NMR Spectroscopy
Properties of nuclei spin, I
quadrupole moment, Q
magnetic moment, µ N
Nuclear Shell Theory assumes that nucleons (protons +
neutrons) occupy sets of orbitals like electrons in atoms.
•
Parallel sets of orbitals for protons and neutrons
•
Spin-orbit coupling is very strong, so j-j coupling applies
rather than L-S coupling.
•
“Minimum Multiplicity” rule. “Spins” (actually mj values)
couple whenever possible.
A “configuration” (n5jx) for 16O nucleus in its ground state
protons: {1s½21p3/241p½2}
neutrons: {1s½21p3/241p½2}
Note conventions:
•
n (principal quantum number) is the analog of (n - 5) for
electron configurations.
•
s, p, d..... for 5 = 0, 1, 2... as for electrons.
•
subscripts j = 5 + s, 5 + s – 1, ..... 5 – s as for electrons
•
superscripts, total occupancy (no. of mj values)
1
“Magic numbers” 2, 8, 20, 28, 50, 82, 126 correspond to closed
shells of nucleons
Nuclei with magic numbers of nucleons are especially stable
Examples: doubly magic nuclei 4He, 16O, 40Ca, 208Pb
Stable nuclei with an odd number of protons (Z) and an odd
number of neutrons (N) are uncommon.
Examples: 2H (0.015%), 6Li (7.5%), 10B (19.9%), 14N (99.6%)
Nuclear Spin, I is the total angular momentum.
because of the minimum multiplicity rule,
•
I = 0 for nuclei with Z and N both even
•
I = half integer (value of j for the unpaired nucleon) for
nuclei with Z odd and N even, or Z even and N odd.
•
For odd-odd nuclei I = an integer (jP + jN or jP – jN or jP +
jN – 1)
2
10
14
H (I = 1),
B (I = 3),
N (I = 1)
2
Nuclear Electric Quadrupole Moment, Q is a measure of the
charge distribution in the nucleus (nuclear “shape”).
1 2
Q = ∫ r ( 3 co s 2 θ − 1) ρ ( r ) d τ
e
'(r) is the charge density in element P
Q = 0 for nuclei with I = 0 or ½. Such nuclei are spherical
because they have closed shells of nucleons, or an odd nucleon
in an “s-like” orbital (j = ½).
For nuclei with I 1,
Q > 0 (prolate spheroid, “american football”) or
Q < 0 (oblate spheroid, “hamburger”)
since the odd nucleon occupies a non-spherical (p, d, f...)
orbital.
Examples (Q, in 10-28m2)
2
H
2.73 ×10–3
17
O
–2.6 × 10–2
23
Na
0.12
59
Co
0.40
181
Ta
3
3
Nuclear Magnetic Moment, µ N arises from the spin and orbital
magnetic moments of the unpaired nucleons.
Classically, a free proton should have a (spin) moment
eh
µP =
4 πm P c
= nuclear magneton, N
(N = 5.05 × 10–27J T–1)
Experimentally, µ P = +2.79 nuclear magnetons
Classically, a free neutron has µ neutron = 0
Experimentally, µ neutron = –1.91 nuclear magnetons (!)
If the odd proton is in a p, d. f,... orbital there will also be an
orbital contribution to µ N. There are no orbital contributions
from neutrons.
Examples (µ N in nuclear magnetons)
1
H
Be
27
Al
95
Mo
9
+2.793
–1.177
+3.641
–0.952
4
The relationship between µ N and I can be expressed in two ways
γ N Ih
µN =
2π
where
factor.
N
or
µ N = gNNI
is the magnetogyric ratio, and gN is the nuclear g-
For NMR spectroscopy the first of these is commonly used, e.g.
for 1H
γN
2 πµ N ( 2 π )( 2 .7 9 )(5.0 5 × 1 0 −2 7 )
=
=
= + 2 6 .7 5 1 × 1 0 −7 rad . T −1 s −1
−3 4
Ih
0 .5 ( 6 .6 3 × 1 0 )
For other above examples the ’s are (in 107 rad T-1s-1)
9
Be
Al
95
Mo
27
–3.759
+6.971
–1.780
5
In an applied magnetic field, B0, the nuclear Zeeman effect
generates (2I+1) equally-spaced levels corresponding to MI
values of +I to –I.
If
> 0, MI = +I is the lowest level; if
< 0, MI = –I is lowest
The resonance condition (spacing between Zeeman levels) is
h
∆E = h ν = γ
B0
2π
(or E = gNB0)
where is known as the Larmor Frequency.
