Journal of Magnetic Resonance 238 (2014) 8–15
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Heteronuclear dipolar coupling in spin-1 NQR pulsed spin locking M.W. Malone, K.L. Sauer ⇑ George Mason University, Fairfax, VA 22030, United States
a r t i c l e
i n f o
Article history: Received 9 August 2013 Revised 15 October 2013 Available online 31 October 2013 Keywords: Nuclear quadrupole resonance Spin-locking Dipolar coupling Broadening
a b s t r a c t We investigate theoretically and experimentally the role of broadening due to heteronuclear dipolar coupling in spin-1 nuclear quadrupole resonance pulsed spin locking. We find the experimental conditions where heteronuclear dipolar coupling is refocused by a standard multipulse sequence. This experimental condition allows us to extend our previously reported ability to measure the homonuclear dipolar coupling of powder samples to include substances that have heteronuclear coupling. These results are useful for designing substance detection algorithms, and for performing sample characterization. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Nuclear quadrupole resonance (NQR) is a spectroscopic technique that exploits the orientation of the nuclear quadrupole moment in the local electric field gradient (EFG). In NQR and the closely related nuclear magnetic resonance (NMR) spectroscopy, the application of a resonant rf pulse causes the nuclear magnetic moment with spin I of a target nucleus to oscillate at a frequency characteristic to the substance of interest. This makes the technique useful for substance detection, especially in the field, since no static field, as in NMR, is required. However, NQR measurements are possible only for crystalline materials with nuclei that have a quadrupole moment, i.e. nuclei with spin > 1/2, such as the spin-1 14N nuclei that are the target of this research. While there are exceptions, NQR signals are typically weak, due to the commonly low resonance frequencies. Additionally, the signal from a single pulse frequently decays on the order of milliseconds. This is due to line broadening mechanisms which are a function of the static crystalline environment investigated here, and T 1 processes that we do not investigate. However, just like the multipulse sequences of NMR[1,2], such as the spin-lock spin-echo (SLSE) [3], a series of refocusing pulses after the initial excitation pulse can selectively refocus the broadening mechanisms to extend the NQR signal in time [4,5]. The class of multipulse sequences we investigate begin with an excitation pulse h0 to create the initial signal, followed by a series of refocusing pulses h1 , with duration t p , shifted in phase by 90° from the excitation pulse. These SLSE sequences are expressed, for N repeat units, as h0y ðs h1x sÞN , where 2s is the time between the refocusing ⇑ Corresponding author. E-mail address: [email protected] (K.L. Sauer). 1090-7807/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2013.10.014
pulses, s0 s þ t p =2 is useful later. The signal is acquired stroboscopically at time s after each refocusing pulse, and the strength of the refocusing pulse varies depending on the desired signal response. It should be noted that unlike in NMR the effective pulse angles h will not be consistent over a powder sample, under consideration here, but will vary from one crystallite to another [6]. Therefore the Hamiltonian is not cyclical and the average Hamiltonian theory [1,2,7,8] cannot be used to analyze NQR response to resonant or near-resonant pulses [9]. Much of the theoretical work in multipluse NQR has focused on the long term behavior of the signal [10–15]. In contrast, our work focuses on the initial signal behavior by focusing on three broadening mechanisms to explain the linewidth of the NQR signal. The first, EFG inhomogeneity, is an inhomogeneous broadening mechanism caused by variations in the local electric field of the nuclei in the sample. Its size is a function of the sample’s history, from its manufacturing process to its handling [16]. The second, homonuclear dipolar coupling, is a homogenous broadening mechanism that should be constant across samples of the same substance. Our previous work [17,18], rooted in the theoretical work of Cantor and Waugh [9], revealed the experimental conditions under which a SLSE triggered a strong response due to homonuclear coupling. This response manifested itself as a rapid Gaussian decay in the initial echo train, and allowed us to measure the homonuclear dipolar coupling strength. While this result has also been obtained in NMR [19,20], the typical NQR substance is a powder, and the corresponding random orientation of the crystallites with the applied fields means the observed uniform response is not obvious. This report addresses the role of a third broadening mechanism to a SLSE sequence: heteronuclear dipolar coupling, another homogenous broadening mechanism. While the other two broadening mechanisms are first order effects, the heteronuclear
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M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
interaction makes only a second order-contribution [21], at least in the case of non-axial symmetry where the NQR energy levels are non-degenerate, an assumption we make in this paper. Nevertheless theoretical calculations [22] indicate that in some samples the heteronuclear interaction may be the dominant broadening mechanism. To our knowledge this is the first treatment of the effects of this broadening mechanism in NQR multi-pulse sequences. Here we focus on detection of spin-1 nuclei and their interaction with spin-1/2 nuclei; one should be able to make similar arguments for other spin nuclei besides 1/2, as long as their transition frequencies are not close to the NQR frequencies. By finding the conditions where heteronuclear broadening is refocused, we show that the strong homonuclear response can still be observed for substances with heteronuclear broadening. While our previous work used sodium nitrite NaNO2, this project used p-chloroaniline ClC6H4NH2 because its nitrogen to hydrogen dipolar coupling was expected to contribute more to the linewidth than the nitrogen to sodium dipolar coupling of sodium nitrite. This is because the second order broadening hDx2 iNK of heteronuclear coupling is proportional to c4K , where cK is the gyromagnetic ratio of the other spin species, and falls off faster than the distance separating the nitrogen and the other nuclei raised to the sixth power [22]. Since hydrogen has a gyromagnetic ratio approximately four times the value of sodium [23], and the nitrogen-sodium bond length in sodium nitrite[24] is approximately 2.5 times the nitrogen–hydrogen bond lengths in the NH2 group in pchloroaniline [25,26], p-chloroaniline should have much stronger heteronuclear coupling. Furthermore, theoretical calculations have already been made for the second moments of both the homonuclear[18,17] and heteronuclear broadening [22] in p-chloroaniline. There are, however, two sites for the nitrogen in the unit cells, such that while the NQR frequencies are identical for both sites, the EFG principal axes frames (PAFs) are rotated by 90 degrees from each other [22]. The x-axes remain aligned, and it is for this reason that we focus on the xx transition, the largest NQR frequency of a spin-1 system. 2. Theory To understand heteronuclear coupling’s response to a multipulse sequence, the progression of a system modeled as a single spin-I (spin-1) nucleus, with angular momentum I, coupled to a set of N K spin-K (spin-1/2), each with angular momentum KðkÞ is examined through the evolution of the corresponding density matrix qðtÞ. The quadrupole Hamiltonian hHQ is
