Pulsed Nuclear Magnetic Resonance Experiment

Transcription

1 Pulsed Nuclear agnetic Resonance Eperiment in LI Dept. of Phsics & Astronom, Univ. of Pittsburgh, Pittsburgh, PA 56, USA arch 8 th, Introduction. Simple Resonance heor agnetic resonance is a phenomenon found in magnetic sstems that possess both magnetic moments and angular momentum. he term resonance implies that we are in tune with a natural frequenc of magnetic sstem, here corresponding to the frequenc of groscopic precession of the magnetic moment in an eternal static magnetic field. Nuclear agnetic Resonance (NR is a ver important one among them, where the magnetic sstem is a nucleus. Since the nuclear magnetic resonance frequencies fall tpicall in the radio frequenc region, we use the term radio frequenc. hrough studing the nuclear magnetic resonance of a nucleus, we will get enough information about processes at the atomic and nuclear level, so NR is also a ver significant tool to eplore the tin internal structure of macromolecule with high resolution in modern science. Generall, we consider a nucleus that consists of man particles coupled together so that in an given state, the nucleus possesses a total magnetic momentum µ and a total angular momentum. In fact, the two vectors ma be taken as parallel, so that we can write: µ γ, where γ is called the gromagnetic ratio. For an given state of a nucleus, knowledge of the wave function would in principle enable us to compute both µ and. In quantum mechanics, both µ and are treated as operators, and we can define h I, where I is a dimensionless angular momentum operator, which has eigenvalues of either integer or half-integer. As we know, an component of I (for eample commutes with I, so we can specif simultaneous eigenvalues of I both I and I. Let us call the eigenvalues I and m respectivel. Of course, m ma be an of the I + values, I, I,..., I. So we can get an component of the operator µ and I along an arbitrar ' direction, which is: < Im µ Im' > γh < Im I Im' > ' '

2 Now if we introduce an eternal magnetic field, which will produce interaction energ µ. aking the field to be along the -direction, we have a ver simple Hamiltonian: Η γ h I with the allowed Eigen energ E γ h m, mi, I-,, -I. So the interaction between the magnetic field and the nucleus will result transitions between different energ levels as spectral absorption. ost commonl coupling used to produce nuclear magnetic resonance is an altering transverse magnetic field applied perpendicular to the static field at -direction. If the amplitude of the altering field is ', we get an additional perturbation term in the previous Hamiltonian of Η per γh ' I cos( ω t. he operator I has matri elements between states m and m', as < Im I Im'> will vanish unless m ' m ±. Consequentl, we will get that the allowed transitions are onl between the adjacent levels and give the absorption frequencω : h ω E γ h, or γ ω.. otion of Single Spin Sstem Now we start our discussion with the motion of a spin in an eternal magnetic field, and will produce a torque on the magnetic momentum µ of amount µ. So the angular momentum will satisf the following equation: Since d µ µ γ, we have d µ µ γ. his equation will hold no matter whether or not is time dependent, it tells us that at an instant, the changes in µ are perpendicular both µ and. If is time independent, the angle between µ and does not change, the vector µ therefore generates a cone (please refer to Figure.

3 µ Fig. Consider a general vector time function F( t i F ( t + j F ( t + k F ( t, here are corresponding to the unit aes vectors. Originall, the should be constant, but now we think the can change with time. As their lengths are fied, the can at most rotate. We assume that the rotate with an instantaneous angular velocit Ω, then we have d i Ω i, same with the unit vector j and k. i, j, k So the total time derivative of F(t is therefore d F( t i df + F d i df + j + F d j df + k d k + F df i + df j δ F + Ω F δt df + k + Ω ( i F + j F + k F Where the coordinate sstem δ F represents the time rate of change of F with respect to the δt i, j, k. using this formula, we can rewrite the equation of motion for µ in terms of a coordinate sstem rotating with an arbitrar angular velocit Ω :

