lecture 1. general introduction, semi-classical description of...

Macromolecular NMR Spectroscopy BCH 5886T.M. Logan Spring, 2005

Lecture 1. General Introduction, Semi-Classical Description of NMR.This course will provide the foundation for understanding NMR experiments and relating

NMR-observables to molecular structure. We will start by describing the NMR phenomenon. A formal description of the response to an rf pulse by an equilibrium system of nuclear spins requires quantum mechanical calculations of angular momentum changes, which we will not get into in great detail in this course. Instead, we will use two models. The first is a convenient, semi-intuitive, model for describing pulsed NMR, called the semi-classical treatment. In this model, nuclei are considered as classical bar magnets that have quantized (or discrete) orientations in the presence of an external magnetic field (hence, the name semi-classical). Although widely used, this model is limited to descriptions of basic NMR phenomena, and can not be used to describe the vast majority of 2D and 3D NMR experiments. Therefore, a second model, using a more formal language for understanding NMR experiments, called the Product Operator Formalism, will be presented later to facilitate understanding those more interesting experiments.

1.1. Equilibrium Magnetization within the Semi-Classical Model.NMR spectroscopy involves probing transitions between nuclear spin energy levels using

pulses of radio-frequency light. The energy spacing, and consequently the frequency of light that is absorbed, is strongly dependent on the specific chemical environment of each nucleus. This sensitivity to environment makes the resonance frequency a sensitive and powerful fingerprint of molecular structure. Therefore, the first step in a study of NMR spectroscopy is to understand how the environment influences the observed resonance frequency and how this resonance frequency is measured in and NMR experiment. The simplest way to approach this problem for a single nucleus is via the semi-classical (or vector) model.

In the semi-classical model, the nucleus is considered to be a charged particle of spherical shape with a magnetic dipole moment, µ (a vector; see Figure 1). In the absence of an external magnetic field, B0, the magnetic moment is randomly oriented. However, in the presence of a magnetic field, two things happen. First, the magnetic moments become aligned (or adopt specific orientations) either parallel or anti-parallel to the magnetic field. More precisely, we would say that the projection of the magnetic dipole moment onto the z-axis takes discrete, or quantized, values. Adopting only two orientations is correct for spin-1/2 nuclei like 1H, 13C, or 15N, but other orientations are allowed for nuclei having larger spin angular momenta. Finally, limiting these vectors to discrete positions is non-classical but this restriction is required to match the quantum mechanical realities of nuclear angular momentum (more on this later). On the

Figure 1. In the semi-classical model, nuclei are pictured as bar magnets that are induced to precess in the presence of an external magnetic field, B0. Parallel and anti-parallel orientations are shown in the left and right, respectively. Precession of the magnetic moment about the B0 field is indicated by the circular arrows.

B0

µ

B0

µ

1


other hand, the x- and y- projections of µ are still randomly oriented in the presence of B0. This behavior is indicated in Figure 1 for a single nucleus.

The second effect of the external magnetic field is to exert a torque on the magnetic moment which causes the magnetic moment to rotate, or precess, about the axis defined by B0. Precession generates an angular momentum for the particle. The time-dependent motion of the

angular momentum vector, , is written as

[1.1]

where µ is the magnetic moment vector of the nucleus. The angular momentum is proportional to

the magnetic moment, , where the constant or proportionality is known as the magneto-gyric (or gyromagnetic) ratio and takes a well-known value for different nuclei (see Table I).

We can describe the motion of the magnetic moment in the presence of an external

magnetic field by substituting for in Eqn 1.1, giving

. [1.2]

This equation of motion is important because the precession rate, ω, (or Larmor frequency) is equal to the resonance frequency of a given nucleus,

. [1.3]

The resonance frequency is related to the energy gap between the two states that we are probing by

[1.4]

Table 1: Properties of some nuclei of interest in NMR.

nucleusgyromagnetic

ratio, γ(107 rad/Ts)

NMR Frequency

(MHz)Spin Natural

Abundance

1H 26.7519 100.0 1/2 99.985

2H 4.1066 15.351 1 0.015

13C 6.7283 25.144 1/2 1.108

15N -2.7126 10.133 1/2 0.37

31P 10.8394 40.481 1/2 100

e- -1.76x104 65,918 1/2 100

J

tdd J µ B0×=

γJ µ=

J

tdd µ µ γB0×=

ω γB0–=

∆Eh

------- ω γB0–==

2


where is Planck’s constant divided by 2π. In the case of NMR, we are probing the difference in energy between the two orientations of µ along B0, Figure 2.