Possible measurable NMR parameters are:
•
•
•
•
•
Magnetic shielding (chemical shift, )
Indirect spin-spin (J-) coupling
Direct dipolar (D-) coupling
Quadrupolar coupling
Relaxation times (T1, T2) and their consequences, e.g. NOE
In liquid and gas phases rapid molecular tumbling means that
only average values of and J-coupling can be observed.
Average values of D- and quadrupolar coupling = 0.
6
Chemical Shift
In a particular chemical environment the nucleus experiences an
effective magnetic field
Beff = B0(1 – ))
where ) is expressed in ppm
The resonance condition becomes
0 = B0(1 – ))/2%
The chemical shift is defined as
/ppm 106(sample – ref)/ref
when ) << 1,
= )ref – )sample
Since 1972 the convention has been to report chemical shifts in
terms of frequency. Thus > 0 signifies a less shielded
environment than the reference (previously this would have
been considered a “low-field” signal with < 0.)
7
Understanding chemical shifts
Two contributions to ), diamagnetic and paramagnetic
) = )(d) + )(p)
Diamagnetic term: Applied field B0 causes circulation of
electrons in filled shells to induce a magnetic field opposed to
B0, i.e. )(d) > 0. More electrons, greater )(d).
σA
(d)
Zµ
e2 A
=
P
nµ
1 2 πm ∑
n 2a 0
µ
where Pnµ is the charge density on the µ-th orbital, and Zµ is the
effective nuclear charge for the µ-th orbital.
8
Paramagnetic term: a result of the second-order Zeeman effect, a
distortion of the valence electron orbitals, achieved by mixing
with excited electronic states.
Note, this stabilizes the ground state in the magnetic field and
leads to a negative shielding of the nucleus.
σ( p ) = −( co n s tan t ).
(
1
. r −3
∆E
−3
+
P
r
u
np
nd
Du
)
E is the HOMO-LUMO energy gap, Pu and Du are the relative
imbalance of the valence electrons about the nucleus, and
r −3
1Z 
=  eff 
3  na 0 
3
This is the so-called Average Excitation Energy (AEE) model.
Usually simplified to
σ
(p)
A r −3
=−
∆E
9
Chemical shifts for most nuclei are dominated by )(p).
Exceptions: 1H, 7Li, 23Na ..... No valence-shell p-electrons. For
these nuclei chemical shift range is small (10-20 ppm) and can
be understood in terms of )(d).
Two “remote” effects recognized for protons.
1.
2.
Neighboring atom anisotropy.
Ring currents
Chemical shift range for other light p-block nuclei dominated by
paramagnetic term
11
B
200
27
Al 270
13
C
650
29
Si
400
14
N
930
31
P
700
17
O
~1000
33
S
600
19
F
800
35
Cl 820
Note:
Increasing r–3np across periods ;
After Group 14 possibility of n %* transitions with small E;
â
e.g. (14N)
NO2– 250 ppm from NO3–
10
Remote effects upon
•
Neighboring atom anisotropy. The shielding depends upon
the orientation of the molecule with respect to B0.
In a molecule HX when the H– X bond is aligned with B0
the diamagnetic circulation around X (its diamagnetic
susceptibility, 3 (X) ) will increase shielding of the proton.
When H–X is perpendicular to B0, 3(X) will decrease the
shielding of the proton. For a rapidly tumbling molecule in
solution the resultant contribution to the (decreased)
shielding of the proton is
)H = – R–3(3 (X) – 3(X))
E
Clearly, if the susceptibility of X is isotropic there is no
effect upon the shielding of H.
If the shielding of X is dominated by the paramagnetic
contribution (for example if X is a metal atom) the same
equation would apply, but the 3’s would be positive, and in
fact 3 = 0). Thus the remote effect leads to an increased
shielding for H. This effect accounts for the negative ’s (-5
to -60 ppm) for H atoms directly bound to metal atoms.
“Remote diamagnetic effect”
“Remote paramagnetic effect”
â decreased shielding
â increased shielding
11
•
Ring currents. Magnetic-field-induced circulation of
electrons over loops of atoms generates enhanced
(dia)magnetic susceptibility and greater shielding (smaller
) for nuclei inside the loop, less shielding (larger ) for
nuclei outside the loop.
The amount of shielding/deshielding depends upon the
orientation of the molecule in the magnetic field, but does
not average to zero.
Classical example: benzene and other aromatic molecules.
