Table Of ContentMulti-spin dynamics of the solid-state NMR Free Induction Decay
H. Cho,1 T. D. Ladd,2 J. Baugh,3 D. G. Cory,1 and C. Ramanathan∗1
1Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2Department of Applied Physics, Stanford University, Palo Alto, CA 94305, USA
5 3Institute for Quantum Computation, University of Waterloo, Waterloo, ON, Canada
0 (Dated: February 2, 2008)
0
2 Wepresent anew experimental investigation of theNMR free induction decay (FID)in alattice
of spin-1/2 nuclei in a strong Zeeman field. Following a π/2 pulse, evolution under the secular
n
dipolarHamiltonianpreservescoherencenumberintheZeemaneigenbasis,butchangesthenumber
a
of correlated spinsin thestate. The observed signal is seen todecay as single-spin, single-quantum
J
coherences evolve into multiple-spin coherences under the action of the dipolar Hamiltonian. In
4 order to probe the multiple-spin dynamics during the FID, we measured the growth of coherence
2
orders in a basis other than the usual Zeeman eigenbasis. This measurement provides the first
directexperimentalobservationofthegrowthofcoherentmultiple-spincorrelationsduringtheFID.
r] Experiments were performed with a cubic lattice of spins (19F in calcium fluoride) and a linear
e spin chain (19F in fluorapatite). It is seen that the geometrical arrangement of the spins plays a
h significantroleinthedevelopmentofhigherordercorrelations. Theresultsarediscussedinlight of
t
o existing theoretical models.
.
t
a PACSnumbers: 75.40.Gb,76.60.-k,82.56.-b
m
-
Solid-state NMR is an ideal test-bed for studying co- j and k is given by
d
n herent quantum dynamics in a large Hilbert space. In
o this work, we experimentally investigate the many-spin D = γ2~2 1−3cos2θ , (3)
c dynamics of the NMR free induction decay. The free in- jk r3 jk
[ jk (cid:0) (cid:1)
duction decay (FID) is the response of the spin system
1 followingaπ/2pulse. Inasolidlatticeofspin-1/2nuclei where γ is the gyromagnetic ratio, rjk is the distance
v inastrongmagneticfield,thisevolutionisdominatedby between spins j and k, and θjk is the angle between the
8 the secular dipolar Hamiltonian. This is a classic prob- externalmagneticfieldandinternuclearvector~rjk. Since
7 leminspindynamicsandhasbeenwell-studiedsincethe theHamiltonianistime-independent,theformalsolution
5 early days of NMR1–11. At the magnetic fields typically oftheLiouville-vonNeumannequationyieldsthedensity
1
used in NMR experiments (< 20 T), the spin system is matrixofthespinsystemattimetfollowingthepulseas
0
highly mixed at room temperature, and its equilibrium
5
0 state is represented by a thermal density matrix. If the ρˆ(t)=e−iHˆintt/~ρˆ(0)eiHˆintt/~ . (4)
/ external Zeeman field is much stronger than the inter-
t An exact solution to this many-body problem has not
a naldipolarfieldsofthesample,thenormalizeddeviation
m density matrix in thermal equilibrium can be approxi- beenfound,buttheequationcanbeexpandedinapower
series to examine the short time behavior of the system:
