physics 352: lab manual for experimental modern...
TRANSCRIPT
Physics 352: Lab Manual forExperimental Modern Physics
W. J. Kossler, C.F. Perdrisat
January 13, 2010
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Contents
1 Optical Pumping 9
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Setting up the Optical Pumping Apparatus . . . . . . . . . . . . . . . . . . . 12
1.3 Cross-section Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Absorption of Rb resonance radiation by atomic Rb . . . . . . . . . . 12
1.4 Experiments on Low Field Resonances . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Measurement of the Nuclear Spins . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Low Field Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Sweep Field Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4 Main Field Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.5 Sample Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.6 Low field Zeeman effect: . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Higher Field Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.1 Quadratic Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.2 Sample Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Experiments on Transient Effects . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6.1 Transient Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Nuclear Magnetic Resonance, NMR 31
2.1 Introduction to NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 T1 and T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 T1, the Spin-lattice relaxation time . . . . . . . . . . . . . . . . . . . . 34
2.2.2 T2, the Spin-spin relaxation time . . . . . . . . . . . . . . . . . . . . . 34
2.2.3 T2* and magnetic field inhomogeneity . . . . . . . . . . . . . . . . . 34
2.2.4 The reason that T1 is slower than T2 . . . . . . . . . . . . . . . . . . . 35
2.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.3 T1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.4 T2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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3 The Mossbauer Effect 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Energy Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 Isomer Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Quadrupole Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.3 Zeeman Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.2 Zeeman Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.3 Quadrupole Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.4 Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.5.5 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Useful data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Alpha Particle Scattering 474.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Energy Loss by Charged Paticles . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 Charged Particle Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5.1 Energy Loss Measurements . . . . . . . . . . . . . . . . . . . . . . . . 494.5.2 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 γ-ray Studies 515.1 Objectives of the Experiment: . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.1 Photoelectric Cross-section . . . . . . . . . . . . . . . . . . . . . . . . 525.3 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 Procedure: First Week . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.5 Measurements: Second Week . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.6 Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.6.1 Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.6.2 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.6.3 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.6.4 Total Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.7 Differential Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.7.1 The Thomson Cross-section . . . . . . . . . . . . . . . . . . . . . . . . 605.7.2 Heuristic Derivation of Pr . . . . . . . . . . . . . . . . . . . . . . . . . 615.7.3 Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.8 NaI Detector and Photo-multiplier Tube . . . . . . . . . . . . . . . . . . . . . 625.8.1 Classification of Radionuclear Decays . . . . . . . . . . . . . . . . . . 62
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Thursdays, 2-5 Rm 303 of Millington Hall
Instructors
C. F. Perdrisat Rm 302 221-3572 [email protected]
Travis Horrom [email protected]
You will carry out 4 labs over the course of the term; you can choose them from a listof 5 labs. Each lab will take three weeks, the first two of which will be primarily datataking and the third will be analysis and writeup. Each lab will be carried out by a teamof between 1 and 3 students.
The Labs
1. γ-ray and X-ray Measurements (γ − r)
2. α Scattering and Energy Loss (α − s)
3. Optical Pumping (OPu)
4. Nuclear Magnetic Resonance(NMR)
5. Mossbauer Effect.(ME)
Schedule
Jan. 21 Organization28 γ − r α − s OPu NMR ME
Feb 4 γ − r α − s OPu NMR ME11 γ − r α − s OPu NMR ME. 18 α − s OPu NMR ME γ − r First Lab due25 α − s OPu NMR ME γ − r
Mar. 4 α − s OPu NMR ME γ − rMar 11 Spring Break
18 OPu NMR ME γ − r α − s Second Lab due25 OPu NMR ME γ − r α − s
Apr. 1 OPu NMR ME γ − r α − sApril 8 NMR ME γ − r α − s OPu Third Lab due
15 NMR ME γ − r α − s OPu22 NMR ME γ − r α − s OPu Fourth Lab Due
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Lab reports are due the week following the end of lab work. You will be expectedto do four out of the five labs; your choice!. The lab reports will use the format ofreport template.tex. To handle Latex files you have accounts on the Camelot Linux cluster.You will be able to use this cluster from your windows boxes if you use the availablesoftware throughW&M IT (Xwindows client and F-secure ssh client.) Youmay also workdirectly in Room 151. If you have a MAC running OS X, there is an available Xwindowsclient. It works quite well. The default printer is in Room 151. If you want to print locallyto your home computer you will need to print to a file and then transfer the file. Thephysics senior projects also use Latex and an appropriate template, so the use of Latexhere should be useful later. Further, it is quite useful in Physics Laboratories around theworld. If you have any trouble please see me, I’ll be happy to help.
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Grading Policy
The schedule will be flexible so that if you cannot be present for one week it will bepossible to make up that time. Beyond one week special arrangements will have to bemade. To repeat: Lab reports are due a week after the end of labwork.Approximate Grade Ranges:A 9-10B 8-9C 7-8D 6-7F < 6Grading is fairly generous in this course, but only so if you submit thewriteups on time.
The lab report is due on the Thursday one week after the last day of that lab. Writeupshanded in after that will have 1.5 points deducted. Writeups will not be accepted if theyare more than one week late.
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Chapter 1
Optical Pumping
1.1 Introduction.
This optical pumping lab might be considered a ”Course” in Atomic Physics with thetopics:
• Optical Pumping of Rubidium Atoms, 85Rb and 87Rb
• Explore Magnetic Hyperfine Interactions of Rubidium
• Observe Zero-Field Transitions
• Confirm Breit-Rabi Equation
• Observe Double Quantum Transitions
• Study Rabi Oscillations
• Measure Optical Pumping Times
• Study Temperature Dependence of Atomic Parameters
You should do as many of the experiments as time allows.In optical pumping circularly polarized photons are generated, sent through an ab-
sorbing gas, and then to a detector. See Fig. 1.1Right circular polarized photons carry one unit of angular momentum. To understand
the nature of optical pumping suppose the absorbing atoms can be represented by thesimple spin system shown in Fig. 1.2. Absorbed photons will take the absorbing atomfrom the M=-1, ground state to the excited state, which then decays equally to all three Mstates of the ground state. Since there is no path to take theM=0, andM=+1 ground statesback to the M=-1 stater, the state which becomes depleted. The depletion of the M=-1state as a function of time is shown in Fig. 1.3. Since absorption can only take place forthe M=-1 state, the gas of absorbing atoms becomes more transparent and allows more
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Figure 1.1: Apparatus arrangement for optical pumping.
F=1
F=0
M −1 0 1
Figure 1.2: A simple spin system for which the ground state has total angular momentum1, and the excited state has total spin 0. The excitation by right circular light is shown asare the three decay modes of the excited state.
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0 2 4 6 8 10t(decay/excitation times)
0
0.2
0.4
0.6
0.8
1
Pro
babi
lity
F=1, M=-1 Ground State
F=0, M=0
Figure 1.3: The time dependence of the populations of the ground state and excited state.It is assumed that the rate of excitation from the ground state equals the rate of decay fromthe excited state.
Isotope Nat.Abund.(%) I µ/µN J F(lower) F(higher)85Rb 72.17 5/2 1.35303 1/2 2 387Rb 27.83 3/2 2.75124 1/2 1 3
Table 1.1: Properties of the Rb isotopes.
light to reach the detector; this increase of light transmission is the signature of opticalpumping.
The actual system that you will be studying is Rb. Some of the characteristics of Rbare in Table 1.1. While this is clearly more complicated than the simple case of Fig. 1.2,for right circular light the result is nearly as simple. The maximum M state for bothisotopes will become much more heavily populated. See, e.g., Franzen and Emslie[?] andthe discussion in the TeachSpin Manual.
To determine that the system has been optically pumped one needs to find a meansof de-pumping the maximumM state. Upon being de-pumped the gas will absorb morereadily and less light will be detected. One means of de-pumping is to arrange that themagnetic fieldwhich is applied goes to zero. ThenM is no longer a good quantumnumberand collisions can easily change the orientation of the atom. A second means to de-pumpis to induce by an RF magnetic field, transitions from the maximumM state.
