PHYS 3701: Electrons and photon Laboratory...

PHYS 3701: Electrons and photon LaboratoryLaboratory Manual (version:16.4)

Julie Roche∗

Physics and Astronomy DepartmentOhio University

November 21, 2016

Contents

1 Laboratory general directions 3

1.1 General conduct in the laboratory . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Guidelines to prepare yourself to perform an experiment: Run plan . . . . . 4

1.3 Reporting experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Statistics and treatment of experimental data 7

2.1 Physical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Reading and homework assignments . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The Hall Effect 13

3.1 Physical concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Experimental setup and procedure suggestion . . . . . . . . . . . . . . . . . 15

3.3 Lab objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Preliminary questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Electron diffraction 17

4.1 Physical concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Lab objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4 Preliminary questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

∗Prof. Brune also contributed to this manual

1

version:16.5 PHYS3701, Ohio University

5 The Millikan’s oil drop experiment 255.1 Physical concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Lab objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 Preliminary questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 The Franck-Hertz Experiment 316.1 Physical concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Experimental setup and procedure suggestion . . . . . . . . . . . . . . . . . 356.3 Lab objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.4 Preliminary questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7 The Photo-electric effect 397.1 Physical concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.2 Experimental setup and procedure suggestion . . . . . . . . . . . . . . . . . 407.3 Lab objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.4 Preliminary questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8 Interferometry 458.1 Physical concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458.2 Experimental setup and procedure suggestion . . . . . . . . . . . . . . . . . 468.3 Lab objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478.4 Preliminary questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

9 Numerical simulation 519.1 Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.2 Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.3 Physical concept used in this lab: π0 decays detected in a Electro-magnetic

calorimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539.4 Lab objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.5 Preliminary questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

A Version history of this lab manual 61

2

1 Laboratory general directions

1.1 General conduct in the laboratory

These directions are to be considered a guide to your laboratory work rather than a recipeto be followed blindly.

- Come prepared: Before the lab session, prepare by reading this manual and doinganalysis in between sessions. Bring the necessary supplies and submit your reportpromptly. Also, if you ask the instructor for help between sessions bring your notes asit is much easier to talk about your data then to talk in general terms of the experiment.

- Be safe: Be careful with the equipment as you take data in order to protect yourselfand the equipment. Especially, you will be using equipment which poses some hazardssuch as high-voltage power supplies. Listen to safety instructions and heed them. Alsoif a piece of equipment is not working even after you have followed all instructions, becareful what you fiddle with! Some fiddling is good, but if you are planning anythingmajor (like taking a piece of equipment apart), it is EXTREMELY IMPORTANT toask the instructor first. Also it is good lab practice not to eat or drink in the lab: youmight damage the equipment and/or eat or drink something you don’t intend to!

- Understand the purpose of the experiment: Think of the purpose of the experiment sothat you know why you are doing each step. A scientist or engineer knows what is to bemeasured before the experiments begins (i.e., Are you measuring a constant, verifyinga law?). In addition, professional scientists and engineers are paid for their work andconsequently do not do things without a good reason. Do not depend upon yourpartner to do your thinking for you. Plan your laboratory activities for the greatesteconomy of time and effort. Often a few minutes of preliminary manipulation of theapparatus will save an hour of repeating the experiment due to taking worthless datahurriedly. Laboratory work is a joint venture. Carry your share of the responsibility.Don’t leave the lab without making sure you understand your data and have recordedsufficient information to allow the data to be analyzed later. If you are unsure of aprocedure, ask a laboratory assistant.

- Record data completely and honestly: Record what you observe, not what you hope itshould be (or what someone else says it is). Keep a good laboratory book and recordyour data and the steps you take! Write down data in a neat and organized way.Don’t think it is a trivial task. Feel free to ask your instructor for advices on how toaccomplish this. The goal is to be able to read and understand what you did afteryou leave the laboratory. Do not be tempted to falsify data (in this course or later inyour career). Sometimes there may be pressure to do this. Those who succumb to thepressure are usually found out and the ramifications are very serious. These incidentsare damaging to society and the credibility of the scientific professions. The careers ofthe scientists responsible are typically severely damaged by these incidents.


- Cross-check your data: There are a variety of honest mistakes that an experimentalistcan make in recording or in analyzing data. One possible mistake is to assume thatan observed signal is due to the sample when it is actually due to the instrumentation.Another mistake is to reach the wrong conclusion in analyzing data. It may be difficultto avoid this pitfall. Perhaps the best advice is to urge caution, especially if you havesome doubts. Publishing a mistake and then publishing a retraction is embarrassingand may damage one’s career. For this lab, if the final result has intolerably largeerrors (more than 25% unless otherwise indicated), repeat the experiment after findingout what is wrong. If you find that the error cannon be reduced, you are expected toexplain why this is the case. You are not being graded on the “correctness” of yourdata.

- Investigate: Look out for refinements in technique and equipment which indicate acarefully and well-performed experiment with the least error possible. For example,in an experiment with pulleys and a force table, one refinement is to make sure thatall pulleys are the same size. Feel free to innovate and improve the apparatus andprocedure if you can. Record these changes in your report. Also, when doing anexperiment, it is important for the experimentalist not to accept the data as correctwithout adequately questioning it. Always be curious.

1.2 Guidelines to prepare yourself to perform an experiment: Runplan

The following questions are typically answered before hand by someone well prepared toperform an experiment. In this class, you are expected to answer these questions during thefirst session dedicated to an experiment. After discussing with your lab partner, write downyour plan and turn in your answers to the instructor before leaving class.

- What are the goals of this experiment?

- What is the method used to achieve these goals? Write down a list of steps detailingthe sequence of your work.

- What are the potential sources of error in this measurement? What will I do to reducethis as much as possible?

- What quantities will I measure? How will the data be analyzed? You may want toprepare tables that you will fill during the data taking.

1.3 Reporting experimental work

In addition to planning and thinking about the experiment to be done and optimizing theequipment and technique as indicated, communicating your results and idea to other is animportant skills.

4


Scientists do present reasoned arguments supported by experimental evidence. Those ar-guments include elements such as plots, tables, numerical results with uncertainties anddiagrams.

For this class, you are asked to present a short 7-minutes presentation of your work andbe prepared to answer questions about your work for an additional 5 minutes. You are askedto prepare a maximum of 5 slides that motivate your experiment, present your measurementand your results, and evaluate the significance of your results. Bring print-outs of your pre-sentation1. The grading scheme used to evaluate your presentation is shown on figure 1. Thelab presentation will be scheduled with 15 minutes intervals on the dates listed on table 1and will take place in the instructor office.

The following resources will help you prepare your presentations. While these pagesare for full-blown presentation, many of the advices presented her apply well to your shortpresentation.

- How To Make an Oral Presentation of Your Research, UVa,http://www.virginia.edu/cue/presentationtips.html

- Powerpoint tips for science talks, C. Elliot, U of Illinois,http://people.physics.illinois.edu/Celia/PPT-Tips.pdf

- Celia’s tip for science talks,http://people.physics.illinois.edu/Celia/ScienceTalks.pdf

1You can use whichever software you want to produce your sides. For the printout, you have up to 4slides on a page

5

Student: Lab:

your pts weight total

A: Opening statement 0.075The topic/purpose of the study is presentedThe significance of your result is statedB: Physical principle 0.25The physical law under study is correctly explained

C: Experimental methods 0.25Physical principles of the measurement are correctly describedSufficient details are given to understand how the study was conductedEfforts to maximize precision are explainedD: Data and analysis 0.25Relevant data are presented in an appropriate format (table, plot, etc..)Analysis is correctly carried out and well reasonedRelevant equations are explained and correctly usedE: Discussion and conclusion 0.075Results are quoted with proper precision, significant figures, and unitsThe interpretation of the result is correct, data driven and evidence basedThe conclusion is clear and a concise summation of important points.F: Presentation skills 0.1Slides are free of spelling and grammar mistakes and easily readableThe presenter connects with the audienceThe speech is of good quality (volume, rate, varied tone, distracting filler)Visual aids are used when appropriateThe presentation respects timing and slides instructions

G: Presentation (A+B + C +D + E + F ) /100H: Lab participation and data taking /100

Final grade ((H +G)× 0.5) /100

Comments:

Table 1: The grading system for the PHYS 3701 lab presentations.

2 Statistics and treatment of experimental data

2.1 Physical concepts

There is no such thing as a perfect experiment. Each measurement contains some degree ofuncertainty due to the limits of instruments and the people using them. In physics, these ofuncertainties are called errors (think of the error as the difference between the measured valueand the true underlying value). We will consider two distinct error concepts: accuracy andprecision. Precision refers to how close together a group of measurements actually are to eachother. It is often called random error or statistical error. The accuracy of the measurementrefers to how close the measured value is to the true or accepted value, after averaging overmany measurements so that the error due to precision is negligible. The accuracy is oftencalled the systematic error. Precision has nothing to do with the true or accepted value ofa measurement, so it is quite possible to be very precise and totally inaccurate. In manycases, when precision is high and accuracy is low, the fault can lie with the instrument.2

2.2 Reading and homework assignments

The following book is the required reading for this class3.P.R. Bevington and D.K. Robinson, ”Data reduction and error analysis for the physicalsciences,” ISBN 0-07-247227-8.

- For the second class: Read Chap 1 and the first part of chapter 2.

- For the third class: Read the end of Chap 2 and Chap 3.

- For the fourth class: Read Chap 4.

- For the fifth class: Read Chap 5.

- For the sixth class: Read Chap 6.

Homework 1: work out exercises 1.1, 1.2, 1.4, 2.8 and 2.13Homework 2: work out exercises 3.5, 3.6, 4,5 and 4.12 (skip t-distribution question).

