Principles of NMR Spectroscopy[1]

8/3/2019 Principles of NMR Spectroscopy[1]

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Principles of NMR Spectroscopy

1. What is NMR Spectroscopy?

NMR is an acronym for Nuclear Magnetic Resonance. NMR spectroscopy is a powerful tool foridentifying nuclei based on the interaction of electromagnetic fields with a sample in a magnetic

field. The technique has developed from an interesting physical curiosity in the 1940s into oneof the most important methods of spectral identification in chemistry, biochemistry, and

medicine.

The goals of these notes are to provide an intuitive understanding of the NMR

phenomenon. Underlying concepts will be emphasized over mathematical formalism.

2. Background

2.1 Classical Angular Momentum

Just as linear momentum represents the tendency of an object move in a straight line, angularmomentum represents the tendency of an object to move in angular motion or rotate. Since

momentum has both a magnitude and a direction, it is a vector quantity. Linear momentum isrepresented by a vector in the direction of motion. Angular momentum, however, is represented

through use of the Right Hand Rule (RHR): when the fingers of the right hand are curled in thedirection of circular motion, the thumb points in the direction of the angular momentum

vector. Mathematically, this is because the angular momentum vector is the cross-product of

the position and linear momentum vectors

(1)

There are two kinds of classical angular momentum: orbital angular momentum and spin angularmomentum. Orbital angular momentum is the circular motion of an object about a point, like the

earth orbiting about the sun once a year. Spin angular momentum is the spinning motion of anobject about its own axis, like the earth spinning about its axis once a day.

Figure 1. Orbital vs. spin angular momentum.

Exercise 1: Where do the angular momentum vectors lie for each case in Figure 1?


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2.2 Quantum Mechanical Angular Momentum

Just as energy is known to be quantized at the atomic level, angular momentum is quantized. For

example, an electron in an atom may only have orbital angular momentum quantum numbers l=0, 1, 2, ... (which are more commonly denoted s, p, d, ...) and orbital magnetic quantum

numbers ml

= l,

l+1, ...,

l(which lead to px, py, and pz, for

l=1 and to the five dorbitalsforl=2). Also, the electron only has spin angular momentum quantum numbers = 1/2 and spin

magnetic quantum numbers ms = 1/2, 1/2 (which are also known as spin down and spin up).

All types of angular momentum obey the same quantum mechanical rules. In quantum

mechanics, an angular momentum vector is restricted to having a magnitude of

(2)

where

L = 0, 1/2, 1, 3/2, ... (3)

is the angular momentum quantum number and

= h/2T

(4)

is called h-bar and equals Plancks constant divided by 2T. One component of the angularmomentum vector, which is conventionally chosen to be the z-component, is restricted to havingvalues of

(5)

where

mL = L, L+1,

..., L

(6)

is the magnetic quantum number.

Since the magnitude of the angular momentum vector is greater than its maximum projection onthe z-axis

(7)

it is not possible for to lie along the z-axis. Rather is always tipped away from itsquantization axis.


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A consequence of the Heisenberg Uncertainty Principle is that only one component of angularmomentum can be precisely specified. Defining the z-component of angular momentum results

in the x- and y-components being not well-defined, and the direction in which tips away fromthe z-axis is therefore not known. It can be shown that the expectation, or average, values of

the x- and y-components of angular momentum are each zero

(8)

A useful picture for representing the different angular momentum quantum states is to imagine

2L+1 vectors with the same quantized magnitudes of but different quantized

projections on the z-axis of and undefined projections on the x- and y-axes.

Figure 2. Quantized angular momentum vectors forL=1/2 and L=1.

Exercise 2: Calculate the angle that the L=1/2, mL=1/2 state lies away from the z-axis. (Hint:

Recall that cosine equals adjacent over hypotenuse for a right triangle.)

2.3 Nuclear Spin

Spin is a fundamental property of particles, analogous to mass and charge. It is the angularmomentum that is intrinsic to the particle, rather than the angular momentum arising from the

overall motion of the particle in space. The spin quantum numberIof a nucleus depends on thenuclear species, and has been observed to follow the pattern in Table 1.