Note: If you quantize along the photon’s propagation axis, but apply a magneticfield opposite to that direction, then the energies of the M levels will be inverted. Or,
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conversely, if you quantize along the positive field direction, then the angular momentumof the photon is reversed and the maximally populated state will have the most negativevalue.
1.2 Setting up the Optical Pumping Apparatus
To connect the electronics of the device, first plug the Lamp power into the back panelconnector. Then plug the blue Thermocouple into lower front panel and blue heaterbanana plugs. Next, plug Black plastic Pre-amp power and Detector BNC into lowerfront panel. Plug in Vertical Field banana plugs into lower front panel. Finally plug inthe Horizontal Sweep Field banana plugs into the lower front panel. Now, turn on thepower switch on the back panel entry module. The temperature regulator will displaythe current cell temperature. Check that the set point of the regulator is 50C. Push thescroll key twice. SP will be displayed for 1.5 seconds and then the value of the set pointwill be displayed. If the value is not 50C, push the up or down keys till it is. Pushthe scroll key again twice. PROC will be displayed for 1.5 seconds and then the currenttemperature. After these settings, the lamp will take a few minutes to warm up. Usually,it takes 10-20minutes for the cell oven to stabilize. To obtain maximum optical alignment,the room lights are recommended to be off for the alignment. Set the preamp gain for10MΩ. On the detector amplifier set the gain = 1, gain mult. = xl, time constant = 100ms,meter multiplier = xl, and DC Offset = 0. Use a card to block the lamp and make sure thissignal is from the lamp and not the room lights. If the signal is off scale. change the metermultiplier to x2. The Pre-amp gain will need to be changed to 3MΩ if the signal is still offscale.
1.3 Cross-section Experiments
1.3.1 Absorption of Rb resonance radiation by atomic Rb
In this first experiment you will make an approximate measurement of the cross-sectionfor the absorption of rubidium resonance radiation by atomic rubidium. The measuredvalue will then be compared with the geometric cross-section and the value calculatedfrom theory.The apparatus shouldbe arrangedas shown inFigure 1.3.1. The linearpolarizer and the
quarter wave plate should be removed since they will not be needed for this experiment.The cell heater should be off, and the apparatus allowed to come to equilibrium. Itmay benecessary to insert a neutral density filter before the absorption cell to prevent saturationof the detector amplifier.Set the cell heater to 300 K, and allow thermal equilibrium to be established. It will
take about 30 minutes for the temperature to become stable. Measure the intensity of theoptical signal taking care to record all amplifier gain settings. Repeat the measurement in
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Table 1.2: Density of rubidium atoms over solid or liquid rubidium as a function oftemperature
Temperature, K Density, atoms/cubic meter290 3.3 X 1015
300 1.1 X 1016
310 2.9 X 1016
320 7.5 X 1016
330 1.8 X 1017
340 4.3 X 1017
350 8.3 X 1017
360 1.5 X 1018
370 3.7 X 1018
380 6.3 X 1018
390 1.2 X 1019
400 2.4 X 1019
temperature increments of 10 K, taking care that thermal equilibrium is reached betweenreadings. Repeat the series of measurements a few times, increasing and decreasing thetemperatureDetermine the density of atomic rubidium in the cell as a function of temperature from
Table 1.2 and fit the data to an equation of the form
I = ae−bρ (1.1)
where ρ is the density of atomic rubidium in the cell. From the value of b determinethe cross-section for the absorption of rubidium resonance radiation by atomic rubidium.Compare your result with the calculated value of the cross-section and with the geo-
metrical cross-section.It can be seen from the plot that above a density of about 200 X 1016 there is no further
decrease in the intensity of the transmitted light. Ideally the cell should be optically thick,and no light should be transmitted. The light that is transmitted does not fall within theabsorption profile of the rubidium in the cell, and hence gets through the cell and causesthis background.This radiation comes from the wings of the emission line and from the buffer gas in
the discharge lamp. In order to correct for this a constant detector output voltage must besubtracted from all readings, and the plot and fit will be limited to the first seven points.The result is shown in Figure 1.5.
I = 1.36e−0.040ρ (1.2)
The length of the absorption path was about 2.5 cm giving a result
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Table 1.3: Sample Data. Output voltage versus temperature.
Temperature, K Detector Output, Volts300 1.57310 1.31320 1.06330 0.72340 0.52350 0.24360 0.17370 0.14380 0.13390 0.12400 0.12
Figure 1.4: Plot of Sample Data
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Figure 1.5: Plot of Sample Data with background correction. You may want to use asemi-logarithmic display instead
0.025σ × 1016 = 0.040 (1.3)
and
σ = 1.6 × 10−16m2 (1.4)
This can be compared with the result calculated from the equations in section 2C of theTeachSpin Optical Pumping Manual, using a Doppler width at 350 K of about 550 MHz,and a center frequency of about 3.77 X 1014Hz. This corresponds to a center wavelength of795 X 10-9 m. The resulting maximum cross-section is σ0 = 15 × 10−16m2. A more detailedcalculation of the cross-section is in the literature [?], and a value of about 10 X 10-16 m2 isgiven there. The geometrical cross-section is about (10-10)2 = 10-20 m2. Notice that theresonant cross-section is much larger than that normally associated with atomic scatteringprocesses. As a point of interest the value of the absorption cross-section for sodiumresonance radiation in atomic sodium is 12 X 10-16 m2 [?].
Care needs to be taken in the interpretation of these results, since the cross-sectionsinvolved are somewhat ambiguous. The cross-section is a function of the frequencydistribution in the absorption profile of the rubidium atom, and the intensity of theabsorbed light will depend on the relationship of the intensity profile of the incident lightto the absorption profile.
Therefore the measured result should be considered to be only approximate. Theseconsiderations are discussed in detail in the literature [?]. Themain point here is to realize
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that the cross-section for absorption of resonance radiation by an atom ismuch larger thanwhat is usually taken as a measure of the geometrical cross-section.
The measured cross-section is about 10 times smaller than that calculated from theory.However this is not unreasonable considering the sources of error in the experiment. Oneof the largest of these is the rapid variation of the density of rubidium atoms in the cellas a function of temperature. This dependence, as shown in the Table 1.2, was calculatedfrom graphical data contained in the book by Margrave[?], and is subject to considerablesystematic uncertainty.
1.4 Experiments on Low Field Resonances
In all of the following experiments of this lab it will be necessary to apply aweakmagneticfield along the optical axis of the apparatus. In order to do this satisfactorily, the apparatusmust be located where the local residual magnetic field is as uniform as possible. Theproposed location should be surveyed with a compass to check for gross inhomogeneityin the local field, and the orientation of the horizontal component of the residual fieldshould also be determined. All iron or steel objects should be removed from the vicinityof the apparatus. The instrument should be placed on a table made with no magneticmaterial, such as the one supplied for this experiment by TeachSpin.
The optical axis of the apparatus should be oriented such that the horizontal componentof the residual field is along this axis. The apparatus should be set up as shown in Figure1.1, and the interference filter reinstalled. Be sure that the linear polarizer is ahead ofthe quarter wave plate in order to obtain circularly polarized light, and that the two areoriented properly.
In order to observe the zero-field transition, no RF is applied,and the magnetic fieldis swept slowly around zero. This is accomplished by varying the current in the sweepwindings. The current through the main horizontal field coils should be set to zero.Adjust the current in the vertical compensating coils to achieve minimum width of thezero field transition. Also check the orientation of the apparatus along the horizontalcomponent of the residual field by rotating the apparatus about the vertical axis andsetting for minimum line width.
Set the cell temperature to 320 K and allow thermal equilibrium to be established. Itis most convenient if the output of the optical detector is observed on the vertical axis of astorage oscilloscope, and a signal proportional to the current in the horizontal axis sweepcoils is displayed on the horizontal axis. As will be shown later, optical pumping is a slowprocess, and during these experiments it will be necessary to use a very slow sweep ratefor the magnetic field current.
