Fourier Transform Multiple Quantum Nuclear Magnetic
Resonance
BY
GARY
DROBNY,
ALEXANDER PINES, STEVEN
SINTON,
DANIEL
P.
WEITEKAMP AND
DAVID
WEMMER
Department
of
Chemistry,
University
of
California,
Berkeley,
California,
U.S.A.
Received 18th December, 1978
The excitation
and
detection
of
multiple quantum transitions in systems
of
coupled spins offers,
among other advantages,
an
increase in resolution over single quantum n.m.r. since the number
of
lines decreases as the
order
of
the transition increases. This paper reviews the motivation for de-
tecting multiple
quantum
transitions by a Fourier transform experiment
and
describes
an
experi-
mental approach to high resolution multiple
quantum
spectra in dipolar systems along with results
on some protonated liquid crystal systems. A simple operator formalism for the essential features
of
the time development
is
presented
and
some applications in progress are discussed.
The
energy
level
diagram
of
a system
of
coupled
spins 1/2 in
high
field
is
shown
schematically
in
fig. I.
The
eigenstates
are
grouped
according
to
Zeeman
quantum
number
mi
with
smaller
differences
in
energy
within
a
Zeeman
manifold
due
to
the
couplings
between
spins
and
the
chemical shifts.
For
any
eigenstate
10
of
the
spin
Hamiltonian
H
(in
frequency
units)
HI
i)
=
(t)i
I
i)
lz I
i)
= m i I
i).
(I)
The
single
quantum
selection rule
1
of
the
low
power
c.w.
experiment
and
the
one
dimensional
Fourier
transform
experiment
arises because
<i
I
lx
I
j)
vanishes unless
(2)
Simple
combinatorial
considerations
show
that
the
number
of
eigenstates decreases
as
lmi
I increases
and
the
number
of
transitions
decreases as
lqiil
increases.
The
highest
order
transition
possible is
the
single
transition
with
lql = 21 where I is
the
total
spin.
For
a system
of
N spins 1/2,
transitions
up
to
order
N
are
possible.
Detection
of
multiple
quantum
transitions
in
c.w.
experiments
is well
known.
2
-
4
Extension
to
high
order
transitions
is
not
promising,
since
the
transitions
observed
are
a sensitive
function
of
r.f. field
strength.
This
leads
to
difficult spectral
simulations
and
experimental
problems
of
saturation
and
sample
heating.
The
alternative
time
domain
experiment
is
the
determination
of
multiple
quantum
transition
frequencies by following
the
time
development
of
multiple
quantum
coherences
point
by
point.
5
-
7
This
work
treats
a class
of
such
multidimensional
N
N=--:z
N
-z-•
1
l
•
I
I
I
l
N
-2+2
0~
JL_2
2
.!.
-1
2
.,
I
l
I
I
I
I
N
2--~-------------~-----
-j
FIG.
I.-Schematic
representation
of
the high field energy level diagram
of
coupled spins 1/2. Broken
arrows indicate the forbidden types
of
transition observed in Fourier transform multiple quantum
experiments.
P
1
Pz
P3S
q=O
ITI
t1
IT!
q=1
q=2
q=3
q=4
q=5
q=6
frequency
(l!.w =
5.967
kHz)
FIG.
2.-Multiple
quantum spectrum
of
benzene (
15
mol
%}
in p-ethoxybenzylidene-n-butylaniline
(EBBA)
at
20
oc.
The three pulse sequence was PI =
n/2,~"
P2
x/2:x,
P3 =
x/2x.
The magnitude
spectra obtained for
11
values
of
r spaced
at
0.1
ms intervals from 9.6 to 10.7 ms were added. The
value
of
t
1
ranged from 0 to 13.824 ms in 13.5
ps
increments for each
r.
A single sample point was
taken
at
tz = r after PJ. One half
of
a symmetrical spectrum is shown.
DROBNY,
PINES,
SINTON,
WEITEKAMP
AND
WEMMER
51
experiments
in
which
the
irradiation
consists
of
pulses
at
the
Larmor
frequency.
