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University of California at Berkeley Physics 111 Laboratory OPT2003.4

Version 2003.4 Copyright © 2003 The Regents of the University California. All Rights Reserved of 20

OPTICAL PUMPINGRevision 2003.4

Copyright ©2003 The Regents of the University of California. All rights reserved.

Watch the Optical Pumping Video first then Watch the Error Analysis video

Student's Name

Partner's Name

Pre-lab Discussion QuestionsIt is your responsibility to discuss this lab with an instructor on the first day of yourscheduled laboratory period. This signed sheet must be included as the first page ofyour report. Without it you will lose 1/3 of a letter grade. You should think about andbe prepared to discuss at least the following questions before you come to lab:

1. What is optical pumping? Draw qualitative energy level diagrams for Rb85 and Rb87showing fine, hyperfine, and Zeeman splittings. Show the transitions between theselevels that are important to this experiment. For our rubidium system, what is thepumping process? Where is the pumped level? Where is the rf transition?2. Why do we modulate the external magnetic field? How would we take data if thefield were not modulated? Does the amplitude of modulation affect the resonancefrequency?3. What data will you take, and what plots will you make?

Staff Signature Date

Completed on the first day of lab? (circle) Yes / No

Mid-lab Questions

On day 2 of this lab, you should have successfully produced a plot of frequency versuscurrent for at least one rubidium isotope, and have made an estimate of the earth'smagnetic field. Show them to an instructor and ask for a signature.

Staff Signature Date

Completed on the second day of lab? (circle) Yes / No

INCLUDE THIS SHEET AS THE FIRST PAGE OF YOUR REPORT

UC Berkeley Physics 111 Advanced Undergraduate Laboratory OPT2003.4


Physics 111 Advanced Lab Student Evaluation of ExperimentNow that you have completed this experiment, we would appreciate your comments. Please take a few moments

to answer the questions below, and feel free to add any other comments. Since you have just finished the experimentit is your critique that will be the most helpful. Your thoughts and suggestions will help to change the lab andimprove the experiments.

Please be as specific as possible, using both sides of the paper as needed, and turn this in with your report.Thank you!

Experiment name:_________________________________________ Date: ________

How was the write-up for this experiment? How could it be improved?

How easily did you get started with the experiment? What sources of information were most/least helpful ingetting started? Were the reprints appropriate? Did the Pre-lab discussion help? Did you need to go outside thecourse materials for assistance? What additional materials could you have used?

What did you like and/or dislike about the experiment?

Would you recommend this lab to fellow student? Why or why not?

What advice would you give to a friend just starting this experiment?

If the course materials were available over the internet (WWW, FTP, etc), would you (a) have access to themand (b) would you prefer to use them this way?

Please circle the abbreviations of the other labs you have done.

ATM BBC BRA COM CO2 GMA HAL HOL JOS LIF

LLS MNO MUO NLD NMR OPT RUT SHE XRA

Overall quality of this experiment?1 2 3 4 5Poor Average Good



OPTICAL PUMPINGRevision 2003.4 August 2003

I. References

References 1. and 2. are the most informative, but Bloom in particular does not discuss pumping ofRubidium, but for simplicity describes pumping of a hypothetical atom.

1. †R.L. De Zafra, “Optical Pumping”, American J. of Physics. 28, 646 (1960). #QC1.A42. †A.L. Bloom, “Optical Pumping”, Scientific American, October 1960, p.72. #T1.S53. †N.F. Ramsey, Nuclear Moments, Wiley, 1953. #QC174.1.R314. C.M. Sitterly, Atomic Energy Levels Vol. II, National Bureau of Standards, 1952.

#QC453.M65. J.R. Taylor, An Introduction to Error Analysis, Oxford, 1982. #QA275.T386. L. Lyons, A Practical Guide to Data Analysis for Physical Science Students, Cambridge,

1991. #QC33.L9Check out the Reprints for this experiment;† Contained in the Optical Pumping Reprints, located in the Physics Library.

II. Introduction

Measuring the energy levels of atomic, molecular, nuclear, and particle systems is alarge part of experimental physics. The technique of optical pumping is used to measureatomic energy level differences with great precision. This experiment uses optical pumpingto measure the splitting of rubidium atomic energy levels, when the atom is placed in amagnetic field. It is so easy to make these measurements that you can use the opportunityto consolidate what you know and understand about atomic physics and quantummechanics. You can get a solid appreciation of physics and how elegant it is from thissimple experiment. It also gives you an idea of how grubby an actual laboratory set up iscompared to how slick the physics looks in a textbook.

