[1] Pulsed NMR Apparatus Manual, Teach Spin Inc. (Copy available in the lab)

[2] C. P. Slichter, Principles of Magnetic Resonance, Springer, New York, 1996. (Copy available in the lab)

[6] N. Bloembergen, “Encounters in Magnetic Resonances”, World Scientific Series in 20__th_ Century Physics 15_, World Scientific,1996.

In this experiment you will study the phenomenon of pulsed nuclear magnetic resonance. Specifically, your goals for this experiment include the following:

• to learn to use the Pulse Programmer apparatus and the large, water-cooled electromagnet;
• to establish a resonance in samples of mineral oil and teflon, and to use these measure the gyromagnetic ratio of the proton and of fluorine nuclei, respectively;
• to measure the _T_2 relaxation time in mineral oil using the Hahn spin-echo method; and
• to measure the _T_1 relaxation time in mineral oil using the inversion-recovery method.

Here we wish to examine the effect of magnetic fields on protons and other particles having magnetic moment and angular momentum. It is noteworthy that the underlying physics of nuclear magnetic resonance (NMR) is very similar to that of electron spin resonance and optical pumping.

Consider a loop of positive current I whose path encloses area A, as in Fig. 1.

{ ${https://wiki.uchicago.edu/download/attachments/142050624/fig_1.png?version=1&modificationDate=1448385119000&api=v2}$ Figure 1: Magnetic dipole moment of a current loop.

The area enclosed by the loop may be considered a vector where the magnitude is the area and the direction is given by the right-hand rule. (With the fingers pointing in the direction of positive current flow, the thumb points in the direction of the resulting area vector, which is the same direction as the magnetic dipole moment vector.) The magnetic dipole moment is therefore given by

{ ${/download/attachments/141459859/eqn_1.png?version=1&modificationDate=1451404534000&api=v2}$

(1)

If we place a magnetic dipole moment in an external magnetic field, the dipole will experience a torque given by

{ ${/download/attachments/141459859/eqn_2.png?version=1&modificationDate=1451404540000&api=v2}$

(2)

where { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ is the magnetic field. This torque tends to twist the dipole to align parallel to the field; this is the lowest energy state. Therefore, it takes energy

{ ${/download/attachments/141459859/eqn_3.png?version=2&modificationDate=1451408747000&api=v2}$

(3)

to twist the dipole away from the orientation parallel to the field.

<blockquote> NOTEBOOK: </HTML>

• If an object having a magnetic moment (but no angular momentum) is free to move, how will it move in the presence of the magnetic field?
• If the object is given angular momentum, parallel or anti-parallel to its magnetic moment, and is placed it in a magnetic field, how will it move?

</blockquote></HTML> To help us answer the second question above, carefully consider Fig. 2.

{ ${https://wiki.uchicago.edu/download/attachments/142050624/fig_2.png?version=1&modificationDate=1448386979000&api=v2}$ Figure 2: Torque acting on an object with angular momentum and magnetic moment in a magnetic field (after Eisberg and Resnick, Quantum Physics).

The magnetic field acts on the magnetic dipole moment to produce a torque, given by Eq. (2). This torque also acts on the angular momentum (since the angular momentum and magnetic dipole moment are here assumed to remain co-linear). Thus, the torque on the angular momentum gives rise to a change in the angular momentum { ${/download/attachments/141459859/dL.png?version=1&modificationDate=1451404603000&api=v2}$ during the time d_t_ such that

{ ${/download/attachments/141459859/eqn_4.png?version=1&modificationDate=1451404552000&api=v2}$

(4)

The change d_L_ causes { ${/download/attachments/141459859/L.png?version=1&modificationDate=1451404590000&api=v2}$ to precess through an angle ω_d_t, where ω is the precession angular velocity. Note from Fig. 2 that

{ ${/download/attachments/141459859/eqn_5.png?version=1&modificationDate=1451404561000&api=v2}$

(5)

such that

{ ${/download/attachments/141459859/eqn_6.png?version=1&modificationDate=1451404567000&api=v2}$

(6)

where

{ ${/download/attachments/141459859/eqn_7.png?version=1&modificationDate=1451404572000&api=v2}$

(7)

This precession is referred to as _Larmor precession _and the frequency of Eq. (7) is known as the Larmor frequency. Eq. (7) is often re-stated as

{ ${/download/attachments/141459859/eqn_8.png?version=1&modificationDate=1451404577000&api=v2}$

(8)

where γ is called the gyromagnetic ratio.

If available (ask an instructor or TA), take a few minutes to use the TeachSpin Magnetic Torque apparatus to explore these ideas. This apparatus consists of a cue ball with a small magnet embedded at its center. 

1. Turn on the air supply and the current to the Helmholtz coils. Without spinning the ball, observe how it is affected by the magnetic field. How is the ball’s magnetic dipole moment aligned relative to the ball’s black handle? - Turn down the magnetic field and twist the black handle to spin the ball. (I.e., give the ball some angular momentum.) What is the direction of the angular momentum relative to its magnetic dipole moment? - While the ball is spinning, turn on the magnetic field. How does the ball move now? - Reverse the direction of the ball’s angular momentum. How does the ball’s motion change in the B field? - Try to explain what you observe by referring to Fig. 2.

