Van Bistrow, Department of Physics, University of Chicaqgo. Experiment VI. Electron Spin Resonance

Transcription

1 Expeiment VI Electon Spin Resonance Intoduction In this expeiment we will study one classical ßpaticle and one quantum mechanical paticle. In paticula, we will choose paticles having the common popeties of angula momentum and magnetic moment. The objective is to study how these paticles behave in extenally applied magnetic fields. The classical expeiment should illuminate the concepts used late in the quantum mechanical system. Classical System Fo the classical paticle, you will use a spinning billiad ball, containing a magnet embedded at its cente. The objectives ae to: a. place the ball in a magnetic field and detemine the ball s magnetic moment b. add angula momentum to the ball and obseve its motion c. detemine the elationships among the motion of the ball and the angula moment and magnetic moment. Classical Theoy Magnetic Moment Conside a loop of positive cuent I whose path encloses aea A, as in Fig

2 µ A I Fig. 1 Magnetic Dipole Moment of a cuent loop. The aea enclosed by the loop may be consideed a vecto: The magnitude of the vecto is just the aea. The diection of the vecto aea is given by the ight-hand ule, with the finges pointing in the diection of positive cuent flow. Then the thumb points in the diection of the esulting magnetic dipole moment. Then the magnetic dipole moment is given by: µ = A I (1) If we place a magnetic dipole moment in an extenal magnetic field, the dipole will expeience a toque given by whee B is the magnetic field. τ = µ B (2) Question 1: If the object having the magnetic moment (but no angula momentum) is fee to move, how will it move in the pesence of the magnetic field? Question 2: 64

3 If the object is given angula momentum, paallel o anti-paallel to its magnetic moment, and is placed it in a magnetic field, how will it move? To help us answe question 2, caefully conside Figue 2. Fig. 2 Toque acting on an object with angula momentum and magnetic moment in a magnetic field. (afte Eisbeg and Resnick, Quantum Physics) The magnetic field acts on the magnetic dipole moment to poduce a toque, given by eq.(2). This toque gives ise to a change in the angula momentum d L duing the time dt such that 65

4 τ = d L dt. (3) The change dl causes L to pecess though an angle ω dt, whee ω is the pecession angula velocity. Note fom Fig. 2 that dl = L sinθ ωdt thus, τ = µbsinθ = dl dt = ωl sinθ theefoe, ω = µ B. (4) L Eq.(4) is often e-stated as whee γ is called the gyomagnetic atio. ω = γb (5) Appaatus We will use the TeachSpin Magnetic Toque appaatus fo this ßclassical pat of the expeiment. The appaatus consists of: a. a contol box fo supplying cuent to the magnetic field coils, setting the diection of the fields poduced, tuning on the ai supply and contolling the stobe light. b. a pai of coppe wie coils which can poduce a magnetic field in the vetical diection. The elation between the cuent in the coils and the magnetic field poduced is: B = (1.36 ± 0.03) 10 3 Tesla/amp. (6) 66

5 c. a cue ball with an embedded magnet, a small black handle fo spinning the ball. The magnetic dipole is aligned paallel to the axis containing the black handle. d. an aluminum od with a steel tip fo holding a sliding plastic mass fo changing gavitational toque on the ball. e. venie calipes to measue the position of the sliding mass f. balance fo weighing the sliding mass. g. Movable index to indicate a stating and stopping position of pecession. h. Rotating saddle to povide a otating magnetic field, pependicula to the steady, vetical field. i. Stopwatch to measue the pecession peiod. Expeimental Pocedue We will test eq.(4) using the following pocedue. Measue µ. Fo this pat of the expeiment keep the angula momentum of the ball equal to zeo. Use gavitational toque balancing the magnetic toque to detemine µ. To do so, 1. Place the aluminum od in the cental hole of the cue ball s black handle, with the od s magnetic end inseted into the ball. 2. Place the sliding plastic mass on the aluminum od. 3. Set the magnetic field to point up, with the field gadient tuned off. 4. Adjust the cuent in the magnetic field coils so that the magnetic toque just balances the gavitational toque: 67

6 µ B = mg (7) whee is the distance fom the cente of the ball to the cente of the sliding mass, m, and g is the acceleation due to gavity. 4. Daw a sketch of the ball showing the vectos in eq. (7) and demonstate that eq.(7) educes to: Eq.(8) suggests a technique fo detemining µ : µb = mg. (8) 5. Move the sliding mass to about 10 positions along the aluminum od and, at Question 3: each position, detemine the magnetic field needed to balance the ball. Do you eally need to measue diectly to do this expeiment? Estimate uncetainties in you measuements and detemine µ, with its uncetainty. Adding angula momentum You may povide angula momentum to the ball by spinning it using the black handle. Recall that, fo a unifom, solid sphee, the angula momentum L is given by: L = 2 5 MR2 Ω, (9) whee M is the mass, R is the adius, and Ω is the spin angula velocity of the sphee. Note that fo you sphee, the angula momentum and magnetic moment ae paallel. 1. Measue the mass and adius of the ball. 2. Tun on the magnetic field and set it to some intemediate value. 68