This description given so far is adequate for a single nuclear spin. However, in general a typical NMR sample contains nearly Avagadro’s number of spins (usually millimolar sample concentration). In the presence of an external field, each of these 1020 magnetic dipoles have random orientations along the x- and y-axes, but one of two orientations along the z-axis. The collective behavior of these spins is given by the vector sum of the individual magnetic vectors, known as the bulk, or net magnetization vector, M0, shown in Figure 3. Random orientation along x and y gives a net zero projection along these axes (Mx=My=0); enhanced population in one orientation over the other (governed by the Boltzman distribution) leads to a net vector in one direction (Mz = M0), which is parallel to B0 for 1H, 13C, and 31P, but anti-parallel for electrons and 15N. Note that this is traditionally drawn as “up”, i.e., aligned along the positive z axis, where this direction represents the lower energy. The interaction between the net magnetization vector and the static magnetic field is described by an equation analogous to Eqn. 2 by substituting M for µ,

[1.5]

Finally, the energy of the bar magnet orientation is proportional to its projection onto the external

magnetic field, .

h

B0

parallel

anti-parallel

∆E

∆E hω hγB0–= =

Figure 2.

tdd M M γB0×=

Energy M– B0⋅ M B0 θcos–= =

x

y

zFigure 3. The collective behavior of a large number of spins (left) is represented as a single vector (right) called the bulk, or net, magnetization vector, M.

3


1.2 Effects of RF Pulses on Bulk Magnetization.Brief pulses of radio-frequency (rf) light are used to excite the spins in an NMR experiment.

As in any other form of spectroscopy, excitation induces transitions between adjacent energy levels, which we will discuss more fully in a later section. In the semi-classical model, the rf pulses excite the system by rotating the net magnetization vector, M. More precisely within the framework of this model, we would say that the magnetic component of the radio-frequency EM radiation applied to the sample generates a torque on the net magnetization vector and rotates it about the axis of the applied pulse. The angle through which the spins are rotated, generally called the “flip angle”, θ, is dependent on the total power of the rf pulse, which is given by

[1.6]

where γB1 is the intensity of the applied pulse (field strength), and τp its duration. The intensity and duration of the pulse are experimental variables set by the spectroscopist. Typical values corresponding to a 90° flip angle for 1H’s on the 500 MHz spectrometer at the Magnet Lab are γB1= 25 kHz and τp=10 µsec. It is important to mention two things. First, the linear relationship between power, duration and flip angle is important. A 90° rotation would also be obtained for a 12.5 kHz field (half the pwer) applied for 20 µsec (twice the duration). On the other hand, doubling the pulse duration (20 µs) at a given power (γB1= 25 kHz) rotates the spins through 180°. Second, the power of a pulse is easily specified by the spectroscopist, but actual power transmitted to the probe is not easily predicted as it depends on many variables, and must be (re)calibrated for each sample. Once calibrated at a given power, the response is generally quite linear using modern spectrometers. Some of the factors affecting the power transmitted to the probe are probe-specific (such as probe Q or quality factor and tuning; more on these topics in Chapter 3); some are sample specific (such as sample dielectric or salt content; organic solvents typically transmit the rf power with lower loss than do aqueous solvents). Probe factors are generally stable over large periods of time (except for the tuning, which is done every time a new sample is inserted into the probe), and 90° pulse widths are typically constant (or show only slight variation) over extended time periods. If you experience a significantly different 90° pulse length from what you are used to, this may indicate some problem with the instrument.

A 90° flip angle (or 90° pulse) will rotate magnetization through 90°, as shown in Figure 4. The sense of rotation is clockwise for positive flip angles (by convention, although not by rigorous mathematics), and the effects of pulses are (to a first approximation) cummulative. The two pulses that are used most often in NMR experiments are the 90° and 180°. The 90° pulse along

90° x pulse 90° x pulse

Mz My -Mz

Figure 4.

θ γB1τp=

4


the x-axis rotates Mz → My, or -My → Mz, while a 180° x pulse rotates Mz → -Mz, or My → -My, etc. These pulses are shown in Figure 5. Finally, since pulses apply torque to the bulk magnetization vector, it follows that a pulse applied along the x axis would leave unchanged any component of the net magnetization vector on the x-axis. In other words, Mx is un-affected by a 90° or 180° x pulse; similarly, My is unaffected by pulses along the y-axis.

1.3. Mathematical Description of Pulses: The Rotating FrameIn Section 1.2, the effect of the rf pulses on an isolated spin was described using English. Of

course, being physical scientists, we long for a more formal and rigorous description of this process. In this section, we describe the effect of pulses on the bulk magnetization using a more quantitative, mathematical picture. This requires us to work in the rotating frame of reference. Therefore the first part of this section describes the transformation into the rotating frame, and the effects of rf pulses are discussed in the second part of this section.

Minimally, what we are doing is appying a second external magnetic field to the spins. Therefore, the motion of the bulk magnetization can be described using a modified form of Eqn. 5 that includes the additional magnetic field arising from the pulse which includes the additional magnetic field of the pulse (B1). Since B0 and B1 are vectors, the magnetization will precess about the vector sum of the two fields. This motion begins to get complicated due to the rapid precession about B0, so we will change our coordinate system to simplify the math. Specifically, we will change from the “lab frame” to the “rotating frame”.

