# P301 Lecture 11 “Scattering experiments”

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P301 Lecture 11 “Scattering experiments”
The rate at which you detect particles has to be proportional to the incident flux (particles/cm2.sec), and it is a rate (particles counted in a certain direction per unit time). Conventionally, therefore, the scattering probability is expressed as a “cross Section” (i.e. has dimensions of an area). s(q) = {Z21Z22e4/[256p2e2oK2sin4(q/2)]} (4.13) in T&R, [rewritten as a cross-section per nucleus, the count rate in the detector is this cross section times the incident flux (Ni), times the number of nuclei per unit area (nt), integrated over the angular acceptance of the detector]

P301 Lecture 13 “Fine Structure constant”
a= {e2/[2eohc} (or e2/4peo’hbar’c) lc/2pao = a (Ratio of Compton wavelength to Bohr radius, almost) 2|Eo|/mec2 = a2 (the Hartree is the natural energy unit for Chemistry, and it is a2 * the rest energy) re = e2/(mc24peo)= a2ao (the classical radius of the electron, This shows up in cross sections for light scattering off of electrons.) Note: As we saw in class, these ideas provide pretty remarkable agreement with experiment for hydrogen, but they don’t provide the spectra for heavier elements (more than one electron), and they are very much ad-hoc (appear to be conjured up just to agree with the data, there is no other reason to expect them to be true, and they don’t lend themselves to generalization in any obvious way.

P301 Lecture 12 “Bohr’s Hypotheses”
Bohr formulated the following ad hoc model: Atoms exist only in certain stable “stationary states” The dynamic equilibrium of these stationary states is determined by the laws of classical physics, but the way the atoms interact with the electromagnetic field of light waves is not The emission and absorption of EM waves by atoms takes place ONLY in conjunction with a transition between two stationary states, with the frequency of the emitted light being determined according to the Planck hypothesis: |E1 – E2 |= hf The (orbital) angular momentum of the electron in a stationary state can only take on values given by integral multiples of Planck’s constant divided by 2p Ln = nh/2p It is this last hypothesis that is the truly new (revolutionary) idea from Bohr himself, the other three are pretty much inescapable and/or had been provided by someone else already. Note: As we saw in class, these ideas provide pretty remarkable agreement with experiment for hydrogen, but they don’t provide the spectra for heavier elements (more than one electron), and they are very much ad-hoc (appear to be conjured up just to agree with the data, there is no other reason to expect them to be true, and they don’t lend themselves to generalization in any obvious way.

P301 Lecture 13 “JITT question”
What is the key similarity, and what is the key difference between the inelastic collisions discussed in the context of the Franck-Hertz experiment and those you studied in Physics I? Like any inelastic collision, energy is not conserved; however, a certain quantized level of energy must be obtained by the electron before an inelastic collision can occur. [several expressed something like this; energy is in fact conserved, it is only KE that is not conserved.] Similarity: momentum is conserved in both while KE isn't ; Difference: the type energy loss is different [many got this similarity, though few encapsulated the whole answer as succinctly. What is the difference in the energy loss?] I

Franck-Hertz Experiment
Accelerate electrons through a gas-filled tube and measure the current getting to the anode. What happens if electrons scatter off the atoms?

Franck-Hertz Experiment
If the scattering is elastic, then nothing dramatic happens, you measure average transfer of charge from cathode to anode (per unit time), and this is essentially independent of the electron’s path).

Franck-Hertz Experiment
What if the collision is inelastic?

Franck-Hertz Experiment
If the electron loses so much energy that it does not arrive at the grid with enough energy to climb the potential to the anode, it cannot contribute to the anode current, and Ic drops.

Franck-Hertz Experiment

Franck-Hertz Experiment
Fig. at right shows optical emission from Neon gas in a F-H tube. In regions where the electrons have enough energy to excite the neon atoms, the atoms emit visible light when they relax back to their ground state (from about 19eV to about 16.7 eV)

Franck-Hertz Experiment
Fig. at right shows the energy levels for Hg (from the instructions for the F-H experiment in our P309 lab). In the grounds state of Mercury, there are two electrons in the 6s level, and the other levels shown are unoccupied.