hHQ ¼
eQ 6Ið2I 1Þ
h
i V xx 3I2x I2 þ V yy 3I2y I2 þ V zz 3I2z I2 ; ð1Þ
where e is the charge of the proton, and Q is the nuclei’s quadrupole moment. The second derivatives of the electric potential V are found ^; ^z axes of the PAF of the EFG tensor at the nucleus, using the ^x; y with jV zz j > jV yy j > jV xx j [27,28]. These axes also define the angular momentum operators Ii ¼ I ^i. For the sample of interest, p-chloroaniline, there are two different PAFs within each unit cell, however the ^x axes are parallel [25]. For spin-1, the three eigenstates of HQ can be expressed in terms of the eigenstates of Iz jmi ¼ mjmi [27]
jm ¼ 1i þ jm ¼ 1i pffiffiffi ; 2 jm ¼ 1i jm ¼ 1i pffiffiffi ; and jxi ¼ 2 jzi ¼ jm ¼ 0i:
jyi ¼
ð2Þ
three sets of fictitious spin-1/2 vector operators [29]. For the hxx ¼ Ey Ez transition, HQ can be defined by these operators in three equivalent ways [30,9]:
HQ ¼ xx Ix3 þ
1 xy xz Iy3 Iz3 ; 3
ð5Þ
with cyclical permutations of x; y; z providing the other ways. The operators
1 1 Ix ¼ ðjyihzj þ jzihyjÞ 2 2 1 1 ðIy Iz þ Iz Iy Þ ¼ ðijyihzj ijzihyjÞ 2 2 1 1 2 2 I Iy ¼ ðjyihyj jzihzjÞ; 2 z 2
Ix1
ð6Þ
Ix2
ð7Þ
Ix3
ð8Þ
with cyclical permutations of x; y; z providing the remaining definitions for Iyi and Izi , are the fictitious spin-1/2 operators of an angular momentum Ix in a space ^x1 ; ^x2 ; ^ x3 with Ix ^xi ¼ Ixi . In addition, ðIy3 Iz3 Þ in Eq. (5) is conventionally relabeled as[9]
1 Ix4 Iy3 Iz3 ¼ ð1 3jxihxjÞ; 2
ð9Þ
where 1 is the identity operator. Clearly, Ix4 commutes with Ix1 ; Ix2 , and Ix3 , which themselves commute like angular momentum operators. During the refocusing pulses, the radio-frequency Hamiltonian b and frequency xrf is for a pulse of strength B1 direction B hHrf , where
b cos xrf t c I þ c Hrf ðtÞ ¼ B1 B I K
! NK X ðkÞ K :
ð10Þ
k¼1
For our sample, K corresponds to hydrogen with gyromagnetic ratio cK , and I to nitrogen with gyromagnetic ratio cI . In these experiments the applied field has a frequency xrf close to the NQR transition xNQR , so only the interaction with I is important; as our example, we take xNQR ¼ xx . The heteronuclear dipolar Hamiltonian between the spin-1 nu HIK , where cleus and the N k spin-1/2 nuclei is h
HIK ¼
NK X KðkÞ Dk I;
ð11Þ
k
and
Dk
l0 4p
cI cK h
1 ð1 3^rk ^rk Þ; r 3k
ð12Þ th
and rk ¼ rk ^rk is the displacement vector between the spin-1 and k spin-1/2 nuclei. For convenience, the operator I is expressed as
^ þ ðjyihxj þ jxihyjÞ^z; I ¼ ðjyihzj þ jzihyjÞ^x þ ðijxihzj þ ijzihxjÞy
ð13Þ
so that it is clear that HIK only connects states of different quadrupole eigenenergies. The homonuclear dipolar Hamiltonian between HDK is similarly defined, as is homonuclear the spin-1/2 nuclei h dipolar Hamiltonian between the spin-1 nuclei hHDI . In order to fo HDI is dropped from cus on the effects of heteronuclear coupling, h this model, having already been treated in detail in our previous work [17]. However, we discuss its contribution to the signal in Section 4. 2.1. Evolution
ð3Þ ð4Þ
Excluding the degenerate case, the three corresponding eigenenergies Ey ; Ex ; Ez lead to three possible transition frequencies xNQR and
To find the evolution of qðtÞ due a series of repeat units ðs h1x sÞN , we begin with the Liouville equation
._ ¼ i½.; H;
ð14Þ
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M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
H is given by where the total Hamiltonian h
H ¼ HQ þ HDK þ HIK þ Hrf ðtÞ:
ð15Þ
When the rf is off, such as between pulses, the Hamiltonian is static and the evolution of the density matrix is
qðtÞ ¼ eiHðtt0 Þ qðt0 ÞeiHðtt0 Þ
D¼
exp isjxbihxbjDxb
ð29aÞ
n s o exp i ð1 þ Ix4 ÞðDyb þ Dzb Þjbihbj 3 exp isIx3 ðDyb Dzb Þjbihbj ;
ð29bÞ ð29cÞ
ð17Þ
where jbihbj shows that the state of the K spins is preserved. The terms in Eqs. (29a), (29b) can be dropped because they will not contribute to the evolution of the signal, reducing D to
ð18Þ
D¼
X
n jwn ihwn jðtt0 Þ
eiHðtt0 Þ ¼ e n X jwn ihwn jein ðtt0 Þ : ¼
X b
ð16Þ
where H ¼ HQ þ HDK þ HIK and qðt0 Þ is the density matrix before the free evolution. In principle H can be diagonalized to a basis jwn i with eigenvalues n , allowing the free evolution operator to be written as i
where the last expression uses Eq. (22), H0 ¼ H0 DHQ , and h:c. is the Hermitian conjugate of the evolution operators to the left of P q~ ðt0 Þ. The dipole operator D a;b jabihabjeiDab s simplifies:
n
X eisIx3 ðDyb Dzb Þjbihbj :
ð30Þ
b
It is not computationally viable, however, to diagonalize H because of the large number of particles involved. Using perturbation theory, approximate expressions for jwn i and n are found by taking hH0 ¼ hðHQ þ HDK Þ as the unperturbed and hHIK as the perturbation. This a reasonable assumption when cK cI , as in our sample where the ratio of cK =cI 14. Because HIK only joins states with different quadrupole energies, non-degenerate perturbation theory can be used. Given the eigenstates of H0 ; H0 jabi ¼ ab jabi, where a corresponds to an eigenstate of HQ and b to an eigenstate of HDK , the surviving lowest order correction to the energy will come from
H1 ¼ DHQ þ Hrf ðtÞ;
n ab þ Dab ; where
ð19Þ
Hrf ¼ x1 Ix cosðxrf tÞ
ð20Þ
b In the interaction representation of h H0 , the with x1 ¼ cI B1 ^ x B. perturbation is given by
XX jhjljHIK jabij
2
Dab ¼
j–a
ab jl
l
;
XX hjljHIK jabi jjli : ab jl j–a l
2 IK jabi 1 MHz. Therefore, the ratio hjljH ab jl is on the order 10 , making jwn i jabi. This approximation permits the free evolution operator eiHs to be approximated in terms of jabi:
X jabihabjeiðab þDab Þs :
ð22Þ
m
To solve for the evolution due to the rest of the repeat unit it will be beneficial to use the interaction representation of hH0 , where
H0 HQ þ HDK ;
ð23Þ
HQ is the reduced quadrupole Hamiltonian and h
H Q ¼ H Q þ DH Q
xrf H xNQR Q xNQR xrf DH Q ¼ HQ : xNQR
DHQ ¼ DxIx3 þ Dx0 Ix4 ;
ð32Þ
with Dx xx xrf ; Dx ¼ 0
1 ðxy xz Þðxx xrf Þ , xx 3
and
ð33Þ
¼ DHQ þ x1 Ix1 ;
ð34Þ
where in the last line we have used the secular approximation. The evolution of the density matrix during the pulse is
e
e
q~ ðt þ tp Þ ¼ ei H 1 tp q~ ðtÞei H 1 tp ;
ð25Þ
0 e ei H 1 tp ¼ eiðDxIx3 þx1 Ix1 Þtp eiDx tp Ix4 :
ð36Þ
Using Eqs. (28) and (30) the complete evolution due to a repeat unit can now be written as
q~ ðt þ 2s0 Þ ¼
X eiðDyb Dzb ÞsIx3 jbihbj
ð37aÞ
b
eiDxsIx3 eiðDxIx3 þx1 Ix1 Þtp eiDxIx3 s e X 0 0 eiðDyb Dzb ÞsIx3 jb ihb j
ð37bÞ ð37cÞ ð37dÞ
b0
q~ ðtÞh:c: ð26Þ
ð35Þ
where
iDx0 ðt p þ2sÞIx4
ð24Þ
HQ ¼
where
ð21Þ
As discussed below, the experimental NQR linewidth is on the order of 100 Hz, while the NQR frequency is on the order of
eiHs
ð31Þ
e 1 ¼ DHQ þ eiH0 t Hrf ðtÞeiH0 t H
while for the wavefunction it is
jwn i jabi þ
In contrast, during the pulse, the contribution of HIK is ignored since, in practice, Hrf is much stronger. This assumption was found to be valid for HDI in our previous work [17]. The perturbing Hamiltonian hH1 is then given by
ð37eÞ
a period s of free evolution, the density matrix in the interaction representation becomes
The exponential operator containing Ix4 will not contribute to the evolution of the signal, and is dropped: it only includes the identity operator and the states not addressed by the rf pulse. As a result the only remaining operators are the fictitious angular momentum operators, the product of which can be represented ^ b found as a single rotation operator of angle hb and direction n through the following relationships[31]
q~ ðtÞ ¼ eiH0 t eiHs eiH0 t0 q~ ðt0 Þh:c:
cos
While H0 and H0 have the same eigenstates, jabi, they will have different eigenvalues: ab for H0 and ab for H0 . In the interaction rep~ a eiH0 t Ha eiH0 t . After ~ ðtÞ eiH0 t .ðtÞeiH0 t and H resentation of hH0 ; .