Mathematically, the transformation to the rotating-frame is accomplished in the following way. First, the equation of motion for M in the rotating and lab frames are related by

[1.7]

where ω0 is the angular velocity at which the frame of reference is rotating. Substituting Eqn. 1.5

Mz My

Mz-My

-MzMz

My -My

90° x

90° x

180° x

180° x

;

;

Figure 5.

tdd M⟨ ⟩

rot tdd M⟨ ⟩

labM ω0×+=

5


for and re-arranging gives

[1.8]

which says that the motion of the bulk magnetization in the lab and rotating frames can be considered in the same way, with the difference coming in the nature of the magnetic field. Whereas the spins precess about B0 in the lab frame, in the rotating frame, the magnetization rotates about a reduced field, Bred, defined as

, [1.9]

giving

[1.10]

Although this equation resembles Eqn 5, it doesn’t appear that we have simplified the situation much. However, we can set the angular velocity of the rotating frame to be anything we desire. If

the angular velocity of the rotating frame is chosen such that the reduced field

becomes zero in Eqn 1.9, and

. [1.11]

tddM

⎝ ⎠⎛ ⎞

lab

tdd M⟨ ⟩

rotM γ B0

ω0γ

------+⎝ ⎠⎛ ⎞×=

Bred B0ω0γ

------+=

tdd M⟨ ⟩

rotM γBred×=

ω0 γB0–=

tdd M 0=

LAB FRAME. Magnetization shows complicated oscillatory behavior.

ROTATING FRAME. Magnetization shows simpler behavior.

x x’

y’y

Figure 6. Lab (A,C) and rotating (B,D) frame pictures of precession. In the Lab frame, the coordinate system is stationary and M moves within it. In the rotating frame, the coordinate system moves with M, and there is no relative motion.

0 1 2 3 4 5 6 7 8 9 10-0.8

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1

0 1 2 3 4 5 6 7 8 9 100

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1

A B

DC

6


This is a very significant result because it means that, in the rotating frame, the bulk magnetization is time-independent, i.e., stationary. In other words, if we set the angular velocity of the rotating frame equal to the Larmor frequency of the spins (see Eqn 1.4), the spins appear stationary. Conceptually, what we have done is quite simple. Imagine describing the motion of someone on a merry-go-round as you stand in the park. Their motion can be described by a sum of sin and cos functions. However, add a couple of other people to the merry-go-round and have them move relative to one another and you get a complicated mathematical expression that describes their motions. This is the sitation that we have in NMR when working in the laboratory frame. If you are only interested in the motion of each person relative to the others, it would be much simpler to describe their motions if you were also on the merry-go-round, i.e., if you were rotating with them. In this coordinate system, the precession due to the movement of the merry-go-round (B0) would be eliminated, and you could concentrate on the motion of the people (magnetization vector(s)).

The choice of the rotating frame frequency is trivial if we have a single spin - it is set to that spins Larmor frequency. However, if we have two spins, each with slightly different Larmor frequencies, ωa > ωb, then, in the rotating frame, we can chose ω0 = ωa, ωb, or some other

frequency, say . In these cases, the apparent precession rates for ωaand ωb are

shown in Figure 7. If the rotating frame is rotating at a frequency equal to the Larmor frequency of ωa, ωa appears stationary, and ωb appears to precess in a counter-clockwise direction as its frequency is less than ωa. More quantitatively, Bred is not zero but is very close for ωb; therefore this spin precesses (slowly) around the residual field. The situtation is reversed if the rotating frame rotates (or is “on resonance”) with ωb: in this case ωa appears to precess in a clockwise manner relative to ωb. If the rotating frame is taken to be mid-way between ωa and ωb, both vectors precess: ωa in a clockwise and ωb in a counter-clockwise sense. The fact is that it really doesn’t matter where we set the rotating frame so long as it is close to the resonance frequency of the spins whose behavior we are trying to describe. In practical terms, the rotating frame frequency is taken to be the carrier frequency of the spectrometer (we will discuss this further in a later chapter). For now, know that the carrier frequency is typically set in the middle of the spectral region we are interested in.

Finally, the rotating frame transformation can be thought of as a mathematical trick that

ωa ωb+( ) 2⁄

ωa = ω0 ωb = ω0 ω0 = (ωa + ωb)/2

a

b

a

b

a

b

ωb = (ω0 - ωb) ωa = -(ω0 - ωa)

Figure 7.

7


scales the size of the B0 field. If then ; if (e.g., on resonance) then

; and for then .

Now, back to the effects of the rf pulses. As we stated above, the time-dependent motion of M0 in the rotating frame is given by Equation 1.11, with Bred given by Equation 1.10. In the presence of an applied rf pulse, B1, the effective field, Beff, is the vector sum of the two fields

[1.12]

Plugging Eqn 1.12 expression into Eqn 1.11 gives an expression for the motion of M0 during a pulse,

. [1.13]

In the rotating frame, and . Eqn. 1.13 reduces to

[1.14]

which says that M0 rotates about B1, which is what we stated previously. Now, in the pictoral description given above, if B1 is applied along the x-axis, M0 is rotated to My, but according to the mathematics, vector rotation follows the right hand rule: a pulse oriented such that it’s value increases along positive x-axis will rotate M0 onto -My (Figure 8a), which is backwards from the rotation shown in Figures 4 and 5. Traditionally, the rotation is taken to be clockwise; a pulse oriented along x is drawn as increasing along the negative x axis (see Figure 8b). We will follow convention, if not physics, when using the vector diagrams (the difference between our model and more rigorous mathematics is the orientation of the pulse and ultimately makes no significant differnece in the results). Be aware that vector diagrams are models of the spin system, and we can modify our model to be convenient as long as we are internally consistant.