2.2.1 Important formulas

Straight average: This method applies when one performs a set of N measurements ofa given quantity xi with equal precision ∆xi = ∆x for each measurement. For this sample,

2This section is based on a document of the Fordham Preparatory School.3Another excellent reference is: .R. Taylor, “An introduction to error analysis, The study of uncertainties

in physical measurements,” ISBN 0-935702-75


the mean value is x, the standard deviation of the sample is σx and should be equal to ∆x,and the precision on the mean is ∆x:

x =1

N

∑xi σx =

√1

N − 1

∑(xi − x)2 ∆x =

σx√N. (1)

Weighted average: This method applies when one performs a set of N measurements ofa given quantity xi with different precisions ∆xi for each measurement. For this sample, themean value is x, and the precision on the mean is ∆x:

x =

∑(xi/∆x

2i )∑

(1/∆x2i )∆x =

1√∑(1/∆x2i )

. (2)

χ2 minimization: A numerical comparison between the observed distribution of measure-ments xi with precisions ∆xi and a function y can be defined via χ2:

χ2 =∑ (xi − y)2

∆x2i(3)

The number of degrees of freedom ν is defined to be the number of data points minus thenumber of free parameters in the function y. The “best fit” is determined by minimizing χ2.The minimum value of χ2 should be approximately equal to ν. The reduced χ2 is defined tobe χ2

ν = χ2/ν.

Linear fit: Suppose a set of measurements of xi, yi and ∆yi, from which one to extract alinear function of the form y(x) = ax + b. One way to extract the parameters a and b is tominimize the χ2 between the data and the function. The solution for the least-squares fit ofa straight line is:

a =EB − CADB − A2

b =DC − EADB − A2

(4)

(∆a)2 =B

DB − A2(∆b)2 =

D

DB − A2(5)

where

A =∑ xi

(∆yi)2B =

∑ 1

(∆yi)2(6)

C =∑ yi

(∆yi)2D =

∑ x2i(∆yi)2

(7)

E =∑ xiyi

(∆yi)2F =

∑ y2i(∆yi)2

(not used). (8)

8


Note that in many real life experiments, the measured quantities are xi±∆xi and yi±∆yi.How to deal with errors on both x and y is a non-trivial and sometimes controversial issue.For the purpose of this class and in this case, if on supposes yi = f(xi), one should translate∆xi into a ∆yi error by replacing ∆yi with√

(∆yi)2 + (∂f

∂xi∆xi)2. (9)

2.3 Tutorial

This exercise is extracted from “An introduction to error analysis: The study of uncertaintiesin physical measurements,” 2nd edition, J.R. Taylor, University Science Book.Systematic errors sometime arise when an experimenter unwittingly measures the wrongquantity. Here is an example.Gravity can be measured using a pendulum made of a steelball suspended by a light string. Gravity g can then be extracted using the following formula:

g = 4π2l/T 2 (10)

where l is the length of the pendulum and T is its period. A student tries to measure gusing this method. He records six different lengths of the pendulum l and the correspondingperiods T . He estimates that his precision on the measurement of the length is 0.1 cm andon the periods is 0.0005 s. His data are presented in table 2.

Length l (cm) 51.2 59.7 68.2 75.0 79.7 88.3Period T (s) 1.4485 1.5655 1.669 2.098 1.8045 1.896

Table 2: Measurements of the length and the period of the pendulum

1. For each pair (li, Ti), he calculates gi. He then calculates the mean gm of those values,their standard deviation σm and their standard deviation of mean ∆gm, assuming allhis errors are random. What is his answer for gm ± ∆gm? He now compares hisanswer with the accepted value g = 979.6 cm/s2 and he is horrified to realize that hisdiscrepancy is nearly 10 times larger than his uncertainty. Confirm this sad conclusion.

2. Studying his results, the student notices that the values gi he extracted from his resultsseem to increase as the period T increases which might be the sign of systematic bias.He decides to check if his observation is correct.

First he computes the precision of each of his measurements gi, i.e.,

(∆g)2 =

(∂g

∂l∆l

)2

+

(∂g

∂T∆T

)2

(11)

9


Using his set of gi and ∆gi values, he recomputes gw±∆gw using the weighted averagemethod. How does gm and gw compare? Why are they different? Which methodshould be used to compute the average value of g? Explain your answer.

Then he plots gi ±∆gi as a function of T 2 and draws the line corresponding to gw onhis plots. Reproduce his plot. Finally, he computes the χ2 of gw compared to his dataand concludes that indeed there is bias in his data. What is the χ2, the reduced χ2

and the probability of his data being independent of T 2? Why would you say, like thestudent did, that there is a bias in his data?

3. Having checked his calculations and found them to be correct, the student concludesthat he must have overlooked some systematic errors. He is sure there was no problemwith the measurement of the period T and that the problem must be in his measure-ment of l. So he asks himself: How large would a systematic error in the length lhave to be so that his result would be the accepted value 979.6 cm/s2? Show that theanswer is approximately 1.5%.

4. In order to make a maximum usage of his data, the student decides to extract the bestvalue of his systematic error on the length l by performing a linear fit of his data. Firsthe rewrites Equation 10 to take into account his supposed systematic error l0 such thathe measured l when he should have measured l − l0:

T 2g + 4π2l0 = 4π2l (12)

Then he plots 4π2l versus T 2 such that the slope of straight line is g and the offset is4π2l0. Reproduce this plot, don’t forget the error bars. Using the linear fit method,extract gl ±∆gl, as well as l0 ±∆l0. How does the reduced χ2 of this fit compared tothe case where l0 was ignored? What does it mean?

5. This result would mean that the student’s length measurement suffered a systematicerror of about a centimeter – a conclusion he first rejects as absurd. As he stares at thependulum, he realizes that 1 cm is about the radius of the steel ball and that the lengthshe recorded were the lengths of the string. Because the correct length of the pendulumis the distance from the pivot to the center of the ball, his measurements were indeedsystematically off by the radius of the ball (see Figure 1). He, therefore, uses a caliperto find the ball’s diameter, which turns out to be 2.00±0.01 cm. Make the necessarycorrections to his data and compute his final result for gf with its uncertainties. Don’tforget to take into account the precision on the ball radius. Justify your choice to usethe straight average method or the weighted average method. What about the χ2 ofthe corrected values of g?

10


l−l0l

Figure 1: A pendulum consists of a metal ball suspended by a string. The effective lengthof the pendulum (l) is the length of the string (l-l0) plus the radius of the ball.

11


12

3 The Hall Effect

The Hall Effect was discovered by E. Hall in 1879 while working on his doctoral thesis. Hall’sexperiments consisted of exposing thin gold leaf (and, later, using various other materials) ona glass plate and tapping off the gold leaf at points down its length. The effect is a potentialdifference (Hall voltage) on opposite sides of a thin sheet of conducting or semiconductingmaterial (the Hall element) through which an electric current is flowing. This was createdby a magnetic field applied perpendicular to the Hall element. The ratio of the voltagecreated to the amount of current is known as the Hall Resistance and is a characteristic ofthe material in the element. In 1880, Hall’s experimentation was published as a doctoralthesis in the American Journal of Science and in the Philosophical Magazine. One veryimportant feature of the Hall Effect is that it differentiates between positive charges movingin one direction and negative charges moving in the opposite. The Hall Effect offered the firstreal proof that electric currents in metals are carried by moving electrons, not by protons.The Hall Effect also showed that in some substances (especially semiconductors), it is moreappropriate to think of the current as positive ”holes” moving rather than negative electrons.

3.1 Physical concept

The Hall Effect comes about due to the nature of the current flow in the conductor. Currentconsists of many small charge-carrying ”particles” (typically electrons) which experience aforce (called the Lorentz Force) when in the presence of a magnetic field (see figure 2) .When a perpendicular magnetic field is absent, there is no Lorentz Force and the chargefollows an approximate ’line of sight’ path. When a perpendicular magnetic field is present,the path is curved perpendicular to the magnetic field due to the Lorentz Force. The resultis an asymmetric distribution of charge density across the Hall element perpendicular to the’line of sight’ path the electrons would take in the absence of the magnetic field. As a result,an electric potential is generated between the two ends.

For a metal or for a n- or p-type semiconductor4 at normal temperature, electric conduc-tion is due to a single carrier type with charge q and mobility µ. If a magnetic field ~B isapplied along the axis z as on Figure 3, then the carrier of charge q experiences a LorentzForce :

~F = q( ~E + ~v × ~B) (13)

If the definition of carrier mobility is expanded to include magnetic forces, the drift velocityis given by :

~v = ±µ( ~E + ~v × ~B) (14)

where the plus and minus sign is used for holes and the minus sign is used for electrons. Forthis configuration of fields, the drift velocity components are related by

vx = ±µ(Ex + vyBz) vy = ±µ(Ey − vzBx) vz = ±µEz (15)

4You may want to refresh your knowledge on semiconductors


Figure 2: Hall effect diagram, showing electron flow (rather than conventional current).Legend:1. Electrons (not conventional current!), 2. Hall element, or Hall sensor, 3. Mag-nets, 4. Magnetic field, 5. Power source. In drawing ”A”, the Hall element takes on anegative charge at the top edge (symbolised by the blue color) and positive at the lower edge(red color). In ”B” and ”C”, either the electric current or the magnetic field is reversed,causing the polarization to reverse. Reversing both current and magnetic field (drawing ”D”)causes the Hall element to again assume a negative charge at the upper edge. This figure istaken from Wikipedia.

The components of the current density ~J = nq~v are thus:

Jx = ±µ(nqEx + JyBz) Jy = ±µ(nqEy − JzBx) Jz = ±µnqEz (16)

If the current is supplied by a DC source of current attached to Faces 1 and 2 of the sample,then in the steady state we must have Jy = Jz = 0. This implies that there is a component

of ~E transverse to the current flow:

Ey =JxBz

nq(17)

This is accounted for physically by the buildup of a static charge distribution on Faces 3 and4 which produces an electric field along the y axis that compensates for the forces of ~B onthe carriers. The Hall constant RH is defined by

RH =EyJxBz

=1

nq(18)

We can restate this definition in terms of experimentally measured quantities by referringto Figure 3 and noting that the Hall voltage VH = V3 − V4 is expressed in term of Ey as

14


Vt

l

xy

z

2

34

1w

B

Figure 3: A Hall sample is a conducting sample through which a current I is flowing. Forthe Hall voltage to appear, the sample needs to be in region where a magnetic field B exists.The current I flows from face 1 to face 2. The resulting Hall Voltage appears between face 3and 4, that is perpendicularly to both the current I and the magnetic field B. This figure istaken from Preston.