!
¼
X ~ ðt0 Þh:c:; jabihabjeiDab s eiDHQ s q a;b
ð27Þ ð28Þ
hb h2 h2 ^1 n ^2 ¼ cos h1 cos sin h1 sin n 2 2 2
hb h h h h h h ^ b ¼ sin 2 n ^ 2 þ 2 sin 1 n ^ 1 cos 1 cos 2 sin 1 sin 2 n ^1 n ^2 sin n 2 2 2 2 2 2 2
ð38Þ ð39Þ
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M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
^1 ¼ ^ where h1 ¼ Dx þ ðDyb Dzb Þ s, n x3 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ x1 þDx^ x3 x 2 1 2 ^2 ¼ tp ^ b lies h2 t p x1 þ Dx , and n . This means that n h2 ^ ^ in the x1 x3 plane. The final expression is
X q~ ðt þ 2s0 Þ ¼ eihb Ix n^b jbihbj q~ ðtÞh:c:;
ð40Þ
and the average xx signal from the spin-1 nuclei with the same ^x of their PAFs is NI n o bX b ¼ ^x B hI Bi Tr IxðiÞ q NI i¼1
¼
NI n o bX ^x B ~ Tr eI ðiÞ x q NI i¼1
ð53Þ
¼
NI h n o n o i bX ^x B ðiÞ ðiÞ ~ cos xrf t Tr Ix2 2 Tr Ix1 q q~ sin xrf t ; NI i¼1
ð54Þ
b
with the evolution over N pulses being
X q~ ðt0 þ 2Ns0 Þ ¼ eiNhb Ix n^b jbihbj q~ ðt0 Þh:c:
ð41Þ
b
As will be discussed in Section 2.3, before the application of a ~ ðt0 Þ that produce signal can be exrepeat unit, the components of q pressed as
q~ ðt0 Þ ¼
3 XX cj Ix ^xj jbihbj
¼
ð42Þ
j¼1
b
X b
^b ½ðc1 ^x1 þ c3 ^x3 Þ n ^b þ n ^ b? ½ðc1 ^x1 þ c3 ^x3 Þ n ^ b? I x ½n
þ c2 ^x2 jbihbj;
ð43Þ
^ b? n ^b ^ where n x2 , and the coefficients cj for a SLSE are given in Section 2.3. Recognizing that [18] ^ 0 h:c: ¼ Ix ½cos hn ^ 0 þ ð1 cos hÞn ^ ðn ^n ^ 0 Þ þ sin hðn ^0 n ^ Þ eihIx n^ Ix n
ð44Þ
it is possible to see that after N repeat units with finite h, as in Eq. (41), that the only terms of the density matrix that contribute to the signal are
q~ ðtÞ ¼
X ^ b Þðc1 ^x1 þ c3 ^x3 Þ n ^ b jbihbj: ðIx n
ð45Þ
b
The Ix3 term makes no contribution to the observed signal, and as explained later phase cycling results in c3 ¼ 0 [18]. Therefore, the effective density matrix reduces to
q~ ðtÞ ¼
X ^ b Þ2 ; c1 Ix1 jbihbjð^x1 n
ð46Þ
b
which we call the projection model due to its simple dependence on ^x1 n ^b . The above is for a single spin-1 nucleus. Expanding the density matrix to include all N I nuclei with similar PAFs
q~ ðtÞ ¼
NI X X i¼1
ð52Þ
when solved in the interaction representation of hH0 . From the proðiÞ jection model, the locked signal only contains the Ix1 term, allowing ðiÞ the Ix2 term to be dropped. Further, the double summation in Eq. (47) can be replaced by a double integral over two normalized distribution functions, f ðdxÞ for the electric field gradient distribution and gðDy Dz Þ for the distribution in heteronuclear dipolar coupling: b ¼ c1 cos xrft ^x B b hI Bi
Z
dðdxÞf ðdxÞ
2
^b Þ : dðDy Dz ÞgðDy Dz Þð^x1 n
ð55Þ This double integral is equivalent to
b ¼ c1 cos xrft ^x B b hI Bi
Z
dðd-Þhð-Þ
ð56Þ
; d-Þ; h2 ðDx ; x1 ÞÞ2 ; ^ b ½h1 ðDx ð^x1 n where d- ¼ dx þ Dy Dz , and the linewidth function h ¼ f g, is the convolution of f with g. In the above, we emphasize the ^ b ; h1 , and h2 on the broadening d- and functional dependency of n and x1 . The broadening from the two the pulse parameters Dx different mechanisms contribute in the same way to the final signal. In our previous work we showed how the 90–180 SLSE should refocus broadening due EFG inhomogeneity [17]. That EFG and heteronuclear dipolar coupling contribute in the same way to the signal allows us to extend that conclusion to broadening due to heteronuclear coupling. The above solution, Eq. (56), is for a single crystal. To obtain the powder averaged signal the integrab would be made, with again tion on the direction cosine ^x B b x1 ¼ cB^x B. 2.3. Initial density matrix
2 ðiÞ ^ bðiÞ : c1 Ix1 jbihbj ^x1 n
ð47Þ
b
þ dxðiÞ where In Eqs. (38) and (39), we can replace Dx with Dx
¼x x xrf ; Dx
ð48Þ
x is the average NQR frequency, and dxðiÞ is the deviation from this x average by a particular nucleus i. For hard pulses, i.e. x1 dxðiÞ , the ^ bðiÞ of Eqs. (38) and (39) is defined by rotation axis n
As shown in Fig. 1, subtle differences in the timing of the sequence before the application of the repeat unit can have big impacts on the SLSE signal behavior. To derive an expression for the density matrix before the repeat units as they occur in our experiments, it is assumed that the sample is in thermal equilibrium and at the high-temperature limit. In this case the initial density matrix qð0Þ is taken as the reduced density matrix [9]
qðt < 0Þ ¼ xx Ix ^x3 : ^x3 Þ ^ 2 ¼ tp ðx1 ^x1 þ Dx h2 n ðiÞ ðiÞ ^ 1 ¼ s Dx þ dxðiÞ þ ðDyb Dzb Þ ^x3 : h1 n
ð49Þ ð50Þ
Significantly, the broadening mechanisms dx and ðDyb Dzb Þ do not appear separately in the above rotation, but are summed together. ðiÞ
The sample is excited by the same coil used to detect the signal. In this case the signal is proportional to
ð51Þ
ð57Þ
b cosðxrf t þ p=2Þ, which The rf pulse during excitation is B ¼ Be B differs in phase by p=2 compared to the refocusing pulse in Eq. (10). Similar to the derivation in Eq. (34), in the interaction representation of hH0 , the perturbing Hamiltonian of the excitation pulse is given by
e 1 ¼ DHQ þ eiH0 t Hrf ðtÞeiH0 t H ¼ DHQ þ x1e Ix2 ;
2.2. The observable
b ¼ TrfI B b qg; hI Bi
Z
ð58Þ ð59Þ
b and the secular approximation is used in where x1e ¼ c1 Be ^x B, the last line. The evolution of the density matrix from the excitation ^ 0 ¼ ^x3 , and pulse with length t e is given by Eq. (44), with n ^ ¼ ðx1e ^x2 þ Dx^x3 Þt e . The product x1 te is chosen in order to be hn
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M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
The coefficient in front of Ix1 is the c1 of Eq. (56). The excitation pulse is such that for Df ¼ 0; h p=2, while for increasing values of jDf j; h increases. For large Df , the second term in the square brackets compensates for what would be a loss of signal. This explains why the adjusted sequence produces the stronger signal in those cases since it creates a larger initial signal that can be spinlocked. 3. Experimental procedure
Fig. 1. The 90–90 SLSE signal on the left (red dots) due to a standard sequence has a stronger response as a function of off-resonance condition Df ¼ Dx=ð2pÞ than a nearly identical sequence on the right (black dots). The only difference is that the first free evolution of the left sequence has been adjusted to sR ¼ s te =2. As discussed below, this reduced free evolution ensures that the density matrix is suitable for pulse spin locking at large off-resonance conditions. The thick lines are the predicted signal strengths of the projection model Eq. (46), and agree quite well with the data. The thicker shaded lines correspond to the density matrix before application of the repeat unit, given in Eq. (62). Data is from the S1 sample of pchloroaniline, taken at xx with s0 ¼ 390 ls, tp ¼ 110 ls, and averaged over 63 ms. Both signals are normalized to the same value, and both sequences are symmetrical about Df ¼ 0. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