As with the hypothetical classical moment above, protons also have collinear magnetic moment and angular momentum, generally denoted { ${/download/attachments/141459859/mu.png?version=1&modificationDate=1451405148000&api=v2}$by and { ${/download/attachments/141459859/J.png?version=2&modificationDate=1451405401000&api=v2}$. In the absence of a magnetic field, the protons have no preferred alignment direction. However, when we apply a magnetic field to a proton's magnetic dipole moment, the proton feels a torque, which is also applied to the proton's angular momentum { ${/download/attachments/141459859/J.png?version=2&modificationDate=1451405401000&api=v2}$. Just as in the classical model, the proton's angular momentum vector would then precess around the { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ field as in Fig. 2. However, the presence of a well-defined vector precessing around { ${https://wiki.uchicago.edu/download/attachments/142050624/B.png?version=1&modificationDate=1448385119000&api=v2}$implies knowledge of the _x- _and _y-_components of the vector, but quantum mechanics tells us that these components are really indeterminate. Instead, all we can say is that the projection of the proton's angular momentum vector onto the { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$field is quantized in units of { ${/download/attachments/141459859/hbar.png?version=1&modificationDate=1451405499000&api=v2}$. Thus, the direction of { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ has unique physical significance, and is referred to as the quantization axis. This precession of the magnetic moment about the quantization axis is referred to as Larmor precession just as in the classical case, and the precession frequency is given by Eq. (8), with the gyromagnetic ratio γ now a purely quantum mechanical quantity.

The precession cone angle (the angle between { ${/download/attachments/141459859/J.png?version=2&modificationDate=1451405401000&api=v2}$ and { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$) of the classical dipole moment can have any value. However, in the quantum mechanical case of protons, the precession cone angle (in this simplified model) can take on only two values as shown in Fig. 3. { ${/download/attachments/141459859/Fig_3.png?version=1&modificationDate=1451407412000&api=v2}$Figure 3: The { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$field and the two cones in which the angular momenta of the protons may precess. Note that the x and y components of the angular momenta are undetermined. <blockquote> NOTEBOOK: </HTML>

• In the Fig. 3, which cone represents the higher energy state?

</blockquote></HTML>

When we apply a rotating { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ field to an ensemble of precessing protons, in the same direction and at the same frequency as the precession (in this simplified model), the precession cone angle will change. This change gradually creates a superposition of the two energy states and thus causes the protons to pass from one energy state to the other. The difference in energy between the two states is given by

{ ${/download/attachments/141459859/eqn_9.png?version=1&modificationDate=1451407728000&api=v2}$

(9)

where ω is the angular velocity of the precession and of the rotating { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ field at resonance. With some care, the resonance condition can be demonstrated at the Magnetic Torque apparatus. Try it!

Slichter [2] gives excellent classical and quantum mechanical descriptions of nuclear magnetic resonance and pulsed NMR.

Let us denote the proton's magnetic moment as { ${/download/attachments/141459859/mu.png?version=1&modificationDate=1451405148000&api=v2}$and its angular momentum as { ${/download/attachments/141459859/J.png?version=2&modificationDate=1451405401000&api=v2}$, which are related by the vector equation

{ ${/download/attachments/141459859/eqn_10.png?version=2&modificationDate=1451408641000&api=v2}$

(10)

where γ is the gyromagnetic ratio. The nuclear angular momentum is quantized in units of { ${/download/attachments/141459859/hbar.png?version=1&modificationDate=1451405499000&api=v2}$ as

{ ${/download/attachments/141459859/eqn_11.png?version=1&modificationDate=1451408712000&api=v2}$

(11)

where { ${/download/attachments/141459859/I.png?version=1&modificationDate=1451407889000&api=v2}$ is the “spin” of the nucleus. The potential energy E of a magnetic dipole in an external magnetic field { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ is

{ ${/download/attachments/141459859/eqn_3.png?version=2&modificationDate=1451408747000&api=v2}$.

(12)

If the magnetic field _B_0 is in the z-direction, then the potential energy of the dipole is

{ ${/download/attachments/141459859/eqn_13.png?version=1&modificationDate=1451408831000&api=v2}$

(13)

Quantum mechanics requires that the allowed values of _I_z, (namely, _m_I) be quantized as

For the proton, with spin I = 1/2, the allowed values of Iz are simply

Thus, there are only two energy states for a proton residing in a constant magnetic field B_0. These are shown in Fig. 4. The energy separation Δ_E between the two states can be written in terms of an angular frequency or as

{ ${/download/attachments/141459859/eqn_16.png?version=1&modificationDate=1451408545000&api=v2}$

(16)

{ ${/download/attachments/141459859/fig_4.png?version=1&modificationDate=1451408234000&api=v2}$ Figure 4: The two energy states of a proton (spin 1/2 particle) in a magnetic field.

As we shall see later, applying an oscillating magnetic field of frequency ω, orthogonal to { ${/download/attachments/141459859/B0z.png?version=1&modificationDate=1451410051000&api=v2}$ which satisfies Eq. (7), will give rise to flipping protons from one of the two energy states to the other. This is the fundamental resonance condition. For the proton,

so that the resonant frequency is related to the constant magnetic field for the proton by

an equation worth remembering.