4 δ µ δ µ Ω + Ω µ µ γ, or further µ ( γ + Ω µ γ ( +. δt δt γ From this equation, we can see that the motion of µ in the rotating coordinate obes the same equation as in the laborator frame if we just replace the actual magnetic field b the effective magnetic field Ω e +. γ We can now readil solve for the motion of µ in a static field b choose a specific Ω, which makes e. hat is, Ω γ k. So in this rotating referent frame, δ µ we have, µ remains fied with respect to i, j, k, but in the laborator frame, δt γ rotates at angular frequenc Ω k. γ is called Larmor Frequenc. Now we can reach a surprise conclusion that the classical precession frequenc Ω is identical in magnitude with the angular frequenc needed for magnetic resonance absorption, as found b view of quantum mechanics. k µ Nerveless, instead of having just one single spin, if we have a group of spins with moments µ, for the kth spin, their total magnetic moment µ can be written as: µ k µ k k. If the spins do not interact with each other, it is easil to prove that the d µ equation µ ( γ still holds. So the eperimentall determined bulk magnetiation is simpl the epectation value of the total magnetic moment. herefore, the classical equation correctl describes the dnamics of the magnetiation, provided the spins ma be thought of as not interacting with one another. 3. Alternating agnetic Fields Now let s consider the effect of the altering magnetic field along the -direction (generall, we create this altering field b using a AC-coil: ( t cos( ω t i

5 And this magnetic field can be divided into two rotating components, each of amplitude, one rotating clockwise and the other rotating anti-clockwise. We denote the two rotating fields b R and L : R L j sin( ω t [ i cos( ω t j sin( ω t [ i cos( ω t + ] ] Near to the resonance regionω ω, one of the components will contribute heavil to transition absorption, the other can be neglected since its frequenc is different b ω. Here we assume we onl have the field L, and the component of frequenc along the -ais, so ω ω k. We ma therefore write the transverse magnetic filed as: ω ( t [ i cos( ω t j sin( ω t ] eanwhile we have a static filed along -direction, k d µ So we get the equation: µ γ[ k + ( t] have: So in a rotating frame with angular velocit referent to the lab frame Ω ω k, we δ µ ω µ [( γ + Ω + ( t] µ γ [( k + i ] δt γ δµ ω Now µ γeff, where eff k ( + i k + i, δt γ ω γ he equation eff k + i tells us in the rotating frame, the magnetic moment precesses in a cone of fied angle about the direction of at angular frequenc ω γ eff. We notice that the motion of the moment is periodic. If it is initiall oriented along the -direction, it will returns to that direction periodicall. As we increase its angle with the -direction, its magnetic potential energ in the laborator changes. All the energ it takes to tilt µ awa from is returned in a complete ccle of µ around the eff

6 cone. here is no net absorption of the energ from the altering field but rather alternatel receiving and returning of energ. Z Y X Fig. If the resonance condition is fulfilled eactl ( ω γ, the effective field is then simpl i, so the magnetic moment that is parallel to the static filed initiall will then precess in the - plane. (please refer to Figure.. hat s to sa, it will precess but remaining alwas perpendicular to. Periodicall, it will be lined up opposed to. If we induce the field as a pulse with duration time t instead of a continuous wave front, the magnetic moment would precess through an angle θ γ t w. If we choose a tw which will satisf θ π, the pulse will simpl invert the moment, such a pulse is π referred to in the literature as a 8 degree pulse. If θ, the magnetic moment is turned from the -direction to the -direction, and the pulse is called 9 degree pulse. Without other effection, following the turn-off the 9-degree pulse, the moment would then remain at rest in the rotating frame, and hence precess in the laborator perpendicular to the static field. ut in fact that is not true, the magnetic moment will go back to the initial -direction after enough long time due to relaation, it will bring the spin sstem back into oltmann equilibrium. w 4. Equilibrium agnetiation and loch Equation In practice, we can t measure the magnetic momentum for a single spin; instead, we can onl measure the macroscopic magnetiation for the whole sample. Now we will tr to understand the equilibrium distribution of spin sstem under the uniforml constant magnetic field along the -ais.

