DEVELOPMENT OF A. Justin Daniel King ********* Submitted in partial fulfillment. of the requirements for

DEVELOPMENT OF A MÖSSBAUER SPECTROMETER By Justin Daniel King ********* Submitted in partial fulfillment of the requirements for Honors in the Department of Physics and Astronomy UNION COLLEGE June, 2006 i

ABSTRACT KING, JUSTIN Development of a Mössbauer Spectrometer. Department of Physics and Astronomy, June 2006. A Mössbauer spectrometer was developed for use in an upper-level undergraduate experimental physics course. Mössbauer spectroscopy involves recoilless resonant absorption of gamma rays and utilizes the Doppler effect in order to use these photons as a sensitive probe. A 57 Co source provided 14.4 kev gamma rays which were used to probe samples of stainless steel and natural iron. A Doppler shift in the gamma ray energy was achieved by creating relative motion between the source and the absorbers. The gamma rays passing through the absorber were counted and Mössbauer spectra were acquired. In stainless steel, the linewidth was measured as 1.05 ±.04 10 8 ev, and the isomer shift as 1.20 ±.05 10 8 ev. In natural iron, the isomer shift was measured as 5 ± 1 10 9 ev. The g-factor of the excited state of natural iron was determined to be.157 ±.003, which is consistent with the accepted value of.1549, and the magnetic field strength at the iron nucleus was determined to be 32.4 ±.1T. ii

Contents 1 Introduction 1 2 Theory 1 3 Experimental Procedure 8 4 Data Analysis 22 5 Conclusions 25 Appendix A 28 References 40 iii

1 Introduction The goal of this research is to develop a Mössbauer spectrometer for use in an upper-level undergraduate physics course. Mössbauer spectroscopy is a very precise method of measuring nuclear properties, and it is an ideal experiment for undergraduates because it allows them to quantitatively measure quantum mechanical effects which they learn about in introductory and intermediate level courses. Working on this project has been beneficial for myself because I plan on going to graduate school to earn my Master of Arts in teaching for physics, and this project has given me experience creating demonstrations and developing curricular materials. I will begin by explaining the theory behind Mössbauer spectroscopy and the quantum mechanical effects that we will be measuring. Next I will explain how our Mössbauer spectrometer operates and how it came to be. Finally, I will explain how we obtained our results. 2 Theory Mössbauer spectroscopy is a method for measuring small shifts in nuclear energy levels with high precision. This method involves the recoilless emission and absorption of gamma rays and utilizes the Doppler effect in order to use these gamma rays as a sensitive probe. 2.1 Resonant Absorption A quantum system can undergo a transition when it absorbs or emits a photon of a specific energy. The energy of the photon is dependent upon the difference in the energy levels of the transition. The energy of the ground state is absolute, but the 1

energy of the excited state is not a precisely defined quantity. Due to the Uncertainty Principle, the natural line width is given by Γ = h 2τ, (1) where Γ is the natural line width of the excited state and τ is the lifetime of the state. However, when a photon is emitted by a free system not all the transition energy is carried away by the photon. This is because the system recoils due to conservation of momentum, and takes with it some of the transition energy in the form of kinetic energy. If we call the energy of the transition E T and the recoil energy E R, then the energy of the emitted photon can be expressed as E γ = E T E R. (2) We can rewrite this expression in terms of the recoil momentum as E γ = E T p2 R 2m. (3) Due to conservation of momentum, the momentum of the photon is equal to the momentum of the recoilless nucleus, so that where p γ is the momentum of the photon. p R = p γ = E γ c, (4) By combining Equations (3) & (4) we can write the expression for the energy of the emitted photon as E γ = E T 2 E2 γ 2mc 2. (5)

Likewise, when the photon is absorbed the absorbing system recoils. Therefore, the distributions of the emission and absorption energies are separated by twice the recoil energy. The probability of resonant absorption is proportional to the overlap of these distributions. In atomic systems this probability is very high because the recoil energy is small compared to the natural line widths. In nuclear systems, however, the recoil energy is much larger than the natural line widths, and therefore the probability of resonant absorption is very small. 2.2 Recoilless Emission and Absorption In 1958, Rudolf Mössbauer showed that for atoms bound in a lattice, a nucleus doesn t recoil individually [1]. The recoil momentum, therefore, is taken up by the entire lattice, which has a very large mass. From Equation (5) we see that as m, E γ E T. Therefore, when the nucleus is embedded in a substrate the recoil energy is negligible. This concept applies to absorption of photons as well. 2.3 Doppler Shift When there is relative motion between the emitter and the absorber there is a Doppler shift in photon energy. The energy of the photon is given by the Lorentz transformation as E γ = 1 (E 1 + β 1 β 2 γ + vp γ ) = E γ, (6) 1 β 2 where β = v c, E γ and p γ are the energy and momentum of the emitted photon, v is the relative velocity between the emitter and the absorber, and E γ is the resulting Doppler shifted photon energy [2]. For β 1 we take the first order of the binomial 3