Figure 1.6 shows the zero field resonance and the Zeeman resonances at a frequencyof 0.0134 MHZ.
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Figure 1.6: Zeeman resonances and zero field resonance at very low magnetic fields.
1.4.1 Measurement of the Nuclear Spins
There are two isotopes of rubidium, and they have different nuclear spins. We are goingto pretend that we don’t know their values, so we can measure them. In order to dothis we must measure the gF values from which the spins can be calculated. This can bedone by measuring a single resonant frequency of each isotope at a known value of themagnetic field. The magnetic field will be determined approximately from the geometryof the field coils. Since nuclear spins are either integral or half-integral we need only anapproximate value of the field.We will use only the sweep field coils for this purpose, and their parameters are in
Table 1.4:
Table 1.4: Field from the Coils
Mean radius = 0.1639 m
B(gauss) = 8.991 × 10−3IN/R11 turns on each side
where I is the current in amps, N is the number of turns on each side, and R is the meanradius of the coils. The coils satisfy the Helmholtz condition. At the sweep monitorterminals on the front panel, a voltage is presented that is numerically equal to the currentin amps (the current passes through a one ohm resistor). Use this voltage as a measureof the sweep coil current.
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First the residual magnetic field at the location of the absorption cell must be deter-mined. Disconnect the main field coils so that there can be no current through them.Adjust the current in the sweep coils to center on the zero field resonance, andmeasure thecurrent. From this and formulas of Table 1.4 calculate the value of the residual magneticfield. Be sure that there is no RF being applied.An RF signal can now be applied to the RF coils, and its amplitude set to an arbitrary
value. Later this amplitude will be adjusted for optimum transition probability. Thefrequency of the RF should be set to about 150 KHz. Sweep the horizontal magnetic fieldslowly increasing from zero, and search for the Zeeman resonances. Measure the currentat which each resonance occurs.An oscilloscope should be used to measure a signal proportional to the RF current at
the connector on the cell holder. This signal is developed across a 50 ohm resistor thatis in series with the RF coils, and therefore it is proportional to the amplitude of the RFmagnetic field.Measure the characteristics of the RF transitions as a function of the amplitude of the RF
magnetic field, and determine the value that provides optimum transition probability[?].The remaining data in this section should be taken using that value of RF magnetic
field.
1.4.2 Low Field Zeeman Effect
With the main coils still disconnected, measure the transition frequencies of each isotopeas a function of sweep coil current, and plot the results to determine that the resonancesare indeed linear in the magnetic field. From the slope of the plots determine the ratio ofthe gF-factors, and compare the measured ratio with that predicted by theory.
1.4.3 Sweep Field Calibration
For the remainder of the experiment it will be necessary to have a more precise value ofthe magnetic field than can be obtained from the geometry of the coils. In this sectionwe will calibrate the sweep coils using the known gyromagnetic ratio values, gF = µ/Iµ0,where µ0 is the Bohr magneton, and the previous measurements.From the previous measurements calculate the value of the magnetic field for each
isotope from the resonance equation, and plot the magnetic field vs the current in thesweep coils. Fit the data to a straight line using a linear regression to obtain an equationfor the magnetic field vs current.It will now be necessary to make a calibration of the main field coils.
1.4.4 Main Field Calibration
Connect up the main coils so that their field is in the same direction as that of the sweepcoils. The current control for the main coils is too coarse to allow the resonances to be
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centered well using it alone. It will be necessary to use both the main coils and the sweepcoils for this calibration. The voltage presented by the main coil monitor on the frontpanel (which is developed across a 0.5 ohm resistor) is one half of the main coil current inamps. Use this voltage as a measure of the main coil current.Use both sets of coils to make measurements at resonance frequencies up to about 1
MHz, and use the sweep coil calibration to correct the measured fields for the residualfield. Plot the data on a linear plot, and use a linear regression to obtain the best fit.
1.4.5 Sample Data
Residual magnetic field
The zero field resonancewas determined to be at a sweep field current of 0.323 amp. Fromthis and the above coil parameters the residual field is 0.188 Gauss. Since the rest of theexperiment will be done with the magnetic field oriented opposite to the residual field,the above number must be subtracted from the values calculated from Equation 4B-1.
Nuclear spins
At an RF frequency of 150 KHz the measured currents for the two isotopes were 0.836 and0.662 amp corresponding to magnetic field values of 0.504 and 0.400 Gauss. From eachof these values a residual field of 0.188 Gauss must be subtracted yielding 0.316 and 0.212Gauss.The resonant frequencies are determined from
ν = gFµoB/h (1.5)
resulting in gF values of 0.34 and 0.51. From Equation 2B-4 the corresponding nuclearspins are I = 5/2 and I = 3/2 with theoretical gF values of 1/3 and 1/2 respectively.
1.4.6 Low field Zeeman effect:
The slopes of the two plots are in the ratio of 0.430/0.287which gives a value of 1.498. Thetheoretical ratio is 1.5.
1.5 Higher Field Experiments
1.5.1 Quadratic Zeeman effect
The RF resonances of both isotopes will now be studied as the applied magnetic field isincreased into a region where the energy level splitting is no longer linear in B. Each ofthe zero field energy levels splits into 2F + 1 sublevels, whose spacing is no longer equal.In this region there are 2F resonances whose splittings can be resolved . Thus for I = 3/2
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Figure 1.7: Low field Zeeman Effect.
Figure 1.8: Sweep field calibration.s
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Freq.MHz
Total field,Gauss
Sweepcurrent,amp
Main cur-rent, amp
B fromsweepcoils,Gauss
B frommain coils,Gauss
Isotope
0.2000 0.2858 0.321 0.0322 0.0047 0.2811 Rb87
0.2000 0.4287 0.316 0.0492 0.0017 0.4270 Rb85
0.3003 0.4291 0.306 0.0500 -0.0045 0.4336 Rb87
0.3003 0.6437 0.313 0.0740 -0.0002 0.6439 Rb85
0.4002 0.5719 0.197 0.0740 -0.0716 0.6435 Rb87
0.4002 0.8578 0.662 0.0740 0.2148 0.6430 Rb85
0.5002 0.7148 0.205 0.0900 -0.0667 0.7815 Rb87
0.5002 1.0722 0.785 0.0900 0.2906 0.7816 Rb85
1.0001 1.4291 0.121 0.1786 -0.1185 1.6482 Rb87
Figure 1.9: Sweep field calibration.
21
Table 1.5: Rb87: Front Panel settings:Output gain = 20 X 10
ν = 4.9874 MHzRC = 100 msec
RF amp gain = 3on dialSweep time = 100 secs
Main field current = 0.820 ampMain field = 7.117 Gauss
there are a total of six resonances with ∆F = 0 and ∆M = ± 1, and for I = 5/2 a total of ten.These can all be observed. Their relative intensities depend on the pumping conditions.The magnetic field at which these resonances can be observed can be approximately
determined from the resonance equation,
ν = gFµ0B/h, (1.6)
and the current for the main field coils set from the previous calibration.The energy levels and hence the more exact frequencies can be determined from the
Breit-Rabi equation: Eq:1.7
W(F,M) = − ∆W
2(2I + 1)−µIIBM ± ∆W
2
[
1 +4M
2I + 1x + x2
]1/2
(1.7)
where x = (g j − gi)µ0B/∆W, and gi = − µiIµ0. W is the interaction energy and ∆W is the
hyperfine splitting[?].Start with the main field current at zero, and set the sweep current to the center of the
zero field transition. Then set the main field current to the desired value, and use thesweep field to observe the resonances. For a given frequency, measure the sweep fieldcurrent corresponding to each resonance, and calculate the total magnetic field. If thefirst frequency that you try does not yield resolved resonances go to a higher frequency.
1.5.2 Sample Data
The observed spectrum is shown in Figure 1.10 and the calculated spectrum from theBreit-Rabi equation is shown in Figure 1.11 for Ru87.The absorption intensities in Figure 1.11 have been adjusted to match the observed
spectrum. The Breit-Rabi equation can not be directly solved for x and hence B, but it canbe easily solved by a computer program such as Maple or Mathematica. The results inFigure 1.11 were obtained using Maple 5.The resonances occur at fields shown in the following table:
There is a systematic difference of 0.009 Gauss or about 0.14% between the calculatedand measured total field values.