7 As we know, the energ of the magnetic dipole interaction is: E γ γ If the energ difference between two adjacent states is ver small, we can consider the distribution is continuous (Classical oltmann Statistics. E γ cosθ So the probabilit that an spin particle in the orientation ( θ, ϕ in space is given γ b: P( θ, ϕ C Ep( E C Ep( cosθ, where C is the normalied k k constant. γ Generall, in room temperature, we have <<, then from the equation k Ω π π γ d P( θ, ϕ C dϕ ( cos sin dθ Ep θ θ, we get C. k 4π hus we get: π < > d ϕ π cosθ γ dθ sinθ ( + 4π k γ cosθ 3k So in thermal equilibrium, onl the -component of the magnetiation is non-ero, the -component and -component both should be ero due to the aimuthal smmetr. γ < > i + j+ k 3k For a sstem with N non-interaction spins, we have the total angular momentum: N < γ > N 3k So the equilibrium magnetiation for the spin sstem should be: γ N γ 3k

8 Now if we induce an altering magnetic field (as we discussed before (t, the magnetiation will deflect from the equilibrium value. d µ From the previous equation µ ( γ, we know there should be a similar equation for magnetiation, here, we include the thermal relaation. So the magnetiation must wish to be parallel to, the - and - components must have a tendenc to vanish. hus d d d + γ + γ ( ( + γ ( his set of equations is called loch Equations. Here we introduce times, is called longitudinal relaation time which is different from which is the transverse relaation time. Now, we want to figure out what s the solution of the loch Equations when the altering transverse magnetic field is turned on. In order to make the calculation easier, we immediatel transform to the coordinate frame rotating at angular velocit Ω ω k, ω taking along the rotating -ais and denoting b b. γ So ( t + + hen in rotating coordinate and b + d γ + d γ b d γ ( b

9 In eperiment, we choose value low enough to avoid the saturation that will kill the resonance absorption signal. As we know, and must vanish as, and is in a stead state, it differs from to the order of. herefore, we replace b in the last two equations, and neglect the transient term, we get χ ( ω ω ( ω + ( ω ω χ ( ω + ( ω ω Nγ Where χ is the magnetic susceptibilit, which is equal to. 3k So we can see in the rotating referent frame with angular velocitω, the magnetiation is a constant. herefore, in the lab frame, it must rotate at frequenc ω. picall, we choose the orientation of the coil along the fied X-direction, we can calculate the time dependent component of magnetiation along the coil direction. X referring to Figure.3, we get cos( ω t + sin( ω t X Y X Fig. 3 If we regard the magnetic field as a linear filed, we will get X ( t X cos( ω t X hen we see both of and are proportional to, and we can write X ' " t ( χ'cos( ω t + χ"sin( ω t X ( χ + jχ X cos( ω t X (

10 Where the quantities χ ' and χ" should be: χ ' χ" χ ( ω ω ( ω + ( ω ω χ ( ω + ( ω ω It is convenient to regard both X (t and (t as being the real parts of the X X X χ( ω χ' χ" C C X ( t ( ω X ( t X ( t Re{ χ( ω X comple functions C (t and C (t. hen if we define C X where I and write Iωt Iωt ( t e, we find χ, and e }, X I. Ordinaril, if a coil of inductance L is filled with a material of susceptibilit χ, the inductance is increased to L +, since the diamagnetic material will increase the ( χ magnetic flu b factor + χ. Similarl, the comple susceptibilit produce a flu ( change from the original value L, L L[ + χ( ω] Denoting the coil resistance in the absence of a sample as R, the coil impedance when including sample becomes Z I Lω ( + χ' + Lωχ" + R. he real part of the susceptibilit χ ' therefore changes the inductance, where as R the imaginar part χ" modifies the resistance. he fractional change in resistance is: R R L ωχ ". R R If we assume that uniform magnetic field occup a volume of magnetic energ produced b an altering current, whose peak value is L i X V While, we can get the average power dissipated in the nuclei is: P i R i L ω χ" V i, the peak stored, is