Useful for assigning spectra of porphyrins – protons close
to axis perpendicular to ring become more shielded.
Inorganic chemistry example. Diamagnetic mixed valence
heteropolyanions. Compare (31P) = –13ppm for
[PMoV2MoVI10O40]5– with -3.9 ppm for [PMoVI12O40]3–
12
Indirect (Scalar, J-) Coupling
The observation of J-coupling between two nuclei proves the
existence of a bond (or sequence of bonds) that is long-lived on
the NMR time-scale.
Contributions to J are transmitted via electron density in the
molecule and are not average to zero through molecular
tumbling
Mechanisms contributing to J (nucleus A observed, coupled to
nucleus B)
1.Spin-orbital effects. The magnetic field of the nuclear dipole
of B in mI = +½ (say) interacts with the orbital magnetic
moment of an electron surrounding B which in turn affects the
magnetic field around A.
2. Dipolar coupling. Polarization of paired electron density in
the molecule by the nuclear magnetic moment of B affects the
nuclear moment of A. Unlike the direct dipolar coupling of
nuclear moments, this coupling does not average to zero upon
molecular rotation and tumbling.
3. Fermi Contact (dominant effect for couplings involving
protons). Direct interaction of nuclear spin moment with
electron spin moment adjacent to nucleus. Only possible if bond
has some s-character.
13
J-coupling is not observed if
•
B relaxes at a rate which is fast compared to the value of J
This is often (but not always) true if B is a quadrupolar
nucleus, e.g. 14N in NH3, amines
•
B undergoes fast exchange.
Example, 17O spectrum of H2O, singlet in water, 1:2:1
triplet in organic solvents.
Relaxation Times.
Spin-lattice (longitudinal) relaxation time, T1
Net magnetization M0 of an ensemble of N nuclei of spin I, in a
magnetic field B0 (M0 and B0 aligned along z-axis by
convention) is given by the Curie law
N 2 γ 2 ! 2 I ( I + 1)
M0 =
B0
3kT
Rate at which Mz approaches M0 is given by
∂M z −1
= (M z − M 0)
∂t
T1
14
Several methods are available for measuring T1. Common
version is the Inversion Recovery Fourier transform
(IRFT) method.
In its simplest form, IRFT uses the pulse sequence
[PD – %x – - – (%/2)x – AT]n
PD is a pulse delay and AT is the computer acquisition time; - is
a variable time delay. PD+AT should be about 5 times the
maximum estimated value of T1 (At 5T1 Mz = 0.99M0). The %
(180() pulse inverts the magnetization from +z to –z. If the %/2
pulse is applied immediately (- = 0) the magnetization is aligned
along –y and the signal is inverted with respect to a normal one%/2 -pulse experiment which rotates the magnetization onto +y.
Spin-spin (transverse) relaxation time, T2
After a 90( pulse B1, the magnetization, originally along z, is
rotated into the xy-plane, along the positive y-axis. If the field
B0 is perfectly homogeneous, the decay of magnetization in xy is
governed by T2.
M
∂M x
M ∂M y
=− x;
=− y
∂t
T2 ∂t
T2
Field inhomogeneities lead to a more rapid relaxation governed
by T2* (effective spin-spin relaxation time).
15
Linewidth, ν1 =
2
γ∆B 0
1
1
=
+
πT 2* πT2
2π
In practice, any mechanism that contributes to T1 relaation also
contributes to T2 relaxation, so that T2 T1.
Relaxtion Mechanisms for nucleus I in a diamagnetic molecule.
1.
Dipolar relaxation. Molecular tumbling of magnetic dipoles
of the same (I1) or different (S) type of nuclei generate
fields that fluctuate at the Larmor frequency of I. For nuclei
in the same molecule as I, 1/T1 is affected by an r–6 term.
Generally the intramolecular contribution is dominant.
Nuclear Overhauser Effect, NOE. If the Boltzmann
equilibrium of spins S is modified (generally by irradiation
of S to cause saturation) an non-equilibrium population
distribution is induced in the I spins, leading to a change in
signal intensity. If the relaxation of I occurs by dipole
interaction with S and the saturation time of S is long
compared to T1, the change of signal intensity is given by
S/2 I . To the extent that other mechanisms contribute to
the relaxation of I the observed change will be less than the
maximum.
Typical proton-proton enhancements are 0.01 – 0.3, less if
internuclear separation > 5Å.