22
Figure 1.10: Observed spectrum of Rb87 at optimum RF power.
Figure 1.11: Calculated spectrum of Rb87.
23
Table 1.6: Resonances
Sweep Field Current Sweep Field Total Field from Total Field from BR(amp) (Gauss) calibration (Gauss) eqn. (Gauss)0.292 -0.013 7.104 7.0950.310 -0.002 7.115 7.1060.321 0.004 7.121 7.1130.339 0.016 7.133 7.1240.355 0.025 7.142 7.1340.373 0.036 7.153 7.145
Table 1.7: Rb85: Front Panel settings:gain = 20 X 10
ν = 3.3391 MHzRC = 100 msec
RF amp gain = 3 on dialSweep time = 100 secs
Main field current = 0.820 ampMain field = 7.117 Gauss
The Rb87 spectrum taken under the same conditions as above, except at higher RFpower, is shown in Figure 1.12. The double quantum transitions, which occur midwaybetween the single quantum transitions, are shown. Notice that the single quantumtransitions have become broader because they are being over driven by the higher RFpower.The resonances occur at fields shown in the following table:There is a systematic difference of 0,005 Gauss or about 0.07% between the calculated
and measured total field values.The Rb85 spectrum taken under the same conditions as above except at higher RF
power is shown in Figure 1.15. Here again, the double quantum transitions, which occurmidway between the single quantum transitions, are shown. As with Ru87, the singlequantum transitions have become broader because they are being over driven by thehigher RF power.
1.6 Experiments on Transient Effects
In order to observe transient effects it is necessary to either turn the pumping light offand on rapidly or turn the RF on and off while tuned to the center of a resonance. Herewe will do the latter while tuned to the center of a low field resonance, and observe the
24
Figure 1.12: Observed spectrum of Rb87 at higher RF power showing double quantumtransitions.
Figure 1.13: Observed spectrum of Rb85 at optimum RF power.
25
Figure 1.14: Calculated spectrum of Rb85.
Table 1.8: Resonances
Sweep Field Cur-rent, amp
Sweep Field, Gauss Total Field fromcalibration,Gauss
Total Field from BReqn., Gauss
0.318 amp 0.003 7.120 7.1150.344 0.019 7.136 7.1300.369 0.034 7.151 7.1460.395 0.050 7.167 7.1620.421 0.066 7.183 7.1780.446 0.081 7.198 7.193
26
Figure 1.15: Observed spectrum of Rb87 at higher RF power showing double quantumtransitions.
transmitted light intensity as a function of time.
1.6.1 Transient Phenomena
A square wave pulse of about 0 to +5 volts amplitude is connected to the RF modulationinput on the front panel, and the frequency of the square wave set to about 5 Hz. Thefalling edge of the square wave should be used to trigger the sweep of a storage scope,and the output of the detector monitored. The following data was taken at a resonancefrequency of 0.3 MHz.
The RF amplitude was taken as the voltage across the 50 ohm resistor in series withthe RF coil. A typical result is shown in Figure 1.16. The upper trace shows the waveformthat is gating the RF, and the lower shows the resulting optical signal.
When the RF is on all of the Zeeman levels are mixed, no optical pumping takes place,and the transmitted light intensity is a minimum. Turning off the RF allows pumping tobegin, and the light intensity increases exponentially until a maximum value is reached.The time constant of this exponential is a measure of the optical pumping time. Thecharacteristic value of the time will be found to be proportional to the intensity of thepumping light.
When the RF is turned on transitions will occur between the Zeeman sublevels andthe population of the levels will be driven toward equilibrium. If the rise time of the RFenvelope is short enough the populations will overshoot giving rise to the ringing shown
27
Figure 1.16: Time dependence of the transmitted light intensity vs. RF amplitude.
in Figure 1.16. The ringing damps out, and the light intensity approaches that for theun-pumped cell.Figure 1.17 shows an expanded region of Figure 1.16 in the region of where the RF is
turned on. It can be seen that the ringing is damped out followed by a longer dampingtime before the light returns to the un-pumped value.Further expansion of the region around the RF turn on time yields a result shown in
Figure 1.18. Here the ringing can clearly be seen, and its period measured. According tothe earlier discussion this period should be linearly proportional to the reciprocal of theamplitude of the RF, since it corresponds to a precession of F about the RF magnetic field.Figure 4D-4 shows this to be the case for both isotopes where the fit has been done byregression analysis in SigmaPlot.At a given value of the RF magnetic field the ratio of the periods of the ringing goes
inversely as the gF factors, and the above data shows that this ratio is 989/641 = 1.54 to becompared with a theoretical value of 1.50.
28
Figure 1.17: Expanded region where the RF is turned on.
Figure 1.18: Expanded region where the RF is turned on.
29
Figure 1.19: Period of ringing vs. peak RF volts.
30
Chapter 2
Nuclear Magnetic Resonance, NMR
2.1 Introduction to NMR
Copies of the TeachSpin NMRManual[?] will be by the apparatus. Please look over thesebefore the first of the lab sessions.We will be doing NMR with protons. A proton has a spin I=1/2, and a nuclear
magnetic moment µp=2.79277 µN. The nuclear magneton µN= 5.05038 ×1−−24 ergs/gauss.In a magnetic field the energy of the two spin orientation states are:
H = −µ ·H (2.1)
If the spin of the proton is aligned along the y axis and the magnetic field is along the zaxis, then the torque µ ×H induces precession around the magnetic field as shown in Fig.2.1.The rate of rotation is: ωL = µH/~. Now suppose one views this precession in a
coordinate system which is rotating about the magnetic field with rate ωR in the samedirection as the spin does. Evidently the spin will now appear to rotate with rate: ωs =ωL−ωR. This has the effect of reducing the applied field in the rotation coordinate system.If the ωR = ωL then there appears to be no applied field and the spin appears to be static.This discussion appears to be classical, but in fact yields the same results as the quantummechanical one[?].Now suppose one applies an rf field with ωr f = ωL = ωR. If one decomposes the rf
field into rotating components, one of these components will be rotating with the rotatingcoordinate system and thus will appear to be static. (The other component has very littleeffect and will be ignored in this discussion.) In the rotating system, the nuclear spin willnow precess about the co-rotating component of the rf magnetic field.If the rf stays on for a time so that the spin just ends up in the x-y plane, this is called
a π/2 pulse and is often the first step in a pulsed nmr experiment.In order to produce a spin-echo one first pulses the spins into the x-y plane, whereupon
the spins dephase due to local field variations. After a delay, a 180 degree pulse is applied.This causes the spins that precessed fastest to be rotated so as to lag, and those that
31
z
x
yµ
H
Figure 2.1: Rotation of a nuclear spin in a magnetic field.
Figure 2.2: The effective field in a rotating coordinate system.
32
Z
X
Y
A B C
D E F
Figure 2.3: The formation of a spin echo using the Carr-Purcell pulse sequence in therotating frame. In A, spins are aligned and produce a net magnetization in the plus Zdirection, parallel to the external field. An rf pulse of magnetic field precesses the spinsby π/2 about the X axis to be along the Y axis as shown in B. Variations in local fieldscause the dephasing as shown in C. A second rf pulse, twice as long causes the spins toprecess about the X axis by π, leading to D. Now, the same local field variations cause thespins to re-phase as in E, producing and echo. In F one sees the spins again dephasing.Taken from Carr and Purcell[?] .
precessed slowest to lead. Following this pulse the various spins will re-phase in just thetime that the 180 degree pulse was delayed from the 90 degree pulse. This may be seen inFig. 2.3.
2.2 T1 and T2
This has been taken from Wikpedia.
Different physical processes are responsible for the relaxation of the components ofthe nuclear spin magnetization vector M parallel perpendicular to the external magneticfield, B0 (which is conventionally oriented along the z axis). These two principal relaxationprocesses are termed T1 and T2 relaxation respectively.