Mass Number Atomic Number Nuclear Spin (I) Example

Odd even or odd 1/2, 3/2, 5/2, ... I(1H) = 1/2

Even even 0 I(12

C) = 0

Even odd 1, 2, 3, ... I(2H) = 1


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Table 1. Nuclear Spin Quantum Numbers.

Exercise 3: Observation of an NMR spectrum requires that a nucleus have nonzero

spin. Explain why carbon NMR spectra are weak, but still observable. (Hint: The molar mass ofnatural carbon is 12.011 rather than 12.000.)

Attempts to rationalize the existence of nuclear spin have been made using the model of aspinning charged particle. Classically, it is known that a moving electric charge induces a

magnetic field. Since the nucleus has finite diameter and a positive charge, it would generate amagnetic field as it spins about its axis. However, a quantitative analysis of this model yields a

value for the intrinsic magnetic field about an electron twice as large as it should be (which isaccounted for by the Landgfactor), and the model cannot explain the observation that the spin

of an electrically neutral neutron is 1/2 (see the Table 1). While the picture of a spinning nucleusis classically appealing, spin angular momentum is best treated as a quantum mechanical

phenomenon.

Figure 3. Classical model of nuclear spin.

2.4 Nuclear Magnetic Moment

The nuclear spin quantum numberIgives rise to nuclear spin angular momentum . The

nuclear spin angular momentum in turn gives rise to a nuclear magnetic moment according to

(9)

where K is the gyromagnetic ratio (or more correctly the magnetogyric ratio, whose nameoriginates as the ratio of the magneto = magnetic object over the gyric = turning

object). Each nuclear species has a different value ofK, which is experimentally determined.

Since the nuclear magnetic moment vector is directly proportional to the nuclear spin angular

momentum vector, the nuclear magnetic moment obeys the same rules of quantization as angularmomentum,

(10)

and


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(11)

where

mI = I, I+1,...,I

(12)

Exercise 4: Draw a picture of nuclear magnetic moment vectors forI=3/2 analogous to thepicture of spin angular momentum vectors in Figure 2. Label the magnitude and z-projection of

each vector.


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Nucleus Spin (I)Natural

Abundance

/ %

Magnetogyric

Ratio (K)

/ 107 kg1sA

Relative

Frequency (R)/ MHz

1H 1/2 99.985 26.752196 100.00

2H 1 0.015 4.106625 15.35

13C 1/2 1.10 6.72828 25.15

15 N 1/2 0.366 2.712621 10.14

17O 5/2 0.037 3.62808 13.56

19F 1/2 100.0 25.18147 94.13

29Si 1/2 4.67 5.319 19.88

31

P 1/2 100.0 10.8394 40.52119

Sn 1/2 8.58 10.0318 37.27

Table 2. Spin, natural abundance, magnetogyric ratio, and relative frequency for some common

NMR nuclei.

Exercise 5: Calculate Qz for a single proton with mI=1/2 in SI units. (Hint: The quantum

numberIis unitless, and J = kgm2s2

.) [Answer: Qz = 1.4106v1026

m2A]

3. Effect of a Magnetic Field

3.1 Nuclear Energy Levels

In the absence of external fields, there is no preferred orientation for a magnetic moment. In the presence of a magnetic field , however, the energy of a magnetic moment depends on itsorientation relative to the field lines. Classically,

(13)

The energy is a minimum when the magnetic moment is aligned parallel to the magnetic fieldand a maximum when it is anti-parallel to the magnetic field.


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Figure 4. Low energy and high energy configuration of a magnet in an external field.

The energies of the above two cases may be calculated to be

Elow = QB0

(14)and

Ehigh = QB0

(15)

where Q is the magnetic moment andB0 is the external magnetic field strength.

The energy of a magnet at an arbitrary orientation between these limits is calculated bymultiplying the projection of the magnetic moment along the field direction by the field

strength. Assuming that the quantization axis is taken to be the direction of the magnetic field,then the energy of the mI nuclear spin state is

(16)

from which it is seen that the energies of different mI states of a nucleus are not degenerate in an

external magnetic field. Hence, the m quantum number is called the magnetic quantum number.

