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University of Ljubljana
Faculty of mathematics and physics
Physics department
SEMINAR
Nuclear magnetic resonance
in condensed matter
Author: Miha Bratkovič
Mentor: prof. dr. Janez Dolinšek
Ljubljana, October 2012
Abstract
The seminar outlines basic principles that are important in nuclear magnetic
resonance spectroscopy. Essential model of pulsed NMR is described along with
relaxations after. Various interactions with nucleus influence the spectrum. Special
focus is being put on quadrupole effect in first and second order corrections. Dipolar
coupling, J-coupling and chemical shift are briefly described to give an overview of
major interactions of NMR sprectroscopy.
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Contents
1 Introduction 2
2 NMR: Basic principles 2
3 Quadrupole effect 5
4 Other NMR interactions 12
5 Conclusion 14
References 15
1 Introduction
Nuclear magnetic resonance, (NMR), was discovered in 1945 by Bloch and Purcell,
who received Nobel prise for this discovery. Since then the development of NMR as a
technique parallels the development of electromagnetic technology and
advanced electronics [1, 2]. NMR principles and applications are today fundamental
tool in medicine, spectroscopy and material science.
2 NMR: basic principles
2.1 Nucleus in homogenous magnetic field (classical view)
All nuclei have the intrinsic quantum property of spin. The overall spin of the
nucleus is conventionally determined by the spin quantum number I. Beside the
angular momentum, �,nucleus also possesses magnetic moment which is oriented in
the same direction and determined by
� = ��, (1)
where γ is the gyromagnetic ratio (depending on the nucleus). In the magnetic field
��, which will be aligned with the z-axis, there is a torque imposed on magnetic
moment
= � × �� = �� × ��. (2)
The torque is proportional to the time derivative of angular momentum
���� = �� × ��. (3)
Equation (2) has a solution in the form of precession. The Larmor frequency is
precession frequency of the nuclear magnetic moment around the magnetic field
�� = ���. (4)
Magnetic moment per unit volume is called magnetization.
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� = ��∑��, (5)
thus it follows
���� = ��× ��. (6)
When we impose short magnetic field ��,of Larmor frequency , in the x direction
(perpendicular on ��), the precession occurs and there is an angle, �, between
directions of magnetization and ��(Fig. 1).
The azimuth angle of magnetization is
dependent upon intensity of impulse
��, and its duration, T. Typically the
signal will be such, that the starting
angle of precession will take place at
� = �/2or � = �. This is the reason the
impulses are called �/2 pulse or � pulse.
Let us consider �/2 example. After the
pulse is over the magnetic moment is
affected only by the outer magnetic field
��. One would expect for precession to
be going on forever under such
conditions. However, precession of
single magnetic moment is also affected
by inner randomly changing magnetic
fields of magnetic moments of other nuclei and electrons. That is why the
magnetization direction is returning to its thermodynamic equilibrium direction
along the external field.
2.2 Relaxation principles, ��and ��
It is useful to take a look from a rotating (with Larmor frequency) polar
coordinate system on Fig.2. In the
ideal precession case, the magnetic
moment would have a static direction
from Larmor rotating systems
perspective. It turns out that magnetic
moments loose their polar
orientation, and eventually they
randomly disperse in all directions.
Projection of magnetization in �’ !’ plane is exponentially decreasing with
a time constant "#, which is called
transverse relaxation time. Synchronic
precession can be considered to be
just one of possible states with same
Fig. 1: Precession around static field is
a result of impulse of radio frequency
magnetic field [3].
Fig. 2: Rotating polar coordinate system.
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energy. In time, system will occupy all possible states.
Because magnetic moments loose their phase coherence, we introduce
another � pulse which turns them around so they are now gathering back together.