expansion and write the expression for the change in photon energy due to the motion as E = E γ E γ = βe γ = v c E γ (7) as an expression for the change in photon energy due to the motion [2]. By varying the relative velocity between the emitter and absorber, we can scan a range of photon energies. In Mössbauer spectroscopy we measure the absorption rate as a function of velocity, which can be converted to energy shift. Analysis of the absorption spectrum yields information about the structure of the nucleus. 2.4 Use of 57 Fe in Mössbauer Spectroscopy In our Mössbauer spectroscopy experiment we use 57 Fe. Our γ-ray source is 57 Co, which decays to 57 Fe via electron capture as shown in Figure 1. When the iron decays from the I=3/2 to the I=1/2 state a 14.4-keV gamma ray is emitted, which is the photon of interest. For absorbers we use foils of stainless steel and natural iron. The resolution of the spectrometer is characterized by the ratio E. The first E excited state of 57 F e, has a natural line width Γ = 2.4 10 9 ev, and the transition produces a photon of 14.4-keV. Thus, the ideal resolution of our apparatus is 1.7 10 13. The actual resolution of the spectrometer is approximately the measured linewidth of the stainless steel. 2.5 Isomer Shift If the chemical environment of the iron nuclei in the source and in the absorber are different, then the electron densities around the nuclei will be different. The electromagnetic interaction between the electrons and the nucleus depends on the 4

Figure 1: Decay scheme of 57 Co. The cobalt undergoes electron capture, yielding an electron neutrino and an 57 Fe atom in an excited state. The iron undergoes a number of nuclear transitions, one of which yields a 14.4-keV gamma ray. This is the gamma ray of interest. electron density, so if the host materials are different there will be a shift in the resonance energy from the source to the absorber, which can be seen in Figure 2 [3]. Figure 2: Isomer shift due to differences in the chemical environments of the emitter and absorber. We see this effect when we take a Mössbauer spectrum of stainless steel. The iron nuclei created in the cobalt have a different host material than the iron nuclei in the stainless steel. This causes a shift in the absorption spectrum of the stainless steel. 2.6 Nuclear Zeeman Effect The nucleus of an iron atom in natural iron and certain other iron compounds is in a strong magnetic field caused by the electrons of the atom and of neighboring atoms. Since both of the nuclear energy levels of the 14.4-keV transition have spin and associated magnetic moments, they experience hyperfine splitting caused by the 5

interaction of the magnetic moment of the nucleus with the magnetic field, as shown in Figure 3. These new levels are the magnetic substates of the energy levels, which are degenerate in the absence of a magnetic field, and this results in more possible nuclear transitions. The transitions that are allowed are determined by the quantum selection rule, which states that m I = 0, ±1 [2]. Figure 3: The isomer shift and nuclear Zeeman effect in 57 Fe. The change in energy caused by the interaction of a magnetic moment with an external magnetic field can be expressed as E = µ B, (8) where µ is the magnetic moment and B is the strength of the magnetic field. This is the expression holds for any system. When quantum mechanics is applied to the case of a magnetic moment of a nucleus interacting with an external magnetic field, the change in energy can be written as ( ) mi E = gµ N B z, (9) I where g is the g-factor or gyromagnetic ratio, µ N is the nuclear magneton, B z is the component of the magnetic field in the direction of the magnetic moment, m I is the 6

magnetic substate, and I is the spin of the state. This expression gives us the shift in energy due to the nuclear Zeeman effect. The energy of the resulting transitions is E = (E e + E e ) (E g + E g ), (10) where E e and E g are the energies of the excited and ground states in the absence of a magnetic field, and E e and E g are the shifts in the energy levels due to the nuclear Zeeman effect. This expression can be rewritten as E = E e E g + E e E g. (11) If we set E e E g = E 0, (12) and E e E g = E T, (13) the expression becomes E = E 0 + E T, (14) where E 0 is the energy of the transition if the levels were not split, and E T is the energy shift caused by the nuclear Zeeman effect. 7

2.7 Design of Mössbauer Spectrometer A schematic of a Mössbauer spectrometer is shown in Figure 4. It consists of a source, an absorber, and a detector. The source and the detector are stationary, and the absorber moves at constant velocity. We are able to vary the velocity of the absorber, and it is able to move toward and away from the source. We get the gamma ray count rate from our detector and we plot this as a function of the absorber velocity. When the Doppler shifted gamma rays have energy that corresponds to a nuclear transition energy, the photons will be absorbed and then re-emitted in all directions. At these velocities we will see a decrease in the gamma ray count rate in the detector. Figure 4: A basic Mössbauer spectrometer. 3 Experimental Procedure During the course of developing the Mössbauer spectrometer we worked with three different setups. All utilize the same essential electronics, but the method of Doppler shifting the gamma rays differs. The procedures are very similar, as are the data, but there are advantages and disadvantages to each system. 8

