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Chem 3500 Fall 2006 P. Hogan
Magnetic Resonance
NMR spectroscopy is used to study the magnetic properties of nuclei.
Only nuclei that possess a spin have magnetic properties and have a
quantum spin number, I. During this course we will study two different
nuclei with a quantum spin number I = ½, the 1 H nucleus and the
13
C
nucleus. A charged nucleus that has a spin will posses a magnetic moment
µ. Equation 1 defines the relationship between the quantum spin number and
the magnetic moment.
µ = ?hI/2p
(eq. 1)
In the above relationship, h is Planck’s constant (h = 6.2608 X 10-34), and ?
is the gyromagnetic ratio of the nucleus has units of 107 T-1 s -1 where T is
units Tesla. The gyromagnetic ratio of 1H is 26.75, and the gyromagnetic
ratio of 13C is 6.73.
The quantum spin number I also determines the number of possible
spin states that a nucleus can possess. There are 2I + 1 spin states . Thus the
1
H and 13C nuclei each have two spin states corresponding to I = ½ and I = -
½. In the absence of external magnetic field, these spin states are equal in
energy and equally populated. If a sample of nuclei are subjected to a
magnetic field B0 (units Tesla) along a direction designated as the z axis, the
energies of the nuclei with two different spin states will no longer be equal
in energy. The will be a slightly higher population of nuclei aligned with the
direction of the magnetic field B0 (+ z) than against (-z) . The force of B0 on
the magnetic moment causes the magnetic moment vector to move in a
circular orbit about the z axis (coincidental with B0), a motion called
precession. The orbits of the two populations are shown below in figure 1.
The angle and magnitude of the vector quantity µ is a function of the atomic
spin number.
z
B0
z
µ
Iz = +1/2
Iz = -1/2
µ
Figure 1. The precession of the magnetic moment with a +z component is shown on the right while the
precession of the magnetic moment with a – z component is shown on the left.
Precession of the magnetic moment around B0 occurs with an angular
frequency ? 0, also called the Larmor frequency. The Larmor frequency can
be expressed in terms of linear frequency ?0 (eq. 2) The energy difference
between the lower energy spin state (Iz = +1/2) and the higher energy spin
state (Iz = -1/2) is given by Planck’s relationship (eq. 3)
? 0 = 2 p ?0
(eq. 2)
? E = h?0
(eq. 3)
The magnitude of the angular frequency ? 0, is proportional to B0. The
constant of proportionality between ? 0 and B0 is ?, the gyromagnetic ration
of the nucleus. This relationship is given in equation 4.
? 0= ?B0
(eq. 4)
This quantity can also be expressed in terms of linear frequency (eq. 5)
?0 = ?B0/2 p
(eq. 5)
Thus the energy difference between the two spin states can be written in
terms of the gyromagnetic ratio and a magnetic field of value B0 (eq. 6).
? E = ? h B0/2 p
(eq. 6)
From equation 5, we can see that the frequency of precession is directly
proportional to the magnitude of the magnetic field B0, and consequently
(see eq. 6) the energy difference between the two spin states is also directly
proportional to the magnitude of the magnetic field B0.
An NMR experiment is conducted by application of a circularly
polarized radiofrequency field B1 into the xy plane. The radiofrequency field
B1 will consist of both an electric component and magnetic component. The
magnetic component of B1 will interact with bulk magnetization of the
sample. Bulk magnetization is illustrated in figure 2.
z
z
M
y
x
y
x
Figure 2. The picture on the right is a representation of the magnetic mo ments of a collection of nuclei in
the absence of a magnetic field with magnitude B0 . There is no net magnetization within the samp le. In the
picture on the right, the collection of nuclei have been exposed to a field B0 along a direction designated
along the z axis. More nuclei have magnetic moments possessing a +z component than nuclei possessing
magnetic moments with a –z component. Thus, there is a net bulk magnetization M of the sample in the +z
direction.
Figure 3 illustrates a vector representation of the circularly polarized
radiofrequency field B1.