close to 90° for the majority of the crystallites contributing to the signal [6]. In order to isolate the echo train and reduce transient pulse effects, successive experiments in which Be is inverted are ~ ðt e Þ is effectively subtracted [18]. Therefore q
q~ ðte Þ ¼ xx Ix
x1e te h
sin h^x1 þ
Dx t e ð1 cos hÞ^x2 ; h
ð60Þ
which means that for certain values of h which occur at large offresonance conditions the Ix ^x1 component of the effective density matrix that produces the locked signal is small, while the Ix ^x2 component that does not produce signal is large. The time between pulses in the echo train is 2s, or if measured from the center of the pulses 2s0 . However, there is some ambiguity in how far the excitation pulse should be separated from the first refocusing pulse. In practice, the midpoint of the excitation pulse to the midpoint of the first refocusing pulse is also set to s0 . This means that the time between the excitation pulse and the refocusing pulse must be reduced to sR ¼ s te =2. For delta function pulses, or for long values of s, the difference between s and sR is inconsequential. For short enough values of s, however, this adjustment provides advantages for large Df , as shown in Fig. 1. To incorporate this sR into the theory above only requires a small adjustment to the initial density matrix. This is done by performing a negative rotation after the excitation pulse using the operator
q~ ðt0 Þ eiDxte Ix3 =2 q~ ðte ÞeiDxte Ix3 =2
ð61Þ
0
^ ¼ Dxt e ^x3 =2, and recognizing that only Using again Eq. (44), with h n the Ix1 component gets locked by series of repeat units, the relevant expression for the density matrix before the first repeat unit is
q~ ðt0 Þ ¼ xx
x1e te h
sin h cos h0 þ
Experiments were performed on three 100 g powder samples of 98% purity p-chloroaniline from Sigma–Aldrich. These samples were never removed from their glass jars. All experiments were performed by placing the jars into a home built coil 8 cm in diameter and 20 cm long, located inside a shielded box. The coil diameter was just large enough to hold the jars, and the coil was long enough to ensure a uniform field over the samples, which were only 4 cm tall. The coil was part of a tuned circuit with a quality factor of 30, consistent across all experiments, and was used both to excite the sample and detect the signal. All experiments were carried out at room temperature using a Tecmag-based spectrometer (Tecmag, Houston, TX), and an rf input power of 1000 W or less. The experiments consisted of performing pulse sequences on each sample at various off-resonance conditions. A typical trial consisted of 16 scans at a given off-resonance condition in order to provide a useful signal to noise ratio (SNR). Before a trial it was necessary to measure the exact NQR frequency with an FID to ensure that the correct off-resonance condition was being used for the SLSE. This measurement was accurate to within 12 Hz. For the actual off-resonance condition experiments, two standard SLSE sequences were used: the 90–90 and the 90–180. Both sequences begin with an optimal pulse, the so-called 90 pulse, where h0 ¼ cI B1 te ¼ 119 because this produces the strongest signal for a powder [6]. For the 90–90, the refocusing pulses are also 90° pulses, which our previous work revealed refocus homonuclear dipolar coupling for the majority of the nuclei. For the 90–180, h1 ¼ cN B1 tp ¼ 2 119 , and our previous work addressed how this sequence largely refocuses EFG broadening [17]. Three samples of p-chloroaniline were used. Initially all three samples had the same FWHM linewidth as the first sample, S1. In order to create samples with a range in broadening due to EFG inhomogeneity the second and third samples, S2 and S3, were melted by submerging their jars in water warmed above 74 °C. Once the samples had entirely melted they were allowed to recrystallize. For S2, the heat source of the water was simply turned off and sample and water were allowed to cool slowly. This only increased the linewidth slightly from 117 Hz to 124 Hz, as shown in Table 1. For S3, after melting, the jar was placed in a bath of room temperature water and gently rocked as the sample recrystallized. Interestingly, an audible cracking noise was observed as the sample solidified. This process greatly increased the linewidth of the sample to from 117 Hz to 190 Hz, however, over the course of two weeks the linewidth then fell to 160 Hz, where it remained for the duration of data collection. For this reason we distinguish between S3, the sample immediately after recrystallization, and S3 , the sample after the linewidth stopped dropping.
Dxt e ð1 cos hÞ sin h0 Ix1 : h
ð62Þ
4. Results If broadening due to heteronuclear coupling behaves like EFG inhomogeneity, as argued above, then the experimental parameters under which homonuclear dipolar coupling appear for a substance with heteronuclear dipolar coupling should be the same as before [17]: perform a 90–180 SLSE, at jDf j ¼ 41s0 , and for s
13
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
Table 1 The NQR signals from numerous resonant FIDs were averaged together and fit to a Voigt function, Eq. (64). This was done on two samples of sodium nitrite (Narrow and Broad) and three samples of p-chloroaniline (S1, S2, S3, and S3 ) that varied only in their broadening due to EFG inhomogeneity. Individually fitting the lineshapes revealed the Gaussian width’s agreed within 10 Hz for the same substance and transition frequency. The Lorentzian components, however, varied with the full width at half maximum linewidths, showing that EFG inhomogeneity manifests itself as a Lorentzian line shape, while the homogenous broadening mechanisms are well represented by a Gaussian lineshape. The results in this table are from fitting corresponding lineshapes simultaneously to the same Gaussian width, with the expected rG calculated from the expected broadening due to homogenous dipolar coupling only. The results show p-chloroaniline has an additional source of Gaussian broadening, seemingly due to heteronuclear coupling. Sodium Nitrite
P-Chloroaniline
xx
FWHM ± 2 Hz cL 5 Hz Observed rG 5 Hz Expected Homonuclear rG Hz Observed/homonuclear (%)
xy
xz
N.
B.
N.
B.
N.
B.