- mated as
d
n ρˆ(t) = ρˆ(0)+ it ρˆ(0),Hˆ −
o ρˆ(0)=− Iˆjz . (1) ~ h inti
:c Xj t2 ρˆ(0),Hˆ ,Hˆ +... (5)
v 2~2 int int
hh i i
i
X Followingaπ/2pulse, the spinsevolveunderthe secular In an inductively detected NMR experiment (in which a
r dipolar Hamiltonian coil is used to measure the average magnetization), the
a
observed signal is given by
1
Hˆint =Xj<kDjk(cid:26)IˆjzIˆkz − 4(cid:16)Iˆj+Iˆk−+Iˆj−Iˆk+(cid:17)(cid:27) . S(t)=ζhIˆ+i=ζTrnIˆ+ρˆ(t)o . (6)
(2) where Iˆ = (I + iI ) and ζ is a proportionality
The strength of the dipolar coupling D between spins + j jx jy
jk constant. ThPe only terms in ρˆ(t) that yield a non-zero
trace in the above equation and therefore contribute to
the observed signal S(t) are the single-spin angular mo-
∗Authortowhomcorrespondenceshouldbeaddressed. Electronic mentum operators such as Iˆj−, which are single-spin,
address: [email protected] single-quantum coherences. Single quantum coherences
2
are off-diagonal terms of the density matrix (in the Zee- The dipolar Hamiltonian in the x-basis is
man eigenbasis or the z-basis) connecting eigenstates
with ∆m = ±1 (corresponding to coherent superposi- Hˆx = −1 D Iˆ Iˆ − 1 Iˆx Iˆx +Iˆx Iˆx −
tions of these eigenstates). Evaluating the commutators int 2Xj<k jk(cid:26) jx kx 4(cid:16) j+ k− j− k+(cid:17)(cid:27)
in Eq. (5)
3
ρˆ(t) = −1 Iˆj++Iˆj− + 8Xj<kDjk(cid:16)Iˆjx+Iˆkx++Iˆjx−Iˆkx−(cid:17) (9)
2
Xj (cid:16) (cid:17) andnolongerconservescoherenceorderinthisbasis(we
3it Djk −IˆjzIˆk++IˆjzIˆk− − (7) usethesuperscriptxtodenotethattheraisingandlower-
2 Xjk (cid:16) (cid:17) ing operators are defined in the x-basis; these operators
are otherwise assumed to be expressed in the z-basis).
3
t2 DlkDjk IˆjzIˆlzIˆk++IˆjzIˆlzIˆk− +... Thecoherenceordersareencodedbyacollectiverotation
4
Xjkl (cid:16) (cid:17) about the x-axis (which is the effective quantizing axis
inthis basis). Transformingthe density matrix shownin
Substituting Eq. (7) into Eq. (6), it can be seen that Eq. (7) into the x-basis yields
the observable magnetization decays during the evolu-
tion under Hˆint because single-spin, single-quantum co- ρˆx(t) = − Iˆjx− 43it Dij Iˆjx+Iˆkx+−Iˆjx−Iˆkx− +
herenceterms aretransformedto unobservablemultiple- Xj Xjk (cid:16) (cid:17)
spin,single-quantumcoherencetermsbythehigher-order
3
nested commutators. The nth term in the expansion in 8t2 DlkDjk Iˆjx+Iˆlx+Iˆkx−Iˆjx+Iˆlx−Iˆkx−
Eq. (5) has n-spin correlations. Xjkl (cid:16)
shaTpheeroefthhaesFbIeDeninmcuaclhciuthmeoflrueotircidalee(ffCoarFt2t)o1–p8.reCdaicltcituhme Iˆjx−Iˆlx+Iˆkx+Iˆjx−Iˆlx−Iˆkx(cid:17)+... (10)
fluorideisastandardtestsystemforspindynamicsasthe
From Eq. (10), it can be seen that even order multi-
19F(spin-1/2)nucleiare100%abundantandformasim-
ple quantum coherences are created in the x-basis. It is
ple cubic lattice. The main goal has been to reproduce
possible to generate only odd order coherences using a
the decay and beat pattern of the observedtime domain
y-basis encoding for the same initial state.