VH = wEy, and that Jx = I/wt. Thus, the experimental Hall Coefficient normalized by thethickness t of the sample is expressed in terms of measurable quantities as

RH

t=

VHIBz

(m2/C) (19)

Note that the carrier density in this case can be calculated directly from a measurementof RH , and that the sign of RH can be used to determine the sign of the charge carriers.This, of course, means that careful analysis of the signs of I, Bz, as well as VH needs to beperformed.

3.1.1 Thermo-magnetic effects

Because of several thermo-magnetic effects, the measured voltage V34 = V3 − V4 may notbe the ”true” Hall voltage VH (see figure 3). Reference5 gives a concise description of theNernst Effect, the Righi-Leduc Effect, and the Ettingshausen Effect, along with suggestionsfor reducing their respective contributions to measured voltages. In practice the apparatusyou will be using is equipped with a compensating tunable voltage.

3.1.2 To go further

- What is the quantum Hall Effect?

3.2 Experimental setup and procedure suggestion

The Hall apparatus used in this lab is a product of the Leybold Cie. A copy of the operatingmanual for this device is available on the class web site, it describes the apparatus as well assuggest a procedure for the experiment. Make sure to work with the leaflet labelled P7.2.1.4.

5O. Lindberg, Proc IRE 40, 1414 (1952): A brief discussion of the Hall effect in semiconductors and thethermo-magnetic effects that complicates measurements of the Hall voltages

15


3.3 Lab objectives

- Explain how the Hall effect helps determine the polarity of the charge carrier as wellas what does it mean to have various polarity for the charge carriers.

- Verify the proportionality of the Hall voltage with the magnetic flux density and thecurrent through the sample.

- Determine the Hal constant RH and the charge carrier concentration of p-Germanium.Do not forget to quote your results with their error bars. The literature value of RH

for Silver is 8.9 10−11 m3/C, the concentration of charge carrier for Silver is 6.6 1028

m−3. How do your p Germanium compare to the Silver value? How can you explainthe difference (if any)?

- How does the value for p-Germanium compare to the one of n-Germanium?

3.4 Preliminary questions

The following questions should help you prepare for this lab:

- Describe at least 4 industrial applications of the Hall Effect.

- Because of several thermo-magnetic effects, the voltage V34 measured on a semi-conductor,may not be the ”true” Hall voltage VH (see section 3.1.1). For example V34 = VH+Vbias.What are the name of the effects creating Vbias? For your data taking, how will youdeal with this issue?

- Suppose you measure RH/t for a fixed magnetic field B0 and a fixed current I throughthe sample. Suppose you know B0, I and VH to 1% relative precision.6 What is thefinal relative precision on RH/t, that is what is ∆(RH/t)/(RH/t)?

- The number density of free electrons in gold is 5.90 1028 electrons per cubic meter. Ifa metal strip of gold 2 cm wide carries a current of 10 A, how thin would it need to beto produce a Hall Voltage of at least one 1 mV. What would the drift velocity of theelectrons be in this case? (Assume a perpendicular field of 5000 Gauss)

References

The discussion of the Hall Effect is taken up from ”Wikipedia, The Free Encyclopedia” aswell as from “The art of experimental physics”, D. Preston, ISBN 0-471-84748-8.

6 That is ∆B0/B0 = ∆I/I = ∆VH/VH = 0.01

16

4 Electron diffraction

A primary tenet of quantum mechanics is the wave-like properties of matter. In 1924,graduate student Louis de Broglie suggested in his dissertation that since light has bothparticle-like and wave-like properties, perhaps all matter might also have wave-like prop-erties. He postulated that the wavelength of object was given by l = h/p, where p = mvis the momentum. This was quite a revolutionary idea, since there was no evidence at thetime that matter behaved like waves. De Broglie received a Nobel prize in 1929 for hisdiscovery of the wave nature of electron. In 1927, however, C. Davisson and L. Germer,and G. Thomson independently discovered experimental proof of the wave-like properties ofmatter -particularly electrons. Davisson and Thomson won the Nobel Prize in 1937 for theirexperimental discovery of the diffraction of electrons by crystal. Not only was this discoveryimportant for the foundation of quantum mechanics, but electron diffraction is an extremelyimportant tool used to study new material.


De Broglie’s Law states that:

λ =h

p(20)

where λ is the electron wave length, h is the Planck’s Constant and p is the electron mo-mentum. For an electron accelerated through a potential V , the momentum p is given bythe following relation:

p =√

2meV (21)

where m is the mass of the electron and e its charge. The above assumes a non-relativisticapproximation. Substituting in the de Broglie’s relation:

λ =h

p=

h√2meV

=

√h2/2me

V(22)

And when h, m and e are substituted:

λ (nm) =

√1.505

V (V olts)(23)

4.1.1 Bragg’s law

The case of waves (electromagnetic waves such as x-rays or ”matter” waves such as electrons)scattering off a crystal lattice is similar to light being scattered by a diffraction grating.However, the three-dimensional case of the crystal is geometrically more complex than thetwo- (or one-) dimensional diffraction grating case. Bragg’s Law governs the position ofthe diffracted maxima in the case of the crystal. A wave diffracted by a crystal behaves as


if it were reflected off the planes of the crystal. Moreover there is an outgoing diffractedwave only if the path length difference between rays ”reflected” off adjacent planes are anintegral number of wavelengths. Figure 4 shows the extra path length off the electron beamscattering of two parallel planes as well as the diffraction patterns produce by electron beamsof two different crystal structure.

Figure 4: Top: A beam incident on a crystal structure which planes are separated by adistance d. Bottom left: Diffraction pattern when scattering off a single crystal. Bottomright: Diffraction pattern off a polycrystal.

The extra path length of the lower ray may be shown to be 2d sin θ so that maxima inthe diffraction pattern will occur when:

2d sin(θ) = n n = 0, 1, 2... (24)

This is Bragg’s Law. Furthermore, the beam is deflected a total angle 2θ. Thus, suppose oneobserves the diffraction pattern on a screen. The maxima will registered as spots or ringson the screen. The distance of the spot from the incoming beam axis will be R, such that:

R = D tan(deflection) = D tan(2θ) ∼ 2Dθ (25)18


where D is the distance from target to screen. Combining Equations 24 and 25 and takingsin(θ) ∼ θ, then:

R =nλD

d(26)

Note, that obtaining a diffraction maximum requires that two conditions be met. Notonly must the angle of deflection bear an appropriate relationship to d and λ, but also thecrystal orientation must be correct to provide an apparent ”reflection” off the crystal planes.The way the crystals are oriented relative to the incoming beam will thus determine theappearance of the diffraction pattern.

4.1.2 Electron diffraction patterns

In relation to diffraction patterns, it is interesting to consider three types of solid matter:single crystals, polycrystals and amorphous materials.

1. Single crystals:Single crystals consist of atoms arranged in an orderly lattice. Some types of crystallattices are simple cubic, face center cubic (FCC), and body center cubic (BCC).In general, single crystals with different crystal structures will cleave into their owncharacteristic geometry. You may have seen single crystals of quartz, calcite, or carbon(diamond). Single crystals are the most ordered of the three structures. An electronbeam passing through a single crystal will produce a pattern of spots. From thediffraction spots, one can determine the type of crystal structure (FCC, BCC) andthe ”lattice parameter” (i.e., the distance between adjacent (100) planes). Also, theorientation of the single crystal can be determined: If the single crystal is turnedor flipped, the spot diffraction pattern will rotate around the center beam spot in apredictable way.

2. Polycrystalline materials:Polycrystalline materials are made up of many tiny single crystals. Most common metalmaterials (copper pipes, nickel coins, stainless steel forks) are polycrystalline. Also,a ground-up powder sample appears polycrystalline. Any small single crystal ”grain”will not in general have the same orientation as its neighbors. The single crystal grainsin a polycrystal will have a random distribution of all the possible orientations. Apolycrystal, therefore, is not as ordered as a single crystal. An electron beam passingthrough a polycrystal will produce a diffraction pattern equivalent to that produced bya beam passing through a series of single crystals of various orientations. The diffrac-tion pattern will, therefore, look like a superposition of single crystal spot patterns: aseries of concentric rings resulting from many spots very close together at various rota-tions around the center beam spot. From the diffraction rings, one can also determinethe type of crystal structure and the ”lattice parameter.” One cannot determine theorientation of a polycrystal, since there is no single orientation and flipping or turningthe polycrystal will yield the same ring pattern.

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3. Amorphous materials:Amorphous materials do not consist of atoms arranged in ordered lattices, but inhodgepodge random sites. Amorphous materials are completely disordered. The elec-tron diffraction pattern will consist of fuzzy rings of light on the fluorescent screen.The diameters of these rings of light are related to average nearest neighbors distancesin the material.

4.1.3 Diffraction gratings

In optics, a diffraction grating is an optical component with a regular pattern, which splitsand diffracts light into several beams traveling in different directions. The directions ofthese beams depend on the spacing of the grating and the wavelength of the light so thatthe grating acts as the dispersive element. Because of this, gratings are commonly used inmonochromators and spectrometers. When scattering an electron beam off a crystal, theatoms that seat on regular spots on the lattice of this crystal acts like the many slits ofan optical diffraction grating. Figure 5 summarizes the important features of a diffractiongrating light curve when both interference and diffraction effect are taken into account. Agood discussion of the combined effect of diffraction and interference can be found in thebook by Melissinos7.

4.1.4 To go further

The particle electron produces diffraction patterns, thus revealing its wave-like properties.Which experiment shows that light which produces diffraction patterns has particle-likeproperties? Hint: What is the Compton effect?

4.2 Experimental setup

The apparatus is an Electron diffraction tube (555 626) from ”KEP-KLINGER, EDUCA-TIONAL PRODUCTS CORP.”. In order to operate the tube, a 10 kV high-voltage powersupply is used. The tube is mounted on a dedicated stand. The target is a polycrystallinegraphite foil. Knowing the distance from the screen to the target crystal (13.5 cm), an in-vestigation into the crystal structure can be carried out using Bragg’s law. A copy of theoperating manual for this device is available on the class web site, it describes how to operatethe tube safely.

4.3 Lab objectives

The objectives of this experience are :

7”Experiments in Modern Physics”, A. Melissinos and J, Napolitano, ISBN:0-12-489851-3

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Figure 5: Light curves produced by a plane wave passing through one (top row), two (middlerow) or three (bottom row) slits. The left column shows the light curves obtained if one con-siders only interference effect while the right column shows the combined effect of interferenceand diffraction effects. This plots are extracted form the ”hyperphysics” web site.