Figure 5. Energy levels forI=1/2 andI=1 as a function of magnetic field strength. (The relative

scale of the diagrams is arbitrary, as different nuclear species have different values ofK.)

Exercise 6: CalculateEfor a mI=1/2 proton in a 1.41 T field. (Hint: T = Vsm2)


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3.2 Boltzmann Distribution

In the presence of an external magnetic field, different mI nuclear spin states have different

energies. At thermal equilibrium, they will also have different populations according to theBoltzmann equation

(17)

where Nhigh and Nlow are the populations of the upper and lower states

respectively, (E=EhighElow is the energy difference between the two states, kis the Boltzmannconstant, and Tis the absolute temperature.

Exercise 7: Calculate Nhigh/Nlow for protons in a 1.41 T field at 298 K. Calculate Nhigh/Nlow in a7.05 T field. State the dependence ofNhigh/Nlow on magnetic field strength.

In the limit as Tapproaches zero (or(Eapproaches infinity), Nhigh/Nlow approaches zeroimplying that only Nlow is populated. In the limit as Tapproaches infinity (or(Eapproacheszero), Nhigh/Nlow approaches unity implying thatNlow and Nhigh are equally populated. In practice,

the difference between nuclear spin energy levels (Ein achievable fields is much smallerthan kT, implying that Nlow is only very slightly in excess ofNhigh.

An analogy to the thermal population ofNlow and Nhigh is a collection of compasses which lie on

a table. When the table is undisturbed, all the compasses will be in their low energy state andpoint toward the earths north pole. However, if the table is shaken, then some of the compasses

will adopt the higher energy state of pointing toward the south pole. In practice, the thermalenergy of shaking vastly exceeds the magnetic force attempting to align the magnets along the

earths field, resulting in near equal numbers of compasses pointing north and south. Note thatthe shaking causes the compasses to continually switch directions that they are pointing. Thus

the equilibrium between high energy and low energy configurations is a dynamic equilibrium,not a static equilibrium.

Exercise 8: Evaluate the high temperature approximation by calculating

(18)

for protons in a 1.41 T field at 298 K and comparing it to the previously calculated exact answer.3.3 Transition Frequencies

NMR spectroscopy is performed by inducing transitions between adjacent nuclear spin energy

states ((mI = s1). The energy change for a nucleus undergoing an NMR transition is

(19)


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Equation (19) may be interpreted as follows. The difference between angular

momentum z-component of adjacent mI states is . This difference is multiplied by K to obtain

the difference in the magnetic moment z-component. This result is then multiplied by themagnetic field strength to obtain the energy difference between adjacent mI states in a magnetic

field.

Exercise 9: Calculate (Efor a proton in a 1.41 T field.

The frequency R of the electromagnetic radiation used to induce a NMR transition between

adjacent mI levels in external magnetic fieldB0 is calculated from

(E= hR

(20)

which yields

R = (E/h = K

B0/h = KB0/2T

(21)

The units of frequency (R) are cycles/second, which are also called Hertz (Hz). In NMR

spectroscopy, it is often more convenient to use angular frequency ([) with units of

radians/second. Since one cycle equals 2T radians,

[| 2TR

(22)

Since cycles and radians are not SI units, both R and [ have the same SI units (s1

). Thus,frequency (cycles/second) and angular frequency (radians/second) must be distinguished through

careful use of the symbols ofR and [, respectively. The angular frequency of an NMR transitionis more commonly written as

[0 = KB0

(23)

which is the famous Larmor equation. Note that the use of[ eliminates the occurrence of 2T inthe Larmor equation.

Exercise 10: Calculate R and [ for a proton in a 1.41 T field. In what region of theelectromagnetic spectrum does this frequency occur? State why a NMR spectrometer with a

1.41 T magnet is commonly referred to as a 60 MHz spectrometer.4. Generation of the NMR Signal

4.1 Overview

The measurement of a NMR spectrum signal is very different from infrared or UV-VISspectroscopy in which absorption of photons is measured as a function of frequency. It has been

argued that NMR spectroscopy is similar to emission spectroscopy, in that absorption ofradiowave frequencies is detected indirectly by the subsequent emission of radiowaves at the


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same frequency. While this explanation can account for a simple proton NMR spectrum, it doesnot readily explain pulsed experiments that have give NMR its tremendous ability to uncover

couplings and correlations among nuclei in a molecule. Furthermore, counterarguments havebeen made that the small NMR probe would be an very poor emitter and receiver of long

wavelength radiofrequency radiation and the relatively low radiofrequency implies extremely

low absorption and emission probabilities. Hence, it is better to view NMR spectroscopy as amagnetic induction experiment rather than as an experiment involving absorption and emissionof photons.