NMR signal. For example, Engelsberg and Lowe10 mea-
It is useful to consider the dipolar evolution of this
sured up to 14 moments of the FID in CaF , and these
2 highly mixed state using the Liouville space formulation
were found to be in good agreement with theoretically
for multiple-quantum dynamics suggested previously17.
calculated values for the 2nd to 8th moments. The odd
The density operator in Liouville space can represented
moments of the FID are zero,and the evenmoments are
as
given by1
N K
(−1)n ρˆ(t)= g (t)Pˆ (11)
M = Tr{ Hˆ , Hˆ , ...,[Hˆ ,Iˆ ... Iˆ } . Knp Knp
2n Tr Iˆ2 h int h int h int xi ii x KX=0n=X−KXp
x
n o 2n times where Pˆ represents a basis operator which is a prod-
| {z } (8) Knp
uct of K single-spin angular momentum operators, n is
Evaluatingthenestedcommutatorsbecomesincreasingly
the coherence order of the operator and p is a label that
challenging and the higher-order moments are difficult
identifies a particular configuration of spins having the
to calculate. However, it is these higher moments that
sameK andn. TheselectionrulesforthedipolarHamil-
characterize the many spin correlations in the spin sys-
tem. It can be seen that the 2nth moment arises from tonian in the Zeeman basis are given by
the (2n+1)th term in the expansion in Eq. (5), which
∆K =±1, ∆n=0 . (12)
creates up to (2n+1) correlated spins. The main weak-
nessofthemomentmethodliesinthe factthatthe most
A projectionof Liouville spaceonto the two-dimensional
important contribution to the value of the higher mo-
plane spanned by K and n is shown in Figure 1(a). Fol-
mentscomesfromthetailsoftheFID,whichareacquired
lowing a π/2 pulse, the trajectory in the Zeeman basis
withthelowestsignal-to-noiseratio(SNR)intypicalFID
is indicated by the arrows (only positive coherences are
measurements3.
shownhere;the evolutionisperfectlysymmetricforneg-
In this paper, we use a novel multiple-quantum NMR ative n). Increasing numbers of spins are correlated fol-
technique14 to study multiple-spin dynamics during the lowing evolution under the dipolar Hamiltonian, but the
FID.Standardmultiple-quantumtechniques15–18 encode coherence number does not change. Figure 1(b) shows
coherence orders in the Zeeman eigenbasis (or z-basis), thesameevolutioninthex-basiswheretheselectionrules
but coherence numbers are conserved under the secular are
dipolar Hamiltonian in this basis19. In our experiment
we encode multiple-quantum coherences in the x-basis. ∆K =±1, ∆n=0,±2 . (13)
3
(a)z-basis (b)x-basis
7 7
6 6
5 5
4 n 4
3 3
2 2
1 1
0 0
1 2 3 4 5 6 7 1 2 3 4 5 6 7
K K
FIG. 1: Projection of Liouville space onto the two-dimensional plane spanned by K and n, showing the dynamics of the FID
in (a) the Zeeman eigenbasis, and (b) the x-basis. The arrows show the allowed pathsin each case.
π/2)y+φ π/2)y+φ π/2)y+π π/2)y π/2)y
(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) 1
(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) t (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) 4se8q puuelnscee (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) δ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) tlock )x tlock)x+π(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) sity1.15 renormalized intensity0 .015-20 Coherenc0e order (n) 20 00 ..1055-20 0 20
FIG. 2: Pulse sequence used in this experiment. t is the en
evolution time under the secular dipolar Hamiltonian. The nt 0-20 0 20
48-pulse sequence was used to suppress the evolution of the ed I0.5 1
internal Hamiltonian during the x-basis encoding step. A aliz 0.5
magic-echosequencewasusedtoreversethedipolarevolution. m
Adelay(δ=10µs)wasinsertedbeforethemagicechotopush or -20 0 20
N 0
the echo out, in order to minimize any pulse transients and
receiver dead time effects. The duration of the spin locking 1
time in the magic echo sequence is 2tlock where tlock = t+ 0.5
2δ+3µs. 0.5 0
-20 0 20
50 100 t (µs) 150 200 250
Starting from an initial I state (K = 1,n = 0), only
x
even order coherences are observed. In this paper we
FIG. 3: x-basis coherence order distribution at various time
characterize the growth of these coherences.