Once they are again aligned, we can measure maximum signal of magnetization in the
z-direction. The purpose of the second, �, pulse is in the fact that the first signal
happens right after initial �/2 pulse, so it can not be properly measured. The second
pulse also has to occur soon enough, so there is still phase coherence present. The
signal measured is called the spin echo (Fig. 3).
Fig. 3: Signals in radio frequency solenoid; spin echo amplitude is dependent
of time delay, $.
Beside the transverse relaxation, there is also spin lattice or longitudinal
relaxation present, i.e. returning the nuclei to its thermodynamic equilibrium
state. Magnetization can be described accordingly
%& = %'1 exp' �,-... (7)
In this case, the whole energy of the magnetic moments of nuclei changes, so
there has to be interactions nuclei-electron present, therefore spin lattice
interaction [4].
2.3 Simple quantum description
Classical view on NMR is appropriate for intuitive understanding of precession.
However, we must turn to quantum mechanics for any further calculations. First
reason for this is that energy states of nuclei in magnetic field are discrete. The
application of a magnetic field � produces Zeeman energy of the nucleus of amount
��. We have therefore a Hamiltonian
H= ��. (8)
Taking the field to be of magnitude B along the z-direction, we get
H= �ħ�0& , (9)
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where 0&is the z-direction component of a dimensionless spin operator I, defined by
the equation:
� = ħ1. (10)
The allowed energies are
2 = �ħ�3, (11)
where m can take any of the values 3 = 0, 0 1,… , 0. Zeeman levels are illustrated in
Fig. 4 for the case I = 3/2. as is the case for the nuclei of Na or Cu. The levels are
equally spaced.
The operator 0& has matrix elements between
states 3 and 3′, ⟨3′|0&|3⟩, which vanish
unless 39 = 3 : 1. Consequently the allowed
transitions are between levels adjacent in
energy, giving
;2 = �ħ� = ħ�� , (12)
which is again expression for Larmor frequency, mentioned before (4). Since that is
the case, we can expect spectral line to be very narrow, positioned exactly at Larmor
frequency [4]. Magnetization as we said fades out with time constant "# . Signal is
therefore proportional to exp' </"#. =>?'��<.. In a frequency domain (Fourier
transform of signal) the corresponding term is Lorentzian shaped line, with
@AB% ∝ 1/"#.
3 Quadrupole effect
So far we have not considered any electrical effects on the energy of the nucleus. That
such effects do exist can be seen by considering a non-spherical nucleus. Suppose it is
somewhat elongated and is acted on by the charges shown in (Fig. 5). We see that
Fig.5 (b) will correspond to a lower energy, since it has put the tips of the positive
nuclear charge closer to the negative external charges [5]. There is, therefore, an
electrostatic energy that varies with the nuclear charge distribution orientation.
Fig. 4: Energy levels
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Fig. 5: Oval shaped nucleus in the field of four charges, DE on �-axis, -E on !-
axis. Configuration (b) is energetically more favorable, because it puts the
positive charge at the tips of nucleus closer to negative charges E.
To present a quantitative theory, we begin with a description in terms of the classical
charge density of the nucleus, ρ. We shall obtain a quantum mechanical answer by
replacing the classical ρ by its quantum mechanical operator. Classically, the
interaction energy E of a charge distribution of density ρ with a potential V(r) due to
external sources is
2 = FG'H.I'H.JH. (13)
Potential can be expanded in Taylor series about the origin (in center of nucleus):
I'H. = I'0. D ∑�L M�
MNOPQR� D�#∑�L�S
MT�MNOMNUVQR�D. ..
(14)
Index α and β stand for �, ! and W. Next we define IL = M�MNOPQR� and ILS = MT�
MNOMNUVQR�.
Interaction energy can be now written in the form
2 = I� FG'H.JH D ∑IL F�LGJH D �
#!∑ILS F�L�SGJH. (15)
Choosing the origin at the mass center of the nucleus, we have for the first term the
electrostatic energy of the nucleus taken as a point charge. The second term involves
the electrical dipole moment of the nucleus. It vanishes, since the center of mass and
center of charge coincide. Moreover, a nucleus in equilibrium experiences zero
average electric field IL. It is interesting to note that even if the dipole moment were
not zero, the tendency of a nucleus to be at a point of vanishing electric field would
make the dipole term hard to see.