S1
S2
S3
S3
178 51 48 44
364 163
154 37 46 39
289 123
122 25 39 44
173 61
117 27 36 17
124 32
187 73
161 57
109
118
89
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi greater than a minimum s, smin ¼ tp =aeff , with the procedure for calculating the dipolar parameter aeff given in Ref. [18]. For p-chloroanaline, aeff ¼ 2pð12 HzÞ [18], so the time conpexpected ffiffiffiffiffiffiffiffiffiffiffiffi stant due to homonuclear coupling T d ¼ 1:16 ð3=2Þ=aeff ¼ 18:8 ms for the powder sample [17], which was less than the observed T 1 of 57 ± 10 ms. To see if a homonuclear response in p-chloroaniline could be obtained, a series of 90–180 SLSEs were run both at Df ¼ 0 and Df ¼ 41s0 for various values of s. The echo trains were then fit to a function composed of the addition of a Gaussian and exponential functions:
"
1 t SðtÞ ¼ A pg exp 2 Td
s s0
2 #
xx
t þ ½1 pg exp T 2e
! ;
ð63Þ
where A is the initial amplitude at t ¼ 0; T 2e is the long term decay constant, pg is the percent of the signal due to a Gaussian decay, and T d is the width of the Gaussian decay associated with the strength of the homonuclear dipolar coupling. The addition of the ss0 term to the Gaussian component is because there is effectively no evolution due to the dipolar coupling during the finite pulse lengths [17]. The results shown in Fig. 2 are consistent with what we previously observed for sodium nitrite. As s0 increases the Gaussian contribution gets stronger and the value of T d converges to the expected value of 18.8 ms, regardless of the EFG component of the sample. This is shown for both S1 and S3, with the values for both samples converging to within 10% of the expected value. The values of pg are somewhat smaller than that observed for sodium
(a)
212
nitrite [17]. This is attributed to the smaller geff , which is less than half the value for sodium nitrite. In addition, for p-chloroaniline the homonuclear dipolar coupling represents a smaller fraction of the total linebroadening, particularly for S3, as shown in Table 1. The expected second moment due to heteronuclear coupling for p-chloroaniline was predicted to be approximately 1 kHz by Vega, but the linewidths of the samples were as low as 117 Hz, far less than could be expected from Vega’s prediction. This had been noted before for p-chloroaniline [32]. Vega’s expected broadening was then re-derived assuming the protons were free to move rapidly within a sphere about their location [25], since the original prediction was based on a purely static model. This motional averaging was found to decrease the expected broadening by orders of magnitude as the sphere of uncertainty increased in size. This suggests that motion is responsible for the narrower than expected line, but since the calculations were only qualitative no prediction of the second moment was made. The inability to confirm the presence of the heteronuclear coupling from the linewidth alone resulted in uncertainty in declaring that the substance had heteronuclear broadening. However, there are three broadening mechanisms that are important to the initial signal: EFG, homonuclear coupling, and heteronuclear coupling. In principle, one could measure the heteronuclear dipolar coupling after isolating the homonuclear and EFG contributions. In order to do that, the FID lineshapes from all of the samples were independently fit to a Voigt function:
(b)
Fig. 2. The fit parameters obtained from the 90–180 SLSE echo trains using Eq. (63) for various ss at Df ¼ 1=ð4s0 Þ for both S1 (thin black line, squares) and S3 (thick red lines, circles). In (a) the strength of the Gaussian contribution pg is low for values of s below the minimum s where the homonuclear response is predicted to be inhibited, and is consistently higher above that value. In (b) both samples show T d approach the expected value for a powder 18.8 ms, while also remaining distinct from T 2e .
14
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
(a)
(b)
(c)
(d)
Fig. 3. The average 90–90 SLSE signals (black squares) obtained from the first 63 ms of echoes, to maximize SNR, from S2 for various values of s0 and for various off-resonance conditions (Df s0 ). Signals were normalized to the value at Df ¼ 0. The expected contribution to the lineshape due to EFG and heteronuclear broadening (dashed black line) is centered around Df ¼ 1=ð4s0 Þ along the bottom axis. For low values of s0 this distribution in Df s0 approaches a delta function, and the model that ignores line broadening behavior (thin red line) closely tracks the data. At longer s0 , incorporating the lineshape (thick black line) does a better job tracking the data. The linewidth data used here came from FIDs taken during a SLSE sequence, as discussed in the text. For S2 the linewidth narrowed from 124 Hz to 117 Hz; for S3 , from 161 Hz to 136 Hz.
VðmÞ /
Z
1
1
"
1 exp 2
x
rG
2 #
1 2
ðm xÞ þ c
2 L
dx;
ð64Þ
where rG is the second moment of the Gaussian component and cL is the width associated with the Lorentzian profile. The results of the fits, shown in Table 1, showed that the Gaussian component remained consistent between samples, within 10 Hz, while the Lorentzian components fluctuated on the scale of the observed changes in the linewidth. This suggests that EFG broadening has a Lorentzian behavior, since neither of the dipolar coupling broadening mechanisms should vary from sample to sample. As further confirmation the two samples of sodium nitrite from our previous work were similarly examined. The results from all of the experiments are shown in Table 1 and are consistent with the conclusion that EFG broadening has a Lorentzian contribution while the homogenous broadening mechanisms are primarily Gaussian. Since the Gaussian contribution for p-chloroaniline was so much greater than expected from homonuclear dipolar coupling alone, which was not observed for sodium nitrite, we attribute this additional broadening to heteronuclear dipolar coupling strength. For a 90–90 SLSE the dominant response should be due to offresonance condition behavior. Using the projection model of Eq. (46) the 90–90 was modeled for various values of s0 . This produced an expected signal that is independent of broadening mechanisms. To account for the expected broadening due to EFG and heteronuclear dipolar coupling, the known homonuclear response was
subtracted in quadrature from the Gaussian component of the observed lineshape for S2. This new distribution, h of Eq. (56), was then convolved with the projection model. Using the lineshape data obtained with standard FIDs, however, did not produce a model consistent with the data. Surprisingly it was found that by taking an FID during the SLSE sequence, i.e. obtaining an FID acquisition immediately after a repeat unit, that the linewidth was narrower than that observed from a single excitation pulse; while this effect is relatively small for S2, it is still significant. This was observed on both S2 and S3 but only after the second repeat unit; the linewidth after the first repeat unit was the same as a normal FID’s. Using the lineshape from this SLSE-FID, the projection model predicted the observed signal quite well, as shown in Fig. 3, confirming the projection model for the 90–90 SLSE.
5. Conclusion We have theoretically demonstrated that broadening due to heteronuclear coupling and EFG inhomogeneity have the same response to a pulse sequence. In particular they are both refocused under a 90–180 SLSE sequence. By examining p-chloroaniline we see that the homonuclear response, which was previously addressed [17], can still be observed, and at the same conditions observed in the simpler case, i.e. using a 90–180 SLSE, jDf j ¼ 41s0 , and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s P tp =aeff . In order to verify that heteronuclear coupling is present we found that the inhomogeneous broadening of the NQR line
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
due to EFG has a Lorentzian shape, while the homogenous broadening has a Gaussian shape. Since the width of the Gaussian cannot be accounted for solely in terms of the homonuclear coupling width that we observed, heteronuclear coupling must be filling the remainder. We note that heteronuclear coupling strength is much smaller than expected, and we argue that this may be due to the motion of the light hydrogen nuclei in the crystalline environment. The ability to separately determine the contribution of these broadening mechanisms can be used to fingerprint samples, to track their manufacturing origin, and help optimize detection sequences depending on the relative size of these broadening mechanisms. References [1] U. Haeberlen, High Resolution NMR Spectroscopy in Solids: Selective Averaging, Academic Press, New York, 1976. [2] M. Mehring, High Resolution NMR Spectroscopy in Solids, Springer, Berlin, Heidelberg, New York, 1976. [3] E.D. Ostroff, J.S. Waugh, Multiple Spin Echoes and Spin Locking in Solids, Phys. Rev. Lett. 16 (1966) 24. [4] R.A. Marino, S.M. Klainer, Multiple spin echoes in pure quadrupole resonance, J. Chem. Phys. 67 (1977) 3388. [5] D.Ya. Osokin, Pulsed line narrowing in nitrogen-14 NQR, Phys. Stat. Sol. (b) 102 (1980) 681. [6] S. Vega, Theory of T1 relaxation measurements in pure nuclear quadrupole resonance for spins I = 1, J. Chem. Phys. 61 (1974) 1093. [7] O.S. Zueva, A.R. Kessel, Dynamic line narrowing of nuclear quadrupole resonance, J. Mol. Struct. 83 (1982) 383. [8] D.Ya. Osokin, Coherent multipulse sequences in nitrogen-14 NQR, J. Mol. Struct. 83 (1982) 243. [9] R.S. Cantor, J. Waugh, Pulsed spin locking in pure nuclear quadrupole resonance, J. Chem. Phys. 73 (1980) 1054. [10] B.N. Provotorov, A.K. Khitrin, J. Exp. Theor. Phys. Lett. 34 (1981) 157. [11] A. Gregorovic, T. Apih, Relaxation during spin-lock spin-echo pulse sequence in 14 N nuclear quadrupole resonance, J. Chem. Phys. 129 (2008) 214504. [12] M.M. Maricq, Quasistationary state and its decay to equilibrium in the pulsed spin locking of a nuclear quadrupole resonance, Phys. Rev. B 33 (1986) 4501.