11 So we find P ω X χ" V. his equation provides a simple connection between the power absorbed, χ ", and the strength and the frequenc of the altering field. So if we can use some detecting sstem to measure the energ dissipated in the sample during the resonance absorption, we can measure χ ". he following figure (Figure. 4 shows the theoretical relationship between χ ', χ " with the oscillation frequenc of the altering magnetic field. χ ' vs. ( ω ω is called the dispersion curve, and the other one χ " vs. ( ω ω is called the absorption curve. We can see when the frequenc is near to the resonance frequenc, the absorption curve will get close to its peak value. " χ ' χ Fig. 4 using sensitive electronic circuit, we can change both of these two magnetic susceptibilities to electronic signals and measure them b using digital oscilloscope. If we tune it well, we can get clear eperimental signals, where the absorption and dispersion are corresponding to the In-Phase Signal and Out-Phase Signal respectivel. 5. Pulsed NR Eperiment and Free Induction Decas Since the static magnetic field has a small inhomogeneit across the sample in the coil, so from the definition of the Larmor Frequencω γ, we know there will be a distribution of the values of Larmor Frequencies as well. If the distribution function is denoted b g ( ω, the number of particles dn with Larmor Frequencies between ω and ω + d is: ω dn N g( ω dω

12 In order to solve the problem of the time dependence of the net magnetiation, we first estimate the average angular momentum of the spin sstem: (t Instead of using k j k, we have ( t ω ( ω t dn N ( t g( ω dω Now we eamine the response of our sample to a rotating magnetic field of magnitude, angular speed ω and its lasting time t. Let us view the behavior of the spin sstem from the reference frame rotating with we choose along the j direction. (please refer to Figure 5. w, and this time due to convenience, k eff j i Fig. 5 eff where So now the effective field in the rotating frame becomes: eff k j, where ω γ At the resonance case, [( ] <<. j. is a good approimation, we can set ust before the application of the rotating pulsed field, we have j ( j( k, j ω ( t is the so-called isochromat. After we induced the transverse field, each of ω

13 the isochromats will precess about the ais in time t b an amount θ that j ( tw j( (cosθ k + sinθ i. ω So the total angular momentum over the sample is: j ω t w N dω g( ω j((cosθ k + sinθ i ( w Nj( (cosθ k + sinθ i ( (cosθ k + sinθ i where ( Nj(. So we find ( t w ( i for θ 9 ( 9 pulse ( t w ( k for θ 8 ( 8 pulse eff γ t After the pulse turned off ( t > tw, we get k and each isochromat w, so precesses about the k ais at an angular speed of ω γ. hus j ω ( w t j( cosθ k + j( sinθ[cos( ω ( t t i sin( ω ( t t j] w If we redefine t t, and do the integral over dn ( ω, we find t w ω ω ( t w + t ( cosθ k + ( sinθ [ i g( ω cos( ω t d ω j g( ω sin( ω t d ] Generall, g( ω is a smmetric function of ω, so the second integral vanishes, while it predicts that the i j k component of the angular momentum is time invariant and the or (if we choose the rotation field along another orientation components decas to ero slowl. of For simplicit, we assume that the function of g ω is a constant ( ω a < ω < ω + a ( ; other wise, it has value of ero: in the range a So we will get