(*Gauss has been the traditional unit to measure magnetic fields in NMR but the tesla is the proper SI unit, where 1 T = 104 G.)

If a sample of water is placed in a magnetic field in the z-direction, the protons will be in either the high energy state (aligned along -z) or the lower energy state (aligned along +z). We can quantify the ratio of the numbers in each state via the Boltzmann distribution. If _N_1 and _N_2 are the number of spins per unit volume in the higher and lower energy states, respectively, then the population ratio (_N_1/_N_2) in thermal equilibrium is given by

{ ${/download/attachments/141459859/eqn_19.png?version=1&modificationDate=1451414773000&api=v2}$

(19)

Here, T is the absolute temperature, k_B is the Boltzmann constant and Δ_U is the energy difference between the two states. (Note that at room temperature, k_B_T ≈ 1/40 eV.) Thus, a net nuclear magnetization in the z-direction eventually becomes established.

<blockquote> NOTEBOOK: </HTML>

• Calculate the ratio of the numbers of atoms in the two states for a sample at room temperature, with an energy difference corresponding to a frequency of 15 MHz.
• Considering that { ${/download/attachments/141459859/comparison.png?version=1&modificationDate=1452267560000&api=v2}$, does this ratio make sense? * If _N_1 + _N_2 ~ 0.01_N_A ~ 1021, compute the net number (_N_2 - _N_1) aligned in the direction of the field.

</blockquote></HTML> The net magnetic dipole moment { ${/download/attachments/141459859/M.png?version=4&modificationDate=1451419584000&api=v2}$, also referred to as the bulk magnetization of the sample, is the vector sum of the individual proton magnetic moments in the ensemble,

{ ${/download/attachments/141459859/eqn_20.png?version=1&modificationDate=1451415100000&api=v2}$

(20)

For the case of protons precessing about _B_0 we have

{ ${/download/attachments/141459859/eqn_21.png?version=1&modificationDate=1451415275000&api=v2}$

(21)

The thermal equilibrium magnetization per unit volume for N magnetic moments is

{ ${/download/attachments/141459859/eqn_22.png?version=1&modificationDate=1451415360000&api=v2}$

(22)

where N = _N_1 + _N_2.

As described by Slichter (p.8), this thermal equilibrium with the “lattice”, i.e., the surrounding material, cannot occur without coupling of the nuclear spins to the lattice, which provides an energy reservoir. The strength of this coupling also determines the time needed for the magnetization to build up to its equilibrium value along the magnetic field direction.

The Bloch equations (see Slichter [2], pp. 33-36) describe the motion of the magnetization in matter:

{ ${/download/attachments/141459859/eqn_23.png?version=1&modificationDate=1451415886000&api=v2}$,	(23)
{ ${/download/attachments/141459859/eqn_24a.png?version=1&modificationDate=1451415961000&api=v2}$,	(24a)

and

{ ${/download/attachments/141459859/eqn_24b.png?version=1&modificationDate=1451415967000&api=v2}$.

(24b)

Here { ${/download/attachments/141459859/M.png?version=4&modificationDate=1451419584000&api=v2}$ is the bulk magnetization, { ${/download/attachments/141459859/H.png?version=1&modificationDate=1451415497000&api=v2}$ the magnetic field, _T_1 is the longitudinal or spin-lattice relaxation time which characterizes the coupling to the lattice, and _T_2 is the transverse or spin-spin relaxation time which characterizes the decay of the transverse magnetization (_M_x or My) which conserves energy in the static field. (I.e., it doesn't require energy transfer to the reservoir.) One cause for relaxation with time constant _T_2 is local field variations which give rise to variations in the precession frequency of the nuclei. In Eqs. (23) and (24), { ${/download/attachments/141459859/torque.png?version=1&modificationDate=1452196339000&api=v2}$ is the torque on the magnetization due to the total magnetic field. If we now place an unmagnetized sample in a static field , Eq. (23) becomes

{ ${/download/attachments/141459859/eqn_25.png?version=1&modificationDate=1451416037000&api=v2}$

(25)

Integration of Eq. (23) with these initial conditions gives

{ ${/download/attachments/141459859/eqn_26.png?version=1&modificationDate=1451416091000&api=v2}$

(26)

where _M_0 = _ΧH_0 and Χ is the magnetic susceptibility.

The rate at which the magnetization approaches its thermal equilibrium value is characteristic of the microscopic processes in the sample. Typical relaxation times range from microseconds to seconds. The study of these processes is one of the major topics in magnetic resonance.

The classical equation of motion for the net magnetization of a nuclear magnetic moment μ is obtained directly from Eqs. (23) and (24) with 1/_T_1 and 1/_T_2 set equal to zero, (i.e., no coupling),

{ ${/download/attachments/141459859/eqn_27.png?version=1&modificationDate=1451927441000&api=v2}$

(27)

Equation (27) is the classical equation describing the time variation of the magnetic moment of the proton in a magnetic field. It can be shown from Eq. (27) that the magnetic moment will execute precessional motion about { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$, depicted in Fig. 2. Let us assume that we have 1020 protons in our sample and they are in thermal equilibrium, and that their magnetic moments are randomly (equally) distributed in x and y. For this case of random phasing, _M_x = _M_y = 0. Conversely, a non-zero _M_x or _M_y requires a non-random phase relationship among all the precessing spins. For example, we might start the precessional motion with the x-component of the magnetic moments aligned along the x-axis. As we shall soon see, there is a way to create such a transverse magnetization using radio frequency pulsed magnetic fields. The idea is to rotate the net magnetization _M_z into the x-y plane and thus create a temporary _M_x and _M_y. Let's see how this is done.