3.1 Austin Science Associates Setup A Mössbauer experiment was performed in the Department a number of years ago, using equipment purchased in the early 1990 s from a company named Austin Science Associates, which is no longer in operation. 3.1.1 Setup and Procedure Our initial spectrometer utilized a Mössbauer drive and linear motor made by Austin Science Associates (ASA) as a motion source, as shown in Figure 5. A rod runs through the linear motor, and the 57 Co source was attached to one end. The linear motor vibrates the rod along its axis at a constant velocity, which is regulated by the Mössbauer drive. Figure 5: Diagram of the Mössbauer spectrometer that utilized the Austin Science Associates Mössbauer drive and linear motor. The absorber is placed between the linear motor and the detector, which is a krypton gas proportional counter. The Ortec [4] high voltage power supply provides 9

a positive potential on the wire relative to the walls of the detector that sets up an electric field in the detector. When gamma rays enter the detector they collide with the krypton atoms, stripping them of some electrons. These electrons are attracted to the wire, which is at a positive potential relative to the cylindrical housing, and are accelerated toward it. This causes a current pulse to be emitted from the detector and is sent to the preamplifier, where it is integrated into a voltage pulse. The height of this pulse is proportional to the number of electrons collected on the wire, which is proportional to the energy of the gamma ray. This signal then goes through an Ortec amplifier, which simply amplifies the signal to useful voltages, and then the signal is sent to an Ortec single channel analyzer (SCA). The signal is also sent to a multi-channel analyzer PCI card on a computer. The gamma ray energy spectrum was acquired with the MCS-32 software on the computer, and is shown in Figure 6. The signals from the amplifier and the SCA also go to the PC. The SCA emits a square pulse for every input pulse within a voltage range that is set on the front panel. The SCA is very important for this experiment because it assures us that we are only counting the gamma rays of interest. Since the voltage of the pulse is proportional to the energy of the gamma ray, we are discounting any gamma rays outside of our range of interest. The signal from the SCA goes to the computer as well, and the gamma ray energy spectrum gated on the 14.4 kev gamma ray is shown in Figure 7. There are also signals sent between the Mössbauer drive and the linear motor. The Mössbauer drive sends a saw-toothed drive signal to the motor. This signal is essentially a displacement versus time graph and dictates the motion of the motor. Simultaneously, the motor sends a square velocity signal back to drive. This signal shows velocity versus time, and is used by the Mössabauer drive as a gate to block gamma ray signals obtained during periods of changing velocity. 10

Figure 6: Gamma ray energy spectrum from the 57 Co source. Figure 7: Gamma ray energy spectrum gated on the 14.4 kev peak. 11

The square signal from the SCA passes through the Mössbauer drive, which gates the signal based on the velocity of the motor, and into the (Tennelec) counter-timer, which counts the number of pulses for a predetermined time interval. In order to obtain a Mössbauer spectrum, we run the motor at a range of velocities, each for a predetermined period of time, and record the number of gamma rays counted for each velocity. We were able to run the motor at constant velocities ranging from -9.99 mm/s to +9.99 mm/s, at intervals of 0.01 mm/s, and we generally counted for 15 seconds at each velocity. 3.1.2 Data A Mössbauer spectrum for stainless steel is shown in Figure 8. This spectrum clearly shows one distinct dip in gamma ray count of about 45%. In the absence of any electromagnetic effects due to the environment of the nucleus, the dip would be expected at 0.0 mm/s, but is offset a little due to the isomer shift in the steel. There is one point at -0.1 mm/s which appears to be showing some unexpected behavior, but after further investigation this point has been determined to be an anomoly. A Mössbauer spectrum for natural iron is shown in Figure 9. We see six distinct dips in the gamma ray count ranging from 10-15%. These six dips are due to the nuclear Zeeman effect, and each one corresponds to a transition in the iron nuclei. Like the stainless steel, the dips are not symmetric around 0.0 mm/s due to the isomer shift in the iron. 3.1.3 Advantages and Disadvantages One of the great advantages to using this setup was that the Mössbauer drive has a Constant Acceleration setting. Instead of recording data at each velocity for a certain period of time, we could run the motor through a range of velocities during 12

Figure 8: Mössbauer spectrum of stainless steel acquired with the Austin Science Associates Mössbauer spectrometer. Figure 9: Mössbauer spectrum of natural iron acquired with the Austin Science Associates Mössbauer spectrometer. 13

each cycle. The data from the detector and from the Mössbauer drive could be sent into a multi-channel analyzer card on a computer, which could create a Mössbauer spectrum automatically. This would make data acquisition a much easier process, but it might not be as illuminating an experience for the students performing this experiment in the laboratory course. One of the disadvantages of this setup is that it is not very transparent for the students. The motor does not move the rod very far, about 1 cm at most, and it moves very quickly, so it might be difficult for students to grasp the idea of the Doppler shift due to the motion of the source. Also, the velocity scale would need to be calibrated. By looking at Mössbauer spectra acquired by other researchers we knew that the velocity we set on the Mössbauer drive was not the actual velocity. In order to make this setup usable we would have had to develop a method for velocity calibration. The main disadvantage of this setup, however, is that it stopped working. We acquired acceptable data, and this would have been useful for the laboratory course, but unfortunately the Mössbauer drive has ceased operating. 3.2 Speaker-Driven Setup Many Mössbauer spectrometers have been made using a speaker as the source of motion for the Doppler shifting, so we decided to try using a speaker and function generator produced by PASCO Scientific [5], which are primarily used for introductory physics courses. 3.2.1 Setup and Procedure After the Mössbauer drive stopped working we wanted to develop something sim- 14