14 + + So we get g( ω sin( ω t d ω g( ω cos( ω t d ω + a a + a a sin( ω t a cos( ω t d ω sin at d ω at a sin at ω ( t w + t ( cosθ k + ( sinθ i. at Since γ, so the magnetiation has the same time dependent function. Since the orientation of the coil is along the -direction, the magnetiation along that direction in the lab frame should be ( t cos( ω t + sin( ω t, so the sin at average magnetiation along that fied direction is X sinθ. So it will at deca after the pulse as time increases. eanwhile, we also know that the amplitude of Fid signal will be proportional to the magnetiation of the direction of the coil, so we know the fid signal will also deca to ero after the pulsed transverse magnetic field is turned off. Now let us consider the effection of a second pulse of the transverse field. We assume that the second pulse comes at time τ after the first pulse. his time in order to differentiate the two pulses, the rotation produced b the first pulse is labeled b θ, and that b the second is labeled b θ. his time, we still choose the second along the j direction. o simplif future discussion, we think the i and j components of (t have died to ero due to the transverse relaation. So just before the nd pulse comes, we have ( t ( cosθ j. Use the same analsis as we used for discussing the response of the sstem to the first pulse, the response to the second pulse should be: X j i w ( τ + t ' + t ( cosθ cosθ k + ( cosθ sinθ ( i g( ω cos( ω t d ω sin at ( cosθ cosθ k + ( cosθ sinθ i at j g( ω sin( ω t d ω It shows how the angular momentum, or magnetiation of the sample functions nd as time after the pulse. From this equation, we can draw the conclusion that the nd t

15 fid has the same shape as the fid curve, and its amplitude is proportional to cosθ sinθ, instead of proportional to sinθ as the first fid. So if θ 9 or θ 8 or both, the second fid is ero. However, this is true just for the case there is no longitudinal relaation; that is to sa, after the first pulse, the angle between the magnetiation and the static field is unchanged, stasθ. ut for the real eperiment, the ideal case is not true. After the first pulse, the magnetiation wishes to go back to the -direction (the direction of magnetic field, so the angularθ will decreases with time, when the nd pulse comes, it will become θ ( '. τ So we get the first fid and the second fid are the following respectivel: sin at + i F ( t w t ( sinθ at F sin at ( τ + t w + t ( cosθ' sinθ i at As we know, when it is in resonance, the average resonance frequenc range a should be the smallest value, the fid signal will deca to the ero slower, so the wiggle of the fid curve will become smaller (Figure Indication of Resonance. Indication of Resonance Resonance cases. Fig. 6 After finding the Resonance condition, we focus our attention on two specific

16 ( he " 9 τ 9 " pulse sequence(figure 7: θ τ θ γt w 9 o Fig. 7 ore ever, we can see if we add " 9 9 " pulses ( θ θ 9 to the spin sstem, the amplitude of the first fid is the maimal of all the values it could be. he amplitude of the nd fid depends on the dela time from the nd pulse to the st pulse τ. When the dela time τ is a constant, we can tune our detecting circuit until we get the maimum of the second fid, when it shows the second pulse is also a 9 pulse. Using loch Equations, let (since the pulse is turned off, and change magnetiation to, so we have: d d γ d γ Solving this equation, we get: ( t + ( ( e t ( t ( e cos( ω t t (where ω γ So we can see that the longitudinal angular momentum (t will go back to the equilibrium value as function of ( after the pulsed transverse field is turned off; e t

17 the transverse angular momentum ( after pulse ( (t has a similar performance as (t. t (t will deca as from its initial value e he fid curve is: τ F Fid ( t, τ ( e f ( t i sin a( t τ t Where function f (t should be i. a( t τ t So we can also measure the amplitudes of a series of second fid curves with differentτ, there should satisf a relation of ( e. using this method, we can measure the longitudinal relaation time. τ ( he " 9 τ 8 " pulse sequence (Figure 8: τ θ γ t 9 o θ γ t 8 o w w Fig. 8 Now let s discuss about another special case " 9 8 " pulses and the phenomenon of echo. Firstl, we follow a tpical isochromat j ω ( t. At the timeτ, before the application of the second pulse, this isochromat will have accumulated an aimuthal phase ω τ relative to the ais. he inducing of the nd pulse will φ make it nutate b 8 about the j ais, so that its phase immediatel following the second pulse is now φ' π ω τ (please refer to Figure 9, I didn t show it here. As the accumulation of the transverse phase should be proportional to the time, so at the time t τ + t, the phase should be φ" π ω τ + ω t. hus we have the transverse component differential: i