The vector representing the direction and
magnitude of B1 makes a circular orbit around the z axis within the xy plane.
If the angular frequency of B1 is equal to the Larmor frequency of the nuclei,
the precessing nuclei will be under the influence of a constant magnetic field
and energy will be absorbed by the nuclei.
This condition is called
resonance. The bulk magnetization of the sample will be tipped off the z
axis and begin to move toward the xy plane. As long as the field B1 is applied
the magnetization M will rotate around B1. If we choose the laboratory
frame of reference to be rotating with B1 at the Larmor frequency, the vector
representing B1 will appear to remain stationary along the x axis. The
magnetization vector M will now precess around B1 within the yz plane with
the frequency ?B1/2 p (figure 3) .
z
z
?=?B1 /2p
M
M
?0
y
y
B1
B1
x
? = ?0
x
Figure 3. The depiction on the right shows the direction and frequency of the circularly polarized
radiofrequency field B1 . The vector representing the direction and magnitude of B1 orbits the z-axis. If the
angular frequency of this orbit is equal to the Larmor frequency of precession, the magnetization vector
will be tipped away from the z-axis. The depiction on the right shows what an observer in a frame rotating
at the Larmor frequency would observe. The field B1 appears to be stationary, while M precesses around B1 .
If B1 is applied for a duration p ?B1/2 (25% of the time needed for M to
make a full rotation around B1, we say we have applied a 90° pulse. This is
illustrated in figure 4. In a laboratory frame rotating at the Larmor frequency
the magnetization vector M appears to tip and become aligned with the y
axis. In the stationary laboratory frame of reference the bulk magnetization
M, now appears to precess in the xy plane (figure 4).
z
z
z
M
M
B1
x
y
B1
x
y
y
M
x
M
? = ?0
Figure 4. The first picture is in from the perspective of the rotating frame as the pulse of B1 begins. The
middle picture is from the perspective of the rotating frame after the 90° pulse. The third picture is from the
perspective of the stationary frame after the 90° pulse.
A detector placed along the y axis sees an oscillating magnetic field at the
Larmor frequency. Thus the magnitude of the signal varies as a cosine
function with time. If the B1 field is turned off after the 90° pulse, and the
magnetization in along the y-axis is measured over time at the resonance
frequency an exponential decay of y magnetization (My ) over time would be
observed (figure 5). Any process that returns the x and y magnetizations to
their equilibrium values of zero is called spin-spin relaxation and is
generally a first order process with time constant T2.
Eventually the
magnetization vector M returns to it’s equilibrium status in which M
consists of only a +z component. Any process that returns the z
magnetization to it’s equilibrium value of an excess of +1/2 spins is called
spin-lattice relaxation and is generally a first order process with time
constant T1.
z
z
M
y
y
x
? = ?0
M
x
Figure 5. The depiction on the left shows the path of the bulk magnetization vector M after B1 is turned off.
Eventually M returns to it’s equilibrium status (right depiction) in which there are no x and y components
to the magnetization only a small +z component.
If the magnitude of the My is plotted vs. time, the result is called an FID.
(free induction decay, figure 6). An FID can be manipulated mathematically
by using a Fourier transformation which transforms the data into a plot of
magnitude of My vs. frequency.
My
t
Figure 6. An FID of a system containing a single type of nucleus.
The Chemical Shift
Nuclei in different chemical environments have different resonance
frequencies. These resonance frequencies are referred to as chemical shifts.
The resonance frequency depends greatly upon the electronic environment
of the nucleus. The electrons surrounding a given create a local magnetic
field that attenuates the applied magnetic field B0. The relative populations
of the two spin states depend on the energy difference between them. The
population ratio of the lower to higher energy spin states is given by eq. 7.
[Iz =+1/2]/[Iz =-1/2] = e-(?E/kT)
(eq. 7)
In eq. 7, ? E is the energy difference between two spin states, k is
Boltzmann’s constant (1.38065 X 10-23 J/K) and T is temperature in units
Kelvin. Recall from eq. 6 that the enrgy difference between the two spin
states is proportional to the applied magnetic field.