ilar, using some kind of vibrator and a function generator to achieve the motion we needed. We decided upon a PASCO mechanical vibrator, which is essentially a speaker with a protruding rod, and a PASCO function generator, as shown in Figure 10. We set the function generator to send a triangular wave to the speaker, which would represent a displacement versus time graph and give the speaker a constant velocity over certain time intervals. An interferometer would have been developed to measure the velocity of the vibration, but since we did not get this far with the speaker setup the data were taken with respect to the voltage of the signal, which is proportional to the velocity. Figure 10: Diagram of the Mössbauer spectrometer that utilized a PASCO mechanical vibrator(speaker) and PASCO function generator. The function generator and the speaker substitute for the linear motor and the drive signal from the Mössbauer drive, but we needed a way to gate the signal from the detector so that we would only be counting gamma rays that were detected while the speaker was moving at a constant velocity in the correct direction. We found that the TTL output from the function generator was a square wave with the same 15

frequency as the triangular wave sent to the speaker, as seen in Figure 11. Figure 11: Output signals from the PASCO function generator We used a BNC Pulse/Delay Generator [6] to shift the TTL signal left or right by a quarter of a wavelength, and it was then sent to the counter-timer as a gate signal. The detector, preamplifier, amplifier, and SCA operated in exactly the same way as in the ASA setup. Data acquisition for this setup was very similar to the ASA setup. Instead of selecting velocities, we selected voltages, and we ran the speaker for a predetermined period of time at each voltage. The voltages ranged from -0.3 V to 0.3 V, and we counted for 20 seconds at each voltage. 3.2.2 Data The Mössbauer spectrum for stainless steel obtained with the speaker driven setup is shown in Figure 12. It is very similar to the spectrum taken with the ASA setup. There is one distinct dip in gamma ray count of about 42%, and it is slightly offset from 0.0 V due to the isomer shift in the steel. The Mössbauer spectrum for natural iron obtained with this setup is shown in Figure 13. In this spectrum we found that there is something happening at the high positive voltages (velocities). We can see what could be the six dips resulting from the nuclear Zeeman effect, but the two on the right seem deformed. The same deformation was found in subsequent spectra, which means there is some systematic 16

Figure 12: Mössbauer spectrum of stainless steel acquired with the speaker-driven Mössbauer spectrometer. problem with this setup. We have come to the conclusion that this is happening because the oscillation of the speaker is not symmetric about zero voltage, and it is reaching its upper boundary when it is run at high positive voltages. Despite this deformation, we still have a few distinct dips ranging from about 10-14%, as well as an offset from 0.0 V due to the isomer shift in the iron. 3.2.3 Advantages and Disadvantages One of the advantages to the speaker setup is that it was very similar to the ASA setup which had already been configured and had produced good results. We had also considered developing a constant acceleration mode for this setup as well, which would automate the data acquisition. Another aspect of this setup that we liked was that for at least part of the setup we were able to use equipment like the PASCO mechanical vibrator and function generator, which are commonly found in undergraduate physics labs. This would 17

Figure 13: Mössbauer spectrum of natural iron acquired with the speaker-driven Mössbauer spectrometer. make our experiment more easily reproduced by other groups. The disadvantages of this setup are essentially the same as the ASA setup. Had it worked correctly, it would still not be very transparent to the students. We would also have had to develop some system to calibrate the velocity of the speaker, which would make the experiment much more elaborate. And, of course, the speaker failed to vibrate correctly. This problem might have been fixed, but we decided to try a different technique. 3.3 Mössbauer Effect Analyzer Setup We are currently using an old piece of equipment produced by Nuclear Science and Engineering Corporation to provide the Doppler shift. 3.3.1 Setup and Procedure We finally decided on using the Mössbauer Effect Analyzer(MEA), shown in Fig- 18

ure 14, to provide the constant velocity motion needed for the spectrometer. In the MEA, unlike in the other setups, the absorber moves instead of the source. The absorber is attached to a base that is mounted on a screw. At the left end of the analyzer there is a pulley, which is attached by an O-ring to a pulley on the motor. The velocity is determined in the motor by manipulating the gear ratio, and is transferred to the screw via the system of pulleys. The direction of motion is set on the control unit. Figure 14: Diagram of the Mössbauer spectrometer currently in use, which utilizes the Mössbauer Effect Analyzer. Since the motion is achieved here not by vibration but by steady motion in one direction there is no need for a gate signal as in the previous setups. The remaining electronics all serve the same function they did in previous setups. The absorber is moved in one direction through a range of constant velocities for predetermined amounts of time and the gamma ray count is recorded at each velocity. The velocities achieved with the MEA range from -10 mm/s to 10 mm/s, with increments of 0.05 19