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- For the two diffraction rings, measure R versus V , the accelerating voltage. Does yourdata agree with the V dependence predicted by de Broglie’s Law? Hint: To check thevalidity of a relationship, evaluate the χ2 of your data compared to your hypothesis.

- Determine the lattice plane spacings of the target. What is the precision of yourextraction? How do your results compare to literature values? Note: with this devicethat shows only two rings, the lattice spacings accessible for the graphite target arethe ones shown on figure 6

Figure 6: Lattice plan spacing observable with the electron tube used in this experiment.


The following questions should also help you to prepare for this lab:

- How do you steer an electron beam traveling in vacuum to a specific position? In thediffraction experiment, how can you be sure that you are looking at diffraction patternscreated by electrons and not by light?

- What is the χ2 test? What is it used for?

- How do electron microscopes relate to the wave-particle duality properties of electrons?What is the advantage of an electron microscope compared to a visible microscope?

- Suppose you observe the diffraction pattern produced by an electron beam interactingwith a crystal of lattice parameter d1. If the radius of the ring produced by the firstmaximum is R1, what is the radius of the ring produced by the second maxima? Nowsuppose that the crystal you are looking at has two relevant lattice parameters d1 andd2 (see figure 6), suppose d2 = αd1, how does the radius R1 of the first maximum ringproduced by d1 compare to the radius of the first maximum ring produced by d2?

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- An electron and a photon each have kinetic energy equal to 50 keV. What are their deBroglie wavelengths?

- To verify de Broglie’s law you will plot R versus 1/√

(V ). The data will line up on a

straight line. To compute the χ2 , you will first only consider the error on the radiusThen you will add the precision on V according to the last sentence of section 2.2.1.Suppose you are observing a diffraction pattern produced by a lattice spacing d=125µm on a screen 135 cm away from the sample. Suppose you measure R and V to 1%.By how much does the relative precision ∆R/R increases when you include or notthe precision on V ? You do not need to know R and V to answer this question, docompute ∆R/R.

Reference

Lab manual of the Physics Department of the Toronto University, Canada.Lab manual of the Physics Department of the Austin College.

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5 The Millikan’s oil drop experiment

In 1910, Robert Millikan published the details of an experiment that proved beyond doubtthat charge was carried by discrete positive and negative entities, each of which had an equalmagnitude. He was also able to measure the unitary charge (which we now recognize to bethe electron) accurately. Millikan received the Nobel Prize in 1923 ”for his work on theelementary charge of the electricity and on the photoelectric effect.” In order to measure thecharge of the electron, Millikan carefully balanced the gravitational and electric forces ontiny charged droplets of oil suspended between two metal electrodes. Knowing the electricfield, the charge on the droplet could be determined. Repeating the experiment for manydroplets, he found that the values measured were always multiple of the same number. Heinterpreted this as the charge on a single electron. This experiment has since been repeatedby generations of physics students, although it is rather difficult to do properly.

The oil-drop experiment appears in a list of Science’s 10 Most Beautiful Experimentsoriginally published in the New York Times. See

http://physics.nad.ru/Physics/English/top10.htm.


In the experiment, a small charged drop of oil is observed in a closed chamber between twohorizontal parallel plates. By measuring the velocity of the fall of the drop under gravityand its velocity of rise when the plates are at a high electrical potential difference, datais obtained from which the charge of the drop may be computed. In the experiment, theoil drops are subjected to three different forces: viscous, gravitational and electrical. Byanalyzing these various forces, an expression can be derived which will enable measurementof the charge on the drop and determination of the unit charge of the electron.

When there is no electric field present, the drop under observation falls slowly, subjectto the downward pull of gravity and the upward force due to the viscous resistance of theair to the motion of the drop. The resistance of a viscous fluid to the steady motion of asphere is obtainable from Stroke’s Law from which the retarding force acting on the sphereis given by:

Fr = 6πaηv (27)

where a is the radius of the sphere, η is the coefficient of viscosity of the fluid, and v is thevelocity of the sphere. For an oil drop of mass m, which has reached constant or terminalvelocity, vg, the upward retarding force equals the downward gravitational force and

Fr = mg = 6πaηvg (28)

Now let an electrical field, E, be applied between the plates in such a direction as to makethe drop move upward with a constant velocity, vE. The viscous force again opposes its


motion but acts downward in this case. If the oil drop has an electrical charge, q, when itreaches constant velocity, the forces acting on the drop are again in equilibrium and

Eq −mg = 6πaηvE (29)

Solving Equation 29 for mg and equating to Equation 28, one obtains

q =6πaη

E(vE + vg) (30)

The electrical field is obtained by applying a voltage, V , to the parallel plates of the condenserwhich are separated by a distance, d. Therefore,

q =6πaηd

V(vE + vg) (31)

From Equation 31, it is seen that for the same drop and with a constant applied voltage, achange in q results only in a change of vE and

∆q = C∆vE (32)

When many values of ∆vE are obtained, it is found that they are always integral multiplesof a certain small value. Since this is true for ∆vE, the same must be true for ∆q; that is,the charge gained or lost is the exact multiple of a unit charge. Thus the discreteness of theelectrical charge may be demonstrated without actually obtaining a numerical value of thecharge.

In Equation 31, all quantities are known or measurable expect a, the radius of the drop.To obtain the value of a, Stoke’s Law is used. It states that when a small sphere falls freelythrough a viscous medium, it acquires a constant velocity,

vg =2ga2(α− α1)

9η(33)

where α is the density of the oil, α1 the density of the air, and η, as stated previously, is theviscosity of air. Since the density of the air is very much smaller than the density of oil, α1

is negligible and Equation 33 is reduced to

a =

√9ηvg2gα

(34)

Substituting this value of a in Equation 31 gives an expression for q in which all the quantitiesare known or measurable:

q =6πd

V

√9η3

2αg(vE + vg)

√vg (35)

Millikan, in his experiments found that the electric charge resulting from the measurementsseemed to depend somewhat on the size of the particular drop used and on the air pressure.

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He suspected that the difficulty was inherent in Stroke’s Law which he found not to hold forvery small drops. It is necessary to make a correction, dividing the velocities by the factor(1 + b

pa) where p is the barometric pressure (in cm of Hg), b is a constant of numerical value

6.16× 10−6, and a is the radius of the drop. The value of this correction is sufficiently smallso that the rough value of a obtained in Equation 34 may be used in calculating it. Thecorrected charge on the drop is, therefore, given by:

q =6πd

V

√9η3

2αg

(1 +

b

pa

)− 32

(vE + vg)√vg (36)

Using the MKS system throughout, the charge q will be in Coulomb when: b = 6.17× 10−6

p is the pressure in cm of mercurya is the radius in metersη = 1.827× 10−5 N.s/m−2 is the air viscosity at 18oCg is the acceleration of gravityvE is the velocity in m/sd is the distance in mV is the potential difference between plates in voltsα is the density of oil.

//For this experiment, the oil used is the doped with small latex shells. The shells havea 1.02 µm diameter. The density of latex is 1.05 g/ml.

5.1.1 How do the drops get charged?

The oil used is the type that is usually used in vacuum apparatus. This is because thistype of oil has an extremely low vapor pressure. Ordinary oil would evaporate away underthe heat of the light source and so the mass of the oil drop would not remain constant overthe course of the experiment. Some oil drops will pick up a charge through friction withthe nozzle as they are sprayed, but more can be charged by exposing them to an ionizingradiation source (such as an x ray tube).

5.1.2 To go further

- Millikan and his contemporaries were only able to measure integer values of electroncharge? Has anyone succeeded in measuring fractional values of electron charge? Hint:look up fractional quantum Hall effect, or the 1998 Nobel Prize for Physics. Also whatis the electric charge of the quarks ? What are quarks?

- What are alpha particles? What are common sources of alpha particles? How muchmatter stops a beam of alpha particles?

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Figure 7: Sketch of the oil drop apparatus.

5.2 Experimental setup

The oil drop apparatus you will use in this lab is the oil drop apparatus from PASCO ,device #AP8210A. A copy of the operating manual is available on the class web site. Pleaserefer to the manual for operating instructions.

5.3 Lab objectives

Verify that the charge of the droplets are quantized. That is, verify that each droplet charge,Q, is an integer multiple of a common elementary charge e. The more droplets you measure,the better your result; plan on measuring at least 20 droplets with very similar total electriccharge.You should plot the electric charge of each droplet versus the droplet number, once thedroplets have been ordered by their electric charge. Do not forget to include error bars.If this plot fail to reveal the electric charge quantization, you might want to plot for eachdroplet Q/e− n (do not forget error bars). You will have to estimate n for each droplet asthe integer the closest from Q/e.


The following questions should help you to prepare for this lab:

- What are the forces acting on the droplets in the Millikan’s oil drop experiment? Whatis the effect of each of these forces on the droplets?

- In the context of this experiment, what is the role of the atomizer? Does it charge thedroplet?

- The Millikan apparatus has two parallel plates separated by approximately 3 mm anda high-voltage power supply. Each time you start a new observation with zero fieldand a squirt from the atomizer you will see a myriad of droplets falling through the

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field of view. Your problem will be to pick droplet of similar size and total electriccharge of few e. Suppose you use droplet made out of latex. The shells have a 1.02µm diameter. The density of latex is 1.05 g/ml. How much time does it take for sucha little sphere to drop a distance of 1 mm?

- Once you succeed in trapping a little latex sphere, you will need to quickly decide ifit carries a charge of a few electron (e.g. 1-3e). Calculate the time it takes for a littlelatex sphere to rise by 1mm if it is pulled up by a 200 V potential and carries an electriccharge of 1 e. What is this time, if the sphere carries an electric charge of 3 e.

- Suppose you measure the rise time of a droplet to be 7.0±0.1 s, the time down to be 31.0± 0.1 s, the distance the droplet travels up and down to be 0.80 ± 0.01 mm. Supposeyou work at atmospheric pressure and with a voltage of 200V. To which precision canyou evaluate the charge of the droplet? Do not consider the precision on the correctionto Stoke’s law.