? E = ? h B0/2 p
(eq. 6)
This equation can be applied to a bare nucleus, however when a nucleus is
surrounded by electrons the actual magnetic field that the nucleus feels is
counteracted in part by the electrons around the nucleus. Thus the value of
actual magnetic field felt by the nucleus can be written as the following:
Beffective = B0 - Blocal
(eq. 8)
In the above equation Blocal is the magnetic field created by electrons
surrounding the nucleus and Beffective is the actual magnetic field seen by a
nucleus.
Equation 6 can now be rewritten to reflect the actual energy
difference between the spin states based upon the actual magnetic field
Beffective seen by the nucleus.
? E = ? h Beffective/2 p
(eq. 9)
Equation 4 defined the relationship between the Larmor frequency of a
nucleus and the applied magnetic field. This equation can now be rewritten
as the following.
? effective= ?Beffective
(eq. 10)
The new Larmor frequency ? effective is the observed resonance frequency and
is different for every nucleus in a chemically different environment. When a
nucleus is surrounded by an area of high electron density, the Blocal term in
eq. 8 is large. This leads to a lower resonance frequency as seen by eq. 10.
When a nucleus is an area of high electron density, the nucleus is said to be
shielded from the applied magnetic field. When a nucleus is surrounded by
an area of low electron density, the Blocal term in eq. 8 is small. This leads to
a higher resonance frequency as seen by eq. 10. When a nucleus is an area
of low electron density, the nucleus is said to be deshielded from the applied
magnetic field.
In order to standardize and simplify the reported chemical shifts of a
given resonance frequency of a nucleus, an internal standard in an NMR
experiment will typically be chosen. The standard of choice for 1H and 13C
NMR experiments is tetramethylsilane (CH3 )4Si, often abbreviated as TMS.
Silicon is much less electronegative than carbon, and therefore both the
carbons and protons are shielded to a much larger extent that carbons and
protons of most organic compounds.
For this reason the resonance
frequency of the protons and carbons of TMS have been arbitrarily assigned
“0”. The chemical shift, most often abbreviated as the symbol d, is the
difference between the magnitude of the resonance frequency of a measured
nucleus and that of TMS. The resonance frequencies also called chemical
shifts of nuclei are typically not reported in units Hz. Chemical shifts instead
are reported in parts per million (ppm), and are obtained by measuring the
difference in frequencies (in units Hz) between a given nuclei and the
relevant nucleus of TMS and dividing the value of that number by the
operating frequency of the spectrometer (in units MHz).
d (ppm) = ? observed (Hz) – ?TMS(Hz)/operating frequency of the spectrometer (MHz)
The chemical shift of a given measured nucleus is independent of the
operating frequency of the spectrometer. The resonance frequencies
recorded in NMR experiment will typically be plotted in ppm. The signal
corresponding to TMS will be placed at “0” at the right hand of the
spectrum. The ppm numbers will increase from right to left. Signals to the
left of the TMS signal are said to be downfield of TMS. Almost all measured
nuclei will have resonance frequencies downfield of The resonance
frequencies of TMS. In general the right hand side of the spectrum is
referred to as the upfield portion of the spectrum and the left hand side is
referred to as the downfield part of the spectrum.
Essential features of an 1H NMR spectrum
The chemical shift of alkanes and functionalized alkanes (1H NMR)
The extent to which a given proton nucleus is shielded from the
applied magnetic field B0 is dictated by the nature of the carbon to which the
proton is attached and nature of any other substituents bonded to the carbon
atom. As we have seen, due to the electropositive nature of the silicon atom,
the protons of tetramethylsilane (CH3 )4Si are shielded from the applied
magnetic field more than most other protons of organic molecules. To begin
our discussion of chemical shifts, we will focus first on alkanes and
functionalized alkanes.