18 j ω ( τ + t ( cos[ θ ' ( τ + t ]sin[ θ ' ( t ][cos( π + ω ( t τ i sin( π + ω ( t τ j] ( τ + t ω ( t dn N ω ( t g( ω dω ( cos( θ'sin( θ ' [ i g( ω cos( ω ( t τ d ω + j g( ω sin( ω ( t τ d ω ] As we know before, the integral [ i g( ω cos( ω ( t τ d ω + j g( ω sin( ω ( t τ d ω ] sin a( t τ i a( t τ So sin a( t τ ( τ + t ( cos( θ'sin( θ ' i a( t τ If t is different withτ, since a is a significant number, this term will go to ero quickl. So generall, we can see nothing after turning off the nd fid signal. ut sin a( t τ whent τ, the factor will become quite large ----, which is ver a( t τ significant compared to the other cases: t τ, it will increase the probabilit for us to detecting the fid signal. his phenomenon (whent τ is called spin refocusing, the signal shows up at time t t + τ τ is called echo. value Here, the transverse angular momentum ( after pulse. t (t will deca as e from its initial Now, based on the previous discussion, we can estimate the fid curves and echo including the spin-relaation. he echo should be like the following term: t F Echo ( t ' ep( g( t i Where g(t should be sin a( t τ sin a( t τ. a( t τ a( t τ

19 So using echo signal, we can measure the echo peak values of different dela timeτ, then fit the eperimental data, we can obtain the transverse relaation time. τ he height of echo should deca as function of ep(. After this point, we see the main eperiment we can do for pulsed NR:. For the " 9 τ 9 " pulse sequence: τ ( e vs. τ, fid curve fitting to get.. For the " 9 τ 8 " pulse sequence: τ ep( vs. τ, echo curve fitting to get. Eperiment SE-UP his eperiment is evolved a lot of equipments including a complicated structure of circuit, now we will introduce the essential parts briefl. he following block diagram (Figure. shows the main contents. A F E C D G H N I S Fig.

20 A: Pulse Programmer : odulator C: Power Amplifier D: Protection and Coupling Circuit E: Receiver F: Oscillator G: Phase Detector H: Oscilloscope I: Sample in coil N (S: North Pole or South Pole of Permanent agnetic Filed he Oscillator (F outputs a continuous radio frequenc signal. he frequenc can be adjusted b different scales. When we are doing pulsed NR eperiment, we can change the frequenc until we are close to the Larmor Frequencω of the resonance. he Pulse Programmer (A generates D.C pulses that are used to determine when the RF (radio frequenc pulse are applied to the sample to ecite sample from equilibrium state. he programmer can determine the repetition rate of series of identical pulse sequences, the length of each of the D.C pulse ( tw and t w, and the dela time between the two individual pulses. he odulator ( modulates the signal from the Oscillator with the pulses from the programmer. It can be looked as a switch that opens and closes according to the pulse signals from the programmer. he main function of Power Amplifier (C is to take the RF pulses from the modulator and amplifies them from about. volts to volts, but keeping the same shape. he volts out of power amplifier will damage the receiver and cause it to block, so that the signal following the RF pulse is unobservable. he Protection Circuit (D connects the resonant circuit of the probe to the power amplifier during the ecitation pulse, while disconnecting the receiver. Following the ecitation pulse, it connects the resonant circuit of the probe to the receiver and disconnects it from the transmitter. So spurious noise from the transmitter doesn t reach the receiver, and almost all of the signal from the resonant circuit goes the receiver. he Receiver (E is a high gain, low noise, radio frequenc amplifier. It takes the signal coming from the sample coil and amplifies it from µ Volt ~ µ Volt to about. volt, then send it to the phase detector. he Phase Detector (G takes the signal from the receiver and changes it to D.C. It can be viewed as a switch that opens and closes in snchronism with the reference signal from the oscillator. In fact, the output of the phase detector seems as the vector sum of the signal from the receiver and the reference signal. here is one thing I have to mention left the resonance circuit around the sample. When the RF frequenc is tuned to the Larmor Frequenc ω, it uses the power