Equation (27) can be generalized to describe the classical motion of the net magnetization,

{ ${/download/attachments/141459859/eqn_28.png?version=1&modificationDate=1451927655000&api=v2}$

(28)

where { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ is any magnetic field, including time dependent rotating fields. Suppose we apply not only a constant magnetic field { ${/download/attachments/141459859/B0k.png?version=1&modificationDate=1451927719000&api=v2}$, but a rotating magnetic field* of angular velocity ω in the x-y plane so the total field is written as

{ ${/download/attachments/141459859/eqn_29.png?version=1&modificationDate=1451927980000&api=v2}$

(29)

(*What is actually applied is an oscillating field

{ ${/download/attachments/141459859/oscillating_field_1.png?version=2&modificationDate=1452008402000&api=v2}$ but that can be decomposed into two counter-rotating fields

{ ${/download/attachments/141459859/oscillating_field_2.png?version=2&modificationDate=1452008516000&api=v2}$ The field rotating opposite to the precession can be shown to have no practical effects on the spin system and can be ignored in this analysis.)

The analysis of the magnetization in this complicated time-dependent magnetic field can best be carried out in a rotating coordinate system. The coordinate system of choice is rotating at the same angular velocity as the rotating magnetic field with its z-axis in the direction of the static magnetic field. In this rotating coordinate system the rotating magnetic field appears to be stationary and aligned along the x*-axis (Fig. 5).

{ ${/download/attachments/141459859/fig_5.png?version=1&modificationDate=1451928435000&api=v2}$ Figure 5: Rotating coordinate system

However, from the point of view of the rotating coordinate system, _B_0 and _B_1 are not the only magnetic fields. An effective field along the z*-direction, of magnitude { ${/download/attachments/141459859/magnitude.png?version=1&modificationDate=1451929616000&api=v2}$ must also be included. Let us justify this new effective magnetic field with the following physical argument. Equations (28) and (29) predict the precession of a magnetization in a constant magnetic field { ${/download/attachments/141459859/B0k.png?version=1&modificationDate=1451927719000&api=v2}$. Suppose one observes this precession from a rotating coordinate system which rotates at the precession frequency. In this frame the magnetization appears stationary, in some fixed position. The only way a magnetization can remain fixed in space is if there is no torque on it. If the magnetic field is zero in the reference frame, then the torque on { ${/download/attachments/141459859/M.png?version=4&modificationDate=1451419584000&api=v2}$ is always zero no matter what direction { ${/download/attachments/141459859/M.png?version=4&modificationDate=1451419584000&api=v2}$ is oriented. The magnetic field is zero (in the rotating frame) if we add the effective field { ${/download/attachments/141459859/magnitude.png?version=1&modificationDate=1451929616000&api=v2}$ which is equal to { ${/download/attachments/141459859/B0k.png?version=1&modificationDate=1451927719000&api=v2}$. Transforming the magnetic field expression in Eq. (29) into such a rotating coordinate system, the total magnetic field B* in the rotating frame,

{ ${/download/attachments/141459859/eqn_30.png?version=1&modificationDate=1451929899000&api=v2}$

(30)

is shown in Fig. 6.

{ ${/download/attachments/141459859/fig_6.png?version=1&modificationDate=1451929948000&api=v2}$ Figure 6: Precession shown in the rotating frame

The classical equation of motion of the magnetization as observed in the rotating frame is then

{ ${/download/attachments/141459859/eqn_31.png?version=3&modificationDate=1451934414000&api=v2}$

(31)

which shows that { ${/download/attachments/141459859/M.png?version=4&modificationDate=1451419584000&api=v2}$ will precess about { ${/download/attachments/141459859/B_eff.png?version=1&modificationDate=1451930166000&api=v2}$ in the rotating frame. Suppose now, we create a rotating magnetic field at an angular velocity _ω_0 such that

{ ${/download/attachments/141459859/eqn_32.png?version=1&modificationDate=1451934487000&api=v2}$

(32)

In that case { ${/download/attachments/141459859/B_eff_2.png?version=1&modificationDate=1451934557000&api=v2}$ is a constant magnetic field in the x*- direction. Then, the magnetization Mz begins to precess about this magnetic field at a rate Ω = γB_1 (in the rotating frame). Thus, by applying the rotating field for a controlled duration (an RF pulse), it is possible to change the net magnetization through a controlled angle away from the z-axis._ A 90º pulse, for example, is an RF pulse of just sufficient duration to rotate the net magnetization 90º away from the z*-axis, i.e., into the x-y plane. If the rotating field is applied for twice this time, then the net magnetization will be rotated into the –z* direction. If the pulse is left on four times as long the magnetization will be back where it started, with Mz along the z*-axis. These RF pulses are labeled as follows: 90o or π/2 pulse: Mz → My
180o or π pulse: Mz → -Mz
360o or 2π pulse: Mz → Mz

In the laboratory reference frame the magnetization not only precesses about B1 but also precesses about { ${/download/attachments/141459859/k.png?version=1&modificationDate=1451934791000&api=v2}$ during the pulse. It is not possible, however, to observe the magnetization during the pulse. Pulsed NMR signals are observed after the RF pulse has been applied. Because of the orientation of the pick-up coil, wound around the sample vial, the measurable effect is the precession of the magnetization in the x-y plane.