The third term is the so called electrical quadrupole term. We note at this
point that one can always find principal axes of the potential V such that
ILS = 0if Y Z [ (16)
I must also satisfy Laplace’s equation \#I = ∑ILL = 0. In the case of cubic
symmetry all derivatives are zero, the quadrupole coupling then vanishes. This
situation arises, for example, with ]^#_ in ]^ metal. It is convenient to consider the
quantities `LS, defined by the equation
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`LS = ab3�L�S dLSH#eGJH. (17)
This will be useful turned around as
a�L�SGJH = 13 (`LS +adLSH#GJH). (18)
We continue with writing expression for quadrupole energy 2(#), 2(#) = �
f∑(ILS`LS + ILSdLS F H#GJH), (19)
Since V satisfies Laplace's equation, the second term on the right of (10.11) vanishes,
giving us
2(#) = �f∑ILS`LS . (20)
This term is independent of nuclear orientation. For proper quantum mechanical
expression for the quadrupole coupling, we simply replace the classical G and `LS by
their quantum mechanical operators,
Bg = �f∑ILS h̀LS. (21)
With the help of Wigner-Eckart theorem, we can continue
⟨0,3i`LSi0,39⟩ =
= j ⟨0,3 P_# b0L0S − 0S0Le − dLS0#P 0, 39⟩. (22)
We will express constant, j, with matrix element for 3 = 3’ = 0 and Y = [ = W.
k` = ⟨0, 0|`&&|0, 0⟩ = j⟨0, 0i30&# − 1#i0, 0⟩ == j⟨0, 0|0(20 − 1)|0, 0⟩. (23)
Constant is therefore
j = lgm(#mn�). (24)
Quadrupole Hamiltonian is rewritten in quantum mechanical form as
Bg = lgfm(#mn�)∑ILS o
_# b0L0S − 0S0Le − dLS0#p. (25)
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If we express operators 0Nand 0q with operators 0± and apply interaction (25) in
eigenspace where ILS is diagonal, then we get
Bg = lTrgsm(#mn�) o30&# − 1# +
t# b0u# + 0n#ep, (26)
Where kE = I&& and v, asymmetric parameter, defined by
v = �wwn�xx�yy . (27)
These two parameters are the ones which determine the shape of the spectrum.
3.1 First order corrections of quadrupole interaction
When the quadrupole interaction is small compared to interaction nucleus-external
magnetic field, perturbation theory can be applied. However it is often strong enough
that the second order correction is of significant importance. That is why they are
called first and second order quadrupole interactions.
Perturbation theory corrections [6] of Zeeman energy for m-th energy level are
2z = 2z(�) + ⟨3iBgi3⟩ + ∑ i⟨zi{|i}⟩i~�n~�
# +⋯. (28)
Let us consider now just the first order correction 2z(��Q��) = ⟨3iBgi3⟩ for the m-th
energy level of nucleus with the spin 0:
2z = −ħ��3+ ⟨3iBgi3⟩ =
−ħ��3+ lTrgsm(#mn�) (33# − 0(0 + 1)). (29)
When transition between 3 and 3 − 1 levels occurs, the frequency we detect with
nuclear magnetic spectroscopy is
�z,zu� = ~�n~��-ħ = �� + _lTrg
sm(#mn�)ħ (23 + 1)). (30)
The 3 = 1/2 → −1/2quantum transition is called central transition, which is
unaffected by the quadrupole anisotropy to first order. Transitions between other
levels are called satellite transitions (Fig. 6).