15
[13] G.B. Furman, G.E. Kibrik, A.Yu. Poljakov, I.G. Shaposhnikov, NQR multiple-pulse spin-locking as a method of study of molecular motions in solids, J. Mol. Struct. 192 (1989) 207. [14] G.B. Furman, I.G. Shaposhnikov, Multiple-pulse NQR dynamics of spin systems with strong heteronuclear coupling, J. Mol. Struct. 321 (1994) 239. [15] G.B. Furman, Nitrogen-14 NQR multiple-pulse spin-locking in the axial electric field gradient, Z. Nat. 49a (1994) 97. [16] M. Buess, S. Caulder, Factors affecting the NQR line width in nitramine explosives, Appl. Magn. Reson. 25 (2004) 383. [17] M.W. Malone, M. McGillvray, K.L. Sauer, Revealing dipolar coupling with NQR off-resonant pulsed spin locking, Phys. Rev. B 84 (2011) 214430. [18] K.L. Sauer, C.A. Klug, Spin dynamics in the pulsed spin locking of nuclear quadrupole resonance, Phys. Rev. B 74 (2006) 174410. [19] T. Gullion, D.B. Baker, M.S. Conradi, New, compensated Carr–Purcell sequences, J. Magn. Reson. 89 (1990) 479. [20] M.J. Lizak, T. Gullion, M.S. Conradi, Measurement of like-spin dipole couplings, J. Magn. Reson. 91 (1991) 254. [21] G.W. Leppelmeier, E.L. Hahn, Nuclear dipole field quenching of integer spins, Phys. Rev. 141 (1966) 724. [22] S. Vega, Adv. Magn. Reson. 6 (1973) 259. [23] R.E. Sheriff, D. Williams, Nuclear gyromagnetic ratios. III, Phys. Rev. 82 (1951) 651. [24] G.B. Carpenter, The crystal structure of sodium nitrite, Acta Crystal. 5 (1952) 132. [25] J. Trotter, S.H. Whitlow, T. Zobel, The crystal and molecular structure of pchloroaniline, J. Chem. Soc. A (1966) 353. [26] J.H. Palm, The crystal structure of p-chloroaniline, Acta Cryst. 21 (1966) 476. [27] A. Abragam, The Principles of Nuclear Magnetism, second ed., Clarendon, Oxford, 1961. [28] C.P. Slichter, Principles of Magnetic Resonance, third ed., Springer, New York, 1996. [29] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Wiley, New York, 1977. [30] S. Vega, Fictitious spin 1/2 operator formalism for multiple quantum NMR, J. Chem. Phys. 68 (1978) 5518. [31] K.L. Sauer, C.A. Klug, J.B. Miller, A.N. Garroway, Using quaternions to design composite pulses for spin-1 NQR, Appl. Magn. Reson. 25 (2004) 485. [32] R. Ambrosetti, A. Colligiani, P. Grigolini, F. Salvetti, Conformation of the NH2 group in a single crystal of p-chloroaniline as determined by combined Zeeman effects on the 35Cl and 14N nuclear quadrupole resonances, J. Chem. Phys. 60 (1974) 459.
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Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Heteronuclear dipolar coupling in spin-1 NQR pulsed spin locking M.W. Malone, K.L. Sauer ⇑ George Mason University, Fairfax, VA 22030, United States
a r t i c l e
i n f o
Article history: Received 9 August 2013 Revised 15 October 2013 Available online 31 October 2013 Keywords: Nuclear quadrupole resonance Spin-locking Dipolar coupling Broadening
a b s t r a c t We investigate theoretically and experimentally the role of broadening due to heteronuclear dipolar coupling in spin-1 nuclear quadrupole resonance pulsed spin locking. We find the experimental conditions where heteronuclear dipolar coupling is refocused by a standard multipulse sequence. This experimental condition allows us to extend our previously reported ability to measure the homonuclear dipolar coupling of powder samples to include substances that have heteronuclear coupling. These results are useful for designing substance detection algorithms, and for performing sample characterization. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Nuclear quadrupole resonance (NQR) is a spectroscopic technique that exploits the orientation of the nuclear quadrupole moment in the local electric field gradient (EFG). In NQR and the closely related nuclear magnetic resonance (NMR) spectroscopy, the application of a resonant rf pulse causes the nuclear magnetic moment with spin I of a target nucleus to oscillate at a frequency characteristic to the substance of interest. This makes the technique useful for substance detection, especially in the field, since no static field, as in NMR, is required. However, NQR measurements are possible only for crystalline materials with nuclei that have a quadrupole moment, i.e. nuclei with spin > 1/2, such as the spin-1 14N nuclei that are the target of this research. While there are exceptions, NQR signals are typically weak, due to the commonly low resonance frequencies. Additionally, the signal from a single pulse frequently decays on the order of milliseconds. This is due to line broadening mechanisms which are a function of the static crystalline environment investigated here, and T 1 processes that we do not investigate. However, just like the multipulse sequences of NMR[1,2], such as the spin-lock spin-echo (SLSE) [3], a series of refocusing pulses after the initial excitation pulse can selectively refocus the broadening mechanisms to extend the NQR signal in time [4,5]. The class of multipulse sequences we investigate begin with an excitation pulse h0 to create the initial signal, followed by a series of refocusing pulses h1 , with duration t p , shifted in phase by 90° from the excitation pulse. These SLSE sequences are expressed, for N repeat units, as h0y ðs h1x sÞN , where 2s is the time between the refocusing ⇑ Corresponding author. E-mail address: [email protected] (K.L. Sauer). 1090-7807/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2013.10.014
pulses, s0 s þ t p =2 is useful later. The signal is acquired stroboscopically at time s after each refocusing pulse, and the strength of the refocusing pulse varies depending on the desired signal response. It should be noted that unlike in NMR the effective pulse angles h will not be consistent over a powder sample, under consideration here, but will vary from one crystallite to another [6]. Therefore the Hamiltonian is not cyclical and the average Hamiltonian theory [1,2,7,8] cannot be used to analyze NQR response to resonant or near-resonant pulses [9]. Much of the theoretical work in multipluse NQR has focused on the long term behavior of the signal [10–15]. In contrast, our work focuses on the initial signal behavior by focusing on three broadening mechanisms to explain the linewidth of the NQR signal. The first, EFG inhomogeneity, is an inhomogeneous broadening mechanism caused by variations in the local electric field of the nuclei in the sample. Its size is a function of the sample’s history, from its manufacturing process to its handling [16]. The second, homonuclear dipolar coupling, is a homogenous broadening mechanism that should be constant across samples of the same substance. Our previous work [17,18], rooted in the theoretical work of Cantor and Waugh [9], revealed the experimental conditions under which a SLSE triggered a strong response due to homonuclear coupling. This response manifested itself as a rapid Gaussian decay in the initial echo train, and allowed us to measure the homonuclear dipolar coupling strength. While this result has also been obtained in NMR [19,20], the typical NQR substance is a powder, and the corresponding random orientation of the crystallites with the applied fields means the observed uniform response is not obvious. This report addresses the role of a third broadening mechanism to a SLSE sequence: heteronuclear dipolar coupling, another homogenous broadening mechanism. While the other two broadening mechanisms are first order effects, the heteronuclear