A more complete understanding of NMR spectroscopy is achieved if one considers the effect ofapplied magnetic fields on the nuclear magnetic moments in the system. Briefly, an NMR

sample in an external magnetic field is disturbed from equilibrium by application of a transversesecondary magnetic field, and its response to the disturbance is recorded. Furthermore, a

tremendous enhancement of signal-to-noise, as well as the ability to carry out multiple-pulseexperiments, occurs when a pulse of radiowave frequencies is used to disturb the system from

equilibrium. The time dependence of the return to equilibrium is then measured, from whichfrequency spectrum is generated via a Fourier transform. In this section the underlying physical

basis if the NMR phenomena is described. The implementation details in a modern spectrometerare covered in a subsequent section.


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4.2 Net Magnetization Vector

It will be necessary to distinguish between motion relative to the laboratory in which the external

magnetic field originates and motion relative to other coordinate systems. The cartesiancoordinates associated with the laboratory will be denoted X, Y, and Z. If the magnetic moment

vectors of a collection of nuclei are quantized in the direction of the external magnetic field, thenthey have well-defined projections along the Z-axis but are randomly distributed in

theXY-plane. Since there is a slight excess of the lower energy projections aligned parallel to thefield over the anti-parallel projections, there is a slight net magnetization in

the Zdirection. However, there is no preference for the XorYdirection so there are equalnumbers of projections onto this plane pointing in each direction, resulting in no net

magnetization in the XY-plane. The netmagnetization vectoris labeled , and may also calledthe macroscopic magnetization vectoror the equilibriummagnetization vector.

Figure 6. The vector sum of a collection ofI=1/2 nuclear magnetic moments in an externalmagnetic field.

The measurement of the net magnetization vector was a very challenging task due to the small

magnitude of the vector and the very large applied field in the same direction. Bloch and Purcellindependently made this measurement for the proton in 1946, for which they were jointlyawarded the Nobel Prize in physics. However, neither of them would have predicted that

organic chemistry students would routinely make such measurements in order to distinguishamong the same nuclear species in different chemical environments with differences in magnetic

moments of less than 0.1 ppm!

Exercise 11: Calculate the net magnetization of 0.1 mL of water in a 1.41 T magnet at

298K. Compare this value to the earths magnetic field strength of approximately 1gauss. (Hints: Each water molecule contains 2 H atoms, and O has an even number of protons

and neutrons. 104

gauss = 1 T.)

4.3 Larmor PrecessionWhen a rapidly spinning gyroscope is tipped from vertical, it does not fall over but ratherexperiences a torque given byand precesses about the earths gravitational field. Careful

observation of a gyroscopes precessional motion reveals that the precessional frequency isindependent of tip angle and the tip angle remains constant during precession.

When a magnetic moment lies at an angle to an external magnetic field, it experiences a similartorque


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(24)

and it precesses about the external field. The precession frequency is given by the famous

Larmor equation[0 = KB0 (23)

and [0 is the Larmor frequency andB0 is the external magnetic field strength. The Larmorfrequency has two important physical interpretations. It is the precessional frequency of the

nuclear magnetic moment about the magnetic field. It is also the frequency of theelectromagnetic radiation that induces a transition between nuclear spin quantum states in the

magnetic field.

Since each individual nuclear magnetic moment vector is tilted from the external magnetic field,

it experiences precessional motion about the Z-axis. However, as long as the net magnetization

vector lies along the Z-axis, it does not undergo precession.

Figure 7. Precession of I=1/2 magnetic moment vectors in applied magnetic fieldB0.