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Fig. 6: Spectrum of quadrupole interaction of first order for I=5/2. Dotted line
presents theoretic calculation, and full line presents the actual spectrum. It is
apparent that spectrum does not consist from narrow, separated lines, but
from connected peaks with allowed frequencies even between peak positions.
This is a consequence of angular dependence of quadrupole interaction,
described by angle �, which is the angle between external field and interaction
principle axis. Quadruple interaction is heavily dependent upon such orientation [7].
In other words, the first order term splits the spectrum into 2I components of
intensity
@ = |⟨3 − 1|0N|3⟩|#, (31)
where @ is intensity at frequency �z(�)away [4]. We can take a look at asymmetry
parameter influence on (Fig.7). As v varies, the lineshapes of both the central and
satellite transitions change, which can provide useful structural information as v is
related to the local symmetry.
Fig. 7: The effects of the asymmetry parameter (v) on the first order satellite
without the central peak [7]. Peak symmetry, however is conserved in first
order corrections.
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3.2 Second order quadrupole interaction
In order to get the second order correction 2z(�l�) = ∑ i⟨zi{|i}⟩i~�n~�
#(the third term on the
right side of (28)), we have to calculate all matrix elements ⟨3iBgi?⟩. In this case the
central transition changes too, as do all other transitions. The change of frequency is
�z,zu� = �� +2n�# − 2�#
ħ . (32)
A comprehensive energy diagram is presented in Fig.8. Zeeman energy levels are
equidistantly apart. After applying first and second order corrections, levels change
accordingly.
Fig. 8: Energy level diagram for I=5/2 nucleus for Zeeman energy levels and
corresponding first and second order corrections. Here θ is again the angle
between the principal axis of the interaction and the magnetic field. The first order
interaction has an angular dependency with respect to the magnetic field
of 3cos#(� − 1) (the P2 Legendre polynomial), the second order interaction depends
on the P4 Legendre polynomial [7].
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Beside energy levels, line shapes are affected by v, similar to first order case. On
Fig.9 we see the central transition peak will be split in two peaks with small value of
asymmetric parameter. In real experiments such two peaks can be positioned very
close and are blurred by other effects. They often appear as one asymmetric wide
peak. Comparison between measured and theoretical prediction was made for I=5/2
nuclei (Fig.10). Asymmetric parameter is set to v = 0 [7].
Fig. 9: The effects of asymmetry parameter on second order central transition
lineshapes [7].
Fig. 10: Central transition for nucleus of spin I = 5/2 of second order
quadrupole interaction. Dotted line is theoretical prediction, full line presents
what would be measured. Again transition frequency is dependent upon
relative interaction orientation (angle �). [7]
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We can see from (30), that for spin I=½, the quadrupolar correction of first
(consequently the second) order vanishes. Since the coupling of nucleus with
electric field gradient takes place only with half-integer spin larger than ½, all
elements are not subject to such interaction. On Fig. (11), we can see both
kinds of elements marked on periodic table.
Fig. 11: Periodic table; most elements nuclei have spin larger than ½ [8].
4 Other NMR interactions
4.1 Direct dipole coupling
The direct dipole coupling is spin-spin interaction of each spin influencing on
their neighbor through magnetic field. Interaction energy of two magnetic
moments �� and �� is
2� = ��s� o
��∙��Q� − _(��∙�)(��∙�)
Q� p , (31)
where � is radius vector from �� to �� . The dipole coupling is very useful for
molecular structural studies, because it is dependent only on intermolecular distance
(Hn_).
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4.2 J-coupling (indirect coupling, scalar coupling)
J-coupling is the coupling between two nuclear spins due to the influence of bonding
electrons (with spin �) on the magnetic field running between the two nuclei with
interaction energy 2� ∝ �l�}1�. Each nucleus weakly magnetizes electrons, which
generate a magnetic field at the site of the neighboring nuclei and vice versa.