1.4. Precession After the Pulse and Return to EquilibriumWhen the pulse is finished, the net magnetization vector in the xy plane (refered to as

transverse magnetization) experiences a torque do to B0 and precesses (this is true in either the lab or rotating frames; the only difference being the precession frequency, which could be zero if we are exactly on-resonance in the rotating frame). If we could sit one one axis (say, the x-axis)

ω0 0= Bred B0= ω0 ω=

Bred 0= 0 ω0 ω≤ ≤ B0 Bred 0≥ ≥

Beff Bred2 B1

2+=

tdd M t( ) M γBeff×=

Bred 0= Beff B1=

tdd M⟨ ⟩

rotM γB1×=

90°x 90°x

A. B.

Figure 8.

8


and look towards the origin at the transverse magnetization vector, this precession would generate a (co)sinusoidal variation of the magnetization intensity in time, with frequency equal to the Larmor frequency for our spin (cos ωt), and intensity proportional to the number of nuclei having that Larmor frequency. In the absence of any additional forces, this oscillation would continue un-damped, essentially forever (Figure 9A). However, this is a non-equilibrium state, and the system relaxes back towards equilibrium where Mz = M0 and Mx = My = 0. Relaxation is assumed to be a random process (in terms of which individual magnetic moment returns to equilibrium at which time) and is modelled using an exponential function of the form where T is the relaxation time constant, in units of sec. If we multiply the undamped sinusoid by an exponential decay, the result is a damped sinusoid, which is plotted in Figure 9B. These plots provide a “one-dimensional” representation of the trajectory of the bulk magnetizaion after the pulse. The plot in Figure 9b is familiar to anyone having experience in collecting NMR spectra: it is precisely the signal detected, which is termed the “free induction decay”, or FID. Remember, the transverse magnetization precesses about B0 in a circular motion; combine this with the exponential decay due to relaxation, and you generate a helical trajectory in three dimensions, with the radius decreasing in time as the transverse magnetization realigns along the z-axis this is shown schematically in Figure 10. We’ll have more to say about the details of relaxation, and how this signal is detected later.

1.4 A More Mathematical Treatment: The Bloch Equations.The time-dependent motion of the bulk magnetization vector is concisely described by the Bloch Equations, which are phenomenological in nature (this means that they were not derived from

t– T⁄( )exp

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0

0.2

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0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10-1

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0

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1

A

B

Plot: y = cos αx → undamped sinusoid

Plot: y = cos (αx) e(-x/T) → exponentially damped sinusoid

Figure 9.

9


first principles, but were developed based on certain assumptions about the behavior of the system). There are two things that need to be introduced before the Bloch equations can be presented. First, we need a better description for the nature of the rf pulse, B1. An oscillating magnetic (or electric) field of frequency ω0 propagating in one direction, say the x-axis, can be decomposed into two counter-rotating vectors oscillating at ±ω0. Since only one of these fields is in resonance with the rotating frame, the magnetic field component of the rf pulse is represented as

[1.15]

where i, j, and k are unit vectors along the x, y, and z directions of either the lab or rotating frame. Second (this is the phenomenological part) Bloch assumed that the relaxation of the bulk magnetization could be described by a simple exponential, as mentioned above. Combining this exponential relaxation with our previous expression for the motion of the bulk magnetization gives the Bloch Equation:

. [1.16]

M is the bulk magnetization vector, behaving classically, and R is the Relaxation matrix (the hat signifies a matrix). Equation 1.16 can be decomposed to give the following expressions for the projection of the M0 onto the three cartesian axes (in the rotating frame):

, [1.17]

, [1.18]

and

. [1.19]

These equations appear rather intimidating, but can be described simply. First, recall that the bulk magnetization, M, and applied rf field, B1, are vector quantities so the first term on the right hand side of Eqns 1.17-1.19 represents the cross-product of two vectors (recall that for any two vectors, ). Since we are in the

rotating frame, the only time these terms contribute is in the presence of a pulse or a continuously-applied rf field. Second, the amount of magnetization projected onto each of the three Cartesian axes depends on a constant, either T1 for Mz or T2 for Mx and My. These time constants are known as the longitudinal and transverse relaxation times, respectively, and represent the time at which the magnetization is reduced (or increased) by a factor of .

B iB1 ωtcos jB1 ωtsin kB0+–=

tdd M M t( ) γBeff× R M t( ) M0– –=

tdd Mz t( ) γ MxBy MyBx–[ ]

Mz M0–( )T1

-------------------------–=

tdd Mx t( ) γ MyBz MzBy–[ ]

MxT2-------–=

tdd My t( ) γ MzBx MxBz–[ ]

MyT2-------–=

u v× uyvz uzvy–( )i uzvx uxvz–( )j uxvy uyvx–( )k+ +=

1 e⁄

10


The Bloch equations are useful for calculating the trajectory of the bulk magnetization following a pulse, during the application of a selective pulse (generally a low-power, long-duration pluse), and a variety of other situations where there is no transfer of magnetization from one spin to another, i.e., they can not be used to describe the vast majority of multi-dimensional NMR experiments in use today. A graphical example of the magnetization trajectory calculated using the Bloch equations is shown in Figure 10. We won’t use these equations extensively in the class, but but it is important that you have a working understanding of them. In a later section, I will present a more formal description of the Bloch Equations using vectors and matrices, and we will do a couple of simple calculations to see that they are not very complicated. Our calculations will be performed for a single spin, though. For more complicated spin systesm, the Bloch Equations quickly become tedious to do by hand, but luckily, there is a web-based computation tool, Gamma, that can be used.