33
2.2.1 T1, the Spin-lattice relaxation time
The longitudinal relaxation time T1 is the decay constant for the z-component of thenuclear spin magnetization,Mz. For instance, if the z magnetization is zero (e.g. ifM hasbeing tilted into the xy plane by a 90 pulse), then it will return to its equilibrium value,Mz,eq as follows:
Mz(t) = Mz,eq(
1 − e−t/T1)
(2.2)
i.e. the magnetization will recover to 63% of its equilibrium value after one time constantT1.T1 relaxation involves redistributing the populations of the nuclear spin states in order
to reach the thermal equilibrium distribution. By definition this is not energy conserving.Moreover, spontaneous emission is negligibly slow at NMR frequencies. Hence trulyisolated nuclear spins would show negligible rates of T1 relaxation. However, a variety ofrelaxation mechanisms allow nuclear spins to exchange energy with their surroundings,the lattice, allowing the spin populations to equilibrate. The fact thatT1 relaxation involvesan interactionwith the surroundings is the origin of the alternative description, spin-latticerelaxation.Note that the rates of T1 relaxation are generally strongly dependent on the NMR
frequency and so may vary considerably with magnetic field strength, B.
2.2.2 T2, the Spin-spin relaxation time
The transverse relaxation time T2 is the decay constant for the component of M perpen-dicular to B0, designated Mxy, MT, or M⊥. For instance, initial xy magnetization at timezero will decay to zero (i.e. equilibrium) as follows:
Mxy(t) =Mxy(0)e−t/T2 (2.3)
i.e. the transverse magnetization vector drops to 37% of its original magnitude after onetime constant T2.T2 relaxation is a complex phenomenon, but at its most fundamental level, it corre-
sponds to a decoherence of the transverse nuclear spin magnetization. Random fluctu-ations of the local magnetic field lead to random variations in the instantaneous NMRprecession frequency of different spins. As a result, the initial phase coherence of thenuclear spins is lost, until eventually the phases are disordered and there is no net xymagnetization. Because T2 relaxation involves only the phases of other nuclear spins it isoften called ”spin-spin” relaxation.T2 values are generally much less dependent on field strength, B, than T1 values.
2.2.3 T2* and magnetic field inhomogeneity
In an idealized system, all nuclei in a given chemical environment in a magnetic fieldspin with the same frequency. However, in real systems, there are minor differences in
34
chemical environment which can lead to a distribution of resonance frequencies aroundthe ideal. Over time, this distribution can lead to a dispersion of the tight distributionof magnetic spin vectors, and loss of signal (Free Induction Decay). In fact, for mostmagnetic resonance experiments, this ”relaxation” dominates. This results in intra-voxeldephasing, a phenomenon related to the motion of spins in fluids, velocity differentials aswell as directional effects; for example, in tissues, can be related to capillary diffusion.However, decoherence because of magnetic field inhomogeneity is not a true ”re-
laxation” process; it is not random, but dependent on the location of the molecule in themagnet. Formolecules that aren’tmoving, the deviation from ideal relaxation is consistentover time, and the signal can be recovered by performing a spin echo experiment.The corresponding transverse relaxation time constant is thus T2*, which is usually
much smaller than T2. The relation between them is:
1
T∗2
=1
T2+1
Tinhom=1
T2+ γ∆B0 (2.4)
where γ represents gyromagnetic ratio, and ∆B0y the difference in strength of the locallyvarying field.Unlike T2, T2* is influenced bymagnetic field gradient irregularities. The T2* relaxation
time is always shorter than the T2 relaxation time and is typically milliseconds for watersamples in imaging magnets.
2.2.4 The reason that T1 is slower than T2
As a general rule, the following always holds true: T1 > T2 > T2*.If T2 were to be slower than T1, then the magnetizations perpendicular to the initial
direction would have not dephased by the time the sample had returned to equilibrium.This is physically impossible, as once the sample has returned to equilibrium, there is nomagnetization perpendicular to the original direction. Hence, T1 must be greater than orequal to T2.
2.3 Experiments
2.3.1 Getting Started
Observe the spin echo for the mineral oil sample. Tune the apparatus for the largest andcleanest measurement. The lab instructor will help at this stage. It is suggested thatyou make all the different types of measurements first with the mineral oil sample beforeproceeding to the other samples.
2.3.2 Samples
• Mineral Oil, the sample you will start with.
35
• Water and Glycerine, various concentration of glycerine
• Paramagnetic Salts, various concentrations of CuSO4 and/or Fe(NO3)3
2.3.3 T1Measurements
See page 32 of the TeachSpin Manual. Measure T1 by the techniques described after page32.
2.3.4 T2Measurements
Measure T2 using the 2 pulse technique, the multiple pulse technique, and the Meiboom-Gill technique. Be sure to compare your results. These techniques are described on pages34 and 35 of the TeachSpin Manual.
2.3.5 Analysis
Determine how the two constants depend on the characteristics of the samples.
36
Chapter 3
The Mossbauer Effect
3.1 Introduction
In 1957 Rudolf Mossbauer discovered the effect which bears his name. This was reportedin 1958[?]. He was studying resonance fluorescence as a function of temperature. Theprocess of resonance fluorescence is one in which γ-rays are resonantly scattered fromnuclei. At low temperature this process was expected to turn off because the recoils givento the emitting nucleus and to the absorbing nucleus would cause the emitted γ-ray to notbe at the right energy to resonantly interact with the absorbing nucleus. To see why thiswas expected, consider the case of 57Fe, the nucleus for this lab. ( Mossbauer studied 191Ir,a much more difficult nucleus to study.) The pertinent information for 57Fe are shown inTable 3.1. The 14.36 keV γ-ray gives a recoil momentum, Pr, to the emitting nucleus of14.36 keV/c. This corresponds to an energy of recoil:
Er = P2r/(2M(Fe)) = .018eV (3.1)
The absorbing nucleus would also be given a similar amount. Thus there is about .036 eVenergy mismatch. Next consider the width of the emission and absorbing lines:
Γ = ~/τ = 4.7 × 10−9eV
So at low temperature the emission and absorption lines will not overlap and so resonantscattering should not occur. (At higher temperature or by the use of ultra-centrifuges, theenergy mismatch can be compensated for.)
Mossbauer, however, observed an increase in scattering rather than the expected de-crease at low temperature. The explanation of this surprising and Nobel prize winningdiscovery is that for some of the γ-rays the whole of the crystals in which the emittingand absorbing nuclei are embedded take up the recoil and thus the mass, M, in Eq. 3.1, ismultiplied by a number on the order of Avogadro’s number. This makes the recoil energyunmeasurably small.
37
Ground State First Excited StateEnergy(keV) 0 14.36
Spin and parity 12
− 32
−
Magnetic Moment (nm) 0.0903 -0.153Quadrupole moment (barns) 0 0.29Mean lifetime (s) Stable 1.4×10−7α (Int. Conv. Coef.) 9.7± 0.2
Table 3.1: Properties of 57Fe
High Voltage +1950 VDwell 100µSCA =200 20 mm/s max
Table 3.2: Settings for the Mossbauer effect experiment.
3.2 Energy Splittings
The various shifts and splittings are shown inf Fig. 3.1.
3.2.1 Isomer Shift
The ismeric (or isomer) shift is the energy shift an atomic spectral line and gamma spectrallines due to replacement of one nuclear isomer by another. Isomers have the same atomicnumber Z.
Ea − Ee =2
5πZe2[R2is − R2gr][|ψ(0)a|2 − |ψ(0)e|2] (3.2)
Here, Ris/gr are the radii of the isomeric state and ground state respectively and |ψ(0)a/e|2are the electron densities at the nucleus for the absorbing and emitting atoms. For aderivation see Wertheim[?]
3.2.2 Quadrupole Splitting
E =e2qQ
4I(2I − 1)[3m2 − I(I + 1)] (3.3)
where q = 1e∂2V/∂z2, Q = 1
e
∫
ρ(3z2 − r2)d3r and ρ is the nuclear charge density.