7 3. Tun magnetic field gadient switch to the on position. With this setting the cuents in the uppe and lowe coils ae in opposite diections, poducing B = 0 at the cente of the appaatus. 4. Tun on the stobe light and set its fequency to about 5 Hz. Note that in ode to measue this fequency accuately the fequency counte must count fo seveal seconds. It updates evey 10 seconds. 5. Oient the ball so its black handle points towad the stobe light. 6. Spin the ball using the black handle and educe any wobble with you fingenail. 7. As the ball s angula velocity slowly deceases, the white dot on the ball s black handle will begin to appea stationay in the stobe light. You now know the angula velocity of the ball. Quickly set the position make as nea as possible to the black handle, tun off the field gadient and stat the stopwatch. 8. Measue the time it takes fo the ball to pecess one complete cycle. 9. Repeat steps 2 though 7 at 1/2 amp intevals. Estimate uncetainties in all measued quantities. 10. Plot Ω vs. B with uncetainties. 11. Check you expeimental esults fo consistency with eq.(4). Take uncetainties into account. Spin (Flip) Resonance This pat of the expeiment povides a qualitative demonstation of how the ball, having both angula momentum and magnetic moment, behaves in a otating magnetic field, pependicula to the constant, vetical B field. 69

8 Remove the position indicato and install the magnetic field saddle. This saddle povides a field of constant magnitude, which may be otated in the hoizontal plane. 1. With the vetical field set to a maximum, stat the ball spinning with its black handle midway between the ed dots on the saddle. As the ball pecesses in one diection, manually otate the saddle in the othe. Ty to move it smoothly and continuously. What effect does the otating field have on the pecessional motion? 2. With the black handle midway between the ed dots on the saddle, spin the ball again, but this time ty to otate the saddle in the same diection at an fequency diffeent fom the pecession fequency. (This is tough!). How does the otating magnet affect the pecession? 3. Repeat the expeiment with the saddle otating in the same diection at the same fequency as the pecession. This equies some pactice! How does the ball move now? Quantum mechanical system Electon Spin Resonance Theoy The electons in atoms ae bound in discete enegy states. Magnetic fields ae geneated within the atom by the obital motion of the electons aound the atom spin of the electons spin of the atomic nucleus. 70

9 If atoms ae placed in an extenally applied magnetic field, the inteactions of the applied field with the intenal fields listed above cause the enegy levels of the atoms to shift. Similaly, if the atoms ae placed in a solid, the magnetic fields poduced by neighboing atoms will also contibute to enegy level shifts. Electon Spin Resonance (ESR) is a technique fo inducing and detecting tansitions among enegy levels. Enegy level shifts ae induced by application of a known magnetic field, while tansitions among enegy levels is induced by application of electomagnetic adiation of a known fequency. It is found that only fo paticula combinations of magnetic field and fequency ae tansitions induced. Detection is accomplished by measuing the slight decease in enegy in the electomagnetic field which occus when the enegy is absobed duing the tansition. A lage ensemble of atoms is needed to absob sufficient enegy to be detectable. It should be noted that the net enegy shifts ae due to the total field: applied and neaest neighbo. Since we know the value of the applied field, it follows that measuing the fequencies at which esonances occu is a pobe into the details of the envionment of the solid sample at the atomic scale. The detailed study of solids using ESR is complex and beyond the scope of this couse. Theefoe, fo simplicity we will study a much simple system: ßfee electons, not 71

10 bound to an atom. The sample we will use is the molecule DPPH (diphenyl-picihydazyl), which has one, nealy fee, electon pe molecule. States of a fee electon in a magnetic field The electon is a spin 1/2 paticle, which means that if an electon is placed in a steady magnetic field, the electon will pecess about the applied magnetic field with two possible oientations, one as shown in Fig. 2 and the othe with the µ and L vectos evesed elative to B. In these two states, the magnitudes of the components of spin angula momentum paallel to the field (in the z-diection) ae ±h /2. The magnetic moments associated with these states ae µ z = ±gµ B /2, (10) whee µ B = eh/2m is called the ßBoh magneton. Fo the fee electon, fo which all the angula momentum is spin (athe than obital) angula momentum, g = The enegy of a magnetic dipole moment in a magnetic field is given by Thus, the enegy diffeence between the two states is ESR fo the fee electon Enegy pespective E = µ B. (11) ΔE = 2µ z B = gµ B B. (12) If we apply electomagnetic adiation with fequency f, such that hf = ΔE, (13) whee ΔE is given by eq.(12), we should induce tansitions between the two enegy states. 72