1.6 From Precession to the SpectrumSo far, we have described equlibrium (un-excited) magnetization, the effects of rf pulses, and

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

0 0.2 0.4 0.6 0.8 100.20.40.60.8

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0

0.5

Figure 10. The temporal trajectory for M0 is plotted in three dimensions in A, with the projections along the x, y, and z axes shown in B, C, and D, respectively, calculated using the Bloch equation in the rotating frame (with ω0 ≠ Larmor frequency, i.e., offset from resonance). The initial conditions were Mx = My; Mz = 0; T1 = T2 =0.1 sec; offset = 100 Hz. The trajectory is calculated over 1 sec. Note the oscillatory behavior of transverse magnetization components (because the rotating frame is not on-resonance with the spin). Note, also that the longitudinal component does not oscillate.

A.

B

C

D

Mx

My

Mz

Figure 10.

11


how the excited magnetization relaxes back to its equlibrium (Boltzmann) distribution. We need to finish with a brief discussion of the NMR spectrum resulting from all of these manipulations. The discussion will follow from the Bloch Equations, and will not involve detailed use of Fourier transforms.

The signal we detect in an NMR experiment is the transverse magnetization as it precesses about B0 in the absence of an rf pulse (hence the designation as a free induction decay). Without any details, the magnetization vector induces a voltage in the receiver coils of the spectrometer; the intensity of the voltage is proportional to the size of the magnetization vector along the axis of the detector. Therefore, the “signal” is a damped sinusoid, similar to the function plotted in Figure 9b. In the rotating frame, the Bloch Equations for the three components of the bulk magnetization are

. [1.20]

Solving this differential equation by integration gives the Mz value at any time, t,

. [1.21]

Similarly, the transverse components, which together represent the signal (M+(t)) are given by

[1.22]

where Mx and My in integrated form are given as

[1.23]

and

. [1.24]

These equations say that the detected signal is the sum of the projections of bulk magnetization vector (M0) onto the x- (Mx) and y-axes (My). Furthermore, Mx has a cosine modulation while My has a sine modulation at the Larmor frequency, ω; both are attenuated by a time constant, T2. Note for completeness, that the rate of transverse relaxation, R2, is the inverse of the transverse relaxation time constant, T2. The signal is traditionally represented as the sum two components, one being in phase with the transverse magnetization (u(t)), the other being 90° out of phase (v(t)):

[1.25]

These two components are related to Mx and My as

[1.26]

tddMz Mz M0–( )–

T1----------------------------=

Mz M0 1 expt T1⁄–

–( )=

M+ t( ) Mx t( ) iMy t( ) M0exp iωt tT2-----–⎝ ⎠

⎛ ⎞=+=

Mx M0 ωt( )exp t–T2-----⎝ ⎠

⎛ ⎞cos=

My M0 ωt( )exp t–T2-----⎝ ⎠

⎛ ⎞sin=

S t( ) u t( ) iv t( )+=

u t( ) Mx ωtcos My ωtsin–=

12


and

[1.27]

The signal can be Fourier transformed to provide a complex signal composed of two parts, one absorptive (u(ω)), the other dispersive (v(ω)).

[1.28]

where

[1.29]

and

. [1.30]

These two components represent the real (u) and imaginary (v) parts of the signal. They are graphed in Figure 11. Note that they have distinctly different appearances. Specifically, the absorptive line is quite narrow compared to the dispersive line (which extends beyond the range shown in the plot before reaching zero intensity again. Note that the Bloch Equations predict that the linewidth is dependent on the T2 relaxation time constant. This linewidth is given by (1/πT2), and is shown in the figure (as the full width at half height or half-max; note that the absorptive and

v t( ) M– x ωtsin My ωtcos–=

S ω( ) u ω( ) iv ω( )+=

u ω( ) γB0M0R2

R22 ω ω0–( )2+

------------------------------------=

v ω( ) γB0M0ω ω0–

R22 ω ω0–( )2+

------------------------------------=

-100 -80 -60 -40 -20 0 20 40 60 80 100-0.3

-0.2

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0

0.1

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Figure 11. Plots of the absorptive (solid) and dispersive (dashed) parts of a complex signal detected in an NMR experiment (see Eqns 1.29 and 1.30). Plots were made in MATLAB using

and for the absorptive (solid line) and dispersive (dashed line) components, respectively. In these equations, x is the resonance offset term (ω0-ω), and the transverse relaxation rate (R2) is reflected in the numerical constants (these numbers were chosen merely for esthetic reasons, i.e., they made the plots look good.

y 2 4 x2+( )⁄= y x 4 x2+( )⁄=

arbi

trary

inte

nsity

uni

ts

arbitrary frequency units-40 -30 -20 -10 0 10 20 30 40

-0.3

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0

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linewidth

13


dispersive lines have exactly the same linewidth). The linewidth decreases (e.g., line becomes narrower) as the time (rate) constant for decay of transverse magnetization increases (decreases) (Figure 12).