In a magnetic field a nuclear spin is oriented in one possible eigenstate. An
electron nearby tends to be antiparallel to the nuclear spin, owing to the Fermi
interaction between the two particles (Fig. 12). The bond's second electron must be of
opposite spin following Pauli's exclusion principle. The second electron defines the
preferred orientation of the bound
nucleus and gives rise to a small excess of
antiparallel oriented nuclear spins that
are directly bond.
Coupling over more bonds can be
explained by Hund's rule which states that
electron spins close to a nucleus tend to
be ordered in parallel. The information is
thus transported over to the next bond.
Since only s-electrons have finite
probability to be near the nucleus the J-
coupling increases with increasing s-
character of the chemical bond [9].
4.3 Chemical shift
The signal frequency that is detected in nuclear magnetic resonance would be a pure
Larmor frequency if the only magnetic field acting on the nucleus was the externally
applied field.
However when the magnetic field is
applied, it induces currents in the electron
clouds in the molecule. The circulating
electrical currents in turn generate a magnetic
field and the nuclear spins sense the sum of
the applied external field and the induced field
generated by the molecular electrons. This
change in the effective field on the nuclear spin
causes the NMR signal frequency to shift
(Fig.13). Larmor frequency of nucleus is
always diminished by atomic electrons.
The magnitude of the shift depends upon the
type of nucleus and the details of the electron motion in
the nearby atoms and molecules.
Fig. 12: A simple model of J-coupling [10].
Fig. 13: Energy diagram (between
bare and atomic nucleus) of chemical
shift. Rate of change in magnetic field,
�,is called shielding factor [11].
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4.4 Spin Hamiltonian overview
In general Hamitonian is the sum of different terms representing different physical
interactions B = B�� + B�# + B�_ +⋯ . These can be divided in magnetic and electric
interactions. It is convenient if each B�is time independent. Terms that depend on
spatial orientation may average to zero with rapid molecular tumbling. On (Fig.14)
we can see relative importance of different interactions for solids and liquids [12].
Fig. 14: Overview of different interactions; anisotropic liquids and solids have
similar proportions of NMR effects. Hamiltonian of isotropic liquids is without
contribution of quadrupole effect and direct dipole interaction [12]. Interaction
where magnetic moments of electrons are oriented in approximately the same
direction is not included.
5 Conclusion
Although we have mentioned all the major interactions, there are also
additional physical influences on shape and position of spectral lines, for
example Knight shift. Different interactions dominate for different molecules,
the level of anisotropy is often of significant importance. It is necessary to
have all different interactions in mind when dealing with NMR experiments.
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References
[1] http://www.lbl.gov/Science-Articles/Archive/MSD-NMR.html (10.10.2012)
[2] http://www.ssbc.riken.jp/english/contents/nmr/index.html (10.10.2012)
[3] http://users.fmrib.ox.ac.uk/~peterj/safety_docs/pf6img5.gif (20.4.2012)
[4] http://www.phys.ufl.edu/courses/phy4803L/group_II/nmr/nmr.pdf (20.4.2012)
[5] C.P. Slichter, (Principles of Magnetic Resonance, Springer 1996)
[6] http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Time_Ind_PT.htm (10.10.2012)
[7] M.E.Smith, E.R.H. van Eck, Prog. Nucl. Ma. Res. Sp: 34, p159, (1999).
[8] http://www.grandinetti.org/resources/Research/NMR/PeriodicTable.png (20.4.2012).
[9]http://www.chemie.uni-hamburg.de/nmr/insensitive/tutorial/en.lproj/coupling.html
(20.4.2012)
[10]http://www.chemie.uni-hamburg.de/nmr/insensitive/tutorial/img_scalar_coupling.png
(20.4.2012)
[11] http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nmrcsh.html (20.4.2012)
[12]http://www-
mrsrl.stanford.edu/studygroup/2/Files/cw466091_Lecture_8Spin_Hamiltonian.pdf (20.4.2012).
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