3.2.3 Zeeman Splitting
E = −µ ·H = −gIµnmmH (3.4)
Here µ is the nuclear moment and H is the magnetic field at the nucleus.
38
Figure 3.1: Absorption spectra for various characteristic shifts and splittings.
39
3.3 The Uncertainty Principle
The 14.4 keV excited state has an half-life of 98.3(3)ns. Thus its mean-life τ = 141.8(5)ns.The absorption cross-section
σ(E) = σ0(Γa/2)2
(E − E0)2 + (Γa/2)2(3.5)
where Γa = ~/τ and σ0 is:
σ0 = 2πn2 2Ie + 1
2Ig + 1
1
1 + α(3.6)
Since the energy distribution of the emitted photon has a similar shape, the expectedcross-section is then the result of a convolution:
σexpt(E) =
∫ ∞
−∞ω(e)σ(E − e)de (3.7)
where
ω(e) =1
π
Γe/2
(e − E0)2 + (Γe/2)2
resulting in:
σexpt(E) = σ0 ·Γa2· Γa+Γe2
(E − E0)2 +(
Γa+Γe2
)2(3.8)
3.4 Apparatus
A block diagram of the apparatus is shown in Fig. 3.5. A 57Co source is attached to the rodof the electro-mechanical drive. The drive signals can produce a constant acceleration.Following electron capture 57
27Co → electroncapture → (57
26Fe)∗ → (same)groundstate (see Fig.
3.4) and transition to the 14.4 keV, I = 3/2 excited nuclear state, a γ-ray is emitted. Theγ-ray traverses an absorber, and those that are not absorbed proceed to a proportionalcounter. This proportional counter has a thin Be window to allow the γ-ray through,is filled with a special gas, and has a small diameter central wire at high voltage. Theentering γ-ray causes an atom to eject an electron with nearly the full γ-ray energy. Thiselectron ionized further atoms. The electrons drift toward the central electrode and areaccelerated to produce proportionally more electrons (hence the name). The charge ispassed to a pre-amp. and amplifier which produce a voltage pulse proportional to thecharge which is in turn proportional to the γ-ray energy. This pulse is sent to a computercontrolled multi-channel analyzer (MCA) which makes a histogram of pulse heights.
40
0 500 1000Channel Number
0
200
400
600
800
1000
1200
1400
1600
Cou
nts
12
3 4
56
Figure 3.2: Spectrum observed with enriched 57Fe absorber foil.
Figure 3.3: The spectrum observed by Wertheim[?]. This may be used to calibrate thecurrent experiment.
41
Figure 3.4: Decay and absorption scheme for 57Fe. Taken from a web page of Dyar[?]. .
42
Prop. Ctr.
Ortec 142Pre. Amp.
TC 241Amp
S700MoessbauerDrive
Source Abs.
Electro−Mechanica Drive
Vel. Dr.
MSB
Bi.St.
Dir In.
MCA UCS 20
HV
+1950
Figure 3.5: Block diagram of the Mossbauer apparatus.
43
3.5 Measurements
Obtain Mossbauer spectra for:
Enriched 57Fe in Iron A pure Zeeman splitting case, i.e. no quadrupole splitting due tothe symmetry of the site in iron.
Na-Prusside A pure quadrupole case. No magnetic field on the iron nucleus.
Stainless Steel Supposedly no splitting, but you will see an larger width for the singleabsorption dip.
3.5.1 Calibration
Calibrate the mechanical drive by using the measurements of Preston, Hanna, andHeberle[?] for the Zeeman splittings of 57Fe in iron and your data. Preston et al. ob-tain the results shown in Table 3.3
mg me Shift (mm/s)-1/2 -3/2 -5.328-1/2 -1/2 -3.084-1/2 +1/2 -0.840+1/2 -1/2 +0.840+1/2 +1/2 +3.084+1/2 +3/2 +5.328
Table 3.3: Calibration data.
3.5.2 Zeeman Field
Determine the magnetic field at the 57Fe in iron for both the ground and excited states. (Iknow this is cyclical reasoning, since we have already used this information to determinethe calibration.) Assume the nuclear moments are known. Compare this value with330kG.
3.5.3 Quadrupole Splitting
Assuming that the quadrupole moment of the excited state is known, determine theelectric field gradient for Na-Prusside. Determine the distance from an electronic chargewhich would give this electric field gradient.
44
3.5.4 Impurities
What value of local field variation would yield the absorption line-width you observe forstainless steel.
3.5.5 Uncertainty Principle
Fit the Na-Prusside absorption lines to Lorentzian line shapes’:
d(i) = A − B
(i − C)2 + (D/2)2 (3.9)
where d(i) is the counts in channel i. List your values for A, B, C, and D for both dips.From your calibration determine Γexpt. Compare this to that expected for the lifetime ofthe 14.4 keV excited state.Pound and Rebka[?] observed a ∆E/E of 1.13−12. This was in their famous experiment
to observe the gravitational red-shift of photons. How does your value for ∆E/E comparewith theirs. If you equate a mass m with Eγ/c2 what do you expect for the gravitationalred shift for their 22.4 m tower?
3.6 Useful data
Isotopic abundance 2.14(1)Ground State PropertiesIP 1/2−
µ 0.09062(3) nmExcited State PropertiesIP 3/2−
E 14.412497(3) keVEr 1.95883310(4) 10-3 eVαIC 8.20σ 2.56 10-18cm2Q 0.21(2)bT1/2 98.3(3) nsW 0.194(2) mm/s
Table 3.4: 57Fe properties.
45
46
Chapter 4
Alpha Particle Scattering
4.1 Introduction
In this lab you will explore energy loss by charged particles, Rutherford scattering andcharged particle detection. You should do some outside reading.
4.2 Energy Loss by Charged Paticles
The theoretical calculation of the energy loss by ions has a long history and many wellknown theorists have been involved. These include N. Bohr, E. Fermi, and H. Bethe. Agood derivation of this energy loss is given in the book by E. Segre [?]. Most of the energyloss is due to ion-electron scattering. One determines the momentum transfer ∆p givento an electron at a given impact parameter, b, (This is the closest distance that the ion’sundeflected path has with a given electron.) Then the energy given to that electron is:
∆E =∆p2
2me=2
m
(
ze2
bv
)2
(4.1)
where z is the ion’s charge number and v is its speed. One sums this energy loss overelectrons of density N between bmin and bmax, judicially chosen minimum and maximumimpact parameters. This yields:
−dEdx= 4πN
z2e4
mv2lnbmaxbmin
(4.2)
Reasonable values of the minimum and maximum impact parameters then yield:
−dEdx=4πz2e4
mv2N
[
ln2mv2
I(1 − β2) − β2
]
(4.3)
I is the ionization energy of the material and β is v/c. An estimate for I is:
47
Figure 4.1: Alpha-particle range curve[?]
I = Z · 9.1 · (1 + 1.8Z−2/3)eV (4.4)
The range of an ion can be obtained by integrating Eq. 4.3. AnAlphaparticle range-energycurve is shown in Fig. 4.1.
4.3 Rutherford Scattering
In 1909 Geiger and Marsden[?] scattered alpha particles from nuclei and discovered thattherewere considerable large angle scatters. Muchmore than expected, though lower thanat small angles. Rutherford[?] in 1911 showed that this could only be from single largeangle scatters from a heavy, pointlike charged nucleus. This destroyed the raisin puddingmodel of atoms.Rutherford, in the same cited paper derived the scattering differentialcross-section which bears his name:
dσ
dΩ=1
4
(
e2Zz
mv2
)21
sin4(θ/2)(4.5)
48
4.4 Charged Particle Detector
Wewill be using a Si diode chargedparticle detector. Agooddescription of the functioningof these devices is to be found in Melissinos and Napolitano[?]. Please read that section.For here just note that the detector is a back biassed diode which allows electron and holecarrier particles to be produced and subsequently collected in a so-called depleted region.