Extended Discussion: A Closer Look at RF PulsesThere are many more interesting, and practical, details we need to cover regarding rf pulses.

First, we need to discuss the effective bandwidth of an rf pulse. Then we will discuss off-resonance effects. A generalized formalism for the Bloch Equations will be presented that will allow us to calculate the trajectory of M0 for any arbitrary set of conditions. Finally, we will take a brief look at a special pulses used for selective excitation or for extremely broad-banded excitation.

Effective Bandwidth, or How Short Should my 90° Pulse be?We excite the spins by applying a short-duration rf pulse, which we will refer to as a “hard”

pulse, or a “square” pulse. This square, hard pulse effects a broadbanded excitation. In this section, I present a couple of different ways to rationalize how this short square pulse provides broadband excitation. The models will move from handwaving to more rigorous.

Heisenberg’s argument. The simplest way to think about the broadband effect of a short pulse is to remember the Heisenberg uncertainty principle, which says that

. [1.31]

Heisenberg postulated this as a fundamental property of quantum systems, saying that the more precisely the energy of a given system is known (i.e., the shorter the range of possible energies), the longer is the time needed to measure that energy. There are other Heisenberg uncertainy pairs, like position and momentum, and different components of angular momentum, but we won’t get into them. Eqn 1.31 is significant because the energy we are referring to is the

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1

Figure 12. Absorptive lines with transverse relaxation R2=1 (solid) and R2=3 (dashed). The linewidth at FWHM are shown in the inset figure.

R2=1R2=3

∆E∆t constant=

14


resonance frequency of the nuclear spins. In other words, if we have a short range of energies (frequencies) to excite, then we need a long pulse, and vice versa. Hence, a short pulse effects broadband excitation. This also holds true for digitization of the frequency, as we’ll see later. If we want a very high resolution signal, one that accurately reflects the true lineshape of the transition, we need to acquire the signal for a long period.

Sum of Sinewaves Argument. Another way to visualize the broadband effect is the summation of a series of sine waves. As shown in Figure 13, the summation of several sine waves, each having different frequencies but the same phase, results in the formation of a region of high amplitude, having approximately a square shape (not in the figure, but in theory). In the limit of having an infinite number of sine waves added together, we would get a “square wave”. This wave would contain each individual frequency, even though it has non-zero amplitude only for a short time.

Fourier Transform of Pulse Shape. The most precise way to think about the broadbandedness of a square pulse is to use the Fourier transform to connect the time and frequency domains of

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-10 -5 0 5 100

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Figure 13. Summing Cosine waves leads to a pulse of rf. Each figure generated by summing the a cosine function (squared) over the entire range of frequency, each wave having a slightly longer wavelength but identical phase. As the number of waves added together increases, there is constructive interference only at the center of the wave, i.e., at ω=0.

10 waves 100 waves

1000 waves expansionfrom 1000waves

15


the excitation. The frequency-response (or excitation profile) of a square pulse is a sinc(x) function (sinc(x) = sin(x)/x, shown in Figure 14). Here, x is the frequency range; x = 0 indicates on resonance, with frequencies extending symmetrically away from the center of the pulse. The key points of this figure are the center plateau, and the location of the first zero crossing. Although the resolution in this diagram is low, at the center of this function (around x = 0), there is a plateau where the frequency response is relatively flat; in other words, frequencies in this plateau region experience essentially the same rf intensity (and, consequently, flip angle). As frequencies increase from the center, the rf intensity begins to roll off and reaches zero at 1/τp before going negative (I’m not sure what negative intensity means, but you can actually see resonances invert). For a short-duration pulse, say 10 µs, the first null point occurs at 100 kHz. Since we are generally exciting 1H over a 2500 Hz bandwidth (at 500 MHz), we can see that the pulse intensity has diminished only

[1.32]

at the edges of the spectral range.We have already mentioned that the rf pulse applied to the spin system rotates the bulk

magnetization. Let’s take a closer look at this. Equation 1.13 is the formal equation for describing the effects of a pulse on an isolated spin. We simplified that equation by assuming that the rotating frame was rotating at the same frequency as our spin, to give Equation 1.14. Under these conditions, when ω = ω0, we are “on resonance”, meaning that the rotating frame (and hence frequency of the pulse), ω0, equals the Larmor frequency of the spin we are exciting, ω. On the other hand, in a system having multiple spins with different Larmor frequencies, some of these will necessarily have Larmor frequencies that differ from the rotating frame; they will be off-resonance, with resonance offset given by ω - ω0. In this case, the B0 term is not cancelled completely in the rotating frame and the spins rotate about an effective field, Beff, given Eqn 1.12. This effective field is oriented at an angle, θ, to B1 given by

. [1.33]

This is shown schematically in Figure 15. If the spins are on-resonance (or nearly on

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-0.4

-0.2

0

0.2

0.4

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frequency

pow

er

y = sin(π∆ωτp) / (π∆ωτp)

first crossing point at ω = 1/τp

Figure 14.