mm/s, and we generally counted for 20 seconds. In order to calibrate the velocity of the MEA, we set up two photogates a known distance apart, and measured the time for the absorber to pass between them. We found that the velocity of on the MEA dial was accurate to within 1%, and we decided that cailbration was unnecessary. 3.3.2 Data Figure 15: Mössbauer spectrum of stainless steel acquired with the Mössbauer Effect Analyzer. The Mössbauer spectrum for stainless steel obtained with the MEA is shown in Figure 15. There is one distinct dip of about 22%, and it is slightly offset from 0.0 mm/s due to the isomer shift in the steel. The Mössbauer spectrum for natural iron obtained in the setup is shown in Figure 16. This plot clearly shows the six distinct dips, ranging from 4-11%, which correspond to the six transitions in the iron nuclei caused by the nuclear Zeeman 20

Figure 16: Mössbauer spectrum of natural iron acquired with the Mössbauer Effect Analyzer. effect. As seen in all other spectra, there is a slight offset from 0.0 mm/s due to the isomer shift in the iron. 3.3.3 Advantages and Disadvantages The main advantages to this setup are that it is very transparent for students, and that is yields clear and accurate results. The motion in this setup is much more obvious than in the previous setups, and this should help the students grasp the idea of Doppler shifting the energy of the gamma rays. Alternatively, data acquisition with this method can be rather tedious, and although it may be a worthwhile project for the future, we do not currently plan on developing a more automated method. 21

4 Data Analysis In the lab handout for this experiment, which can be found in Appendix A, we ask the students to extract several quantities from the Mössbauer spectra they acquire. For the stainless steel absorber, they are instructed to find the linewidth of the transition and the isomer shift. For the natural iron they are instructed to find the isomer shift, the g-factor of the excited state given the g-factor of the ground state, and the strength of the magnetic field at the nucleus. 4.1 Stainless Steel The Mössbauer spectrum for stainless steel acquired with the MEA setup is shown in Figure 17. In order to accurately determine the centroids of the dips, we utilized the PeakFit software [7], which gives us values for the centroids with uncertainty. Figure 17: Mössbauer spectrum for stainless steel acquired with the MEA setup. We determined the linewidth of the transition using the expression 22

Γ = v fwhm E γ, (15) c where v fwhm is the full width at half maximum of the dip in the stainless steel spectrum, and E γ = 14.4 kev. The linewidth was determined to be (1.05 ±.04) 10 8 ev. The isomer shift in stainless steel was determined using the expression E isomer = v isomer E γ, (16) c where v isomer is the offset from 0 mm/s in the spectrum. The isomer shift in stainless steel was determined to be ( 1.20 ±.05) 10 8 ev. 4.2 Natural Iron The Mössbauer spectrum for natural iron acquired with the MEA setup is shown in Figure 18. Figure 18: Mössbauer spectrum for natural iron obtained using the MEA setup. 23

We will use Equation (16) to determine the isomer shift in natural iron. In order to determine v isomer we averaged the offsets of the three sets of dips. We found the isomer shift of the natural iron to be ( 5 ± 1) 10 9 ev. In Equation (14) we defined the energy shift of the transition to be E T = E e E g, (17) where E T is the energy of the transition, E e is the energy shift of the excited state, and E g is the energy shift of the ground state. If we substitute using Equations (7) and (9) we find v c E γ = g e µ N B z ( MIe I e ) ( ) MIg + g g µ N B z, (18) I g where the subscripts e and g refer to the excited state and ground state respectively. Solving for the velocity at which each peak occurs we find v MIg M Ie = g e a ( MIe I e ) + g g a ( MIg I g ), (19) where a = c E γ µ N B z. We will use these expressions for the velocities of the peaks to find an expression that relates g g and g e v 1 2 3 v 1 2 2 1 2 v 1 2 1 2 which yields the expression v 1 2 1 2 = g ea + g g a + 1 3 g ea g g a 1 3 g ea + g g a 1 3 g ea + g g a, (20) g e = 3g g v 1 2 3 2 v 1 2 1 2 v 1 2 1 2 v 1 2 1 2. (21) 24