4.4 Transverse Magnetic Field

Larmor precession of the magnetization vector is not observed as long as lies in the

direction of the magnetic field . In order to observe precession of about , must betipped away from the Z-axis. This can be accomplished by introducing another magnetic

field that is perpendicular to the Z-axis.


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Figure 8. Coils in a NMR spectrometer which tip the magnetization vector M.

In a NMR spectrometer, the field is generated through the use of a solenoid coil of wire

whose axis lies in the XY-plane. We will define the X-axis to lie along the axis of the coil. Ifelectric current flows through the coils, a magnetic field is generated along the X-axis, and the

magnetic moment is tipped off the Z-axis toward Y-axis in accordance with Eq. (24). If were

a static field, then would precess about the new overall magnetic field at the

Larmor frequency. In an NMR spectrometer,


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Figure 9. Precession of about for stationary and for rotating .

A rotating magnetic field is generated by connecting the coils to a radiofrequency source whose

output voltage varies sinusoidally at frequency [0. An alternating current (AC) will flow in the

coil, producing a linearly oscillating magnetic field. The resulting magnetic field is colinear withthe X-axis but varies in magnitude so as increase along the positive X-axis, reach a maximum,

decrease toward zero, increase along the negative X-axis, reach a maximum in this direction, andreturn to zero. A linearly oscillating vector can be decomposed into the sum of two counter

rotating vectors. Each rotating vector rotates at frequency [0; however, one vector rotates in the

same direction of Larmor precession of and the other in the opposite direction. The

component of the magnetic field rotating with is the field. (It turns out that the other

component has a negligible effect on the net magnetization vector, as it differs too much in

frequency from [0.)

Figure 10. Decomposition of a linearly oscillating magnetic field into two counter rotatingmagnetic fields.

4.5 Rotating Frame of Reference

In order to simplify the motions of the nuclear magnetic moments during an NMR experiment, a

new axis system is introduced which rotates in the same direction and rate as the moments areprecessing. The axis system is called therotating frame, and the axes are labeled with


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lowercase x, y, and z. Recall that the stationary orlaboratory frame axes are labeled withuppercase X, Y, and Z. (Older literature, however, uses x, y, and z for the stationary frame

and x', y', and z' for the rotating frame.) Use of the rotating frame allows one to visualize moreeasily all motion of the magnetic moments other than Larmor precession. An analogy to

understand this is to consider reading the label on a record while it is being played on a

turntable. It is difficult to read the label while you are stationary and the label is rapidly rotating;however, it would be much easier to read the label if you jumped aboard the rotating albumand rotated with it, as the label would appear to not move in your new rotating frame. However,

note that the label could appear upright, upsidedown, or even sideways, depending on when onejumped aboard the rotating frame.

Figure 11. Rotational motion in the laboratory frame appears to stop in the stationary frame.

The motion of the rotating field is quite simple in the rotating frame. Since the rotating

frame rotates at exactly the frequency of the vector , the vector is a stationary in

the xy-plane. The motion of the magnetization vector in the rotating field is also very

simple. The precessional motion of about the z-axis at Larmor frequency [0 stops, and

simply precesses about the stationary vector at frequency [1.

Figure 12. Motion of magnetization vector in rotating frame of reference.

It is quite clear in the rotating frame system that the rotation frequency of must match the

Larmor frequency [0 in order to significantly tip the magnetization vector. Imagine that the


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vector were rotating slightly slower (or faster) than which rotates at the Larmor

frequency [0. Initially, the field would tip the magnetization vector away from the z-axis

in a particular direction. However, the mismatch between rotation frequency of and the

Larmor frequency [0 would soon cause to precess around the z-axis to the side opposite of

its initial tip. The field would then tip back toward the z-axis restoring it to its original

alignment. The only frequencyat which the rotatingmagnetic field can significantly tip the

magnetization vector is the Larmor frequency[0. Thus, it is an excellent approximation to

ignore the component of the linearly oscillating magnetic field that rotates in the opposite sense

as , for its rotational frequency is 2[0 in the rotating frame and its effect on is thereforeneglible.