Suppose a 90o (π/2) RF pulse is applied to a sample in thermal equilibrium. The net equilibrium magnetization will be rotated into the x-y plane where it will precess about { ${/download/attachments/141459859/B0k.png?version=1&modificationDate=1451927719000&api=v2}$. But the net x-y magnetization will not persist indefinitely. For most systems, this magnetization decays exponentially as shown in Fig. 7. { ${/download/attachments/141459859/fig_7.png?version=1&modificationDate=1451939822000&api=v2}$ Figure 7: Decay of the net magnetization

The differential equations which describe the decay in the rotating coordinate system are (from the Bloch equations):

{ ${/download/attachments/141459859/eqn_33a.png?version=1&modificationDate=1451940261000&api=v2}$

(33a)

and

{ ${/download/attachments/141459859/eqn_33b.png?version=2&modificationDate=1451940332000&api=v2}$

(33b)

the solutions of which are

{ ${/download/attachments/141459859/eqn_34a.png?version=1&modificationDate=1451940391000&api=v2}$

(34a)

and

{ ${/download/attachments/141459859/eqn_34b.png?version=1&modificationDate=1451940397000&api=v2}$

(34b)

where the characteristic decay time _T_2 is called the Spin-Spin Relaxation Time. This relaxation time comes from the following mechanism: Each proton experiences a range of magnetic fields produced by its nearest neighbor protons. This range of fields produces a range of precession frequencies, thus de-phasing the ensemble of protons. Therefore, even if all the protons begin in phase (after the 90º pulse) they will soon get out of phase and the net x-y magnetization will eventually go to zero. A measurement of _T_2, the decay constant of the x-y magnetization, gives information about the distribution of local fields at the nuclear sites.

From this analysis it would appear that the spin-spin relaxation time _T_2 can simply be determined by plotting the magnitude of _M_x (or _M_y) after a 90º pulse, as a function of time. This signal is called the free precession or free induction decay (FID). If the magnet's field were perfectly uniform over the entire sample volume, then the time constant associated with the FID would be _T_2. But in most cases it is the magnet's inhomogeneity that dominates the observed decay constant of the FID.

Before the invention of pulsed NMR, the only way to measure the real _T_2 was to improve the magnet's homogeneity and make the sample smaller.

Here is where the genius of Erwin Hahn's discovery of the spin echo plays its crucial role. 

Consider the two-pulse sequence shown in Fig. 8.

{ ${/download/attachments/141459859/fig_8.png?version=1&modificationDate=1451940478000&api=v2}$ Figure 8: A 90º - τ - 180º pulse sequence

How does the magnetization respond to such a pulse sequence? Figure 9 shows the progression of the magnetization in the rotating frame. Note that the z*-axis of the rotating frame, the z-axis of the lab frame and the { ${/download/attachments/141459859/worddav61b886f3ab615e3dc721abcf080c3b5b.png?version=1&modificationDate=1447348745000&api=v2}$ field applied by the electro-magnet are all parallel to each other. { ${/download/attachments/141459859/fig_9.png?version=1&modificationDate=1451940537000&api=v2}$ Figure 9: The response of the magnetization to a 90º - τ - 180º pulse sequence

Study Figs. 8 and 9 carefully. In Fig. 9(a), the net magnetization is in thermal equilibrium and points along the z*-axis, before the 90º pulse. In (b), the magnetization has been rotated into the y* direction by the 90º pulse. In ©, precession about the z*-axis (the applied { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ field) begins, with faster precession of those spins in regions of larger { ${/download/attachments/141459859/B.png?version=3&modificationDate=1451405427000&api=v2}$ field. In (d), the spins are rotated through 180º about the x*-axis by a 180º pulse. In (e), the faster and slower precessing spins re-align along the -y* axis. The 180o pulse allows the x-y magnetization to re-phase to the value it would have had with a perfect magnet. This re-phasing gives rise to a large signal, a spin-echo, at time 2_τ_.

In spite of the re-phasing, some loss of _M_x,y magnetization occurs and the amplitude of the echo is smaller than the amplitude of the FID. This loss of transverse magnetization occurs because of stochastic fluctuation in the local fields at the nuclear sites which is not re-phasable by the 180º pulse. These are the actual T_2 processes of interest. A series of 90º - τ - 180_º pulse experiments, varying τ, and plotting the echo height as a function of time between the FID and the echo, will give us the best measure of _T_2.

The transverse magnetization as measured by the echo amplitude is written as

{ ${/download/attachments/141459859/eqn_35.png?version=1&modificationDate=1451941084000&api=v2}$

(35)

Figure 10 shows the components of the apparatus. We will summarize the role of each component here. For more detail on the operation of the electronics, refer to pages 14-25 of the TeachSpin manual in the laboratory.