Your goals for this lab are to use the method of optical pumping to find the resonancefrequencies of Rb85 and Rb87 for various values of a magnetic field, and to determine theenergy level splittings of these isotopes. You will then determine the nuclear spins of theisotopes, and finally determine the value of the earth's magnetic field at this experimentstation.

GETTING STARTED ON OPT DATA REDUCTION

You will have four sets of data and four straight lines when you plot them withthe variables of frequency and current. The individual points, the slopes of thecurves, and their axial intercepts have an interrelated significance. You have twoequations to work with. One is the Breit-Rabi equation, and the other is the field ofHelmholtz coils as a function of current. The B-field in the B-R equation is the sumor difference of the Helmholtz field and the earth’s field.

Write down the equations, rearrange them, see how the two isotopes fit in, seehow additions or subtractions can help, use both + and – currents.



Once you have determined the nuclear spins, the Breit-Rabi equation is asaccurate as the numerical constant 2.799, since the nuclear spins must be odd half-integers exactly. The Helmholtz coil field is not as accurate as the numericalconstant 0.900 because the radii of the copper wire turns are not all exactly thesame, and the separation of the coils is not exact. By using the Breit-Rabi equation,you can determine the B(i) constant more accurately. For example, draw a lineparallel to the frequency axis at a particular current, both plus and minus. Whatcan you learn from the intersections with the curves? Do the same with a lineparallel to the current or B axis.

Now fit the curves with straight lines; determine their slopes and their errors,and the intercepts and their errors. Check to see that their slopes are consistentwith the values of nuclear spins that you have chosen. From the interceptscalculate the earth’s field, and its errors. Ultimately you can have four values forthe earth’s field for each isotope. Check to see that all eight values fall within therange of your statistical errors.

Say something about which values you think are the most accurate, and why.

III. Preparation

Starting with the articles by Bloom and de Zafra, read through the Optical PumpingReprints. The reprints for this lab are all theoretical, and should be understood beforecoming to lab. Note, however, that not all of the diagrams or discussions are correct for ourexperiment: some of the articles discuss only transitions between hyperfine levels, whilewe have Zeeman splittings as well. Try to keep clear which splittings are which, andwhich are important for our transitions. See, for example, de Zafra p.647.

As you study, here are some terms to understand:

absorption line widthatomic energy levels LS couplingatomic orientation Maxwell-Boltzmann distributionbuffer gas magnetic dipole momentBreit-Rabi equation modulationdegeneracy nuclear spindischarge lamp Paschen Bach effect

quantum numberselectron configuration relaxationelectric dipole transition radiative lifetimeequilibrium distribution resonancefine structure selection rules

Helmholtz coil spontaneous emissionhyperfine structure stimulated emissioninterference filter spectroscopic notationLarmor frequency Zeeman effectlinear and circular polarizers

A. Draw qualitative energy level diagrams for Rb85 and Rb87 showing fine, hyperfine,and Zeeman splittings. Show the transitions between these levels that are important to



this experiment. Everyone in 111-Lab has had 137AB, and so will understand what youmean when you refer to fine, hyperfine, and Zeeman splittings. Do not draw an energylevel diagram of a hypothetical atom as described in "Scientific American."

B. Starting with the Hamiltonian

$ ( )H B B BI J Ext J Ext= − ⋅ + − ⋅r r r r rµ µ (eq. 1)

whererµI is the nuclear magnetic moment,

rµ J is the electronic magnetic moment,

rBJ is the magnetic field at the nucleus arising from the rest of the atom, and

rBExt is an

externally applied field, derive the Breit-Rabi law in the low-field case:

νB I

MHz gaussExt

=+

2 7992 1

./ (eq. 2)

(See Ramsey, for example)

C. Derive the expression for the magnetic field at the rubidium bulb due to theHelmholtz coils:

BNi

atesla= −0 9 10 6. (eq. 3)

where N is the number of turns of the coils, i is the current in amperes, and a is theradius of the coils in meters. Discuss the Helmholtz coil. Why is the field so uniform atthe center, both laterally and longitudinally? How inhomogeneous is the magnetic field atthe bulb? What are the qualitative and quantitative effects of this inhomogeneity? Arethey important in this experiment?