Rotating the axis system at the Larmor precession frequency [0 is extremely useful. WhileLarmor precession is a consequence of the external magnetic field that is necessary to generate a

NMR signal, it contains no useful information beyond the value of the magnetogyricconstant K. Rather, all the interesting information in a NMR spectrum arises from the slightlydifferent magnetic fields that nuclei experience due local environment differences, and this

information is contained in the small differences among their Larmor frequencies rather than thetotal magnitude of the Larmor frequency. It is always easier to see a small signal when it is not

on top of a large background. Hence, it is both mathematically convenient and experimentallynecessary to reference all NMR signals to the Larmor frequency of the external magnetic field.

If is chosen to lie on the x-axis, then tips from the z-axis toward to y-axis according to

Eq. (24). Note that because this is precessional motion, the angle between and (90 in

this case) is maintained (motion remains in the yz-plane in this case). In a pulsed Fourier

Transform NMR spectrometer, is it possible to control the direction in which tips, the rate at

which tips, and the amount of the tip angle. The field is generated by a short pulse of

radiofrequency (RF) into the coils which surround the sample in the probe. The direction of

tipping is controlled by the relative phase between the applied radiofrequency pulse and therotating frame. For example, if the maximum RF intensity occurs when rotating the x-axis lies

along the coil, then the field is along the x-axis. However, if the maximum RF intensity

occurs when the y-axis lies along the coil, then the field is along the y-axis. The rate and

magnitude of the tip is determined by the RF power and the duration of the RF pulse,

respectively. Stronger RF power creates a stronger field that results in a higher[1, and alonger RF pulse causes the precession about precession to continue further. It is the ability to

precisely control the tip of the magnetization vector that makes multiple pulse NMR

experiments possible.


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Figure 13. Generation of along the x-axis and along the y-axis.

Current convention defines an x-pulse as a pulse that induces rotational of about the x-axis

in a positive or Right Hand Rule sense, i.e., ypzpypz. However, the older literature and

many textbooks define an x-pulse in the opposite sense. As long as either convention is usedconsistently, it will predict the same result for NMR experiments as the other convention. Since

both conventions are widely found, one should not worry about the particular choice made anauthor but should note the choice and follow it in that article or book. Careful consideration of

Eq. (24) and the definition of positive rotation reveals that . Thus, the magnetic field

which induces an x-pulse is actually aligned along the negative x-axis for a nucleus with a

positive magnetogyric ratio. The choice of whether to define an x-pulse with respect to therotation vector or the magnetic vector is matter of preference. However, current convention is to

focus attention on rotation of the magnetization vector rather than on the radiofrequencymagnetic field vector. This is known as the Right Hand Convention, because if one aligns the

thumb of the right hand along the axis of the pulse, the fingers curl in the direction thatrotates.


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Figure 14. The effect of an x-pulse and a y-pulse on the magnetization vector components

Exercise 12: What is R1 for a 10 Qs pulse that rotates by 90 for a proton? What field

strength is required for this pulse? Compare the strength of to for this case in a 300

MHz NMR spectrometer? [Answers: 25kHz; 5.87 gauss; 12000:1]

Exercise 13: A typical RF coil in a probe consists of 2 turns of wire with 1 cm radius separated

by 1 cm and has an impedance of 50 ohms. How much power must be delivered to the probe toproduce the above 90 degree pulse? (Hints: A general physics textbook calculation reveals 1 A

of current in a 1 loop coil with 1 cm radius produces a 6v105

T magnetic field. P=VI=I2R.)

We will later see that the rotating frame is more than a mathematical construct to assist in thevisualization of motion of nuclear magnet moments. The rotating frame is actually the reference

signal to the Phase Sensitive Detector in the NMR spectrometer!

4.6 Observation of the NMR Signal

Just as an oscillating electric field in a coil of wire creates an oscillating magnetic field, anoscillating magnetic field causes an oscillating electric field in the wire. The alternator found in

every automobile operates on this principle. The NMR signal is observed with a coil in

the XY-plane similar to the transmitter coil that detects the rotating magnetization vector

. Early NMR spectrometer designs used separate transmitter and receiver coils; however,modern designs use a single transceivercoil that is electronically connected to the transmitter

during the pulse and then to the receiver after the pulse. Although is stationary in the

rotating frame, it is moving at the Larmor frequency in the stationary frame. As , the

projection of on thexy-plane, cuts through the turns of the received coil, it induces a RF

current at frequency [0. The spectrometer looks for an electrical signal at [0 in the receiver coil,typically by comparing the output signal to the input signal with a Phase Sensitive Detector as

discussed in a subsequent section.