Therefore, knowing the ground state g-factor to be 0.09044, we found the excited state g-factor to be 0.157 ±.003, which is consistent with the accepted value of -0.1549. Finally, we want to find the value of the magnetic field strength at the nucleus of the iron atom. In order to do this we must compensate for the isomer shift in the iron using the expression v MIg M Ie v isomer = c E γ µ N B z Solving for the magnetic field we get [( MIg I g ) g g ( MIe I e ) g e ]. (22) B z = E γ ( vmig M Ie v isomer) cµ N [( MIg I g ) gg ( M Ie I e ) ge ]. (23) We found the magnetic field at the nucleus of the iron atom to have a strength of 32.4 ±.1T. 5 Conclusions In this experiment we successfully developed a Mössbauer spectrometer, which is being used in an upper-level undergraduate experimental physics course. Mössbauer spectroscopy is ideal for undergraduates because it allows students to examine quantum mechanical effects which they learn about in introductory and intermediate courses. Mössbauer spectroscopy utilizes the principles of recoilless resonant absorption and the Doppler effect in order to use gamma rays as a very sensitive probe, which we can use to measure certain properties of the nucleus. In the course of developing the spectrometer we tried three different setups before 25

we found one that yielded accurate results and was consistent. In our first Mössbauer spectrometer we used components produced by Austin Science Associates. This setup yielded good data, but eventually ceased to operate properly. In an attempt to recreate he operation of the ASA setup, we substituted a PASCO Scientific mechanical vibrator and and function generator. While this apparatus yielded similar results, there was a problem with the motion of the positive velocity. This caused the positive end of the spectra to be distorted. We settled on an old device that had been used in the Department a number of years ago which was produced by Nuclear Science and Engineering Corporation. This setup yields accurate, consistent results, and operates in a way which is very transparent to the students. In this experiment the students are instructed to obtain Mössbauer spectra of stainless steel and natural iron, and to measure the linewidth and isomer shift of stainless steel, and the isomer shift, the g-factor of the first excited state given the g- factor of the ground state, and the value of the magnetic field strength at the nucleus for the natural iron. We determined the linewidth of stainless steel to be 1.05 ±.04 10 8 ev, and the isomer shift in stainless steel to be 1.20 ±.05 10 8 ev. For natural iron we found the isomer shift to be 5 ± 1 10 9 ev. We determined the g-factor of the excited state of natural iron to be.157 ±.003, which is consistent with the accepted value of.1549 [8]. The magnetic field strength at the nucleus was determined to be 32.4 ±.1T. This experiment has already been performed by students in the Modern Experimental Physics course. It was apparent that the students understood the operation of the apparatus as well as the physical properties that they were examining. Their data was very similar to the data shown above, and their results should agree the 26

values we determined. There is one aspect of the apparatus that will require future work to sustain this experiment. The 57 Co source has a half-life of 270 days, and had a radioactivity of 10mCi when it was purchased. As the source weakens we will need to count gamma rays for longer periods of time in order to obtain sufficient data. This time will increase to a point when the absorber will not be able to run at the higher velocities because it will hit one end of the device. A future project on the Mössbauer spectrometer will be to devise a system to allow the absorber to run continuously back and forth in the apparatus in order to count for longer periods of time. The control unit of the MEA allows for manual and automatic control of the absorber direction. Currently we utilize the manual setting, and select the desired direction. On the automatic setting, the absorber will automatically change direction when it reaches either end of the apparatus. One way to allow for longer count times would be to use two counters, and count for both the positive and negative of a given velocity at the same time, allowing the absorber to run back and forth for as long as desired. A mechanism would need to be developed to determine which direction the absorber is moving and to select the correct counter to send the signal to. 27

Appendix A Mössbuer Spectroscopy Physics 300 Winter 2006 1.1 Background Information Mössbauer spectroscopy is a method for measuring small shifts in nuclear energy levels with high precision. This method involves the recoilless emission and absorption of gamma rays and utilizes the Doppler effect in order to use these gamma rays as a sensitive probe. 1.1.1 Resonant Absorption A quantum system can undergo a transition when it absorbs or emits a photon of a specific energy. The energy of the photon is dependent upon the difference in the energy levels of the transition. The energy of the ground state is absolute, but the energy of the excited state is not a precisely defined quantity. Due to the Uncertainty Principle, the natural line width is given by Γ = h 2τ, (2) where Γ is the natural line width of the excited state and τ is the lifetime of the state. However, when a photon is emitted by a free system not all the transition energy goes into the photon. This is because the system recoils due to conservation of momentum, and takes with it some of the transition energy in the form of kinetic 28

energy. If we call the energy of the transition E T and the recoil energy E R, then the energy of the emitted photon can be expressed as E γ = E T E R. (3) We can rewrite this expression in terms of the recoil momentum as Due to conservation of momentum we see that E γ = E T p2 R 2m. (4) where p γ is the momentum of the photon. is p R = p γ = E γ c, (5) By combining Equations (4) and (5) we find that the energy of the emitted photon E γ = E T E2 γ 2mc 2. (6) Likewise, when the photon is absorbed the absorbing system recoils. Therefore, the distributions of the emission and absorption energies are separated by twice the recoil energy. The probability of resonant absorption is proportional to the overlap of these distributions. In atomic systems this probability is very high because the recoil energy is small compared to the natural line widths. In nuclear systems, however, the recoil energy is much larger than the natural line widths, and therefore the probability of resonant absorption is very small. 29