Roughly 3 eV are required per charge. So a 1 MeV particle has a charge Q = 106
3e. On a
200 pf capacitor this will produce a .25 mV signal.
4.5 Procedure
The sources are both 244Cm.
4.5.1 Energy Loss Measurements
The detector requires about 50 - 70 V to work properly. Be sure that the base of the lightshield is sealed with black tape before voltage is applied. There are two sources: theweaker one, has a thin window.
• Place the thin window source about 1 cm from the detector. Seal the bell jar and lightshield. Evacuate the bell jar.Measure the spectrum seen in the detector as a functionof applied voltage. Plot the peak position andwidth as a function of applied voltage.Plot these.
• Measure the position and width of the peak as a function of pressure in the bell jar.Plot these and compare to theory.
• Bring the bell jar up to atmospheric pressure.Lower the voltage. Remove the bell jar.Remove the thin window source. Place the detector so it points at the collimatedsource. Do not yet put the gold target in place. Re-seal the bell jar and shield. Applythe detector voltage again and determine the peak position and width. Why is itshifted?
4.5.2 Rutherford Scattering
• Remove voltage. Open the bell jar and place the gold target in place. Reclose thebell jar and re-evacuate the jar. Apply voltage.
• Measure the peak position. How much energy is lost in the gold. Can you thusdetermine the gold’s thickness?
• Measure the peak rate andposition as a function of angle. More points in the forwarddirection, but out as far as possible.
49
• Plot and compare to theory. To do this you will need to know the geometry of thesystem. Make appropriate measurements of the distances and apertures and includethem in your report.
50
Chapter 5
γ-ray Studies
In the first week you will measure total cross-sections for various materials and becomefamiliar with the apparatus. In the second week you will study Compton scattering.
5.1 Objectives of the Experiment:
• To appreciate the role of γ-rays in nuclear electromagnetic transitions.
• To become familiar with the experimental methods of γ-ray spectroscopy, startingwith the fundamental mechanisms which occur when a γ-ray enters a detector suchas a NaI(Tl) crystal (sodium iodide doped with thallium), and concluding withan understanding of a Multi-Channel analyzer (MCA), which processes amplifiedsignals from the detector.
• To investigate quantitatively the passage of γ-rays through matter.
There are two dominant scattering processes for γ-rays of the energies we will use here.These are Compton scattering in which the γ-ray scatters from an essentially free electronand the photo-effect in which the γ-ray is absorbed by an atom which then ejects anelectron with all the available energy. These are discussed below.
5.2 Compton Scattering
Klein and Nishina[?] derived the following equation for Compton scattering. The Klein-Nishina cross-section is:
dσ
dΩ=r2c2
(
E′γ
Eγ
)2 (Eγ
E′γ+E′γ
Eγ− sin2 θ
)
(5.1)
Here, Eγ is the incident photon’s energy, Eγ′ is the outgoing photon’s energy, and rc =e2/mc2 is the classical radius of the electron of mass m. One can solve the relativistic
51
kinematic equations for Eγ′ in terms of Eγ and θ, the scattering angle for the photon toobtain:
Eγ′ =Eγ
1 + ǫ(1 − cosθ) (5.2)
where ǫ = Eγ/mc2. Putting this into Eq.5.1 one obtains:
dσ
dΩ=r2c (1 + cos
2 θ)
2
1
[1 + ǫ(1 − cosθ)]2[
1 +ǫ2(1 − cosθ)2
(1 + cos2 θ)[1 + ǫ(1 − cosθ)]
]
(5.3)
Now one would use Feynman diagrams to calculate this cross-section. Klein andNishina did not, since they had not yet been invented. The lowest order Feynmandiagram for Compton scattering is shown in Fig. 5.1
Figure 5.1: Feynman diagram for Compton Scattering. A photon, represented by awigglyline interacts with an electron, this in turn radiates a photon. A more detailed discussionwill have to wait until graduate school.
5.2.1 Photoelectric Cross-section
InAlpha-, Beta- and Gamma-ray spectroscopy edited by K. Siegbahn [?] the dominant K-shellcross-section is given as:
σpe = 1.367 × 10−22(αZ)51
ǫcm2/atom (5.4)
Note that α ≈ 1/137 is the “fine structure constant” and ǫ = Eγ/mc2 the ratio of theincoming γ-ray’s energy to the rest mass energy of an electron. For aluminum σpe is onlyabout 1% of the Compton cross-section and therefore will be ignored in the Comptonscattering part of this lab. For lead with Z=82 the photo-effect cannot be ignored.
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Figure 5.2: Theoretical and experimental differential cross-sections. Taken from Heitler[?]
5.3 Equipment
The general layout is shown in Fig. 5.5.You will use a set of weakly radioactive sources:22Na, 137Cs, and 60Co. See Fig. 5.4 for the energy levels.
Figure 5.3: A NaI(Tl) detector
5.4 Procedure: First Week
Signals Determine the amplitudes, rise-time and fall time of the signals directly fromthe photo-multiplier tube.You may need a 50 ohm terminator at the input to theoscilloscope. Why? .
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Figure 5.4: Nuclear states and their decay characteristics.
Figure 5.5: γ-rays from the source are scattered by the atoms in the plates.
Calibration Determine the photo-peak channel numbers for the two peaks of 60Co, and22Na, and the single peak of Cs. Use the known energies from Fig. 5.4 and make aplot of E versus channel number. Fit this to a straight line: E = a + b · n. Determinethe constants a and b and their errors. You will need this calibration for later use.
Compton Edges Determine the channel number for the Compton edges (see Fig. 5.3).From these and your calibration determine the γ-ray energies of these edges. Com-pare to the theoretical values.
Line Widths Determine the fullwidth athalfmaximum, fwhm, for eachpeak. Youcanuse
the region of interest feature to do this. Make a plot of fwhm vs.√E for all the peaks.
Assume the energy measured is proportional to some number N, and the fwhm is
proportional to√
(N), both with the same proportionality constant. DetermineN for
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Figure 5.6: Pulse height distribution for a gamma-ray incident on a NaI(Tl) detector. Notethe Compton edge, and the width of the photo-peak.
one of the peaks which you identify. This√
(N) behavior is characteristic of Poissonstatistics which is appropriate for counting numbers.
Absorption Cross-sections For 137Cs, 22Na and 60Co place varying thickness of plastic,aluminum, copper, and lead between the sources and the detector. Does the positionof the photopeak change with absorber thickness? Plot the counts in the photopeaksversus absorber thickness. From these determine the absorption cross-sections.
5.5 Measurements: Second Week
• Check the energy calibration
• Determine the number of counts for fixed live-time as a function of angle with thealuminum cylinder in place. You might wish to take more time at the back angles.
• Repeat the previous with the target cylinder removed. Youmight want to try severalshielding configurations with both sets of measurements. Record the shieldingarrangement that you use.
• Determine the peak energy as a function of angle. Compare this to the expectedvalues for Ec from:
1
Ec=1
E+1
mc2[1 − cosθ] (5.5)
, which can be obtained from energy and momentum conservation.
• Determine dσ(θ)dΩ. Plot this as a function of angle and plot the theoretical dσ(θ)
dΩ. This
will take a fair amount of work. Describe and try to explain the discrepancies.
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5.6 Considerations
We are using the apparatus shown in Fig. 5.7. With some dimensions shown in Fig. 5.8.We divide our considerations into:
Figure 5.7: The experimental apparatus.
• Beam Characteristics
• Target Considerations
• Detector Considerations
5.6.1 Beam
We start by determining the number of decays per second of 137Cs there are. The activityN0s at a certain time in the past is given on the source. Note that a Curie is: 3.7 × 1010decays per second. Presuming the half life is τ1/2, the current activity is:
Ns = N0s · (1
2)t/τ1/2 (5.6)
The number of γ-rays per second and steradian is then:
dNsdω=Ns4π
(5.7)
The beam is limited by the aperture shown in Fig. 5.8. Even here there are approxima-tions. The beam is not completely and cleanly collimated since the γ-rays will penetratea little into the collimator and can then scatter back into the beam.