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6 The Franck-Hertz Experiment

In 1914, James Franck and Gustav Hertz performed an experiment which demonstratedthe existence of excited states in mercury atoms, helping to confirm the quantum theorywhich predicted that electrons occupied only discrete, quantized energy states. In 1925,Franck and Hertz were awarded the Nobel Prize on Physics for their discovery of the lawsgoverning the impact of an electron upon an atom. In the Franck-Hertz apparatus, electronsare accelerated and collected after passing through a gas of neon. As the Franck-Hertz datashows (see figure 8), when the accelerating voltage reaches 4.9 volts, the current suddenlydrops, indicating the sharp onset of a new phenomenon which takes enough energy awayfrom the electrons that they cannot reach the collector. This drop is attributed to inelasticcollisions between the accelerated electrons and atomic electrons in the neon atoms. Thesudden onset suggests that the mercury electrons cannot accept energy until it reaches thethreshold for elevating them to an excited state. This 4.9 volt excited state corresponds toa strong line in the ultraviolet emission spectrum of mercury at 254 nm (a 4.9eV photon).Drops in the collected current occur at multiples of 4.9 volts since an accelerated electronwhich has 4.9 eV of energy removed in a collision can be re-accelerated to produce othersuch collisions at multiples of 4.9 volts. This experiment was strong confirmation of the ideaof quantized atomic energy levels.

Figure 8: Sketch of the Franck-Hertz apparatus (left) and graph of typical data (rigth).Thermionic electrons are emited by a cathode (rigth), accelerated in capsule containing mer-cury and collected on a anode (left). The data show the current collected as a function ofthe accelerating voltage, the peaks reveal the first excited state of Mercury.


In the Franck-Hertz experiment, an electron beam is produced by thermionic emission froma filament. The electrons are accelerated, pass through the vapor, and are then retarded(decelerated) by a few volts before collection at the anode. This all takes place in a tube


contained within an oven that controls the tubes temperature and thus the mercury vapordensity. The setup is illustrated schematically in Figure 9.

Figure 9: The Franck-Hertz tube and electrical connections. The cathode is a filament heatedby the current provided from a 6.3 V supply. The voltage Va accelerates the electrons towardthe grid. A voltage, Vg slows the electrons as they travel between the grid and the anode. Thecurrent of electrons reaching the anode is measured by a picoammeter/electrometer.

Consider first what happens as the filament is heated, thus raising the average energyof conduction electrons in the metal. As the temperature increases, a greater fraction ofthe electrons have kinetic energy exceeding the work function WPt of the cathode material(platinum) . Some of these even reach the anode and a small current (e.g., nanoamps) flows.When the accelerating voltage Va is increased from zero, filament electrons are acceleratedtoward the grid and a much greater current reaches the anode. If Vg =0, some of the emittedelectrons reach the anode and the electron current is registered. The path of current in thiscircuit passes through the two power supplies, the filament, the mercury vapor in the tube,and through the ammeter. Not all the electrons make it to the anode: the beam really isnta beam, and the trajectories spread out over a large range of angles. The planar geometrydoes improve the situation. As the accelerating voltage increases, the anode current increasesbecause the electron trajectories are more focused, and the electrons are deflected less byscattering from the atoms of vapor in the tube. The anode current is observed to rise fasterthan linearly with Va.

We now take into account electron collisions with the mercury atoms in the vapor. Elas-tic scattering is one channel that results in deflection of the electrons from their originaltrajectory. In an elastic scattering, the atom recoils in its ground state like a hard sphere,so the electron loses very little energy. Inelastic scattering is also possible. In this process,kinetic energy is lost to excitation or ionization of mercury atoms. The excitation energy isEex ∼ 5 eV, so when (eVa −WPt) < Eex, no such excitation is possible. For electron kineticenergy greater than Eex, it is, in principle, possible for the inelastic process to occur anddecrease the electron kinetic energy by about Eex for each mercury atom an electron excites.

The scattering process is best represented by a cross section σ - the effective area of amercury atom presented to the electrons as they move through a vapor. Imagine looking

32


through the vapor with total thickness t and a number density n atoms/cm3 from the per-spective of an electron with some initial trajectory. The vapor may look partly transparentwith an areal density of nt atom/cm2 in the electrons path. The total fraction of that areataken up by mercury atoms from which the electron can scatter is nt , a measure of theprobability the electron will scatter while propagating through the vapor. The cross sectionis an effective area and depends on the nature of the interaction, initial energy, final energyand direction, and other quantum mechanical features of the process. The elastic and in-elastic cross sections are generally quite different: the inelastic excitation cross section canbe a resonant process with a relatively large cross section.

As an electron passes through the vapor, it gains energy in the accelerating field producedby Va. At the point where the electrons energy exceeds Eex, inelastic scattering becomespossible, and the electron scatters with a short scattering length and loses almost all ofits kinetic energy so that the process begins again. Thus multiple inelastic scatterings arepossible as an electron moves from cathode to grid.

The final energy of electrons that reach the grid is the initial energy, eVa, minus energylost in scatterings. A small retarding voltage between the grid and anode deflects electronswith energy less than eVg+WCu, where WCu is the work function of the anode. This energy isa few eV. Electrons with greater energy reach the anode and the electron current is registeredby the ammeter. For electrons that lose total energy ∆E by scattering, the requirement fordetection (reaching the anode) is ∆E ≤ e(Va−vg−∆W ), where ∆W , the contact potential,is the difference of work functions of the cathode and anode. The anode current is thereforea measure of the integral number of electrons that satisfy this inequality. These raw datamust be analyzed to determine the electron energy loss spectrum and extract Eex.

6.1.1 Energy levels of Mercury

A mercury atom has 80 electrons. For an atom in the ground state the K, L, M, and Nshells of mercury are filled and the O and the P shells have the following electrons: O shell(5s2, 5p6, 5d10) and P shell (6s2). The energy levels of mercury, which are relevant to thisexperiment, are shown in figure 10. In this figure the energy levels are labeled with twonotations:- nl where n is the principal quantum number and l is the orbital angular momentumquantum number designated by s(l = 0) and p(l = 1).- 2S+1Lf , where S, L, and J are the total spin quantum number, total orbital angularmomentum quantum number, and total momentum quantum number.The 1P1 and 3P1 are ordinary states, having lifetimes of about 10−8s before decaying to 1S0

ground state by photon emission. The 3P2 and 3P0 are meta-stable states, having lifetimesof about 10−3s or 105 times as long as an ordinary state. Hence the probability per second ofan electron making a transition from either the 3P2 or 3P0 state to the 1S0 ground state byphoton emission is 105 times smaller than the transition from either the 1P1 and 3P1 stateto the 1S0 ground state. Thus some transitions are forbidden while other are allowed. The

33


allowed transitions for photon emission are indicated by the two arrows on the left of figure10, and the four arrows on the right indicate energy spacing in units of electron volts.

Figure 10: Energy levels of mercury that are relevant to this experiment. Energy levelseparation in electron volts is indicated.

6.1.2 Atomic excitation by inelastic electron scattering

An electron traveling from the cathode K toward the anode A has a mean free path l givenby8 :

l =1√

2πd2n[m] (37)

where d ∼ 302 pm is the diameter of a mercury atom and n is the number of atoms per unitvolume. At the end of one mean free path, the electron has gained a kinetic energy K fromthe electric field E:

K = eEl [J ] (38)

where e is the electron charge and E is the electric field established by the the acceleratingvoltage Va. If l is long, K is large. The number density n is very sensitive to the tubetemperature; therefore l and, hence, K are very temperature sensitive.

8See, D. Halliday and R. Resnick, ”Physics”, Part 1, 3d edition, Wiley, NewYork 1978. Mean free pathis discussed on pages 522-524.

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6.1.3 Loss of kinetic energy in an elastic collision

The loss of kinetic energy by an electron when it collides elastically with a mercury atomis greatest when the collision is head-on. For an elastic head-on collision with the mercuryatom initially at rest, the change in electron energy is given by9:

∆K =4mM

(m+M)2K0 [J ] (39)

where m and M are the masses of an electron and a mercury atom and K0 is the initialelectron energy.

6.1.4 Ionization potential of mercury

The Franck Hertz tube can be connected such that the ionization potential of mercury canbe measured. In this case, the anode is connected so that the anode is maintained at anegative potential of a couple of volts with respect to the cathode. In this condition, it isenergetically impossible for any electron to reach the anode. When the grid potential ismade positive with respect to the cathode, electrons are accelerated to the grid. Some passto through the grid into the space between the the grid and the anode, but all are eventuallypuled back and collected by the grid. However, if the electrons enter the space between thegrid and anode have sufficient energy to ionize the mercury atoms, then the resulting positivemercury ions are drawn to the anode and a positive current is registered. The measure ofthe anode current as a function of the grid to cathode potential difference yields a measureof the ionization potential. The first difficulty is to determine the optimal oven temperature.If the vapor pressure of the mercury is too low, then the electrons entering the grid-to-anoderegion with sufficient energy to cause ionization will have a small probability of collision(long mean free path) and the ion current collected by the anode will be too low. If thevapor pressure of mercury is too high, then the electrons will suffer inelastic collisions in thespace between the cathode and the grid as soon as they energy slightly exceed the energyrequired to raise mercury atoms to their first excited state and will therefore never attainsufficient energy to cause ionization.


In this experiment, we will use a version of the Franck Hertz tube Mounted on Front-Platewith Mercury Filling and Heating Chamber from Klinger KEP (Model KA6040-KA6041),the Franck Hertz tube control unit KA6045, read out by an digital oscilloscope. A copyof the operating manual for this device is available on the class web site, it describes theFranck Hertz tube, the power supply used in this experiment as well as suggestions for thedata taking.

9See K. Rossberg, ”A first course in Analytical Mechanics”, Wiley, New York, 1983. Collision of twoparticles is discussed on page 157-160

35


The Franck-Hertz tube is contained within an oven, which is a metal box with a ther-mostatically controlled heater and terminals for connections to the tube. A thermometercan be inserted through a hole in the top of the oven. The tube has a parallel system ofelectrodes in order to produce a fairly uniform electric field. The distance between the fila-ment/cathode and the perforated grid is much larger than the mean free path of electronsthrough the mercury vapor under normal operating conditions. A platinum ribbon with asmall barium-oxide spot serves as a direct heated cathode. An electrode connected withthe cathode limits the current and suppresses secondary and reflected electrons, making theelectric field more uniform. In order to avoid current leakage along the hot glass wall ofthe tube, a protective ceramic ring is fused in the glass as a feed-through to the counterelectrode. The tube is evacuated and coated inside with a getter which absorbs traces of airthat leaked in during the manufacturing process and over the lifetime of the tube.