{ ${/download/attachments/141459859/fig_10.png?version=1&modificationDate=1451939668000&api=v2}$ Figure 10: Components of PNMR experiment

[A] Electro-magnet:

The electro-magnetic is capable of generating fields up to 10 Tesla between the pole pieces. The strength of the field is determined by the amount of current flowing through the water-cooled coils and magnet power supply. Before turning on the electro-magnet power supply [B], you must first turn on the water supply in the next room. You should see a small, steady stream emptying into the sink.

[B] Electro-magnet power supply:

Controls the current flowing through the coils of the electro-magnet. Only the coarse and fine current controls are used, do not change the settings of the other controls. Before turning on the power supply make sure that both the coarse and fine current controls are set to 0. When turning the power supply off at the end of the day, also make sure that the both current controls are set to zero. Turning the power supply on or off with the current controls not at zero can potentially damage the equipment.

[C] Sample Probe:

The sample probe is a rectangular brass box which slides snugly between the pole pieces of the electro-magnet. The interior of the probe is shown in Figure 10. There is a hole in the top which accepts a sample vial containing a small amount of material to be studied. When inserted to the proper depth, the sample volume will be in the center of the receiver and Helmholtz coils. The receiver coil is used to detect time varying magnetic fields along the vertical axis (x-axis). The Helmholtz coils are used to create time varying magnetic fields along the horizontal axis (y-axis) which will be used to rotate the nuclei in the sample.

{ ${/download/attachments/141459859/worddava31e2eaf9b3ad02b9b13737181e7bc90.png?version=1&modificationDate=1447348747000&api=v2}$ Figure 11: Sample probe

[D] TeachSpin Electronics Rack

The electronics rack houses three different modules. The power switch is located on the right hand side of the back of the rack. The three modules are;

15 MHz Receiver

A low noise, high gain amplifier connected to the receiver coil in the sample probe.

15 MHz Oscillator/Amplifier/Mixer

Sends pulses of 15MHz ac current to the Helmholtz coils in the sample probe. When these pulses turn on, and their duration, is determined by signals from the pulse programmer. Also contains a mixer which is used to compare the frequency of the oscillator signal with the signal induced in the receiver coil.

Pulse Programmer

Allows the user to setup sequences of pulses from the 15MHz Oscillator.

Before attempting to detect a PNMR signal it is instructive to examine how the pulse programmer functions. 

Start by observing single pulses from the pulse programmer. Make the following settings on the pulse programmer:

• A-width: halfway
• Mode: Int
• Repetition time: 10 ms and 10%
• Sync: A
• A: On
• B: Off
• Sync Out: connected to oscilloscope external trigger input
• A&B Out: connected to oscilloscope channel 1 input

Set the storage scope for external triggering on a positive slope. Start with a horizontal time base of 10 μs and a vertical scale of 0.5 V. Make sure that the trigger point is centered on the oscilloscope screen. 

Turn the A-width knob and observe the effect on the pulse.

<blockquote> NOTEBOOK: </HTML>

• What are the maximum and minimum pulse widths that the generator can produce?
• If a pulse width in the middle of this range is to create at 90º pulse, use Eq. (18) to estimate how strong the RF B field must be inside the sample.

</blockquote></HTML> Set the scope's time base to 500 μs and the repetition time of the pulse programmer to 10 ms. Watch the scope while you adjust the variable repetition time between 10% and 100%.

<blockquote> NOTEBOOK: </HTML>

• What is the effect of the repetition time? 

</blockquote></HTML>

Make the following settings on the pulse programmer:

• A Width: arbitrary
• B Width: arbitrary
• Delay Time: 0.10 x 100 (100 μs)
• Mode: Int
• Repetition Time: 10 ms and 100%
• Sync: A
• A: On
• B: On
• Number of B Pulses: 1
• Sync Out: connected to oscilloscope external trigger input
• A&B Out: connected to oscilloscope channel 1 input

These settings should create an A pulse followed by a B pulse delayed by the delay time. The sequence repeats after the repetition time.

Set the scope's time base to 25 μs and make sure the trigger point is centered on the screen. Identify which pulse on the scope corresponds to the A pulse and which is the B pulse by adjusting the A and B pulse widths. Change the pulse programmer sync setting from A to B.

<blockquote> NOTEBOOK: </HTML>

• What effect does the sync have?

</blockquote></HTML> Set the scope time base to 500 μs. Play around with the variable repetition time setting and the delay setting.

<blockquote> NOTEBOOK: </HTML>

• What effect does changing the delay time have?
• What effect does changing the repetition time have?
• By adjusting the A and B pulse widths as well as the delay time, is it possible to make the two pulses overlap? What would be the consequence of this in terms of the behavior of the nuclei in the sample? 
• Is it possible to adjust the repetition time so that the B pulse overlaps with the A pulse of the following A-B pulse pair? What would be the consequence of this in terms of the behavior of the nuclei in the sample?

</blockquote></HTML>

It is important to adjust the sample to the proper depth inside the probe. A rubber O-ring, placed on the sample vial, acts as an adjustable stop for placement of the sample in the center of the RF field and receiver coil. (See Fig. 11.)