Time
proportional
incrementation
of
the
r.f.
phase
(TPPI)
allows
separate
determination
of
the
spectra
of
all
orders
free
from
effects
of
magnet
inhomogeneity.
EXPERIMENTAL
The spectrometer
is
of
pulsed Fourier transform design with super-conducting magnet
(Broker) operating in persistent mode at a proton frequency
of
185
MHz. Phase shifting
was performed
at
185
MHz by a digitally controlled device (DAICO 10000898) under con-
trol
of
the pulse programmer.
Samples were approximately 400 mg sealed
in
6 mm glass tubing after degassing
by
repeated freezing and evacuation. All observations are in the nematic phase. Synthesis
of
4-cyano-4'-[
2
H]
11
pentyl-biphenyl was by the procedure
of
Gray and Mosley.
8
RESULTS
The
spectrum
of
benzene dissolved in a
liquid
crystal
served
as
a
prototype
in
the
development
of
the
single
quantum
n.m.r.
of
complex
spin
systems
in
ordered
phases.
9
The
multiple
quantum
spectrum
of
ordered
benzene
is
shown
in
fig. 2.
The
resolution
is
limited
by
magnetic
homogeneity
and
the
inhomogeneous
linewidth
is
proportional
to
jqj.
P,
p2
rr
p3
s
lrl
t,
I
t,
]rj
T
T
q=O
q=l
q=2
q=3
q=4
q
:5
q=6
I
l.~~-~~L-J~
FIG.
3.--Multiple
quantum
spectrum
of
benzene
at
22
oc
and
TPPI
pulse sequence. The sample
is
the same as in
fig.
2. The pulses are
Pl
=
1t/21f1,
P2 =
1t/2iP
and
P3
=
1t/2x,
where
rp
=
/).wt
1
•
The
increment in
rp
was 29.5 degrees
and
the increment in 1
1
was
10
JlS
for each
of
1024 points. The
magnitude spectra for eight values
of
r between 9
and
12.5
JlS
were added. The magnetization was
sampled
at
t = r.
52
FOURIER
TRANSFORM
MULTIPLE
QUANTUM
N.M.R.
The
spectrum
of
fig.
3 demonstrates
the
use
of
the spin echo to remove inhomogene-
ous line
broadening
and
the use
of
time
proportional
phase increments (TPPI)
to
restore
the
offset. Resolution is limited by
truncation
of
the multiple
quantum
free
induction
decay
and
scale
of
reproduction.
The
actuallinewidth
is
less
than
2
Hz
for
all
orders
and
suffices to resolve all allowed transitions
of
all orders.
An
application
of
the
TPPI
method
to the eight
proton
system
of
an
alkyl-deuter-
ated
cyano
biphenyl liquid crystal
is
shown in
fig.
4. All eight orders
are
observed.
Resolution is limited by truncation. Actual linewidths are
< 100
Hz
in a spectral
width
of
~
40
kHz
for each order.
q=O
q
=1
q=2
q:3
q:4
FIG.
4.-Proton
multiple quantum spectrum
of
4-cyano-4'-[
2
H
1
dpentyl-biphenyl
at
23.3
cc
by
TPPI.
The increments are 22.5 degrees in
cp
and
1.5
fJS in t
1
for each
of
1024 points. The
timer
took
five
values between 0.5
and
1.0 ms
and
the magnetization was sampled
at
64
intervals
of
5
ps
starting at
t2 =
(r
+
0.1)
ms.
DISCUSSION
THREE
PULSE
EXPERIMENT
The
time development
of
the spin system
during
the pulse sequences
of
fig.
2
and
3
is conveniently discussed in terms
of
a spherical tensor
operator
expansion
of
the
density matrix.
For
any time,
p(t) =
.2
c~:x(t)T~C(
(3)
k,rx,q
where
T~
is the
qth
component
of
a spherical tensor
operator
of
rank
k.