9
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
interaction makes only a second order-contribution [21], at least in the case of non-axial symmetry where the NQR energy levels are non-degenerate, an assumption we make in this paper. Nevertheless theoretical calculations [22] indicate that in some samples the heteronuclear interaction may be the dominant broadening mechanism. To our knowledge this is the first treatment of the effects of this broadening mechanism in NQR multi-pulse sequences. Here we focus on detection of spin-1 nuclei and their interaction with spin-1/2 nuclei; one should be able to make similar arguments for other spin nuclei besides 1/2, as long as their transition frequencies are not close to the NQR frequencies. By finding the conditions where heteronuclear broadening is refocused, we show that the strong homonuclear response can still be observed for substances with heteronuclear broadening. While our previous work used sodium nitrite NaNO2, this project used p-chloroaniline ClC6H4NH2 because its nitrogen to hydrogen dipolar coupling was expected to contribute more to the linewidth than the nitrogen to sodium dipolar coupling of sodium nitrite. This is because the second order broadening hDx2 iNK of heteronuclear coupling is proportional to c4K , where cK is the gyromagnetic ratio of the other spin species, and falls off faster than the distance separating the nitrogen and the other nuclei raised to the sixth power [22]. Since hydrogen has a gyromagnetic ratio approximately four times the value of sodium [23], and the nitrogen-sodium bond length in sodium nitrite[24] is approximately 2.5 times the nitrogen–hydrogen bond lengths in the NH2 group in pchloroaniline [25,26], p-chloroaniline should have much stronger heteronuclear coupling. Furthermore, theoretical calculations have already been made for the second moments of both the homonuclear[18,17] and heteronuclear broadening [22] in p-chloroaniline. There are, however, two sites for the nitrogen in the unit cells, such that while the NQR frequencies are identical for both sites, the EFG principal axes frames (PAFs) are rotated by 90 degrees from each other [22]. The x-axes remain aligned, and it is for this reason that we focus on the xx transition, the largest NQR frequency of a spin-1 system. 2. Theory To understand heteronuclear coupling’s response to a multipulse sequence, the progression of a system modeled as a single spin-I (spin-1) nucleus, with angular momentum I, coupled to a set of N K spin-K (spin-1/2), each with angular momentum KðkÞ is examined through the evolution of the corresponding density matrix qðtÞ. The quadrupole Hamiltonian hHQ is
hHQ ¼
eQ 6Ið2I 1Þ
h
i V xx 3I2x I2 þ V yy 3I2y I2 þ V zz 3I2z I2 ; ð1Þ
where e is the charge of the proton, and Q is the nuclei’s quadrupole moment. The second derivatives of the electric potential V are found ^; ^z axes of the PAF of the EFG tensor at the nucleus, using the ^x; y with jV zz j > jV yy j > jV xx j [27,28]. These axes also define the angular momentum operators Ii ¼ I ^i. For the sample of interest, p-chloroaniline, there are two different PAFs within each unit cell, however the ^x axes are parallel [25]. For spin-1, the three eigenstates of HQ can be expressed in terms of the eigenstates of Iz jmi ¼ mjmi [27]
jm ¼ 1i þ jm ¼ 1i pffiffiffi ; 2 jm ¼ 1i jm ¼ 1i pffiffiffi ; and jxi ¼ 2 jzi ¼ jm ¼ 0i:
jyi ¼
ð2Þ
three sets of fictitious spin-1/2 vector operators [29]. For the hxx ¼ Ey Ez transition, HQ can be defined by these operators in three equivalent ways [30,9]:
HQ ¼ xx Ix3 þ
1 xy xz Iy3 Iz3 ; 3
ð5Þ
with cyclical permutations of x; y; z providing the other ways. The operators
1 1 Ix ¼ ðjyihzj þ jzihyjÞ 2 2 1 1 ðIy Iz þ Iz Iy Þ ¼ ðijyihzj ijzihyjÞ 2 2 1 1 2 2 I Iy ¼ ðjyihyj jzihzjÞ; 2 z 2
Ix1
ð6Þ
Ix2
ð7Þ
Ix3
ð8Þ
with cyclical permutations of x; y; z providing the remaining definitions for Iyi and Izi , are the fictitious spin-1/2 operators of an angular momentum Ix in a space ^x1 ; ^x2 ; ^ x3 with Ix ^xi ¼ Ixi . In addition, ðIy3 Iz3 Þ in Eq. (5) is conventionally relabeled as[9]
1 Ix4 Iy3 Iz3 ¼ ð1 3jxihxjÞ; 2
ð9Þ
where 1 is the identity operator. Clearly, Ix4 commutes with Ix1 ; Ix2 , and Ix3 , which themselves commute like angular momentum operators. During the refocusing pulses, the radio-frequency Hamiltonian b and frequency xrf is for a pulse of strength B1 direction B hHrf , where
b cos xrf t c I þ c Hrf ðtÞ ¼ B1 B I K
! NK X ðkÞ K :
ð10Þ
k¼1
For our sample, K corresponds to hydrogen with gyromagnetic ratio cK , and I to nitrogen with gyromagnetic ratio cI . In these experiments the applied field has a frequency xrf close to the NQR transition xNQR , so only the interaction with I is important; as our example, we take xNQR ¼ xx . The heteronuclear dipolar Hamiltonian between the spin-1 nu HIK , where cleus and the N k spin-1/2 nuclei is h
HIK ¼
NK X KðkÞ Dk I;
ð11Þ
k
and
Dk
l0 4p
cI cK h
1 ð1 3^rk ^rk Þ; r 3k
ð12Þ th
and rk ¼ rk ^rk is the displacement vector between the spin-1 and k spin-1/2 nuclei. For convenience, the operator I is expressed as
^ þ ðjyihxj þ jxihyjÞ^z; I ¼ ðjyihzj þ jzihyjÞ^x þ ðijxihzj þ ijzihxjÞy
ð13Þ
so that it is clear that HIK only connects states of different quadrupole eigenenergies. The homonuclear dipolar Hamiltonian between HDK is similarly defined, as is homonuclear the spin-1/2 nuclei h dipolar Hamiltonian between the spin-1 nuclei hHDI . In order to fo HDI is dropped from cus on the effects of heteronuclear coupling, h this model, having already been treated in detail in our previous work [17]. However, we discuss its contribution to the signal in Section 4. 2.1. Evolution
ð3Þ ð4Þ
Excluding the degenerate case, the three corresponding eigenenergies Ey ; Ex ; Ez lead to three possible transition frequencies xNQR and
To find the evolution of qðtÞ due a series of repeat units ðs h1x sÞN , we begin with the Liouville equation
._ ¼ i½.; H;
ð14Þ
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M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
H is given by where the total Hamiltonian h
H ¼ HQ þ HDK þ HIK þ Hrf ðtÞ:
ð15Þ
When the rf is off, such as between pulses, the Hamiltonian is static and the evolution of the density matrix is