points under the evolution of the secular dipolar Hamilto-
The pulse sequence used in this experiment is shown nian in CaF . The multiple-quantum intensities have been
2
in Figure 2. After an initial π/2 pulse, multiple-spin, re-normalized to set the intensity of the zero-quantum term
single-quantum states in the Zeeman basis are created to onein each case.
during evolution under the secular dipolar Hamiltonian
as described in Eq. (7). A φIˆ rotation encodes coher-
x
enceordersinthexbasis,andamagic-echosequence12 is trometerandhome-builtprobe. Thesamplesusedwerea
used to refocus the multiple-spin terms back to observ- 1mm3 singlecrystalofCaF withT ∼7s,andacrystal
2 1
able single-spin, single-quantum coherence terms. The of fluorapatite (FAp) with T ∼ 200 ms. The FAp crys-
1
φIˆx rotation is obtained by applying two π/2 pulses, tal is a mineral crystal specimen from Durango, Mexico.
with phases y + φ and y¯, which results in the prop- Such natural crystals are expected to have a number of
agator exp(iφIˆz)exp(iφIˆx). The initial π/2 excitation interruptions in the spin chains as well as a variety of
pulse is also phase shifted by φ to cancel out the rota- other defects20 as indicated by the short T . All experi-
1
tion about Iˆz. Since it is difficult to apply back to back ments were conducted on resonance. High power 0.5 µs
π/2pulseswithoutadelaybetweenthem(withoutintro- π/2 pulses were used for the 48-pulse evolution suspen-
ducing phase transients or allowing some dipolar evolu- sion sequence, while lower power 1.5 µs π/2 were used
tionduringthepulses),anevolutionsuspensionsequence during the magic-echo sequence as this sequence is more
needs to be used in between the two π/2 pulses. In this susceptible to phase transient errors. The phase φ was
experiment we use a previously described 48-pulse evo- incremented from 0 to 4π with ∆φ=π/32 to encode up
lution suspension sequence13. to 32 quantum coherences for every experiment. A fixed
Theexperimentswereperformedatroomtemperature time point corresponding to the maximum intensity sig-
at 2.35 T (94.2MHz, 19F), using a Bruker Avance spec- nalwassampledforeachφvalueandFouriertransformed
4
with respectto φ to obtainthe coherenceorder distribu-
1
tion at each dipolar evolution time t.
0.2 0.8
Figure 3 shows the coherence order distribution ob-
served for CaF at various time points during the FID. Stot0.6
At short times2, the maximum coherence order (n ) S /n0.4
max
0.2
correspondsto the maximumnumber of correlatedspins 0.15
(Kmax, i.e gKnp = 0 for K > Kmax ). At longer times, ot 00 50 t (µ1s0)0 150
the maximum coherence order observed in the experi- St
ment sets the lower limit of the size of the spin correla- S /n0.1 02
tion, since the SNR of higher order coherences might be 4
6
too low to be observed. 8
10
Figure4showsthegrowthofthedifferentcoherenceor- 12
0.05 14
ders in CaF during the FID. The insetshows the initial
2
oscillation between the zero and double quantum coher-
encesatshorttimes(whichcorrespondstosingleandtwo
spin correlations respectively) due to the resolved near- 0
20 0 20 40 60 80 100 120 140 160
est neighbor coupling at the [100] and [110] directions. t (µs)
This oscillationmay be theoretically understood by con-
sideringthetimedevelopmentofanisolatedpairofspins
FIG. 4: The growth of multiple-spin correlations during the
under the secular dipolar Hamiltonian (in the x-basis):
FID, showing the sigmoidal fit to the initial growth data of
eachx-basiscoherenceorder(n≥4),along∼[110]directionin
1 3Dt
ρ(t) = cos (I1x+I2x)− CaF2. Theinset showsthedynamicsofn=0and2. Thein-
2 (cid:18) 2 (cid:19) tensityofthesignalforeachcoherenceorderS isnormalized
n
i 3Dt with respect to the total signal Stot, in order to compensate
4sin(cid:18) 2 (cid:19) I1x+I2x+−I1x−I2x− (14) for imperfect refocusing of thedipolar interaction.