The protons of the simplest alkane, methane (CH4), have a chemical
shift d = 0.23. Carbon is slightly more electronegative than hydrogen, and
thus there is somewhat less electron density around the hydrogen nuclei of
methane that there is around the hydrogen nuclei of TMS. The hydrogen
nuclei of methane are deshielded relative to those of TMS. Substitution of a
carbon-hydrogen bond in methane with a carbon-carbon bond leads to
further deshielding of resultant hydrogen nuclei. This can be viewed as
replacina hydrogen with a relatively more electronegative group, further
dishielding the protons by removing some electron density from the carbon
to which they are attached. The chemical shift of ethane (CH3CH3) is 0.86.
The methylene group of propane (CH3CH2CH3) is further deshielded
relative to the methyl groups of ethane and has a chemical shift of 1.33.
Finally the methane proton of 2-methylpropane ((CH3)3CH) are found at d =
1.56. In general the protons of simple alkanes will be found in the region
0.2-1.8 ppm.
Substitution of a carbon-hydrogen bond with an electronegative atom
or group will contribute to the deshielding of protons attached to the carbon
under examination. The chemical shift of protons attached to the carbon of
an alcohol functionality for some simple alcohols are shown below. Note
that with increasing carbon substitution, further deshielding occurs.
The chemical shift of the protons of ethereal carbons are similar to those of
their alcohol counterparts, however note that the protons of of the ester
portion of methyl acetate are deshielded relative to those of ethers or
alcohols. The shifts of the methyl halides are shown below. As expected the
relative electronegativity of the halide dictates the relative chemical shift of
the methyl protons. Replacement of carbon hydrogen bonds with additional
carbon-halogen bonds results in further deshielding of the resultant protons.
Alkanes attached to sp 2 hybridized carbons are also deshielded relative to
their unfunctionalized counterparts.
An sp2 hybridized carbon can be
considered an electronegative substituent relative to an alkane, because of
the lower relative energy (more s character) of the sp2 orbitals. Some
characteristic examples are shown below.
Chemical Equivalence and the Number of Signals in an NMR spectrum
Any nuclei that can be exchanged by a symmetry operation are said to
be chemically equivalent. Within organic molecules the there are two
common types of chemically equivalent sets of nuclei: homotopic and
enantiotopic. Nuclei that can not be exchanged by a symmetry operation are
chemically inequivalent. The number of signals in a 1 N NMR spectrum
tells us how many types of chemically inequivalent protons are contained
within the compound being studied. Chemically equivalent nuclei contribute
equally to their respective signal in an NMR spectrum. In the 13C spectrum
the number of signals will tell us how many chemically inequivalent carbon
nuclei are contained within the compound being studied.
Nuclei that can be exchanged by a rotational symmetry are said to be
homotopic. Homotopic nuclei are chemically equivalent. Some examples
of molecules containing sets of homotopic protons and associated rotational
axes are shown below.
Protons of methylene groups that can not be exchanged by rotational
operations, but can be exchanged by a mirror plane are said to be
enantiotopic.
Enantiotopic protons are chemically equivalent.
These
protons are called enantiotopic because replacement of one proton of the pair
by another atom or group produces the enantiomerthat results when the other
proton is replaced by the same group. The geminal methylene protons of
butane are enantiotopic.
If geminal protons of a methylene group can not be exchanged by a
symmetry operation, they are diasterotopic.
Like the above test for
enantiotopicity, replacement of one proton of the pair by another atom or
group produces a diastereomer of the molecule that results when the other
proton is replaced by the same group. Any geminal methylene protons in a
molecule containing a chiral center are diastereotopic.
Diastereotopic
protons are not chemically equivalent. The diasterotopicity test is shown
below on 2-butanol.
Less obvious is the fact that a chiral center is not a necessarily required for
geminal methylene protons to be diastereotopic. Examine the diethyl acetal
of acetaldehyde.
Note that in conformationally mobile systems that chemically inequivalent
protons can switch positions through carbon-carbon bond rotation and
contribute to the same resonance.