P301 Lecture 13 “Moseley’s Law”
Lb La NOTES: Moseley started to catalogue characteristic x-ray energies (and therefore frequencies) using a technique we’ll discuss next week. He developed the above empirical relations for the frequencies, determined that atomic number, not weight, was the relevant parameter to explain the periodicity of the periodic table (e.g. he reversed the positions of Ni and Fe; K and Ar), and predicted the existence of three (and only three) previously undiscovered elements (Z=43, 61, and 75; later: Tc, Pm, and Re) “between Al and Au” Lg Ka Note: In Moseley’s article, he points out that the idea that chemistry (i.e. elemental identity) was determined by the number of positive charges on the nucleus had been put forward by van den Broek, then he points out that the data above provide “the strongest possible support to this hypothesis”. Additional evidence for the Van den Broek hypothesis had come from studies of the radioactive elements, but Moseley’s work was the most definitive and straight forward (even in the absence of any general theory of atomic structure). (note: ¾ = 1 – ¼; 5/36= ¼- 1/9). Note also that the book’s explanation of the terms -1 and -7.4 in the parentheses is not strictly correct, but it gets much of the idea across (see Whitiker Eur. J. Phys (1999). Kb

P301 Lecture 13 “Characteristic X-ray production”
NOTES: Barkla first discovered “characteristic x-rays” in 1909, several years before Bohr, the Braggs, and Moseley did their work. The “shell” model of the atom, which arises from Bohr’s model for H, is very useful even when considering multi-electron atoms The various “shells” (K, L, M, N, etc., corresponding to increasing values for “n” in the Bohr model) are typically split into a few (or several) individual energy levels that are much more closely spaced than the separation between the shells. We will start to explore the smaller splittings later in the course.

How many of you recognize this?

The Structure of DNA: Rosalind Franklin and X-ray Diffraction

Section 5.1: Bragg’s Law

Bragg’s Law 2dsin(q) = nl q d sin(q)
You get constructive interference only if: 2dsin(q) = nl This gets the right answer, but it is slightly unsatisfying (why do you consider planes of atoms rather than the atoms themselves, it doesn’t explain why some plane sets diffract and others don’t, and it doesn’t give you relative intensities of the various reflections, but it is easy to remember and it is very useful as a quick answer to give you much of the right phenomenology.

“Powder” Diffraction

Silicon powder diffraction (Baxter lab)

Laue Diffraction From “Techniques of X-ray Diffraction by B. D. Cullity

Real X-ray apparatus Single crystal setup w 2-D detector
Laue setup with digital “film”

Bragg’s Law 2dsin(q) = nl q d sin(q)
You get constructive interference only if: 2dsin(q) = nl This gets the right answer, but it is slightly unsatisfying (why do you consider planes of atoms rather than the atoms themselves, it doesn’t explain why some plane sets diffract and others don’t, and it doesn’t give you relative intensities of the various reflections, but it is easy to remember and it is very useful as a quick answer to give you much of the right phenomenology.

P301 Lecture 13 “Characteristic X-ray production”
NOTES: Barkla first discovered “characteristic x-rays” in 1909, several years before Bohr, the Braggs, and Moseley did their work. The “shell” model of the atom, which arises from Bohr’s model for H, is very useful even when considering multi-electron atoms The various “shells” (K, L, M, N, etc., corresponding to increasing values for “n” in the Bohr model) are typically split into a few (or several) individual energy levels that are much more closely spaced than the separation between the shells. We will start to explore the smaller splittings later in the course.

P301 Exam I Review Philosophy:
The most important things in this course are developing an understanding and appreciation for how we know what we know about things that are very small or moving very fast. You should develop some understanding of what very small and very fast mean, but you needn’t be overly concerned with memorizing specific constants or formulae (I give constants, you have your own formula sheet). You should be able to understand the key experimental results, their significance in shaping our current view in the world, and how their data are collected and interpreted. You should be able to summarize/describe these in no more than 4 or 5 sentences, and/or have a short unambiguous name for each. You should also understand and be able to use the various formulae we have derived and or presented in this class to quantify the sometimes strange phenomena involved (but recall you’ll have a formula sheet, so memorizing them is not essential).

P301 Exam I Review NO CALM QUESTION FOR FRIDAY!!!
HW4 solutions posted tonight, any answers submitted after solutions are posted will not be graded. Exam Mechanics: Covers material from chapters 1 through section 5.1 1 side of 8.5x11” formula sheet is allowed. It is not to be a general note sheet and I would like it handed in with your answers 5 questions (50 points, but 8 “parts” worth 5 or 10 points each) If a question has multiple parts with answers from earlier parts feeding later parts, if you have the right method on a later part but use an incorrect answer from the earlier part, you get full credit!! A mix of descriptive and computational answers. Tables from the inside front lay-out of the text will be provided. Exam will start at 11:10 and will last to 12:10. Please answer in PEN (Blue or black). Office Hours: Wed. 2:00 to 3:30 Thursday 1:30 to 3:00 Friday 9:45 to 10:45; No office hours Friday afternoon.