Before making any connections, be sure you understand the circuit. Figure 11 showsthe setup used for this experiment, this figure is extracted from the manufacturer manual.The web site gives a list of important caution and tips that you should read before startingyour experiment. When using the scope, couple the channel to DC to provide a good zeropoint for voltage. Also we will use the 60 Hz functionality of the power supply to minimizetemperature correlated drifts of the measured voltages.

Figure 11: The Franck-Hertz tube and electrical connection used in this experiment.

6.3 Lab objectives

To measure the energy of the 1st excited state of Hg and the work function of the heatingfilament.For this lab, you are asked to:

- explain how the Bohr model of the atoms explains the quantization of excited states,36


- use both peaks and valleys of your data to extract your results. Use the first derivateof the excitation plot, in order to evaluate the location of the peak and the valley.

- use the result of your linear regression, to present a residual plot of your data.

Moreover we recently acquired a Neon Franck-Hertz apparatus. Repeat your measurementand determine the Neon 1st excited state energy. Does your measurement match expecta-tions?



1. What is an ion? What is an excited state of an atom? How do those two conceptscontrast? Which energy is larger ionization potential or first excited state, explain.

2. Why does the Franck-Hertz apparatus have a cathode, anode and grid? Why is theretarding voltage needed to observe the first excited state of atoms (not the ionizationenergy)?

3. The energy of the first excited state of the mercury atom is about 5.0 eV above thatof the ground state. What is the maximum energy loss for an electron of 4.0 eV whencolliding with a Hg atom? Same question for a 6.0 eV electron. Hint: think about thethreshold between elastic and inelastic collision.

4. What is the mean free path of electrons propagating in a gas (see equation 37) at 180oC? At this temperature, the vapor pressure of Mercury is 1.1728 kPa.

5. Suppose a voltage V is maintained between a cathode and an anode. The cathode andthe anode are separated by a distance d = 10cm. Suppose and electron is released atrest from a filament attached to the cathode, the electron will be accelerated to theanode. At what distance d′ from the cathode will this electron reach an energy of 5eV if V = 6V, 12V or 24V?

6. The left plot of Figure 12 illustrates the concept used by Bohr to introduce quantiza-tion of the stationary states of electrons in atoms. Explain how this figure relates tothe notion of principal quantum number in the Bohr’s atom10 What is the principalquantum number of the state shown on the figure? If on the left side of the figure,the principal quantum number is n, draw on the right side of the figure, the statecorresponding to the principal number of (n− 1).

7. Draw and label an energy diagram for hydrogen11. On it, show all transitions by whichan electron in the n = 4 state could emit a photon.

10A good explanation can be found under http://hyperphysics.phy-astr.gsu.edu/hbase/Bohr.html#c2Look for the section on ”Angular Momentum Quantization”

11For example, see http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html

37


Figure 12: Representation of a stationary state of a Bohr’s atom.

References

The discussion of theFranck-Hertz Effect is copied from:

- the HyperPhysics web site: http://hyperphysics.phy-astr.gsu.edu/Hbase/hph.html

- The lab manual of the Michigan University: http://instructor.physics.lsa.umich.edu/adv-labs/Franck Hertz/franck-hertz.pdf

- The Franck-Hertz experiment web page at MIT: http://web.mit.edu/8.13/www/07.shtml

- D. Preston and E. Dietz, The Art of Experimental Physics, Wiley, New York, 1991

- ”Physics for scientists and engineers”, R. Knight.

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7 The Photo-electric effect

The photoelectric effect or photoemission is the pro-duction of electrons or other free carriers when light isshone onto a material. Electrons emitted in this man-ner can be called photoelectrons. The phenomenonis commonly studied in electronic physics, as well asin fields of chemistry, such as quantum chemistry orelectrochemistry.


In 1895 Heinrich Hertz observed that ultraviolet light from the sparks of his generator forradio waves he had recently discovered, falling on the negative electrode of his radio wavedetector, induced the flow of electricity in the gap between the electrodes. Pursuing thephenomenon in detail, he discovered the photoelectric effect whereby light of sufficientlyshort wavelength causes the ejection of electrons from a metal surface. 1905 Nobel LaureatePhilipp Lenard made improved measurements and demonstrated by determination of theircharge to mass ratio that the ejected particles are identical with the electrons that hadrecently been discovered by 1906 Nobel Laureate J. J. Thomson in experiments with cathoderays. Further investigations showed that when light falls on a metal surface, electrons areimmediately ejected with a range of kinetic energies. Some metallic surfaces, like the one inthis experiment, can be made to eject electrons when visible light is shined on them. Theinteresting features of this photo-ejection are that:

- For light with frequencies below a certain minimum value, no electrons are ejected nomatter what the intensity of the light.

- For frequencies above the threshold, electrons are immediately ejected no matter howdim the light is.

- The maximum kinetic energy of the emitted electrons depends only on the frequencyof the incident light, not on its intensity.

None of these observations could be easily reconciled with the classical wave theory of light.In 1905, Einstein showed that Max Planck’s quantum theory (that had been developed tosolve problems in a different area of physics), had practical consequences for photoemissionas well. Einstein’s explanation of the photoelectric effect was that light can behave like a”particle” (or a tightly localized packet of waves), called a ”photon”. In the photoelectriceffect, a photon is absorbed by one electron–not by the whole metal crystal. Planck’s theoryrequired that the photon energy be related to the frequency of the light by:

Ephoton = hf (40)


where f is the frequency of the light and h is Planck’s constant. When a photon strikesan electron in a material, the energy of a photon is transferred to the electron. In orderfor the electron to get out of the metal, it gives up some of the acquired photon energy(Elost to metal). The remaining energy is kinetic energy of the free electron: Thus:

Ephoton = Elost to metal +KEelectron. (41)

The kinetic energy of the electron is then:

KEphoton = Ephoton − Elost to metal (42)

It turns out that there is a well defined maximum kinetic energy for the electrons that occurswhen a minimum amount of energy is lost to the metal during emission. This minimumenergy lost is called the work function of the metal, W and it is a characteristic propertyof a metal. In the photoelectric effect, photons strike the metal surface, are immediatelyabsorbed, and if their energy, hf , is greater than W , electrons are immediately emitted fromthe surface with energy:

KEmax, electron = hf −W (43)

7.1.1 To go further

- What are the problems with classical physics that the Planck’s quantum theory triedto solve?

- What is the difference between the photo-electric effect and the photo-voltaic effect?


Figure 13: Apparatus used for the photo-electric effect experiment.

In this experiment, light from a mercury arc source is filtered to isolate various linesof the spectrum. This filtered light is allowed to shine on the anode of the photodiode

40


which is arranged in the apparatus shown in Figure 13. The voltage (across the diode) canbe controlled by the voltage adjustment potentiometer and read by the digital multimeter.Electrons emitted under the influence of the light can be slowed down as they cross thispotential. Recall that the work done to slow the electrons down is qelectronV . The emittedelectrons have a variety of energies up to a maximum. So there is a voltage above which allcurrent is suppressed. This is the ”repelling voltage”:

qelectronVrepel = KEmax, electron = hf −W (44)

Figure 14: Patterns observed on the scope for different configurations of the photo-electriceffect experiment. See Procedure tips in section 7.2.

Procedure tips:

- Allow the lamp, amplifier-power supply, oscilloscope and multimeter to warm up for afew minutes.

- In this experiment the current of emitted electrons is detected by the amplifier-powersupply and displayed on the oscilloscope. Since it is much easier to detect an ACcurrent than a DC current, we make use of the fact that the lamp puts out light whoseintensity varies at 120 Hz. Thus the current of emitted electrons will also have a 120Hz component. We detect this component with an AC amplifier and feed the outputsignal to the oscilloscope which, when swept against a voltage at power line frequency(60 Hz) produces a ?lazy eight? pattern as shown in Figure 14-(a).

- Place the mercury lamp over the photodiode and note the signal on the oscilloscope.The signal should look like that in Figure 14-(a). Make the signal as large as youcan by adjusting the position of the lamp above the photodiode when no filters arein place. You may also have to adjust the controls on the oscilloscope to get a clear,sharp pattern.

- Cover the diode opening with something opaque and note the signal on the oscilloscope.By completely covering the diode opening, we can cause the current to vanish and theresulting signal looks something like Figure 14-(b).

41


- With the diode opening uncovered again, slowly vary the voltage control potentiometeron the amplifier power supply until the zero voltage pattern found in Figure 14-(b) isseen. The voltage will never be negative and will not exceed 2.00 volts. NOTE: If thevoltage control is turned up too high, you may see a trace like that in Figure 14-(c).

7.3 Lab objectives

- The details of what is happening in box in which the Mercury lamp is shinning onare not given to you. Explain what physical quantities are measured on the horizontaland vertical axis of the oscilloscope as well as how they are measured - a diagram isappropriate. Explain why a AC measurement is used instead a DC one.

- Measure the Plank constant and the work function of the metal on which the light isimpinging. What is photocathode made off?



- What are the strongest peaks of the emission line spectrum of Mercury?

- Figure 15 shows the typical photo-electric behavior of a metal as the anode-cathodepotential difference ∆V is varied.

- Why do the curves become horizontal for ∆V > 0V? Shouldn’t the current in-crease as the potential difference increases? Explain.

- Why doesn’t the current immediately drop to zero for ∆V < 0V? Shouldn’t∆V < 0 prevent the electrons from reaching the anode? Explain.

- The current is zero for ∆V < −Vstop. Where do the photo-electrons go? Are nophoto-electrons emitted if ∆V < −Vstop?

- Based on classical physics, one would expect that Vstop would depend on the lightintensity. it doesn’t. Explain what is the idea behind Vstop depending on thelight intensity in the classical picture. Explain how Einstein’s explanation of thephoto-electric effect relates to the observation that Vstop does not depend on thelight intensity.

- What are the threshold frequencies and wavelengths for photo-emission from Sodiumand Aluminum?