1.1.2 Recoilless Emission & Absorption In 1958, Rudolf Mössbauer showed that for atoms bound in a lattice, a nucleus doesn t recoil individually. The recoil momentum, therefore, is taken up by the entire lattice, which has a very large mass. From equation (6) we see that as m, E γ E T. Therefore, when the nucleus is embedded in a massive substrate the recoil energy is negligible. This concept applies to absorption of photons as well. 1.1.3 Doppler Shift When there is relative motion between the emitter and the absorber there is a Doppler shift in photon energy. The energy of the photon is given by the Lorentz transformation E γ = 1 (E 1 + β 1 β 2 γ + vp γ ) = E γ, (7) 1 β 2 where β = v, E c γ and p γ are the energy and momentum of the emitted photon, v is the relative velocity between the emitter and the absorber, and E γ is the resulting Doppler shifted photon energy. For β 1 we take the first order of the binomial expansion to get E = E γ E γ = βe γ = v c E γ (8) as an expression for the change in photon energy due to the motion. If we are able to vary the relative velocity between the emitter and absorber, we are able to scan a range of photon energies. In Mössbauer spectroscopy we measure the absorption rate as a function of velocity, which can be converted to energy shift. Analysis of the absorption spectrum yields information about the structure of the 30

nucleus. 1.1.4 Use of 57 Fe in Mössbauer Spectroscopy In our Mössbauer spectroscopy experiment we use 57 Fe. Our γ-ray source is 57 Co, which decays to 57 Fe via electron capture as seen in Figure 2. When the iron decays from the I=3/2 to the I=1/2 state a 14.4-keV gamma ray is emitted, which is the photon of interest. For absorbers we use foils of stainless steel and natural iron. The resolution of the spectrometer is characterized by the ratio E. The first E excited state of 57 F e, has a natural line width Γ = 2.4 10 9 ev, and the transition produces a photon of 14.4-keV. Thus, the ideal resolution of our apparatus is 1.7 10 13. The actual resolution of the spectrometer is approximately the natural linewidth of the stainless steel. Figure 2: Decay scheme of 57 Co. The cobalt undergoes electron capture, yielding an electron neutrino and an 57 Fe atom in an excited state. The iron undergoes a number of nuclear transitions, one of which yields a 14.4-keV gamma ray. This is the gamma ray of interest. 1.1.5 Isomer Shift If the chemical environment of the iron nuclei in the source and in the absorber are different, then the electron densities around the nuclei will be different. The electromagnetic interaction between the electrons and the nucleus depends on the 31

electron density, so if the host materials are different there will be a shift in the resonance energy from the source to the absorber, which can be seen in Figure 3. Figure 3: Isomer shift due to differences in the chemical environments of the emitter and absorber. We see this effect when we take a Mössbauer spectrum of stainless steel. The iron nuclei created in the cobalt have a different host material than the iron nuclei in the stainless steel. This causes a shift in the absorption spectrum of the stainless steel. 1.1.6 Nuclear Zeeman Effect The nucleus of an iron atom in natural iron and certain other iron compounds is in a strong magnetic field caused by the electrons of the atom and of neighboring atoms. Since both of the nuclear energy levels of the 14.4-keV transition have spin and associated magnetic moments, they experience hyperfine splitting caused by the interaction of the magnetic moment of the nucleus with the magnetic field, as shown in Figure 4. These new levels are the magnetic substates of the energy levels, which are degenerate in the absence of a magnetic field, and this results in more possible nuclear transitions. The transitions that are allowed are determined by the quantum selection rule, which states that m I = 0, ±1. The change in energy caused by the interaction of a magnetic moment with a magnetic field can be expressed as 32

Figure 4: The isomer shift and nuclear Zeeman effect in 57 Fe. E = µ B, (9) where µ is the magnetic moment of the state and B is the strength of the magnetic field. This is the expression holds for any system. When quantum mechanics are applied, this expression can be rewritten as ( ) mi E = gµ N B z, (10) I where g is the g-factor or gyromagnetic ratio, µ N is the nuclear magneton, B z is the component of the magnetic field in the direction of the magnetic moment, m I is the magnetic substate, and I is the spin of the state. This expression gives us the shift in energy due to the nuclear Zeeman effect. The energy of the resulting transitions is E = (E e + E e ) (E g + E g ), (11) where E e and E g are the energies of the excited and ground states, and E e and E g are the shifts in the energy levels due to the nuclear Zeeman effect. This expression 33

can be rewritten as E = E e E g + E e E g. (12) If we set E e E g = E 0, (13) and E e E g = E transition, (14) the expression becomes E = E 0 + E transition, (15) where E 0 is the energy of the transition if the levels were not split, and E transition is the energy shift caused by the nuclear Zeeman effect. 1.1.7 Design of Mössbauer Spectrometer A schematic of a Mössbauer spectrometer is shown in Figure 5. It consists of a source, an absorber, and a detector. The source and the detector are stationary, and the absorber moves at constant velocity. We are able to vary the velocity of the absorber, and it is able to move toward and away from the source. We get the gamma ray count rate from our detector and we plot this as a function of the absorber velocity. When the Doppler shifted gamma rays have energy that corresponds to a nuclear transition energy, the photons will be absorbed and then re-emitted in all directions. At these velocities we will see a decrease in the gamma ray count rate in 34