Observation of the NMR signal requires that the magnetization vector be tipped toward

the xy-plane. This tipping only occurs when the transverse field rotates at the same


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frequency as the Larmor precession frequency [0, i.e., when the RF transmitter frequency istuned to the Larmor frequency. The transfer of RF electrical energy into tipping of the

magnetization vector relies on a close match between the applied RF frequency and Larmorprecession frequency. The physical phenomenon of two oscillating systems transferring energy

most efficiently when their frequencies are the same is called resonance. Hence the

name nuclearmagnetic resonance is a very appropriate term to describe the frequencydependence of coupling RF electrical signal into the tipping of the magnetic moment vector.

Optional Exercise 14: Estimate the NMR signal power (in watts) of a 0.1 mL water sample in a

1.41 T magnet (exercise 11) observed with a typical RF probe (exercise 13).

4.7 Relaxation

The relaxation of net magnetization vector has been ignored until now. Recall that the

equilibrium which produces is a dynamic equilibrium and that thermal energy constantly

causes transitions between nuclear spin states. The equilibrium population among the nuclearspins states is determined by the Boltzmann distribution. When the system is perturbed from

equilibrium by application of , it will relax back toward equilibrium. The specificmechanisms for relaxation are quite complex and will be discussed in a later section. Factorsthat affect the relaxation rate include nuclear species, magnetic field homogeneity, temperature,

and presence of magnetic material (ranging from other nuclei with spin to paramagnetic speciessuch as a dilute aqueous CuSO4 solution).

If is tipped by a short pulse of radiofrequency, it will commonly take several seconds forto return to its equilibrium position along the z-axis. Two distinct kinds of relaxation occur,

each with its own time constant. Relaxation of along the z-axis occurs with time

constant T1, which typically varies from seconds for protons to tens of seconds forcarbon. Since T1 is a measure of the time required for magnetization vector to return to its

equilibrium length and direction, it directly affects when the next radiofrequency pulse can beapplied. The delay before applying the next pulse can vary from zero to 5 T1, depending on the

specific experiment being carried out. Relaxation of the vector magnitude in the xy-plane occurswith time constant T2*, which is typically on the order of a second or two. Since T2* is a

measure of the time required for the magnetization vector to leave the xy-plane, it is a directmeasure of the length during which a signal can be observed. The magnitude ofT2* therefore

affects the length for which a FID should be recorded. A general guideline is to collect the FIDfor 3 to 4 T2*.

4.8 Factors Affecting Sensitivity

The NMR signal depends linearly on the magnitude of magnetization in the xy-plane,. This quantity is governed by four factors for a given nucleus. First, the tip angle of the

magnetization vector affects . The closer the tip angle is to 90, the larger the magnitudeof the magnitization vector is in the xy-plane. Second, the external magnetic field strength

affects the magnitude the magnetization vector , and consequently its projection in

the xy-plane. Third, the sample temperature also affects magnetization vector , as a colder


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sample results in a larger population difference between spin states. Fourth, the number ofnuclei in the coil, which is usually controlled by sample concentration, also directly affects the

magnetization vector .

The NMR signal also depends on the magnetogyric ratio K raised to the third power. This strong

dependence arises from the magnetic moment depending linearly on K, the population differencebetween spin states depending linearly on K, and the efficiency of signal generation depending on

the Larmor frequency [, which is proportional to K.

Exercise 15: Estimate the relative sensitivity of1H and

13C NMR spectra. (Hint: account for

relative abundance, relaxation, and magnetogyric ratio.)

5. Chemical Shift

5.1 Different Nuclei (magnetogyric ratio)

According to the Larmor equation, the precessional frequency of a nuclear magnetic moment is

proportional to the magnetogyric ratio and the magnet strength.