D. What are the effects of the earth's magnetic field?

III. Experimental Procedure

A. Look over the block diagram (Fig. 1) and check the connections of the equipmentcarefully. Make sure you understand what each unit does, and that you know theequipment limitations (e.g. don't run the current higher than 2 amps; don't heat the bulbover 55 C, etc.) Inspect and open carefully the Rb light box to see its construction. Can youexplain how the Rb lamp works? Be particularly careful of the D1 pass filter. It isexpensive to replace! All pieces in this unit are already in their proper places. The largebulb filled with Rb85 and Rb87 should be in the light box or at the other Optical pumpingexperiment.

B.1. Turn on all of the equipment, starting with the System switch (lower right handcorner of the Coil Driver Panel). Set the Rubidium Supply Output Current for the Rb lampto approximately 25 milliamperes using the Adjust Knob. Also make sure that the Type190A Attenuator is set to 5V. You may change this later, but do not use the 10V setting.



He

ate

r (110 VAC

)

HeaterControl Unit

Rb LampPower Supply

Rb Lamp Assembly

Temp Sensor

Circular Polarizer

Reflector

D1 Pass-filter

Attenuator

Freq Meter

RF GeneratorFrequencyControl

Rb85&

Rb87

Rb

RF Oscillator

AC Modulation Unit

DC PowerSupply

Photodiode Detector

"Phase" output

00

0

Y

X

+

Helmholtz Coils

Scope (X-Y Mode)

Current meter

Amp

Ι Ι

V

Current (I)Control

ModulationAmplitude Control 10

(shunt)

O verall E quipm ent L ayout

Figure 1: Equipment Block Diagram



2. PAR 113 PRE-AMP: When you turn on thePRE-AMP, test the battery by pushing the BATTEST switch to the + and – positions. The greenlight should come on both times. Make sure thatthe INPUT ‘A’ DC/GND/AC switch is in the ACposition, and that the INPUT ‘B’ switch is in theGND position. To start, set the Gain to 500, LFRoll-off to 0.1 Hz, and the HF Roll-off to 10 KHz.You may have to change some settings to makeyour signal clear. Also note that if you suddenly lose your signal, i.e. if your scope tracegoes to a flat-line, you should press down on the OL REC (OverLoad RECovery) switchlocated above the Output connector.

3. Heat the sample to 50 C. You may want to then turn the heater off to reduce noise,(noise due to what? ) but remember that you want the temperature between 42 and 50 forgood data. (Why?) Once the box reaches the temperature you set, the heater cycles on andoff to maintain this temperature. The deviation needle shows a deviation even when thesample is at the temperature set by the knob.

4. Starting at zero current, increase the current inincrements of 0.2 amps up to 2 amps, one amp (1.0A) is fineto start, in both the normal and reverse current directions.Use the shunt to measure the current. This shunt is acalibrated resistor and is temperature compensated, itdrops 50mV for 5 Amps of current passing through theshunt. The voltage that develops across the shunt is 10mVper amp of current through the coils. At each step ofcurrent, find the resonance conditions for both isotopes, andrecord the corresponding frequencies. Take care not to setthe modulation amplitude so large that you see resonancesof both isotopes simultaneously. Make sketches of what theresonance signal looks like with proper and improperadjustments of modulation amplitude and phaseCalculatethe magnetic field due to the Helmholtz coils and thecorresponding resonance frequency. You may assume I for

Rb85 = 5/2. Set the coil current using the Voltage and Current controls on the PowerSupply, and set the frequency using the Signal Generator (Fig. 3) and frequency counter(Fig. 4). The Signal Range knob on the Signal Generator must be set properly for thefrequency range you desire. Also set the Field Modulation Powerstat knob to around 10(relative scale).

C.1. Read the appendix "A Few Words About the X-Y Mode of the Oscilloscope" to findout about the PHASE knob of the Coil Driver and to get an idea of how to tell when youhave found resonance. Also consult the reprints; They have many useful figures to helpyou understand the relationship between the displayed signal and the conditions ofresonance.

Figure 2: PAR 113 Amplifier

Figure 3: 190B FrequencyGenerator



Figure 4: Frequency Counter(See Appendix C for details or the Manual fo the Frequency meter)

Search for resonance by slowly varying the frequency around the value that youcalculated. Alternately, you could set the frequency to some fixed value and then vary thecoil current. The oscilloscope pattern would look exactly the same. Why don’t we do this?

If you can’t find a resonance, get help. Once you have found the resonance condition,vary all of the parameters – current, voltage, temperature, phase, whatever variables areunder your control to get an idea of what the signal looks like under various conditions.How sensitive is the resonance is to changing variables? Record the qualitative behavior,and explain.