rf intensityπ∆ωτp( )sin

π∆ωτp------------------------------ π 2500Hz 0.00001s⋅ ⋅( )sin

π 2500Hz 0.00001s⋅ ⋅------------------------------------------------------------------ 1.7%≈= =

θtanω ω0–

γB1----------------=

16


resonance), then Br≈ 0 (because ω ~ ω0), and the spins rotate about an axis parallel (or nearly so) to B1. This is the optimal situation. On the other hand, if the spins are far from resonance, Br ≠ 0, and the spins rotate about an effective field, Beff oriented at an angle, θ, given by Eqn 1.33, relative to B1. In this case, the magnetization is not rotated to lie along the y-axis, but to some position that has non-zero projections onto the x-, y-, and z-axes (see Figure 16). In the limit of being very far from resonance, Bred ~ B0, and, since B0 >> B1, and the magnetization isn’t affected at all by the pulse. We can use this information to estimate the strength of the rf pulse needed to rotate M0 to My and an arbitrary offset. If we specify that θ = 1° for spins 2500 Hz from resonance, then

[1.34]

which corresponds to a 90° pulse width of 1.7 µsec. This short pulse width is usually not obtained using probes commonly used for liquids NMR. Instead, using a typical 8 µs pulse width, the spins end up making an angle of ~4.5° with respect to the y-axis (i.e., they experience a pulse width of ~ 85.5° rather than 90°). Note that these spins are at the extreme of the chemical shift ranges typically found for 1Hs in organic and biological molecules and that the off-

Figure 15. This figure used the Bloch equations to simulate trajectories of M for pulses that are increasingly off-resonance, e.g., for each pulse, Bred increases and approaches B0. It was taken from Freeman’s Spin Chorepgraphy textbook.

B0 + ω/γ= (ω − ω0)/γ

B1

θ

Beff

B0 + ω/γ= (ω − ω0)/γ

B1

θ

Beff

(ω − ω0)/γ ≈ 0 (ω − ω0)/γ ≠ 0nearly on-resonance far from resonance

z

x x

zFigure 16.

γB1ω ω0–( )

θtan--------------------- 2500Hz

1°( )tan------------------- 2500

0.017------------- 147 kHz≈===

17


resonance effects are generally not significant for 1H (or for 31P and 15N, for that matter). It is also important to consider the “null” points in the excitation profile. In most cases, these

null contribute negatively to the pulse sequence and the resulting NMR spectrum, generating weakened or missing resonances. On the other hand, there are many 3D and 4D pulse sequences that take advantage of the known null points to effectively excite carbonyl carbons (at ~ 175 ppm) without affecting Cα spins (located at about 55 ppm). The null points occur at well-defined locations and you should commit the following relationships to memory if you plan to work in NMR:

Null point for 90° pulse:

[1.35]

Null point for 180° pulse:

. [1.36]

In Eqns 1.35 and 1.36, the frequency calculated is the offset, in hertz, from the center of the pulse to the first null point. (note that these nulls are symmetric about the center frequency of the pulse. Also note that the first null point for a 180° pulse is significantly closer to the center of the pulse than for a 90° pulse.

Using the Bloch Equations to calculate the effect of a pulse on some initial magnetization vector.

To calculate the effect of a pulse on the bulk magnetization using the Bloch equations, we proceed as follows:

[1.37]

[1.38]

for arbitrary rotation and offsett, assuming pulse along x axis. In this matrix, ,

, , and . The angle θ is the “offset” angle, and is related to the pulsewidth as

, [1.39]

with B1 representing the field strength of the applied pulse, (use 90° pulse width in microseconds), and ∆ω is the resonance offset, e.g., difference between resonance frequency of your spin and the excitation (carrier) frequency (this is the same as Eqn 1.33 but uses a slightly different notation). The angle α is the flip angle, which is calculated as

Ωnull 15 γB1( ) 0.97τp≈=

Ωnull 3 γB1( ) 0.87τp≈=

Rp θ α x,( , )M 0( ) M t( )=

cθ2 cαsθ

2+ sαsθ cθsθ 1 cα–( )

sαsθ– cα sαcθ–

cθsθ 1 cα–( ) sαcθ sθ2 cαcθ

2+

Mx

My

Mz

M′x

M′y

M′z

=

cθ θcos=sθ θsin= cα αcos= sα αsin=

θ ∆ωγB1--------⎝ ⎠

⎛ ⎞atan=

γB1 1 4 90°×( )⁄=

18


[1.40]

Notice that this equation differs from Equation 1.6 due to the incorporation of resonance offset effects; to convert from radians to degrees, change .

Example 1: 90° x-pulse, on resonance. These conditions mean that θ = 0, and α = 90° (or π/2 for you radian fans). Now we use Eqn

1.36 to determine the values for each of the matrix elements and apply that rotation matrix to our initial magnetization vector

, [1.41]

The pulse rotation matrix takes on the values indicated in Eqn 1.41, and, when applied to Mz magnetization, we have

. [1.42]

In English, we have converted Mz into -My using a 90° pulse along the x axis. Note: this result is BACKWARDS (notice that I said backwards) from that expected using our finger physics; that is ok!. The difference is in the sense of rotation, and makes no real difference for any calculation. The key is to be consistent in any individual calculation so that the relative phase changes are maintained.