P301 Exam I Review Important experiments: Relativity Quantum Mechanics
Michelson-Morley experiment Muon lifetime observations from cosmic rays. Doppler Effect (expanding Universe, binary stars, extra-solar planets, SMOKEY, …). Synchrotron radiation (transformation of angles in relativity). Quantum Mechanics Cathode-ray tube experiments e/m of the electron X-rays: Bremsstrahlung, characteristic Photo-electric effect Franck-Hertz experiment Line spectra of gasses Compton effect Millikan Oil drop experiment Black-body spectrum Discovery of the positron (antiparticles in general) Rutherford scattering Bragg/Laue scattering Moseley’s law Radioactivity

P301 Exam I Review Important ideas: Relativity Quantum Mechanics
Speed of light is a universal constant irrespective of (initial) FoR. Lorentz transformation: time dilation/ Lorentz-Fitzgerald contraction. Relativistic mass, energy, momentum Transformation of angles Doppler Effect Space-time diagrams The invariance of interval Electricity and magnetism are intimately connected Quantum Mechanics Light is quantized (Blackbody radiation and hf=E, Compton and PE effects) Electric charge is quantized and electrons are much lighter than atoms Anti-particles exist Atoms have internal structure and dynamics (electrons, atomic spectra, X-rays, radioactivity, chemistry, Rutherford’s experiment). We can understand atoms in terms of quantize light and angular momentum (Bohr) We can explain the periodic table (sort of) Moseley

P301 Exam I Review Example descriptive questions:
Identify and provide BRIEF descriptions of 3 important experimental results that came out of the study of electric currents in vacuum tubes or such tubes back-filled with dilute gas. Provide an annotated sketch showing the essential elements of the apparatus used by Millikan to quantify individual elementary charges. Identify 3 of the crucial postulates Bohr used in constructing his model of the atom. Identify and briefly describe three observations or phenomena extant prior to Bohr’s development of his atomic model which suggested that atoms had internal structure and dynamics. (Q1 on 2009 test). BRIEFLY describe two important results published by Einstein in 1905. Describe, BRIEFLY, the phenomenon known as the Ultra-Violet Catastrophe and how Planck’s quantum hypothesis avoids this failure of classical theory. (these last two are of the right style, but probably deal with subjects we did not cover in enough detail to be worth more than 5 points on the exam, if they would be asked at all).

P301 Exam I Review Example answers:
Identify and provide BRIEF descriptions of 3 important experimental results that came out of the study of electric currents in vacuum tubes or such tubes back-filled with dilute gas. Line spectra from gas discharge tubes: Each element emitted characteristic pattern of light (sharp lines at specific frequencies) when excited by cathode rays. e/m ratio: Crossed magnetic and electric fields acting on cathode rays allowed Thomson to show that the e/m ratio for these rays was much larger than that for atoms. X-rays: Roentgen discovered new penetrating radiation was released for high enough accelerating voltages Photo-electric effect: light promotes the evolution of electrons from a metal surface, but frequency plays a crucial role in contradiction to predictions of classical physics. Franck-Hertz expt: passing electrons through a tube with some gas can be used to see quantize excitation of the gas atoms (provided you have an accelerating grid combined with a small retarding voltage between the grid and collecting anode).

2-56,61 If F is always perpendicular to v, then F=mga (see problem 2-55). Use this to prove that a relativistic particle moving perpendicular to a magnetic field orbits with a radius R=p/qB (p momentum, q charge of the particle). What B field is needed to hold protons with a kinetic energy of 230 MeV in a 15m circle? (how about 1.5 m, comparable to a medical cyclotron)? 5-DVB The “<111>” family of planes in Si has a d-spacing of nm, through what angle would you expect a Cu Ka1 x-ray (l= nm) to be scattered by this family of planes? 4- 30 EM radiation with l=100 nm is incident upon a hydrogen atom that is at rest and in its ground state. What is the highest state to which this atom (treated within the Bohr model) can be excited? 3- DVB Question: why do they say max. kinetic energy here?