Taking data for this experiment is more straightforward than for any other lab in thiscourse. But the experiment deserves more time and thought than most because itillustrates fundamental ideas about quantum mechanics which you probably have onlyvague notions. Take the time to think about what's going on, and answer any questionsthat occur to you.

2. Starting at zero, increase the current in increments of 0.2 amps up to 2 amps, inboth the normal and reverse current directions. Use the shunt to measure the current.The voltage that develops across the shunt is 10mV per amp of current through the coils.At each step of current, find the resonance conditions for both isotopes, and record thecorresponding frequencies. Take care not to set the modulation amplitude so large thatyou see resonances of both isotopes simultaneously. Make sketches of what the resonancesignal looks like with proper and improper adjustments of modulation amplitude andphase.

3. Turn off the RF generator, and vary the current while looking for a resonance.This is a resonance that occurs for any frequency, including ZERO!

4. Set the oscilloscope for a linear internal sweep in the x-direction, and observe thesignal at resonance. From its shape and length, can you determine what the pumping time



is – the time required to pump up the level, starting from a Boltzmann distribution? Whatmeasurements are required to make this determination?

IV. Analysis See “Error Analysis Notes” in appendix B of this write-up.

A. Make plots of frequency vs current to help you analyze your data, and to illustrateanswers to the following results and questions.

B. From equation (2) we have a relation between ν1 and I1, and ν 2 and I2 where

the subscripts refer to the Rb85 and 87 isotopes. From your data determine the best ratioν ν1 2/ and from this deduce the values of I1 and I2. Their true values are exactly half-

integral.

C. Having now determined the values of I1 and I2, use equation (2) to determine thevalue of B at the bulb for one positive and one negative value of the current in the coil.Compare these values with those calculated from the coil dimensions and current. Whichvalues are more accurate? Why?

D. From your data, and using the results of part C, determine the magnitude of theearth's magnetic field in the direction of the axis of the Helmholtz coil by fittingstraight lines to your data and determining the uncertainties in slopes andintercepts using the theory and techniques of error analysis as illustrated in Lyons,Data Analysis for Physical Science Students, Section 2.9, page 63ff. In otherwords, find the slopes and intercepts and errors using the elements of data anderror analysis. Show clearly how you do this, with a sample calculation. If you useExcel you must show that you know what the program is doing. What is R? Whatinfluences its value? Don’t just use a data reduction code blindly – you don’t learnanything that way.

E. Explain why there is a resonance at zero frequency.

F. What do you need to know and to take account of, in order to make a roughestimate or calculation of the pumping time to compare with your experimentallymeasured value?

Appendix A

A few words about the X-Y mode of the oscilloscope.For most purposes, we wish to watch how voltage signals change with time, so we use

our oscilloscope in the "normal" mode with the vertical deflection input as a voltage and thehorizontal deflection swept out as time. However, in some cases we are really interested inthe relation between the two signals, and time is just a parametric variable relating them.The Optical Pumping set-up is just such a case.

Our two signals are the magnetic field 60 Hz modulation, and the amplified light-signal from the photodetector. (See Appendix D) The PHASE ADJUST control on CoilDriver panel changes the phase of the modulation signal seen by the scope relative to themodulation signal seen by the rubidium sample, and hence relative to the photodetectedlight-signal. Put the amplified photodetector signal into Ch. 1 (Y), and for now put themodulation signal into Ch. 2. With the scope in Dual Trace Mode, trigger on Line Source(60 Hz PG&E power line) and set the time scale to around 2 msec/DIV. You may have toadjust the Level knob to get a stable trace. Now change the phase with the Adjust knob:the modulation signal (Ch. 2) moves because we are changing its phase relative to the 60



Hz line signal that we are triggering on. The detected signal does not, though, because theAdjust knob does not change the phase of the signal that the sample sees. How can you tellyou have met the resonance condition when viewing the signals in this mode?

Now put the modulation signal through the divide-by-10 attenuator, into the EXT.TRIGGER input. Put the scope in the X-Y mode. You should now see the detected signal(Ch. 1) displayed on the y-axis versus the modulation signal (Ext Trigger) on the x-axis.This is called a "Lissajous figure."

Convince yourself that this is the correct picture by remembering what the two signalsversus time looked like and checking how the detected signal changes as the modulatedsignal changes. Can you determine how you can tell when you have found the resonancecondition in this mode? Should there be any symmetry?