Example 2: 180° x-pulse, on resonance. Proceeding as above, we get

, [1.43]

which, when applied to My magnetization, gives:

. [1.44]

α2π------ γB1τp 1 θtan( )2+–=

2π 360°→

cθ2 cαsθ




2+

1 0 00 0 1–0 1 0

→

1 0 00 0 1–0 1 0

001

01–

0=

cθ2 cαsθ




2+

1 0 00 1– 00 0 1–

→

1 0 00 1– 00 0 1–

010

01–

0=

19


Completely general case of rotations by arbitrary rf pulses.For completely general case of rotation with arbitrary phase of applied pulse, Eqn 1 is

modified as follows:

, [1.45]

where

, [1.46]

, , and φ is the relative phase angle. Note that a pulse applied along the x

axis has φ = 0; a pulse along the y axis has φ = 90.

Example 3. 90° hard pulse applied along y axis. The initial conditions are θ=0, α=90, and φ=90. This sets up the following matrix calculation:

[1.47]

Eqn 11 shows the effect of this pulse applied to magnetization initially aligned along the z-axis; the result is magnetization aligned along the Mx axis. Again, the calculated result is BACKWARDS from the result obtained using finger physics or vectors.

A.3. Composite Pulses.The discussion regarding pulse bandwidths shows that not all pulses are as effective as we

would like. While an on-resonance pulse certainly effects exactly the excitation it was designed for, other pulses fall far short of this behavior. For instance, 180° pulses are used in many 1D, 2D, and 3D NMR experiments. Assumining a good 13C 90° pulse width is ~ 12.5 µs (making the 180° pulse 25 µs) will never excite the entire 13C 200 ppm bandwidth on a 600 MHz NMR, let alone at 900 MHz. The non-180° rotation generates off-axis components to the magnetization that can lead to spectral noise, reduced coherence transfer efficiency, multiple-quantum coherence generation, and lowered decoupling. Obviously these are bad things.

To (partially) overcome these problems, NMR spectroscopists use composite pulses. These pulses are exactly what they say they are: rather than being a single pulse of rf, a composite pulse consists of two or more pulses applied in succession without free precession between them. The net rotation of this pulse series or pulse train is either a 90° or 180° pulse. One of the most common composite pulses is designed to overcome the offset effects of a single 180° pulse (e.g., the composite pulse has a significantly broader excitation bandwidth than the single inversion pulse). This pulse is applied as

90°(X)180°(Y)90°(X).

Rzˆ φ( )Rp θ α x,( , )Rz

ˆ φ–( ) M 0( )⋅ M t( )=

Rzˆ φ( )

cφ sφ– 0sφ cφ 00 0 1

=

cφ φcos= sφ φsin=

0 1– 01 0 00 0 1

1 0 00 0 1–0 1 0

0 1 01– 0 0

0 0 1

001

100

=

20


The effect of this composite pulse can be visualized using the vector notation we introduced earlier in this chapter or by using the Bloch equation methods introduced in the Appendix.

The initial 90° pulse rotates Mz onto My, but not completely (Figure 16). The magnetization rotates about an effective field, and generates transverse magnetization having both Mx and My components (we will assume that Beff is sufficiently strong that there is little residual z component). The 180°(Y) pulse rotates the vector in the transverse plane by converting the Mx component into M-x but leaves the My component unchanged. The last 90°(X) pulse completes the rotation with the same imperfections as the first 90° pulse, but the intermediate 180° pulse effectively refocuses this trajectory to align along M-z (see Figure 17).

There are many other composite pulses, most of which are more involved than this simple, but very effective, composite 180° rotation. Be aware that composite pulses have been designed for specific purposes and that they generally will not have the designed effect when used out side of this context. For example, this composite 180° pulse is designed to invert z magnetization. If it is used, instead, to invert x- or y- magnetizations, the results will be very differnet than anticipated, and probably not beneficial to the outcome of your experiment.

Section on shifting pulses? selective pulses?

SummaryIn this chapter, I introduced some rudimentary concepts of nuclear magnetism that allow us to being analyzing simple pulse sequences with some accuracy. The concepts that I expect you to

Figure 17. Composite pulse trajectories calculated using the Bloch equations for pulses that have different Bred. Again, these figures are taken from Freeman’s Spin Choreography textbook.

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know are:

• transformation into the rotating frame• origin of the bulk magnetization vector• effects of pulses on the bulk magnetization vector• relaxation behavior of transverse and longitudinal magnetization• the relationship between pulse length (or intensity) and rotation angle• the relationship between pulse length and excitation profile

For those inclined towards more in-depth analysis, you should have at least a working knowledge of the Bloch equations and be aware of the matrix formulation for Bloch equation calculations.

Additional ReadingBloch Equations:

Protein NMR Spectroscopy, Cavanagh et al. Multidimensional NMR in Liquids, van de Ven.

Composite Pulses:Spin Choreography, Freeman

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lecture 1. general introduction, semi-classical description of...

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