16
If I and S have opposite signs, the NOE will be negative,
e.g. maximum proton-induced NOE’s for 29Si and 15N are –
2.5 and – 4.9 respectively. (Possibility of loss of signal in
some cases.)
2.
Scalar interactions. A rapidly-relaxing S collapses S-I
scalar coupling and increases T2 of I.
3.
Spin-rotation. Small rapidly rotating molecules in mobile
liquids or the gas phase have magnetic moments
proportional to their angular momentum, as a result of the
lag of electrons adjusting to the new nuclear positions.
Collisions cause rearrangements of these moments after a
characteristic time interval and cause relaxation. Relaxation
increases as temperature increases.
4.
Magnetic Shielding Anisotropy. If I has ) g ) molecular
tumbling generates fluctuating field at I. Expressions for
1/T contain B02. Thus lines for e.g. square-planar 195Pt or
for linear 199Hg become broader in high-field
spectrometers.
17
5.
Quadrupolar Interactions. For all quadrupolar nuclei this is
the dominant relaxation mechanism. For solutions,
1
3 π2 2 I + 3  e 2 q Q   η2 
=
⋅ 2
⋅
 1 +  τ c

T 1 1 0 I ( 2 I − 1)
h  
3
2
where Q is the quadrupole moment, q is the electric field
gradient at the nucleus, is the assymmetry parameter for
nuclei in non-axial environments [ = (qxx – qyy)/qzz].
-c is the rotational correlation time of the molecule.
For a given nucleus, the linewidth of the NMR signal will
therefore depend upon the magnitudes of q and -c. Smaller
values will lead to narrower lines. Since molecules tumble
faster in non-viscous solvents, always best to record spectra
of quadrupolar nuclei at higher tempratures.
The electric field gradient is determined by the distribution
of electron density around the nucleus. To first order,
tetrahedral and octahedral molecules with central atoms
with electron configurations that correspond to empty, halffilled, or filled subshells should have q = 0. Since the linewidth depends upon the square of q, slight distortions can
produce significant effects.
Example: 51V (I = 7/2, Q = 0.03 × 10–28 m2)
line-widths (Hz): VO43–
2
V(O2)43– 300
18
Some Nuclear Quadrupole Moments
I-127
Mn-55
Bi-209
Co-59
As-75
Nb-93
Cu-63
Al-27
Ga-71
Mo-95
Na-23
Cl-35
Be-9
B-11
V-51
O-17
Cs-133
0
0.2
0.4
0.6
0.8
Q / barns (negative values in red)
19
Editing NMR Spectra
Decoupling – the use of RF radiation to cause nucleus X to
undergo rapid spin state changes so that observed nucleus A
cannot reliably “know” the state of the X spin. Amount of power
required for decoupling increases as the magnitude of J
increases. Can lead to heating of sample.
Heteronuclear decoupling. Most common application to use
broad band proton decoupling to simplify spectra of nuclei such
as 13C, 31P etc.
Such decoupling can also induce NOE and modify intensities.
These effects can be separated by the use of “gated decoupling”,
the switching on and off of the decoupler during the pulse
sequence. Discrimination is possible because decoupling is
essentially instantaneous and NOE depends upon the build-up of
population differences (decoupler must be on during relaxation
delay for a time that is long compared to T1).
20
Decoupler Status
Relaxation
delay
Acquisition
time
Result
on
on
decoupling + NOE
off
on
decoupling only
(no NOE)
inverse gated
decoupling
on
off
NOE + coupling
gated
decoupling
off
off
coupling (no NOE)
Selective and off-resonance decoupling provide information that
is more effectively obtained by Polarization Transfer methods.
These methods involve magnetization transfer from an abundant
nucleus (typically 1H) to a rare nucleus such as 13C. They can be
used to observe, selectively, subspectra containing only methyl,
methylene, methine, or quaternary carbons.
Insensitive Nuclei Enhanced by Polarization Transfer (INEPT)
has been replaced by
Distortionless Enhancement by Polarization Transfer (DEPT)
21
The DEPT pulse sequence is
1
H
13
C
t D – ( ) y
(%/2)x — tD – (%)x –
(%/2)x – tD – (%)x –
tD – Acquire
The delay tD is set to (2JCH)–1 and the intensities of the methine,
methylene, and methyl resonances depend upon the width of the
pulse. See Figure
Unlike NOE, DEPT enhancements are always positive and are
larger, S/ I.
e.g. maximum effect for 29Si – {1H} is 5.03 (DEPT) vs –1.52
(NOE).
22