(cid:0) (cid:1)
where D is the strength of the pairwise coupling. In an
extended spin system, this oscillation is rapidly damped [110] and [100] in good agreement with the ratios of α
n
by leakage from isolated pairs to higher order correla- showninthefigure. ThevaluesofC areseentodecrease
n
tions. The higher order coherences (n ≥4) are seen to as coherence number increases, independently of crystal
followasigmoidalgrowthcurve. Thehigherordercoher- orientation. This is expected as the total polarization
ences develop later in time, and this progressive growth is conserved, and the signal is progressively spread over
leads to a saturation of the intensities of the lower order increasingly larger regions of the system Hilbert space.
coherences, consistent with the model in Figure 1. As Figure 6 shows the onset time tonset of each of the x-
n
t increases, imperfect refocusing of the dipolar evolution basis coherences for different orientations of the crystal.
under the magic echo sequence results in a decay of the Physically, the onset time corresponds to the time re-
observedsignal. Inorderto removethis decaythe inten- quiredforaspecificcoherenceordertobecomeobservable
sity for each coherence order is normalized with respect in the experiment. While a first glance at Eq. (7) would
to the total signal measured at that evolution time. seemtosuggestthathigher-ordercorrelationsshouldde-
We have fit the initial growth of each coherence order velop as tn, it is the geometry of the spin system (the
to the following sigmoidal function values of D ) which ultimately determines the rate at
ij
which the spin correlations grow. The onset times de-
C
n
S (t)= . (15) pend on the rate at which the correlations are spreading
n 1+e−αn(t−tonnset) through the spin system, which in turn depends on the
In Figure 5 we plot the variation of α and C as a value of the dipolar couplings. Thus the rate is expected
n n
function of coherence order n. The parameter α rep- to be fastest (shorter onset time) with the crystal ori-
n
resents the underlying rate at which the different coher- ented along the [100] direction and slowest (longer onset
enceordersaretransformed,andshouldbedominatedby time) for the crystal oriented along the [111] direction,
the strength of the dipolar couplings involved. It is seen in agreement with the experimental data. The variation
that α does not vary with n, suggesting that the near- of onset times with coherence order is also observed to
n
neighbor interactions dominate the dynamics here. The depend on the dimensionality of the spin system. The
mean dipolar coupling strength depends on the crystal variationis sub-linear in the cubic CaF2 system and dis-
orientation,so we measuredα withthe crystaloriented playsanapproximaten2/3 dependence. TheinsetinFig-
n
along the [111], [110] and [100] directions with respect ure 6 shows that the onset times obtained for FAp, the
to the external field. The mean dipolar coupling can be quasi-one-dimensionalspinsystem,varylinearlywithco-
estimated by summing |D | over the 26 nearest neigh- herence number, in marked contrast to the results from
1j
bor spins foreachcrystalorientation. Fora simple cubic CaF2.