Spin-Spin Coupling and the Coupling Constant: Simple First Order
Patterns
Spin-spin coupling is a phenomenon observed when spin information
is transferred from one nucleus to another. In the context of organic
molecules, generally three types of coupling are observed; one bond
couplings such as the coupling between a proton and the carbon nucleus to
which is attached, two bond couplings such as coupling between protons on
the same carbon, and three bond couplings like couplings between protons
of vicinal carbons. Four bond couplings can be important in the context of
aromatic systems and will be considered later. When spin information is
transferred between chemically inequivalent nuclei, the resultant resonances
of the respective nuclei are split into multiplets. Multiplets are described
according to the number of signals associated with nucleus.
In order to understand why coupling will cause multiplicities with
regards proton resonances, examine a hypothetical molecule in which two
protons in very different chemical environments are coupled to one another.
We will label one proton HA and the other proton HX (the reasons for this
nomenclature will be apparent later). Each proton has a spin of ½ and can
exist two spin states: those whose magnetic moments are aligned with B0
which we will designate the +½ spin state, and those whose magnetic
moments are aligned against B0 which we will designate the -½ spin state.
The relative population of these two spin states only differ in the parts per
million, and for the purposes of this discussion will be equal. Consider that
half of the HA protons in the sample have HX neighbors that posses the +½
spin state and the other half of the HA protons in the sample have HX
neighbors that posses the ½ spin state. The result is that there are two
magnetically and energetically different populations of HA, and therefore the
signal corresponding to HA is comprised of two resonances of equal intensity
reflecting the two spin states corresponding to HX.
A similar analysis
reveals that HX will also appear as two resonances of equal intensity
reflecting the two spin states corresponding to HA. The multiplicity of HA
and HX are referred to as doublets. The frequency distance between the two
peaks of the doublet in Hz is the coupling constant J and is independent of
the magnitude of B0. The actual chemical shifts of HA and HX will be at the
middle of their multiplicities respectively.
Iz (HX) = +1/2
Iz (HX) = -1/2
Iz (HA) = +1/2
JAX
JAX
d HA
Iz (HA) = -1/2
??
d HX
In general the chemical shift of a simple first order multiplet will
reside at the center of the multiplet. A multiplet will be a simple first order
multiplet if the ratio of ??/J = 8, where ?? is the distance between the
midpoints of the coupled multiplets in Hz. Note that ?? is dependent on the
field strength of the spectrometer. The consequences are that when high
magnetic field spectrometers are used, the spectra is more likely to have
simple first order charcteristics. The number of peaks in the multiplet is n +
1, where n is the number of neighboring protons with the same coupling
constant. In a simple first order multiplet all the neighboring protons have
the same coupling constant, therefore the distance between the individual
peaks of the multiplet in Hz will be the coupling constant. The intensities of
the individual peaks of the multiplet are given by Pascal’s triangle.
n
n+1
Relative line
Multiplet name
intensities
0
1
1
singlet
1
2
1:1
doublet
2
3
1:2:1
triplet
3
4
1:3:3:1
quartet
4
5
1:4:6:4:1
pentet
5
6
1:5:10:10:5:1
sextet
6
7
1:6:15:20:25:6:1
septet
Pople notation
A spin system is defined as a set of multiplets coupled to one another
but do not couple outside the spin system. Proton spin systems are often
separated from one another by heteroatom bonds or quaternary carbons. In
the Pople notation system, multiplets are now called sets.
Each set
represents a group of chemically equivalent protons and is designated by a
capital letter A, B, C …..Z. If ??/J = 8, the coupled sets are weakly coupled
and designated by the well separated letters AX. If ??/J < 8, The coupled
sets are strongly coupled and are designated by the neighboring letters of the
alphabet AB. In general weakly coupled sets can be analyzed bya first order
method and strongly coupled sets can not, however if an AB system has
??/J = 3, first order analyses can often be applied without significant error.
If there is more than one chemically equivalent proton contained within the
set the letters are give subscripts that identify the number of protons in each
set. An ethyl group (CH3CH2-) would be given the notation A3X2. If we
introduce another weakly coupled set into the spin system, we use the letter
M to denote another set well separated from A and X. A substituted propane
(CH3CH2CH2-X) would be designated A3M2X2.