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Figure 5.8: The experimental apparatus showing dimensions.
5.6.2 Target
Here we consider the probability of scattering into some solid angle. If we break the beaminto small cones of solid angle dω , see Fig. 5.9, which intersect the target and have a pathof length l(~ω), then the number of scatterers is approximately ρl(~ω)R2dω, where ρ is thevolume density of scatterers and R is the distance from the source to the target center.The fraction of the incident particles into dω that scatter into dΩ is then:
f =dNs(~ω)
dωdω ·
l(~ω)R2dωρ
R2dω
dσ
dΩ. (5.8)
which, after integrating over the target (dω), is approximately:
dNddΩ=Ns4πR2
0.86Dπ(
D
2
)2 dσ
dΩ(5.9)
where D is the diameter of the target rod and Nd is the number into the detector..
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d
l
ω
Figure 5.9: The target geometry.
5.6.3 Detector
One needs to determine the solid angle, δΩ of the aperture of the detector, the totalprobability of detection of a γ-ray of a particular energy (obtained from Fig. 5.10 ),Ptot(E) and if one only counts the photo-peak, the fraction of the spectrum which is in thephoto-peak: fpp = Ppp/Ptot(obtained from Fig. 5.11). The detected counts are then:
Ndet =dNddΩ
δΩPtot fpp (5.10)
5.6.4 Total Cross-section
Consider the situation depicted in Fig. 5.12. If the target is very thin we can assume thatmost of the particles go through and are then detected. Let Ninc be the number incidenton the target and Ndet be the number detected. If the target is not present then we assumeNdet = Ninc. If the target is very thin Ndet ≈ Ninc and the probability that a scattering hasoccured is:
P =Ninc −NdetNinc
= 1 − NdetNinc
(5.11)
The fraction, f , of this area of the target taken up by scatterers is:
f = Ntgts(cm−2) · σ(cm−2) (5.12)
where Ntgts(cm−2) is the number of scatterers per cm2, and σ is the area taken up by each
scatterer. Evidently:
P = f (5.13)
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Figure 5.10: Absorption in NaI
So:
1 − NdetNinc
= Ntgts(cm−2) · σ(cm2) (5.14)
σ(cm2) =1 − Ndet
Ninc
Ntgts(cm−2)(5.15)
Now suppose one has a detector which will detect any scattering that occurs, but notparticles which don’t scatter. then:
σ(cm2) =Ndets
NincNtgts(cm−2)(5.16)
5.7 Differential Cross-section
In general the scattering will be different to different angles. Consider Fig. 5.13.
We may define dσdΩδΩ to be a small area associated with each scatterer which leads to a
particle scattered into the detector. With this definition:
dσ
dΩ=
NdetsNincNtgts(cm−2)δΩ
(5.17)
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Figure 5.11: Peak to total ratios
5.7.1 The Thomson Cross-section
This cross-section corresponds to the case of a photon scattering from a charge. Usuallyonly scattering from electrons need be considered. A heuristic derivation of the radiatedpower, Pr, from an accelerated chargewill first be given. Then the incoming photon powerwill be associated with n~ω, where n is the number of incoming photons per second. Theoutgoing power will be then n’~ω, with the same frequency which is the case for lowfrequency photon scattering. The Thomson cross-section is:
σT = n′/n = Pr/Pi (5.18)
Beam of Particles
Target
Detector
Figure 5.12: An incident beam of particles strikes a target and some scatter. If no scatteringoccurs it is assumed that the particle will reach the detector and be detected.
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Beam of Particles
TargetDetector
θ
δΩ
Figure 5.13: An incident beam of particles strikes a target and some scatter into a detectorat an angle θ and φ subtending a solid angle dΩ.
5.7.2 Heuristic Derivation of Pr
The classical radiated power from an accelerated charge is derived in many Electricityand Magnetism texts, e.g., Classical Electrodynamics by J. D. Jackson[?]. What follows isnot rigorous, but does yield the right result and can easily be used to re-derive the result.
One knows that the radiated power is proportional to the square of the electric field inthe radiation field. This electric field is certainly proportional to e, the charge in question.
One knows that a stationary charge does not radiate. Neither does a constantlymoving charge in vacuum. Thus we expect the radiation field to be proportional to a, theacceleration of the charge.
Thus, the radiated power is:
Pr ∝ e2a2 (5.19)
But, power should have units of e2/(ls). Note that this is in Gaussian units where φ = e2/r,a set of units it might be useful to learn about. If you want you could re-derive the resultshere in SI units. In Eq. 5.19 the right hand side has units: e2l2/t4. Thus, to make the righthand side have the right units one needs to divide by (l/s)3. The only thing around withthe units of l/t is the speed of light: c. So, now the right hand side is e2a2/c3. We arealmost finished deriving the power from an accelerated charge. One needs to note thatif the acceleration is along the z axis, one can only observe the motion along the x and ydirections, i.e., 2/3 of the three directions. Thus we are left with the final result:
Pr =2
3e2a2/c3 (5.20)
This is correct, in Gaussian units.
5.7.3 Cross-section
The incoming power is:
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Pi =E2i
4πc = n~ω (5.21)
where Ei is the incoming photon’s electric field and ω is the photon’s angular frequency.The acceleration, a = eEi/m where m is the mass of radiating charge. So, we can re-writeEq. 5.20 as:
Pr =2
3
e2E2ie2
m2c4c =2
3(e2
mc2)24πPi (5.22)
So, finally we get:
σT =8π
3r2c (5.23)
where rc = e2/mc2, the so-called classical radius of the electron. Approximately, rc =
2.8 · 10−13cm.
5.8 NaI Detector and Photo-multiplier Tube
ANaI(Tl) scintillatormounted on a photo-multiplier tube is shown in Fig. 5.14. Photons ofenergy hνγ enter the scintillator crystal. Inside this crystal Compton scattering and photo-electric processes occur. In Compton scattering some of the incident photon’s energy istransferred to an electron and a lower energy photon. In a photo-electric essentially allof the photon’s energy is transferred to an electron. A nucleus takes up the momentumnecessary for energy momentum conservation. The Compton scattered lower energyphoton may also scatter again further transferring energy to electrons.
The electrons discussed above travel through the NaI crystal exciting the Na and Iatoms. These atoms de-excite, emitting photons. These photons excite the Tl atoms, whichthen de-excite with photons, some of which enter the photo-multiplier tube striking thephoto-cathode to produce electrons, for example,Nc. (The photo-cathode is very near theentrance to the tube.) These electrons are accelerated by an applied electric field and strikethe first dynode, D1. The dynodes are electrodes held at ever increasing potential. Uponstriking the dynode, more than one electron is ejected, say n, are ejected. This accelerationand ejection process is repeated for 10 or so dynodes. Thus the number of electrons whichfinally reach the anode ( the last dynode ) is thenNa = Nc×n10. The charge: e ·Na producesa voltage at the SIG output.
5.8.1 Classification of Radionuclear Decays
In 1896 Becquerel made the first observation of radioactivity. Investigations in the earlydecades of this century soon led to a classifcation of radioactive materials as α, β, and γemitters. The process of spontaneous fission was first observed much later.
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Figure 5.14: Photo-multiplier tube assembly with NaI scintillator.
α emitters tend to be isotopes of transuranian nuclei (mass number A > 208). Unstablein their nucleon configuration, they decay to a daughter nuclide of mass (A-4) plusan alpha particle. The α “ray” (or particle) was soon identified as a fully ionizedhelium nucleus which departs the scene of decay with about 5 to 10 MeV of kineticenergy.
β emitters tend to be unstable nuclear isotopeswhich have a low tomediummass numberA. The beta particle was determined to be an electron (or positron): n→ p+ e−+νbar.
γ emitters span the full range of the nuclear chart. γ rays from these sources representthe photons emitted in electromagnetic transitions as an excited nucleus makes atransition from an upper to a lower energy level.
Check out the “Chart of the Nuclides” in room Millington 303 to identify α, β, and γemitters.
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