the detector. Figure 5: A basic Mössbauer spectrometer. 1.1.8 References For more information see the following: Krane, Kenneth S., Introductory Nuclear Physics Melissinos, Adrian C. and Jim Napolitano, Experiments in Modern Physics Mossbauer Effect: Selected Reprints, American Institute of Physics 1.2 Procedure for Mössbauer Spectroscopy The following section explains how to perform the experiment. There are important notes at the end of each section, so please read them carefully. A diagram of the experimental setup is shown in Figure 6. 1.2.1 Setting Up the Experiment 1. Affix the 57 Co source to the apparatus. 2. Turn on the Mössbauer Effect Analyzer Control Unit, the oscilloscope, the NIM bin, and the high voltage power supply (HVPS). 35

Figure 6: Diagram of the Mossbauer Effect Analyzer system we use in this experiment. 3. Settings for electronics: (a) Counter-Time: Count/Stop switch - Count, Timer switch - 0.1 sec (b) Amplifier: Gain - 0.5, Course Gain - 100, Shaping Time - 0.5 (c) SCA: Upper Level - 3.9, Lower Level - 2.5, Mode - NORM (d) Delay Amplifier - 0.5 µsec switch ON 4. Set the voltage on the HVPS to +1800V. 5. View output from the SCA and the Delay Amplifier on the oscilloscope. (a) Oscilloscope settings: Ch1-2V, Ch2-2V, M - 500ns (b) Make sure that 0.5 µsec is the correct delay for the peak from the Delay Amplifier to fall as closely within the SCA signal as possible. 6. Connect the output from the SCA to the Gate In input on the MCA, and the output from the Delay Amplifier to the SCA In input on the MCA. 36

7. Run MCS-32 program. (a) Go to Acquire SCA Sweep... (b) Click Yes when asked about sweeping SCA Input. (c) Verify that the 14.4-keV peak is isolated. i. You can verify this by disconnecting the SCA Output from the Gate Input on the MCA and comparing the spectra. ii. The 14.4-keV peak is the last peak. (d) If the peak is not isolated accurately adjust the Upper and Lower Level dials on the SCA. 8. Affix absorber to the holder on the apparatus. Notes: 1. THE 57 Co SOURCE IS RADIOACTIVE. Always be careful when working with or handling it. Keep it away from your body while handling it, and make sure to use the lead bricks to create a wall between the source and yourself while performing the experiment. 2. The window of the detector is very sensitive and will break easily. DO NOT touch it, and be careful when working near it. 1.2.2 Taking Data 1. Select the desired velocity on the motor. 2. Select the desired direction on the control unit. 3. Push the Reset button on the Counter-Timer to begin counting. 37

4. When the Counter-Timer has stopped counting, record the velocity and gamma ray count in a spreadsheet. The velocities for the absorber moving away from the source are defined to be negative. 5. Repeat for the range of velocities of interest. Notes: 1. The smallest accurate velocity increment is 0.05 mm/s. 2. When the O ring is in the outer pulley grooves the dial indicates the velocity directly. When the O ring is in the inner pulley grooves the dial indication divided by 2 is the velocity (i.e. 8 mm/s on the dial is actually 4 mm/s). 3. For velocities under 5 mm/s, the O ring should be in the inner pulley grooves. 4. For best results, try to keep the absorber the same distance from the detector for all velocities, especially for small velocities. 5. The time the Counter-Timer will count for is determined by: 0.1 sec top dial setting bottom dial setting. 1.2.3 Finishing Up 1. Set the voltage on the HVPS to 0V. When the voltage runs down turn off the HVPS. 2. Turn off the NIM bin, the oscilloscope, and the Mössbauer Effect Analyzer Control Unit. 3. Return the 57 Co source to its lead capsule and return this to its lead housing. 38

1.3 Data Analysis Measure the absorption spectrum on stainless steel and extract the linewidth and the isomer shift in energy. Measure the absorption spectrum on natural iron and extract the isomer shift, the g-factor of the first excited state given the g-factor of the ground state (0.09044 ± 0.00007), and the value of the magnetic field strength at the nucleus. 39

References [1] R.L. Mossbauer. Gammastrahlung in ir 191. Z. Physik, 151:124, 1958. [2] Adrian C. Melissinos and Jim Napolitano. Experiments in Modern Physics. Academic Press, 2003. [3] Mossbauer Effect: Selected Prints. American Institute of Physics, 1963. [4] Ortec. http://www.ortec-online.com/. [5] PASCO Scientific. http://www.pasco.com/. [6] Berkeley Nucleonics Corporation. http://www.berkeleynucleonics.com/. [7] Systat Software Inc. http://www.systat.com/products/peakfit/. [8] N.J. Stone. Table of Nuclear Magnetic Dipole and Electric Quadropole Moments. Oxford. 40