this course

University: Texas State University

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Phys 1125
Wave Optics:
Diffraction Gratings and DNA
Introduction
Figure 1: X-ray diffraction pattern of B-DNA
(the hydrated form of the molecule) taken by
Rosalind Franklin.
One of the triumphs of 20th-century biology was
the determination of the double-helix structure of
DNA (deoxyribonucleic acid). James Watson and Fran-
cis Crick proposed the structure in 1953 (and received
the Nobel Prize in 1962), and the key to their discovery
was the X-ray diffraction pattern of DNA produced by
the crystallographer Rosalind Franklin (who, sadly, died
of ovarian cancer in 1958 and did not share in the No-
bel Prize, which is never awarded posthumously). The
diffraction pattern is reproduced in Fig. 1.1
Prior to her experiments it was known that DNA is a
long fibrous polymer made up of phosphate groups, de-
oxyribose sugar, and the four bases (each bonded to a
sugar and a phosphate) thymine (T), adenine (A), cyto-
sine (C) and guanine (G). Franklin’s experiments with
hydration DNA had shown that the phosphates were on
the outside of the molecule. With that information, plus
her knowledge of diffraction, Franklin was able to un-
derstand the structure of the molecular as revealed by
the pattern. In this lab activity, you will learn how to
interpret the pattern as she did.2
Learning Goals
In this lab, you’ll explore some of the quantitative aspects of diffraction gratings. In particular, by the end
of this activity, you should be able to:
•Use a diffraction grating to determine a wavelength
•Determine the largest order of interference observable for a given wavelength and diffraction grating.
•Relate key features of a diffraction pattern to aspects (e.g., pitch of a helix, offset of two helices and
a double helix) of the crystalline structure.
Approximate timing
•Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 min.
•Multi-slit interference and gratings . . . . . . . . . . 30 min.
•Diffraction gratings in biology. . . . . . . . . . . . . . . 15 min.
•X-ray diffraction of DNA . . . . . . . . . . . . . . . . . . . 45 min.
•Applications and extensions. . . . . . . . . . . . . . . . . 10 min.
1R.E. Franklin and R.S. Gosling, “Molecular configuration in sodium thymonucleate,” Nature 171, 740-741 (1953).
2Adapted from the UNC “Physics Activities for Life Sciences” activity set which was in turn inspired by an activity
developed by the Institute of Chemical Education at the Univ. of Wisconsin-Madison http://ice.chem.wisc.edu, and which is
described in A.A. Lucas, Ph. Lambin, R. Mairess, and M. Mathot, “Revealing the backbone structure of B-DNA from laser
optical simulations of its X-ray diagram,” J. Chemical Education 76, 378-383 (1999).

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