21 from the power amplifier to create an oscillating magnetic field to ecite the sample. It can also pick up a signal from the precessing magnetiation to rela to the receiver. Eperimental Data and Discussion In our eperiment, we used three different concentrations of following are data table and plots. Cur solution. he Concentration.375 ( Cur 9 Deg ~ 9 Deg Sequence 9 Deg ~ 8 Deg Sequence Dela ime (ms Amplitude of nd Pulse (mv Dela ime (ms Amplitude of Echo (mv Concentration.5 ( Cur 9 Deg ~ 9 Deg Sequence 9 Deg ~ 8 Deg Sequence Dela ime (ms Amplitude of nd Pulse (mv Dela ime (ms Amplitude of Echo (mv

22 Concentration.75 ( Cur 9 Deg ~ 9 Deg Sequence 9 Deg ~ 8 Deg Sequence Dela ime (ms Amplitude of nd Pulse (mv Dela ime (ms Amplitude of Echo (mv Longitudinal Relaation ime easurement for.375 Cur (9~9 Pulse Sequence Amplitude of nd Pulse (mv Relaation ime (ms.375 Fit 39 Where we get the fit curve is V Fid ( τ ( e. τ

23 Longitudinal Relaation ime easurement for.5 Cur (9~9 Pulse Sequence 4 Amplitude of nd Pulse (mv Relaation ime (ms.5 Fit 75 Where we get the fit curve is ( τ ( e V Fid τ Longitudinal Relaation ime easurement for.75 Cur (9~9 Pulse Sequence Amplitude of the nd Pulse (mv Relaation ime (ms.75 Cur 3 Where we get the fit curve is ( τ 5 ( e V Fid τ

24 ransverse Relaation ime easurement for.375 Cur (9~8 Pulse Sequence Amplitude of Echo (mv e -.49 R Relaation ime (ms.375 τ 4. 8 Where we get the fit curve is: V ehco ( τ 38.9 e. ranseverse Relaation ime easurement for.5 Cur (9~8 Pulse Sequence Amplitude of Echo (mv e -.43 R Relaation Ime (ms.5 τ Where we get the fit curve is: V ehco ( τ 34. e.

25 ranseverse Relaation ime easurement for.75 Cur (9~8 Pulse Sequence Amplitude of Echo (mv e R Relaation Ime (ms.75 τ Where we get the fit curve is: V ehco ( τ 6.93 e. After fitting our eperimental curves, we obtained the longitudinal relaation time and transverse relaation time for the three different samples: Concentration of Cur Longitudinal Relaation ime (ms ransverse Relaation ime (ms Adjustment and tuning of the apparatuses of this eperiment is reall a challenge job, we tr our best to improve the signal, but it turns out that we couldn t finall make it as good as we desired. uning both of the fid curves and echoes need a lot of diligence and patience, Professor Donna Naples (Facult ember in Dept. of Phsics & Astronom, Univ. of Pittsburgh also afforded me lots of help which I reall feel appreciated, however, unfortunatel the whole sstem seems not working consistentl and appropriatel. So the best suggestion here is to update the whole eperiment set up, and make it work well. Looking at our eperiment data, we can see that as the concentration of the salt solvent increases, both of the longitudinal and transverse relaation times decrease. Since our data are not good, so it is quite difficult for us use it to make a plot of Relaation ime vs. Solution Concentration. he most valuable thing of doing this eperiment is to get a chance to review all the phsics details in pulsed NR eperiment and give a reasonable analsis and estimation.