nuclear magnetic resonance in condensed...

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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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