[0 = KB0 (23)

Different nuclei have different magnetogyric ratios (see Table 2). Thus for a given magnetic

strength, the frequency at which nuclei of different elements precess depends on their identityand differs vastly.

It is common practice to characterize magnet strength by a precessional frequency, e.g., a 300MHz magnet. Such terminology assumes a specific nucleus, typically a proton. Thus in a 300

MHz spectrometer (for protons), the precessional frequency of carbon is 75 MHz.

Figure 15. Dependence of precessional frequency on magnet strength on the magnetogyric ratio.

5.2 Inequivalent Nuclei (shielding)

All atoms in a molecule are surrounded by electrons that occupy core and valence orbitals. The

permanent magnetic field induces a current in the surrounding electrons, which in turn

generates an induced magnetic field . According to Lenzs Law, the induced field is

proportional to the permanent magnetic field but is opposite in direction


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= W

(25)

where the proportionality factorW is called the shielding constant.

Figure 16. Electrical current flow from surrounding electrons induces to oppose .The local magnetic field experienced by a nucleus is the sum of the permanent field and theinduced field

= +

(26)

Nuclei in different chemical environments are called inequivalent nuclei, and they experience

slightly different local fields and therefore precess at slightly different Larmor frequencies

[0 = KBloc = K (B0Bind) = K (B0WB0)

= KB0 (1 W)(27)

Nuclear magnetic resonance frequencies are always reported with respect to a standard. Thestandard for protons and for carbon is tetramethylsilane (TMS), Si(CH3)4. If the frequency

difference is reported in Hertz, then it is proportional the external field strength. Given nucleus

A and standard B with Larmor frequencies [A and [B, the chemical shift of A with respect to B

is

RAR& = ([A[B) / 2T = [KB0 (1W%) / 2T] [KB0 (1W&) / 2T] = KB0 (W&W%) /

2T(28)

Thus, the energy spacing between two inequivalent nuclei is proportional to magnetic field

strengthB0. This increase in spectral separation with field strength is one of the main advantagesof using higher strength magnets in modern NMR spectrometers. (The other advantage is thehigher sensitivity achieved due to the greater population difference between spin states.)

It would be quite inconvenient to report nuclear magnetic resonance frequency differences inHertz because the strength of every magnetic is different. Dividing by the frequency of a

standard, which is also dependent on field strength, eliminates this dependence on magnetic fieldstrength.


22/23

(29)

since WB


23/23

Figure 18. Induced field can cause chemical shift anisotropy.

Exercise 16: Consider the proton in chloroform, Cl3CH, and the chemically equivalent protonsin tetramethylsilane (TMS), Si(CH3)4. Which proton has the higher Larmor frequency? Explain

your reasoning. (Hint: Consider the electronegativity of the other atoms in each molecule.)

[Answer: Chlorine is more electronegative than silicon. Thus, the proton in chloroform is lessshielded and has a higher Larmor frequency.]

Exercise 17: The CHCl3 proton resonance is observed at 436 Hz in a 60 MHz spectrometer and

at 2181 Hz in a 300 MHz spectrometer relative to TMS. Calculate the chemical shift H in ppm ineach case. [Answer: 7.27 ppm in each case]

5.3 Equivalent Nuclei (spin-spin coupling)

Nuclei with spin can interact, or couple, with each other. Since each nucleus can be thought ofas a small magnet, the orientation of that magnet has an effect on the local magnetic field

experienced by other nuclei. This effect is most prominent among chemically equivalent nuclei,giving rise to the N+1 rule for equivalent protons learned in organic chemistry. A proton with N

protons on contiguous carbon atoms splits into N+1 peaks with intensity pattern given by 1:1,1:2:1, 1:3:3:1, ... for N equal to 1, 2, 3, ... Spin-spin coupling will be discussed in much greater

detail in a future section, with extensions to more complicated interactions including interactionsbetween different nuclei.

Figure 19. 1:2:1 intensity pattern arises from one proton coupling with two equivalent protons.

Exercise 18: Qualitatively predict the chemical shift and spin coupling pattern of the ethylgroup, CH2CH3, in ethanol, HOCH2CH3. Justify your assignments.

Principles of NMR Spectroscopy[1]

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