lattice, the ratio of these means is 1:1.45:1.87 for [111], A variety of models have been proposed to describe
5
0.2 180 160
2.5 140 FAp
0.18 αn2 0.2 160 (µs)120
0.16 Ratios of 1.15 Cn00..001.551 140 onset tn1068000
0.14 0.5 5 10 15 0 5 10 15 µs)120 C3ohere4nce o5rder (n6)
αn0.12 Coherence order (n) Coherence order (n) set( 100
n
on
t
0.1 80
CaF
CaF 2
2
0.08 [100] 60 ~[1[10100]]
~[110] [111]
[111]
0.06 40
20
0.04
4 6 8 10 12 14 2 4 6 8 10 12 14 16
Coherence order (n) Coherence order (n)
FIG. 5: The values of α for different orientations in FIG. 6: Onset time of multiple-spin correlations for different
n
CaF2. The left inset shows the ratios α[100]/α[111] (*) and crystalorientationsinCaF2. Theinsetshowstheonsettimes
α∼[110]/α[111](+) . The calculated ratios of the mean dipo- fortheFApsample. OddcoherenceordersfortheFApsample
larcoupling,obtainedbyaveragingover26nearestneighbors, were obtained using a y-basis encoding on the same initial
[100]/[111](solidline)and∼[110]/[111](dottedline)arealso state. ThecontinuouslinesrepresentthebestfitsofEq.(20)
shown. The right inset show the values of C ’s for different to the data, assuming that the same equation is valid for
n
orientation in CaF coherence numberas well.
2
the dynamics under a multiple-quantum Hamiltonian.
The mostcommonly usedmodel involvesa randomwalk Munowitz and Mehring11 used this model to numeri-
among the components of the Liouville space basis set cally simulate the growth of multi-spin dynamics of the
Pˆ ,subjecttotheselectionrulesofthemultiplequan- FID in a 21-spin system. In order to track the devel-
Knp
tum Hamiltonian17. The model replaces the Liouville- opment of correlations among the spins, they defined an
vonNeumannequationbyasetofcoupledrateequations induction time tK over which a particular K-spin coher-
with exponential solutions ence reaches half its maximum value. This parameter is
very similar to the experimentally measured onset times
d ofthedifferentcoherenceordersdescribedhere. Figure7
g =R·g , (16)
dt ofRef. 11showsthe variationofinductiontime with the
number of correlated spins. The numerical results show
where the vector g containes the coefficents g . All
Knp that the variation of the induction time is sub-linear for
possibleconfigurationsthatcontributetoaparticularco-
small numbers of correlated spins (< 10), in agreement
herenceareassumedtobe presentinequalmeasure,and
with the experimental data. For larger numbers of cor-
the resulting growth of the spin system is described by
related spins, however,there is a marked deviation from
a hopping procedure between the allowed points on the
thesub-linearbehavior,astheeffectsofthefinitesystem
lattice (shown in Figure 1). Under this assumption, the
size(21spins)begintomanifestthemselvesinthesimula-
hopping rates are solely determined by the degeneracies
tions. The number of correlatedspins would have to ap-
ofthe coupledstates. The modelthus eliminates anyos- proachthenumberofspinsinthesample(≈1021)before
cillatory solutions and precludes the possibility of quan-
such effects would be observed experimentally. While
tum interference effects playing a role in the evolution.
providing some insight into the growth of the dynamics
The model also ignores the geometrical arrangement of
forthe cubic spinsystems,it is seenthatthe model does
the spin system and the specific distribution of dipolar
notdoaverygoodjobatdescribingthe one-dimensional
couplingcoefficientsresponsiblefordrivingtheevolution.
spinsystem. Itisinthissituationthatthespingeometry
All spin systems display a universal growth kinetics, as
plays a dominant role.
long as the dynamics are scaled by a lattice parameter
thatrepresentsthemeandipolarcouplingstrengthofthe Gleason and co-workers have proposed an alternate
system, similar to the parameter α obtained above. A model to describe the growth of spin correlations that
n
numerical solution of the coupled equations for the mul- emphasizes the geometrical ordering of spins22. By ag-
tiple quantum evolution was observedto yield sigmoidal gregatingoverthe differentconfigurationsandcoherence
growth curves for the various coherence orders17. Limi- orders, i.e. summing over n and p in Eq. (11) the den-
tations to this model have been discussed elsewhere21. sity operator ρˆis expressed as a sum of terms with spin
6
number K and coefficients g , tude of g (t) as given by Eq. (19) strongly resem-
K K
bles the sigmoidal growth characteristics of the multiple
N quantum coherence intensities shown in Figure 4. An
ρˆ(t)= g (t)Pˆ . (17)
K K onset time can be obtained from Eq. (19) by setting
KX=0 [tanh(βtK−1/d)]K =1/2, yielding
where g Pˆ = g Pˆ . Essentially this model η
K K np Knp Knp t = K1/darctanh 2−1/K , (20)
assumes a singlePeffective K-spin operator that incorpo- 1/2 β
(cid:16) (cid:17)
rates all the possible spin and spatial configurations of
the K spins. The resulting model for spin propagation where η is a constant scaling factor. Figure 6 shows the
throughalattice yieldsadifferentialequationforthe co- best fit of Eq. (20) to the experimental data, assuming
efficients g (t) of the form that the model holds true for coherence number as well.