It is up to you to determine which of the modes gives a more precise determination ofthe resonance condition–you might consider using both. Try them for repeatability.

Optical Signal vs Temperature (Rb

Temperature (° Celcius)

0

10

20

30

40

50

60

70

80

90

25 30 35 40 45 50 55

Taken while coolingbulb #75, 4/18/83 HAS

Note: Temperatures for Rb 87 may be different. Why?



Appendix BError Analysis Notes

Used in Optical Pumping.First run through one of the several good texts on error analysis, such as the one by

Lyons. (Louis Lyons, A Practical Guide to Data Analysis for Physical Science Students.#QC33.L9) This will give you a background on the statistical model on which we baseour statements, and define many of the terms used, such as average, true mean, estimatedmean for a sample, variance, standard deviation, random error, systematic errors, andothers. There is no substitute for this reading. To give all details here would require usto write a whole book. We are only going to skim the subject of statistical errors.

ERROR OF A SINGLE PARAMETER

Measurement statistics

As we read about laboratory results, we often see the value of a measured parameteras written something like 2.10 ± 0.05. The value 2.10 in this example is usually anaverage of several measurements, because successive measurements of any particularquantity are never exactly the same. Where does ± 0.05, called the standard deviation,come from, and what does it mean?

When we make a measurement and get a value for a single quantity or parameter, wewant to know how likely it is to be correct. So we measure it again and again, say Ntimes in all, and look at the scatter in the measurements. For example, we might havey = 2.10 as the average of the 5 measurements yi = 2.18, 2.03, 1.95, 2.24, 2.10,

y = Σyi = [2.18 + 2.03 + 1.95 + 2.24 + 2.10]/5 = 2.10N

We then calculate the standard deviation sigma, with the equation

σ = [Σ( y – yi )2] = 0.05

[N(N – 1)]

Loosely speaking, this means that if we make another set of N measurements, theaverage of this new set has a 67% likelihood of falling within a range from 2.05 to 2.15,or more compactly written as 2.10 ± 0.05

Counting statistics



In experiments like Muon Lifetime, we record the number of events or counts in agiven time interval, where the events are random, as in radioactive decay. Suppose weget N counts in a fixed time interval. If we take other samples for the same interval, weget some close but different values of N. After many such measurements, we canconstruct the standard deviation, as described above. We don’t need to do this, becausestatistical considerations specify that if the counts come at truly random times, then the

standard deviation is N. So, we can make a single record of counts and write N ± N

for the counts in a given interval of time. The larger the N, the larger the standarddeviation, but the fractional error grows smaller as the number of counts grows larger.The fractional

error is

1

N. For example, to get a fractional error of 1%, N must be 10 000!

CURVE FITTING WITH TWO PARAMETERS (LINEAR REGRESSION)

In many laboratory experiments we have several parameters, rather than just one, andwe want to fit a smooth analytical curve to the datum points and extract the relevantparameters from this curve. In cases where the curve is a straight line, we want to fit adata set to determine the slope and intercept of the line, each of which has some physicalsignificance.

We start with the equation y = mx + b = f(x). We assume that the independentvariable is x, that successive values of x are equally spaced, and that we record a datumpoint yi at each value xi. We then select a straight line and adjust its slope “m” andintercept “b” to get a “best fit” to the experimental points yi. By definition we get a bestfit when the sum of the squares of the deviations of the fitted curve from the datum pointsat each xi is a minimum. In mathematical terms, adjust m and b so as to get a minimum ofthe sum:

Σ [yi – f(xi)]2 = Σ [yi – (mxi + b)]2

We do this by taking the derivatives of the sum with respect to m and b, setting theequations equal to zero, and solving the simultaneous equations for the “best” values of mand b. This is called a “least-square fit”. These values will have error bars – standarddeviations – that can be calculated with techniques described in the texts. Lyons gives aworked-out example.

APPLICATION OF FITTING A STRAIGHT LINE

Let’s see how these ideas can be used in the Optical Pumping experiment. We havetwo isotopes of rubidium as gas in a bulb placed in a magnetic field and subject toelectromagnetic radiation. The earth’s magnetic field also affects the rubidium energylevels and their populations. We set the current in the coil producing the magnetic field,and adjust the radio frequency to produce a resonance condition. The parameters that we



either want to change, vary, or to measure are the current, the field produced by thecurrent, and the frequency of the applied RF radiation at resonance. The relevantequations are:

BH = 0.9 Ni/a

BH, magnetic field from the Helmholtz coilsi, current in the coils

a, radius of the coilsThe value 0.9 is approximate because the radius a is not exactly the same for each of

the N windings and we have no simpler way of incorporating this fact.