For this first experiment we have chosen mineral oil as a sample because it has both a large concentration of protons, and a short spin-lattice relaxation time _T_1.  We must wait at least 3_T_1, (preferably 6-10 _T_1's) before repeating the pulse train. For a single pulse experiment that means a repetition time of 6-10 _T_1.  Mineral oil has a _T_1 in the range of 20-80 ms near room temperature. Thus, a repetition time of a few hundred milliseconds should be sufficient for the magnetization to reach thermal equilibrium by the start of each new pulse sequence.

To create 90º pulses, set the following:

• A-Width: ~20%
• Mode: Int.
• Repetition Time: 100 ms and 100%
• Number of B Pulses: 0
• Sync: A
• A: On
• B: Off
• Time Constant: 0.01 ms
• Gain: 50%
• CW-RF: On
• M-G: Off

Connect channel one of the scope to Detector Out from the 15 MHz receiver and channel two of the scope to Mixer Out from the 15 MHz OSC/AMP/Mixer. Externally trigger the scope from the Sync Out of the Pulse Programmer. Connect A+B Out from the Pulse Programmer to A+B In on the 15 MHz OSC / AMP / MIXER.

Turn on the cooling water and the power supply for the electromagnet.

While observing the Detector Out and Mixer Out signals on the scope, slowly increase the magnetic field until a FID signal is observed on the scope. A sample FID signal is shown in Fig. 12.

{ ${/download/attachments/141459859/fig_12.png?version=1&modificationDate=1451935105000&api=v2}$ Figure 12: Examples of FID and Mixer signals

<blockquote> NOTEBOOK: </HTML>

• When a FID is visible, what is the relationship between the protons' Larmor precession frequency and the 15 MHz applied to the sample?

</blockquote></HTML> For very precise matching of the two frequencies, adjust the magnetic field or the RF frequency for the zero-beat condition of the mixer output.

At resonance, the A pulse will cause { ${/download/attachments/141459859/M.png?version=4&modificationDate=1451419584000&api=v2}$to precess away from the z-direction. The angle of precession away from the z-axis will be proportional to the duration of the A pulse. If the A pulse width is just right, { ${/download/attachments/141459859/M.png?version=4&modificationDate=1451419584000&api=v2}$ will precess 90º into the x-y plane. After the A pulse is turned off, the remaining B field is just the electromagnet's field, _B_z . Thus, { ${/download/attachments/141459859/M.png?version=4&modificationDate=1451419584000&api=v2}$ will now precess around _B_z (precession in the x-y plane). <blockquote> NOTEBOOK: </HTML>

• Given the orientation of the detector coil in the sample probe, how can we tell when the A pulse is a 90º pulse?

</blockquote></HTML> Vary the A pulse width over its full range and observe the effect on the amplitude of the FID. Adjust the gain on the Receiver to nearly maximize the FID amplitude, with no clipping of the FID.

<blockquote> NOTEBOOK: </HTML>

• Describe the FID characteristics for a 90º, a 180º and a 270º A pulse.

</blockquote></HTML>

Adjust the A width for a 90º pulse. Adjust the Tuning control on the receiver to maximize the amplitude of the FID.

Adjust the receiver gain so the FID amplitude is about 10 to 10.5 V (with no clipping) and the mixer output shows zero beat frequency. Experiment with the height of the sample in its holder. Iterate as needed.

No magnet produces a perfectly uniform field. However, it is possible to find a region of maximum uniformity (the sweet spot). To do so, move the sample probe around in the magnet gap while observing the changing shape of the FID. A uniform field is indicated by a long, smooth exponential decay of the FID. In a perfectly uniform field the exponential decay constant of the FID would be _T_2, the spin-spin relaxation time constant, and would be on the order of 10 ms. Field inhomogeneities can dominate the relaxation time with a time constant of order 0.1 ms.

<blockquote> NOTEBOOK and REPORT: </HTML>

• The oscilloscope is attached to the computer using a serial cable. You can use the “Scope Transfer” software on the computer to save a trace of your FID, and reproduce it in your report. When you have the best signal on the scope, hit “Run/Stop” (in the upper right corner) and use the software to transfer both Channel 1 and Channel 2. Save the data as a text file after the transfer is complete.

</blockquote></HTML>

While at resonance (zero-beat condition) remove the sample tube containing the mineral oil sample and measure the magnetic field using the Hall effect gaussmeter. Calibrate your gaussmeter using the calibration magnets provided. Place the tip of the gaussmeter probe in the same region of the magnet where the sample sat. Note that the flat face of the Hall effect crystal must be perpendicular to the direction of the magnetic field being measured. Make a careful estimate of the uncertainty of this measurement.

<blockquote> NOTEBOOK and REPORT: </HTML>

• From your measured values of resonant frequency and magnetic field, calculate γ for protons. 

</blockquote></HTML> Repeat the measurement for Sample #2 which is Teflon containing fluorine atoms. You will have to readjust the magnetic field slightly to find the resonance. 

<blockquote> NOTEBOOK and REPORT: </HTML>

• From your measured values of resonant frequency and magnetic field, calculate γ for fluorine nuclei.
• Discuss the sources of error between your values of the gyromagnetic ratios and the literature values, _γ_p = 2.675 x 104 rad/(G s) and _γ_F = 2.518 x 104 rad/(G s), respectively. 