K
It is seen that there is excellent agreement at larger val-
ddtgK =−i WKf −1gK−1+WKr+1gK+1 (18) uobestaoinfendfforromthethceubfiict CaraeF237s.y4s7teimn.thTehe[1v1a1l]udesireocftηio/nβ,
(cid:0) (cid:1)
31.13 in the [110] direction and 21.22 in the [001] di-
where the rate constants Wf and Wr correspond to the
rection. Their inverses are in the ratio 1:1.46:1.76 for
forward and reverse rates respectively. Under the as-
[111]:[110]:[100]which is in excellent agreement with the
sumptionthatthespatialgroupingoftheK spinsiscon-
theoreticallycalculatedvaluesshownearlier. Forthelin-
tinuous,andthatonlythenearestneighborcouplingsare
ear FAp system, Eq. (20), which is linear for large K, is
important,theforwardrate(andequivalentlythereverse
observed to be weakly superlinear at these small values
rate) can be expressed as WKf ∝ Dnnns, where D is the of K23.
strengthofthe nearestneighborcoupling,n is the num-
s Theconstancyofα inthe sigmoidalplotsinFigure4
berofspinsonthesurfaceofthespatialgrouping,andn n
n and the good agreement observed between the observed
is the number of neighboring spins coupled to each spin.
onset times and Eq. (20) indicate that the spin dynam-
Newspinsareaddedonthesurfaceofthecorrelatedspin
icsaredominatedbythenearest-neighborinteractionsin
cluster. While n is a constant, the term n would differ
n s thisregime. Thisisnotsurprising,aswearestilloperat-
significantly for spin systems of different dimensionali-
ingintheshorttimeregime. Higher-orderspinprocesses,
ties,andcanbe expressedasn ∝K1−1/d wheredisthe
s if significant, would be expected to manifest themselves
dimensionalityofthespinsystem. Foralinearspinchain
at later times, leading to a deviation from the simple
d=1 and n is independent of K, while for a cubic spin
s model behavior described above.
system, d = 3 and n ∝ K2/3. While this model does
s In conclusion, we have presented a new experimental
notdiscusscoherenceorder,thedimensionaldependence
methodtocharacterizemulti-spindynamicsinsolid-state
doesagreewiththeexperimentalresults,iftheonsettime
NMR free induction decay. The initial creationofcoher-
characterizes the effective rate constant. In the limit of
ences were observed to follow a sigmoidal growth curve,
large K, the rate constants Wf ≈Wr , and the co-
K−1 K+1 with the onset times characterizing the dynamics of the
efficients g (t) are approximately given by
K spin system. These dynamics in turn were critically de-
pendent on the geometrical arrangement of the spins as
K
gK(t)∝iK−1 tanh βtK−1/d , (19) expected.
h (cid:16) (cid:17)i The authors thank D. Greenbaum, P. Capperallo and
where β is proportional to the mean dipolar coupling. T.S.Maheshfor helpful discussions,andthe NSF,ARO
With appropriate choice of normalization, the magni- and DARPA DSO for financial support.
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Zeeman field.Thenon-secularcomponentscan changeco- time t = ηH1−1/d, where η is a constant and Hr is a
K K−1 n
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