10
The label
ex.
completes the specification
of
a complete basis
of
tensor operators.
The
initial equilibrium density matrix
is
(4)
and
immediately
after
a n/2y pulse (w
1
~
H, w
1
tp
1
= n/2)
p(tp1) =
-C5
1
(0)1x
=
Cfl(tpl)Tfl
+ C.2I(tp1)T
~f.
(5)
DROBNY,
PINES,
SINTON,
WEITEKAMP
AND
WEMMER
53
The
Zeeman
quantum
number
q
is
conserved
during
evolution
under
a high field
Hamiltonian.
Thus
neglecting relaxation
p(t)
= L
c~x(t)T~x
(6)
k,a,
± 1
at
any time t
after
the
initial n/2 pulse
and
no
multiple
quantum
coherence has been
created.
After
a
period
of
time
on
the
order
of
the inverse
of
the coupling frequencies,
terms with
k
~
21
will be present. A second
strong
pulse
may
then
rotate
T
=r
into
T~a
with
-k
~
q
~
k.
Thus
at
timer,
after
the second pulse, all orders with
-2/
~
q
~
2/
have,
in
general, been created
and
evolve
during
the
time t
1
at
the eigen-
frequencies
wij
= wi - wJ.
If
the
only observable
measured
is
the transverse magnetization
corresponding
in
the
rotating
frame
to
the
JqJ
= 1
operators
lx
and
!y, it
is
not
possible
to
follow
the
evolution
of
orders with
JqJ
# 1 directly.
Rather,
a
third
pulse
at
time t
1
after
the
second pulse is needed
to
rotate
the various
tensor
components
back
to
T~x
1
•
These
may
then
evolve into the signal observed
at
time t
2
after
this detection pulse.
The
signal
may
be written
then
as
(7)
Viewed as a function
of
t
1
this
is
the
multiple
quantum
free
induction
decay.
It
is
collected pointwise by
variation
of
t
1
on
successive shots
and
is
an
example
of
a multi-
dimensional n.m.r. experiment.
SEPARATION
OF
ORDERS
BY
FREQUENCY
OFFSET
In
order
to
obtain
an
increase in resolution over the single
quantum
experiments
it
is
necessary
to
separate
the
spectra
of
different order. Consider
the
decomposition
of
the
density
matrix
at
time
r(t
1
= 0) into
Zeeman
components
2/
p(r)
= 2 pq(r).
(8)
q =
~2/
By construction,
pq{
r)
= 2
c~a(
r)T~r~.
and
it follows from the definition
of
the
kr~.
spherical
tensor
operators
that
pq(
r,
rp)
= exp (irplz)Pi
T)
exp (
-irp/z)
= exp
(iqrp)pq(
T).
(9)
If
we let
rp
=
11wt
1
,
then
eqn
(9) corresponds
to
a calculation
of
the
effect
of
a
frequency offset
term
-Awlz
in
the
rotating
frame
Hamiltonian.
Thus
a
modulation
by exp (iqAwt
1
)
is caused by
the
offset, shifting
the
spectrum
to
ql1w as seen in fig.
2.
The
difficulty with this
approach
to
separation
of
orders
is
that
the effect
of
magnet
inhomogeneity
is
proportional
to
JqJ.
The
sample volume
at
position
r with offset
11w(r)
contributes a
term
pq(
r, t
1
,
r) = exp (iql1w(r)t
1
)pq(
r + t
1
)
and
integration over
r leads
to
damping
of
the
coherence with a time inversely
proportional
to
JqJ.
HIGH
RESOLUTION
SPECTRA
BY
TIME
PROPORTIONAL
PHASE
INCREMENTS
(TPPI)
The
Hahn
spin echo technique
may
be used
to
remove
the
inhomogeneous
broad-
ening.
An
pulse
at
t
1
/2
changes q to
-q,
allowing a refocusing
of
the
coherence
at
t
1
for all orders. However, this n pulse also removes
the
separation
of
the orders
which was a result
of
the
frequency offset.