- Suppose a wave defined by W (A, f, φ, t) = A sin(t/f + φ).Plot W (A, f = 60Hz, φ = 0, t) versus W (A, f = 120Hz, φ = 0, t),W (A, f = 60Hz, φ = π/4, t) versus W (A, f = 120Hz, φ = 0, t) andW (A, f = 60Hz, φ = 0, t) versus W (A, f = 50Hz, φ = 0, t).

42


Figure 15: Photo-electric current as a function of the voltage applied to a photo-cathode.

References

The discussion of the Photo-electric effect t is taken up from ”Wikipedia, The Free En-cyclopedia”, from the PHYS2001 Laboratory manual of Ohio University, the chapter onQuantization from Knight and from the Photoelectric Effect laboratory manual of the de-partment of Physics of MIT.

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44

8 Interferometry

Interferometry is a family of techniques in which waves, usually electromagnetic, are su-perimposed in order to extract information. Interferometry is an important investigativetechnique in the fields of astronomy, fiber optics, engineering metrology, optical metrol-ogy, oceanography, seismology, spectroscopy (and its applications to chemistry), quantummechanics, nuclear and particle physics, plasma physics, remote sensing, biomolecular inter-actions, surface profiling, microfluidics, mechanical stress/strain measurement, velocimetry,and optometry.

Interferometers are widely used in science and industry for the measurement of smalldisplacements, refractive index changes and surface irregularities. In analytical science,interferometers are used in continuous wave Fourier transform spectroscopy to analyze lightcontaining features of absorption or emission associated with a substance or mixture. Anastronomical interferometer consists of two or more separate telescopes that combine theirsignals, offering a resolution equivalent to that of a telescope of diameter equal to the largestseparation between its individual elements.


Interferometry makes use of the principle of superposition to combine waves in a way that willcause the result of their combination to have some meaningful property that is diagnostic ofthe original state of the waves. This works because when two waves with the same frequencycombine, the resulting intensity pattern is determined by the phase difference between thetwo waves, waves that are in phase will undergo constructive interference while waves thatare out of phase will undergo destructive interference. Waves which are not completely inphase nor completely out of phase will have an intermediate intensity pattern, which canbe used to determine their relative phase difference. Most interferometers use light or someother form of electromagnetic wave.Wavefront splitting versus amplitude splittingA wavefront splitting interferometer divides a light wavefront emerging from a point or anarrow slit (i.e. spatially coherent light) and, after allowing the two parts of the wavefrontto travel through different paths, allows them to recombine. Figure 16 illustrates Young’sinterference experiment . Other examples of wavefront splitting interferometer include theLloyd’s mirror, the Fresnel biprism, the Billet Bi-Lens, and the Rayleigh interferometer.An amplitude splitting interferometer uses a partial reflector to divide the amplitude of theincident wave into separate beams which are separated and recombined. Fig. 17 illustratesthe Michelson interferometers. Other examples of amplitude splitting interferometer includethe Fizeau, Mach-Zehnder and Fabry-Perot.

8.1.1 To go further

- What are the various interferometer cited in this description?


Figure 16: Young double-slits experiment.

Figure 17: Michelson interferometer.

- What industrial applications of interferometry?


Since microwaves are identical in nature to other forms of electromagnetic radiations, theyexhibit phenomena analogous to that of geometrical and physical optics. In this lab you willuse double-slit interference and a Michelson interferometer to determine the wavelength ofthe microwave radiation being used in the lab.

The apparatus consists of a microwave transmitter and a receiver which are attached toa base assembly which is called the goniometer. The goniometer has arms attached to abase which can be rotated about the center of the goniometer. The central portion has aprotractor scale which allows you to measure the angles through which the arms are moved.Each arm of the goniometer is equipped with two scales to measure distances from the axis ofrotation; one of these scales indicates the distance from the axis of rotation to the apertureplane of the transmitter and receiver ?horns?, the other indicates the distance from theaxis of rotation to any accessory mounted in the sliding accessory holder. Arrows on the

46


particular item to be mounted on the arm indicate which scale is to be used. Each of theplates to be used has a small hole above its flat edge, which is to be placed downward in theholders. This hole is to be centered in the ”V” notch of the holders.

The transmitter uses a Gunn diode to produce the oscillations. The frequency of the unitis fixed at 10.5 GHz and it cannot be adjusted.

There are two knobs on the receiver; one is a multiplier control and the other is variablesensitivity. The multiplier scale reduces the gain by increasing the value of the multiplier.This means a signal of 0.5 mAmp on the 10 scale is ten times greater that a signal of 0.5mAmp on the scale of 1, or 0.33 mAmp signal on the 3 multiplier setting is equal to 1.0mAmp signal on a 1 multiplier reading. The variable sensitivity is used to adjust the signalto the desired level to prevent over loading the readout. All readings are taken from the0 to 1 scale of the milli-ammeter. The milli-ammeter reading is proportional to the powerreceived and, hence, to the intensity of the radiation. Turn on the Transmitter and turn themultiplier to 30.

You can also used reflectors shown on Figure 18.

Figure 18: Plates that can be used for the interferometry experiments.

During all experiments avoid placing your hand or any other object within the field whilemaking measurements. Extra objects in the path or touching the setup will cause inaccuratemilli-ammeter readings.

8.3 Lab objectives

- To measure the wavelength of micro-waves using both a Michelson interferometer anda Double-slit apparatus.

- To compare and contrast the precision obtained from both methods.


The following questions should help you prepare for this lab:47


- How can the wavelength of a wave be measured using a Double slit experiment appa-ratus?

- How can the wavelength of a wave measured using a Michelson interferometer?

- What does the detection of gravitational waves announced by the Ligo collaborationin 2016 has to do with interferometry?

- What is the wavelength range, frequency and energy of a microwave? Cite a least twosources of microwave and at least two uses of microwave (beside ovens).

- The ”standard” analytical formula for a double slit experiment is mλ = 2 d sin θ whereλ is the wave length, 2d is the distance between the slits and θ is the angle betweenthe perpendicular to the plane where the slits seat and the observation direction. Thisformula assumes that the distance between the slits (2d) is small and the distanceD between the slits and the observation point is large . This question is about try-ing to quantify the validity of this assumption. Consider an (x,y) plane, slit A islocated at (0,d) and slit B is located at (0,-d), the observation point C is located at(D cos θ; D sin θ). Derive the analytical value for the path difference (AC-BC) thatcreates the interference pattern. Assuming that D >> d, derive the approximate pathdifference to match the ”standard” analytical formula given above 12. Now evaluatethe relative mistake made when considering the analytical formula or the approximateformula for D varying between 10 and 50 cm, θ varying between 10 and 80 degreesand d varying between 1 and 20 cm. (Hint you don’t want to do those calculations onpaper but use a computer).

- Consider the three functions plotted on figure 19. Function f3 is the sum of functionf1 and f2 which are parametrized as:

f1(x) = cos(π

2.x) (45)

f2(x) = 2× (1 − cos(π

10.x)) (46)

The local maxima of f3 are displaced compared to the local maxima of f1: the verticallines on figure 19 show the horizontal values of four local maxima for function f1, theseare not the local maxima for function f3 . Evaluate by how much the local maximaof f3 are displaced compared to the local maxima of f1 for each of the four maximahighlighted by the vertical lines. (Hint: you don’t want to do those calculations onpaper but use a computer).

References

The discussion of the Interferometry is taken up from ”Wikipedia, The Free Encyclo-pedia”, from the PHYS2001 Laboratory manual of Ohio University.

12You might want to keep in mind that when x is small,√

1 + x→ 1 + 0.5x

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-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

2

3

4

5

f1

f2

f3=f1+f2

f1+f2

Figure 19: Functions used in the last question of the preliminary questions for this lab

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9 Numerical simulation

This is the most recent lab implemented in this class. I would very much appreciatefeedback on information that are not clear or missing. Thanks.

A numerical simulation is a calculation that is run on a computer following a programthat implements a mathematical model for a physical system. Numerical simulationsare required to study the behavior of systems whose mathematical models are toocomplex to provide analytical solutions, as in most nonlinear systems. In this lab,you will work with a simple numerical simulation written in C++ and interfaced withthe high-energy-physics software called ROOT13 . The goal of this lab is to illustratethe type of physics problems a numerical simulation can answer not to test you oncoding or your hypothetical knowledge of ROOT. So please do not hesitate to askabout difficulties you might have such that you do not get bug down by those tools.

9.1 Monte Carlo method

Monte Carlo methods (or Monte Carlo experiments) are a broad class of computationalalgorithms that rely on repeated random sampling to obtain numerical results. Theiressential idea is using randomness to solve problems that might be deterministic inprinciple. They are often used in physical and mathematical problems and are mostuseful when it is difficult or impossible to use other approaches. Monte Carlo methodsare mainly used in three distinct problem classes: optimization, numerical integration,and generating draws from a probability distribution.

In physics-related problems, Monte Carlo methods are quite useful for simulatingsystems with many coupled degrees of freedom, such as fluids, disordered materials,strongly coupled solids, and cellular structures (see cellular Potts model, interactingparticle systems, McKean-Vlasov processes, kinetic models of gases). Other examplesinclude modeling phenomena with significant uncertainty in inputs such as the calcula-tion of risk in business and, in math, evaluation of multidimensional definite integralswith complicated boundary conditions. In application to space and oil explorationproblems, Monte Carlo based predictions of failure, cost overruns and schedule over-runs are routinely better than human intuition or alternative ”soft” methods.

Monte Carlo methods vary, but tend to follow a particular pattern:

1. Define a domain of possible inputs.

2. Generate inputs randomly from a probability distribution over the domain.

3. Perform a deterministic computation on the inputs.

13The root web site: https://root.cern.ch


4. Aggregate the results.

For example, consider a circle inscribed in a unit square. Given that the circle and thesquare have a ratio of areas that is π/4, the value of π can be approximated using aMonte Carlo method:

- Draw a square on the ground, then inscribe a circle within it.

- Uniformly scatter some objects of uniform size (grains of rice or sand) over thesquare.

- Count the number of objects inside the circle and the total number of objects.

- The ratio of the two counts is an estimate of the ratio of the two areas, which isπ/4. Multiply the result by 4 to estimate π.