11 Angula momentum pespective Eq.(13) can also be witten hω = ΔE. (14) At esonance it tuns out that ω of eq.(14) is the same as that of Fig.2, i.e., the pecession angula velocity of the electon aound the magnetic field. Thus, photons of angula fequency ω caying angula momentum h, cause the electon s spin to flip. Question 4: Is angula momentum conseved in this pocess? Explain. Gyomagnetic atio of the electon Combining eqs.(12) and (14) gives ω = gµ B h 73 B. (15) It should be noted that, fo a fee electon in a magnetic field, the magnitude of the spin magnetic moment is and the magnitude of the spin angula momentum is µ = gµ B s s +1 ( ), (16) S = s s +1 ( ) h. (17) Thus, the atio of magnetic moment to angula momentum is µ S = gµ B h, (18)

12 which appeas also in eq.(15). Eq.(15) is often e-witten as ω = γb, (19) whee γ is called the gyomagnetic atio. Eq.(19) is analogous to eq.(5) fo the classical case. In both cases γ is the atio: magnetic moment/angula momentum. The facto g is equied by quantum mechanics. Appaatus We use the Daedalon ESR appaatus, consisting of 60 Hz AC powe supply fo Helmholtz coils tunable adio fequency oscillato with fequency and feedback contols Helmholtz coils, connected in paallel such that B=0.48xI, whee B is in milli- Tsesla and I is the sum of the cuents, in Amps, flowing in each coil. Sample pobe, containing DPPH, suounded by a coil Expeimental Pocedue Make the electical connections as shown in Fig.3: 74

13 Fig. 3 ESR electical connections. 1. Adjust the height of the pobe to be the same as the cente of the Helmholtz coils. 2. Slide the pobe though the side slot in the Helmholtz coil suppot, so that the axes of the pobe s RF coil and Helmholtz coils ae pependicula. 3. Set the Helmholtz coil cuent to about 2/3 of its maximum value. 4. Set the scope to display in voltage vs. time mode. Gound both channels 1 and 2 and move thei taces to the cente (zeo volt) line. Then DC couple both channels. 5. Set the tuning fequency to its minimum. 6. Set the oscillato to be maginal by adjusting the feedback to give a maximum signal. Note that if the RF oscillato stops oscillating, the fequency display 75

14 will ead zeo. If that happens, eadjust the fequency and/o feedback contols. 7. Measue V on channel 1 at which esonance occus on channel 2. Note that channel 1 is measuing the ßsense signal, fo which 1 volt is poduced by 1 Amp flowing fom the powe supply (1/2 amp flowing though each coil). This cuent should be used to calculate the magnetic field fom the elation given above. Estimate uncetainties in you measuements. 8. Incease the fequency and epeat steps 7 and 8 though the full ange of fequencies available. 9. Plot the esonant fequency vs. magnetic field. 10. Fom you data, obtain a value fo γ, the gyo-magnetic atio. Is this value consistent with eq.(15)? Question 5: What would you expect to happen to the ESR signal if the RF B field wee applied paallel to the diection of the Helmholtz field? Ty it! Refeence Quantum Mechanics, by Eisbeg and Resnick, p

15 Data Analysis V. Bistow 6/2006 Magnetic dipole moment of cue ball: Balance condition, gavitational toque=magnetic toque o mg = µb = µ mg B (m) 0 Magnetic Moment data y = x R= (m) B*10^-3 (T) Fom the slope of the vs. B plot and the values of m and g, we get: µ = slope mg 77

16 Angula momentum of cue ball: µ = 32.3 m T kg 9.8 m s 2 µ = 0.44 ± 0.02 kg m 2 / s 2 T L = IΩ whee L is the angula momentum, I is the moment of inetia, and Ω is the spin angula velocity of the cue ball. The stobe light fequency (and spin fequency) was set to 5.5 Hz. L = 2 5 MR2 2πf L = kg ( /2)2 m 2 (2π 5.5)s 1 L = kg m 2 /s Gyo-magnetic atio of cue ball ω = γb 78

17 2*pi/T (ad/s) *pi/T pecession vs. B using position indicato y = x R= *pi/T (ad/s) B )T) Fom the ω vs. B plot, γ = 298 ad/s/t. Independently, the atio µ/l goves: Thus, γ is consistent with µ/l at the 6% level. µ L = 0.44 kg m2 /s 2 T kg m 2 /s µ ad/s = 317 L T. 79

18 Spin Flip Obsevations Apply a otating B field, othogonal to the steady (vetical) field. a. Rotate field in diection opposite to pecession: no effect. b. Rotate field in same diection, but at diffeent fequency fom pecession: no effect. c. Rotate field in same diection, and at same fequency: Spin ßflips! RESONANCE! Electon Spin Resonance: Theoy pedicts:: ω = γb = gµ B h B γ = gµ B h = J T J s = ad/sec T ESR data gives: 80

19 Reson. Feq (MHz) 32 ESR data y = x R= Reson. Feq (MHz) B Field (G) Slope gives 2.88 MHz/Gauss Hz ad/sec γ = T = T 81