ν/B = 2.799/(2I + 1)

ν, frequency of applied em radiationI, nuclear spin of RbB = BH + Be, the total magnetic field from the Helmholtz coil and the earth

This equation is called the “Breit-Rabi” [bright-robby] equation.

The purposes of the experiment are1. to determine the nuclear spins of the two isotopes 85 and 87. It is known

that both spins are odd half-integral values, like 1/2, 3/2, 5/2, etc., and2. to measure Be, the earth’s magnetic field

Nuclear spins

How and what data should you take? To determine the nuclear spins, rearrange theequation to get an equation that looks like y = mx + b. Then set a value of the current i,adjust the RF to the resonance value, reset i, etc., until you have a table of pairs of values.In fact, you will end up with four tables, because you should take data for both positiveand negative currents for each of two isotopes. A plot of resonance frequency vs. currentwill give a straight line with a slope dependent on 2I + 1, and an intercept dependent onlyon the earth’s field. [Sketch 4 straight lines]

Another way of determining the spins is to set the current and take the ratio of theresonance frequencies for the two isotopes, at the same current.

In both methods there will be some errors, but it does not matter because we knowthat the spins have exact half-integral values. This is a case in which the use of erroranalysis is unwarranted because the results are unambiguous.

Earth’s Magnetic Field



To determine the value of the earth’s magnetic field, we need to find a way todetermine BH exactly, or to eliminate it from the equations. BH can be eliminated bysetting i = 0 and using the frequency intercepts and the Breit-Rabi equation. As alreadymentioned, the equation for BH and a function of i is not exact.

There is yet another way to determine the earth’s field. There is a reversing switch onthe coil current, to reverse the direction of the field. For the same absolute value of thecurrent, the field must have the same strength, except for possible hysteresis effects.Check for hysteresis by measuring the resonant frequency at a particular current, when thevalue of current is reached from above (run the current to a max, and then come down tothe desired value), and from below (run the current to zero, and then come up to thedesired value).

So, at a particular current, measure the resonance frequencies for both positive andnegative values of current. Add and subtract the two equations, and get exactexpressions for Be and for the more exact parameter for the relation between current andBH.

Errors

How do we treat the errors in the value of the earth’s field? Step back a little, and seewhat data you have and how you can arrange it to make error computation the simplest.Here are several approaches.

1. Look at plots of your data. You should have 4 lines, two for each isotope, onewith positive current and one with negative current. You can do a least-squarefit of each line, calculate the position of the zero-current intercepts, and obtainvalues for the field. Then you can calculate “errors of adjusted coefficients”using the methods given in Lyons and other references.

2. But, by changing negative current points to negative frequencies as well, eachisotope has only one plot line, and presumably the zero crossing is moreaccurate. Then there are only two lines to fit, one for each isotope, and theerrors calculated as described above for adjusted coefficients.

3. A still easier method is to use pairs of points with the same absolute values ofcurrent, and add the two resulting equations. We then have values of theearth’s field without adjusting any coefficients, and the errors can be calculatedas an error of a single parameter, rather than as an error of an adjustedcoefficient. We have

ν+ = [2.799] (BH + Be)[2I+1]

ν– = [2.799] (–BH + Be)[2I+1]

Be = (1/2)(ν+ + ν–)(2I + 1)/2.799



The standard deviation is calculated as described above in ERROR OF A SINGLEPARAMETER Measurement Statistics. Compute the average; compute the differences,square them, add, divide by N(N - 1) and take the square root. Then step back, look at theresults, and see if they look reasonable. If not, you’ve goofed somewhere, and must tryagain.

FITTING AN EXPONENTIAL

Experiments like the Muon Lifetime have data that plot as an exponential. We wishto extract the decay constant (mean lifetime) from an exponential curve. It is no easy taskto fit an exponential curve to a data plot. In most cases we transform the exponential intoa straight line, and then do a least-square fit. The slope is the decay time. Errors areexaggerated at larger values of time, and getting a useful error estimate is complicated.Most texts do not discuss the problems, nor will we. [Draw an exponential and a straightline; note how errors get larger both because N gets smaller, and the log amplifies theerrors on the plot]



Appendix C

Frequency Meter



Appendix D

optical pumping - michigan state university · curves, and their axial intercepts have an...

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