</blockquote></HTML>

The relaxation times of protons in solutions are strongly related to the rate of interactions with nearest neighbors, i.e. the Brownian motion of the particles in the liquid. Higher viscosity solutions have more persistent interactions between particles and thus, typically have shorter relaxation times than low viscosity solutions.

The characteristic decay time T_2 is called the s_pin-spin relaxation time. _T_2 is the time constant associated with the decay of the precessing net magnetization in the x-y plane due to local field interactions that cause the protons in the sample to precess at slightly different frequencies. As described in Appendix A of the TeachSpin manual, in the absence of field inhomogeneities, _T_2 can be determined by measuring the decay of _M_x (or _M_y) after a 90º pulse, i.e., the FID amplitude decay envelope. However, in most cases, (such as ours), magnetic field nonuniformity dominates the FID decay waveform and the sample _T_2 is not measured. Our magnet has a sufficient uniformity to produce at least a 0.1 ms decay time, less than many of the relaxation times we wish to measure.

For measuring _T_2 we will therefore use the multiple pulse, spin echo method due to Hahn which uses a pulse sequence

90º → τ → 180º → τ → “echo”

as described in section 4.3.

Our starting point is the setup we have for the first FID experiment, with a 90º A pulse, which we will use together with a 180º B pulse. The 90º pulse rotates the net magnetization away from the z-axis into the x-y plane where it precesses at the Larmor frequency. We allow the magnetization to decay for a time τ. The magnetization decays due to the differences in precession frequency caused by both the reversible effects of field inhomogeneities and the irreversible effects of nearest neighbor interactions with the reversible effects dominating. We now apply a 180º pulse to reverse magnetization within the x-y plane. At a time τ after the 180º pulse all of the spins that were out of phase due to field inhomogeneites will be back in phase and an echo of the magnetization will appear in the detector where any loss of amplitude (relative to the amplitude of the FID following the initial 90º pulse) is due only to the irreversible nearest neighbor interactions.

1. Establish resonance. - Make sure that Mo is between 10 V and 10.5 V. - Turn off the B pulse and trigger the scope on the A pulse. - Set the A Width to give a 90º pulse. - Maximize the FID signal and find the magnet sweet spot. - Turn off the A pulse temporarily, turn on the B pulse and trigger the scope on the B pulse. - Set the B Width to give a 180º pulse. Turn up the B width through its first maximum (90º pulse) to its first minimum which is a 180º pulse. - Turn both A and B pulses on and set the scope to trigger on the B pulse. - For a short delay time τ, view the echo and tune the B width to maximize the echo.

Measure the amplitude of the echo as a function of delay time. Take data for delay times ranging from the shortest time which will allow the echo to be resolved as separate from the FID, to the longest time for which the echo can be distinguished from the noise. The scope's cursor can be used to display the numerical value of the echo amplitude. _T_2 can be extracted from a plot of the echo amplitude vs. delay time according to Eq. (34).

The viscosity of the mineral oil is strongly dependent on temperature. We also use a generic brand of commercial mineral oil and consequently there is no well established “accepted value” with which you can compare your measurements. The literature in the lab will provide you with a range of time constants for various light and heavy oils that can be used to tell if your results are consistent with others. Beyond that however, you should treat the results of your measurements as you would a new measurement of a material for which the time constants are not known. In your lab report, using a careful analysis of the data and relevant uncertainties, you should strive to convince the reader that you have made an accurate measurement of the time constants for your mineral oil sample. You will find that it takes a fair amount of courage to present an experimental result, for which the “correct” value is not known, based solely on your interpretation of the data.

The setup is similar to a two pulse _T_2 measurement, but now the A pulse is a 180º pulse and the B pulse is a 90º pulse.

The 180º pulse rotates the equilibrium net magnetization { ${/download/attachments/141459859/M0.png?version=2&modificationDate=1451420698000&api=v2}$ through 180º to { ${/download/attachments/141459859/minus_M0.png?version=1&modificationDate=1451420723000&api=v2}$ and then the amplitude of the FID signal following the 90º pulse delayed by τ can be used as a measure of _M_z(t) via

{ ${/download/attachments/141459859/eqn_36.png?version=1&modificationDate=1451420813000&api=v2}$

(36)

1. Establish resonance with the A pulse and set the width to 90º. - Set the A pulse to 180º, the first minimum. - Turn off the A pulse, turn on the B pulse and trigger the scope on the B pulse. - Set the B width to give a 90º pulse by maximizing the first FID signal. - Turn both A and B pulses on and set the scope to trigger on the B pulse. - Optimize the echo by checking the magnet sweet spot and small tuning of the A and B pulse widths

Measure M_z(t)_ (i.e. the amplitude of the FID) for 0.1_T_1 < t < 10_T_1 and find _T_1. Make sure you have measured _M_0, the asymptotic value of M_z. You can measure the relative amplitudes of M_z(t) with the scope's cursor if the waveform is not changing its width. You can determine _T_1 by fitting your data to the functional form of Eq. (36).

When writing your report, consult the rubric and notes below for the appropriate quarter.