- In this procedure the domain of inputs is the square that circumscribes our circle.We generate random inputs by scattering grains over the square then perform acomputation on each input (test whether it falls within the circle). Finally, weaggregate the results to obtain our final result, the approximation of π.

There are two important points to consider here. Firstly, if the grains are not uni-formly distributed, then the approximation will be poor. Secondly, there should be alarge number of inputs. The approximation is generally poor if only a few grains arerandomly dropped into the whole square. On average, the approximation improves asmore grains are dropped.

Uses of Monte Carlo methods require large amounts of random numbers, and it wastheir use that spurred the development of pseudorandom number generators, whichwere far quicker to use than the tables of random numbers that had been previouslyused for statistical sampling.

9.2 Root

ROOT is an object-oriented program and library developed by CERN14. It was origi-nally designed for particle physics data analysis and contains several features specificto this field, but it is also used in other applications such as astronomy and data min-ing. For this laboratory you will use three functions of the software: (1) the randomgenerator, (2) the histogramming functions, (3) the display functions.

Random generatorBy default ROOT provides eight methods to generate random numbers accordingto some basic distributions, namely: Exp(tau), Integer(imax), Gaus(mean, sigma)

14https://en.wikipedia.org/wiki/ROOT

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,Rndm(), Uniform(x1), Landau(mpv, sigma), Poisson(mean), Binomial(ntot,prob). Ad-ditional user-defined distributions can be programed. More information about the Ran-dom generators available under ROOT can be found at : https://root.cern.ch/doc/master/classTRandom.html. The default implementation of the TRandom class is called gRandom. Figure 20shows how to generate an histogram and fill it according to a Gaussian distribution.

Figure 20: Simple implementation of the Random generator in ROOT. The right part of thefigure shows the code that produces the histogram on the left.

9.3 Physical concept used in this lab: π0 decays detected in aElectro-magnetic calorimeter.

Pion are naturally occurring in cosmic shower, they are also important players in theinteractions between protons and neutrons inside nuclei. In particle physics labora-tories, the easiest way to observe pion is to scatter an electron or a photon beam offprotons and neutron:

γ + p→ p+ π0 (47)

The π0 is an unstable particle that decays promptly into two photons. A detector ofchoice for those two photons is a segmented electromagnetic (EM) calorimeter.

EM calorimeters.In particle physics, a calorimeter is a detector that measures the energy of particles.When entering a region with dense material, particles initiate a shower of lower en-ergy particles eventually resulting on a limited number of low energy photons beingcollected, and measured (see Figure 21). The number of low energy photons beingdetected defines the energy resolution of the detector. the shower emits those low en-ergy (optical) photons in a cone which opening angle is material dependent. Typically,

53


calorimeters are segmented transversely to provide information about the direction ofthe initial particle, as well as sometimes longitudinally segmented as the mass of theparticle can be inferred from the shape of the shower as it develops. Figure 22 showsa 16 by 13 blocks EM calorimeter on which the simulation you will be working withduring this lab is based.

Figure 21: Electromagnetic shower created by a high energy photon entering a dense ab-sorber. The wikipedia entry on particle showers is a good starting resource for learning moreabout this topic.

Figure 22: Electromagnetic calorimeter used in the E12-06-114 experiment at JLab. Thelead-fluorite (PbF2) calorimeter is 48×39× 20 cm3.

π0 decays.The π0 decay is isotropic into the center of mass of the π0. In that case each of thetwo photons carry half of the π0 initial energy. Which means that in the lab frame,those decays result in different photons energy and angle configurations as illustratedon Figure 23. If the direction of the Lorentz boost and the direction of the photonsemission in the center of mass frame is nearly perpendicular, the resulting lab framedecay is almost symmetric, i.e. each photon carry about half of the π0 initial energyand both photons are emitted in the forward direction. More generally, the kinematicsof the decays imply an angle α between both photons given by

cosα = 1− m2π0

E1E2

(48)

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which is minimum when the energy of each of the photons in the lab E1 and E2 areboth equal to half of the π0 energy.The calorimeter detects photons in coincidence. When located close to a proton orneutron target, it might detect the two photons resulting from the pion decay or twophotons that are not related to each other. One way to differentiate between thosetwo cases is to reconstruct the mass of the hypothetical pion when two photons aredetected in the calorimeter (see equation 48). If those two photons are indeed resultingfrom one pion, the mass of the reconstructed pion will appear at 134.9 MeV while themass of the reconstructed ”pion” for two random photons will be randomly distributed.Figure 24 shows a typical mass distribution produced by the code you will use for thislab.

Figure 23: Two different π0 decays. When both photons are emitted perpendicular to thedirection of the Lorentz boost in the π0 center-of-mass, the decay is symmetric in the labframe, with both photons emitted in the forward direction and each carrying half of the π0

energy. When both photons hit the detector they leave energy in two easily distinguishableloci. If the direction of one of the produced photon in the center of mass is very close to thedirection of the boost, one photon carries most of the energy in the lab frame; the other isemitted at large angle and has low energy. It is rarely detected in the calorimeter.

9.3.1 The simulation you are working with

The physical problem described by the simulation code you will be using is describedin this section. Suppose a point like source of pions with a EM calorimeter in front ofit, this situation is illustrated by figure 25. The major steps of the simulation are:

- Event generation: A pion of a given energy and ”random” direction is generated.First its decay is computed in the pion center mass. It generates two equal energyback-to-back photons. Those two photons are boosted in the lab frame were theynow have individual but correlated energies and directions. The impact of each ofthose photons on the calorimeter is computed. There is another event generation

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pion mass reconstructed

Entries 15000

Mean 126.3

RMS 29.57

pion mass (MeV)80 100 120 140 160 180 200

50

100

150

200

250

300

350pion mass reconstructed

Entries 15000

Mean 126.3

RMS 29.57

pion mass reconstructed

A B CC

Figure 24: Reconstructed mass of pions in a simulated experiment. Experimentally thethick blue histogram is observed. Through simulation it is easy to show that that histogramhas two components: one with a sharply defined peak at the mass of the π0 (black dottedone) and a continuum underneath it (red dashed one). The region B (in between the middlevertical lines) indicate cuts that allow to identify events that result from the decay of a pion.Region A and C allows to evaluate the amount of events originating from random photoncoincidences that are part of region B.Suppose you measure B events in region B, A events in region A and C events in region C,and that all regions have the same width. The number of real pion decays fully detected inthe calorimeter is then N = B − 1

2(A + C) which is measured with a precision (∆N)2 =

B + 14(A+ C).

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process in the code which corresponds to two photons generated with randomdirection and energy.

- Adding experimental resolution: Each (high-energy) photon is creating a showerof (low-energy) photons in a cone surrounding its impact point. This meansthat it leaves energy in multiple blocks. Energy resolution is applied to the en-ergy deposited in each individual block. This step simulate the resolution of thecalorimeter by transforming the actual energy of the photon E and actual hitposition x and y to a measured energy Emeasured and measured hit positionxmeasured and ymeasured. Based on these measured quantities and using equa-tion 48, the mass of the hypothetical pion from which the two photons could havedecayed is reconstructed.

- Result aggregation: the resulting reconstructed pion mass is accumulated in anhistogram.

9.3.2 How to run the simulation

The code you will be starting with is available on the class web site. Search for ”Pionsimulation code”.To run the code you need to have access to a computer where ROOT is installed. Thereis such a computer in the lab as well as in the Physics department computer lab (lastrow of computer on the right-hand-side of the door).To run the code, open a terminal, Figure 26 shows how to open your root session, loadthe code and run it.

9.4 Lab objectives

For this laboratory, answer the following questions:

- Running the simulation multiple times, check that the precision on the numberof real pion fully detected follows the law described on figure 24. How does thatchange if you choose to work with a fixed seed. Explain.

- Change the rate of random ”pion” to range from that of the real pion to up to 5of that of real pion. Estimate the effect of the random background on the relativeprecision on the number of pions decays fully detected by your calorimeter?

- To optimize the resolution of the pion mass, is it better to change the calorimetermaterial to produce 10 times more photons per energy deposited in the calorimeteror to use a calorimeter with the same total size but with blocks of transversesize twice smaller? Note: If the calorimeter produces ten times more low energyphotons per unit of energy, the energy resolution is reduced by a factor of

√10 ∼ 3.

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Figure 25: Typical event simulated by the code used for this experiment. The top viewshows a π0 particle decaying into two photons. Both photons hit an EM calorimeter (shownside view on the top plot). The bottom plot shows the face of the EM calorimeter with thehit positions of the two photons as well as shadow of the EM shower they create within thecalorimeter. This last view shows that the energy of each decay photon is distributed inmultiple blocks.

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Figure 26: Root session example showing how to load and run the code you will work within this laboratory.

- What is the efficiency of the simulation? That is how many events are useful foryour simulation compared to the number of events you simulate. A low efficiencywill result in a long running time. Modify the code to make it run significantlyfaster.


The following questions should also help you to prepare for this lab:

- What are basic criteria listed in Bevington, that random number generator mustsatisfy?

- Based on Bevington, explain the concept of seed and shuffling for random gener-ator?

- Based on Bevington, what is the difference between the transformation methodand the rejection method for pseudo random generators?

- Lorentz transformations (or Boost) are coordinate transformations between twocoordinate frames that move at constant velocity relative to each other. The mostcommonly discussed is the time t and Cartesian coordinate x,y,z that specify aposition in space. List four other ”four vectors” that can be submitted to Lorentzboots. List their time like component and their space-like components. (Hint:Wikipedia is a good source of info on this).

- Download the code from the class website. Identify the subroutines and makea flow chart of the main routine of the code (void pion simulation()) . You do

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not need to describe the algorithms of the subroutines. Hand producing the flowchart is ok.

- Make a flow chart of algorithms used to create the shower (void shower(...)). Handproducing the flow chart is ok.

- Identify the lines in the program in which random generation is called. For eachcase, briefly explain how the randomly chosen value is used.

Reference

Large portions of this chapters are directly copied from the following resource:The Nature magazine web site: http://www.nature.com/subjects/numerical-simulations

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A Version history of this lab manual

- Version 16:4: Released November 8, 2016.REvision tot he simulation chapter, new APQs for interferometry, introduces theusage for hyperref,

- Version 16.3: Released October 25, 2016.Introduction of the chapter on Numerical simulation.

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