qðtÞ ¼ eiHðtt0 Þ qðt0 ÞeiHðtt0 Þ
D¼
exp isjxbihxbjDxb
ð29aÞ
n s o exp i ð1 þ Ix4 ÞðDyb þ Dzb Þjbihbj 3 exp isIx3 ðDyb Dzb Þjbihbj ;
ð29bÞ ð29cÞ
ð17Þ
where jbihbj shows that the state of the K spins is preserved. The terms in Eqs. (29a), (29b) can be dropped because they will not contribute to the evolution of the signal, reducing D to
ð18Þ
D¼
X
n jwn ihwn jðtt0 Þ
eiHðtt0 Þ ¼ e n X jwn ihwn jein ðtt0 Þ : ¼
X b
ð16Þ
where H ¼ HQ þ HDK þ HIK and qðt0 Þ is the density matrix before the free evolution. In principle H can be diagonalized to a basis jwn i with eigenvalues n , allowing the free evolution operator to be written as i
where the last expression uses Eq. (22), H0 ¼ H0 DHQ , and h:c. is the Hermitian conjugate of the evolution operators to the left of P q~ ðt0 Þ. The dipole operator D a;b jabihabjeiDab s simplifies:
n
X eisIx3 ðDyb Dzb Þjbihbj :
ð30Þ
b
It is not computationally viable, however, to diagonalize H because of the large number of particles involved. Using perturbation theory, approximate expressions for jwn i and n are found by taking hH0 ¼ hðHQ þ HDK Þ as the unperturbed and hHIK as the perturbation. This a reasonable assumption when cK cI , as in our sample where the ratio of cK =cI 14. Because HIK only joins states with different quadrupole energies, non-degenerate perturbation theory can be used. Given the eigenstates of H0 ; H0 jabi ¼ ab jabi, where a corresponds to an eigenstate of HQ and b to an eigenstate of HDK , the surviving lowest order correction to the energy will come from
H1 ¼ DHQ þ Hrf ðtÞ;
n ab þ Dab ; where
ð19Þ
Hrf ¼ x1 Ix cosðxrf tÞ
ð20Þ
b In the interaction representation of h H0 , the with x1 ¼ cI B1 ^ x B. perturbation is given by
XX jhjljHIK jabij
2
Dab ¼
j–a
ab jl
l
;
XX hjljHIK jabi jjli : ab jl j–a l
2 IK jabi 1 MHz. Therefore, the ratio hjljH ab jl is on the order 10 , making jwn i jabi. This approximation permits the free evolution operator eiHs to be approximated in terms of jabi:
X jabihabjeiðab þDab Þs :
ð22Þ
m
To solve for the evolution due to the rest of the repeat unit it will be beneficial to use the interaction representation of hH0 , where
H0 HQ þ HDK ;
ð23Þ
HQ is the reduced quadrupole Hamiltonian and h
H Q ¼ H Q þ DH Q
xrf H xNQR Q xNQR xrf DH Q ¼ HQ : xNQR
DHQ ¼ DxIx3 þ Dx0 Ix4 ;
ð32Þ
with Dx xx xrf ; Dx ¼ 0
1 ðxy xz Þðxx xrf Þ , xx 3
and
ð33Þ
¼ DHQ þ x1 Ix1 ;
ð34Þ
where in the last line we have used the secular approximation. The evolution of the density matrix during the pulse is
e
e
q~ ðt þ tp Þ ¼ ei H 1 tp q~ ðtÞei H 1 tp ;
ð25Þ
0 e ei H 1 tp ¼ eiðDxIx3 þx1 Ix1 Þtp eiDx tp Ix4 :
ð36Þ
Using Eqs. (28) and (30) the complete evolution due to a repeat unit can now be written as
q~ ðt þ 2s0 Þ ¼
X eiðDyb Dzb ÞsIx3 jbihbj
ð37aÞ
b
eiDxsIx3 eiðDxIx3 þx1 Ix1 Þtp eiDxIx3 s e X 0 0 eiðDyb Dzb ÞsIx3 jb ihb j
ð37bÞ ð37cÞ ð37dÞ
b0
q~ ðtÞh:c: ð26Þ
ð35Þ
where
iDx0 ðt p þ2sÞIx4
ð24Þ
HQ ¼
where
ð21Þ
As discussed below, the experimental NQR linewidth is on the order of 100 Hz, while the NQR frequency is on the order of
eiHs
ð31Þ
e 1 ¼ DHQ þ eiH0 t Hrf ðtÞeiH0 t H
while for the wavefunction it is
jwn i jabi þ
In contrast, during the pulse, the contribution of HIK is ignored since, in practice, Hrf is much stronger. This assumption was found to be valid for HDI in our previous work [17]. The perturbing Hamiltonian hH1 is then given by
ð37eÞ
a period s of free evolution, the density matrix in the interaction representation becomes
The exponential operator containing Ix4 will not contribute to the evolution of the signal, and is dropped: it only includes the identity operator and the states not addressed by the rf pulse. As a result the only remaining operators are the fictitious angular momentum operators, the product of which can be represented ^ b found as a single rotation operator of angle hb and direction n through the following relationships[31]
q~ ðtÞ ¼ eiH0 t eiHs eiH0 t0 q~ ðt0 Þh:c:
cos
While H0 and H0 have the same eigenstates, jabi, they will have different eigenvalues: ab for H0 and ab for H0 . In the interaction rep~ a eiH0 t Ha eiH0 t . After ~ ðtÞ eiH0 t .ðtÞeiH0 t and H resentation of hH0 ; .
!
¼
X ~ ðt0 Þh:c:; jabihabjeiDab s eiDHQ s q a;b
ð27Þ ð28Þ
hb h2 h2 ^1 n ^2 ¼ cos h1 cos sin h1 sin n 2 2 2
hb h h h h h h ^ b ¼ sin 2 n ^ 2 þ 2 sin 1 n ^ 1 cos 1 cos 2 sin 1 sin 2 n ^1 n ^2 sin n 2 2 2 2 2 2 2
ð38Þ ð39Þ
11
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
^1 ¼ ^ where h1 ¼ Dx þ ðDyb Dzb Þ s, n x3 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ x1 þDx^ x3 x 2 1 2 ^2 ¼ tp ^ b lies h2 t p x1 þ Dx , and n . This means that n h2 ^ ^ in the x1 x3 plane. The final expression is
X q~ ðt þ 2s0 Þ ¼ eihb Ix n^b jbihbj q~ ðtÞh:c:;
ð40Þ
and the average xx signal from the spin-1 nuclei with the same ^x of their PAFs is NI n o bX b ¼ ^x B hI Bi Tr IxðiÞ q NI i¼1
¼
NI n o bX ^x B ~ Tr eI ðiÞ x q NI i¼1
ð53Þ
¼
NI h n o n o i bX ^x B ðiÞ ðiÞ ~ cos xrf t Tr Ix2 2 Tr Ix1 q q~ sin xrf t ; NI i¼1
ð54Þ
b
with the evolution over N pulses being
X q~ ðt0 þ 2Ns0 Þ ¼ eiNhb Ix n^b jbihbj q~ ðt0 Þh:c:
ð41Þ
b
As will be discussed in Section 2.3, before the application of a ~ ðt0 Þ that produce signal can be exrepeat unit, the components of q pressed as
q~ ðt0 Þ ¼
3 XX cj Ix ^xj jbihbj
¼
ð42Þ
j¼1
b
X b
^b ½ðc1 ^x1 þ c3 ^x3 Þ n ^b þ n ^ b? ½ðc1 ^x1 þ c3 ^x3 Þ n ^ b? I x ½n
þ c2 ^x2 jbihbj;
ð43Þ
^ b? n ^b ^ where n x2 , and the coefficients cj for a SLSE are given in Section 2.3. Recognizing that [18] ^ 0 h:c: ¼ Ix ½cos hn ^ 0 þ ð1 cos hÞn ^ ðn ^n ^ 0 Þ þ sin hðn ^0 n ^ Þ eihIx n^ Ix n
ð44Þ
it is possible to see that after N repeat units with finite h, as in Eq. (41), that the only terms of the density matrix that contribute to the signal are
q~ ðtÞ ¼
X ^ b Þðc1 ^x1 þ c3 ^x3 Þ n ^ b jbihbj: ðIx n
ð45Þ
b
The Ix3 term makes no contribution to the observed signal, and as explained later phase cycling results in c3 ¼ 0 [18]. Therefore, the effective density matrix reduces to
q~ ðtÞ ¼
X ^ b Þ2 ; c1 Ix1 jbihbjð^x1 n
ð46Þ
b
which we call the projection model due to its simple dependence on ^x1 n ^b . The above is for a single spin-1 nucleus. Expanding the density matrix to include all N I nuclei with similar PAFs
q~ ðtÞ ¼
NI X X i¼1
ð52Þ
when solved in the interaction representation of hH0 . From the proðiÞ jection model, the locked signal only contains the Ix1 term, allowing ðiÞ the Ix2 term to be dropped. Further, the double summation in Eq. (47) can be replaced by a double integral over two normalized distribution functions, f ðdxÞ for the electric field gradient distribution and gðDy Dz Þ for the distribution in heteronuclear dipolar coupling: b ¼ c1 cos xrft ^x B b hI Bi
Z
dðdxÞf ðdxÞ
2
^b Þ : dðDy Dz ÞgðDy Dz Þð^x1 n
ð55Þ This double integral is equivalent to
b ¼ c1 cos xrft ^x B b hI Bi
Z
dðd-Þhð-Þ
ð56Þ
; d-Þ; h2 ðDx ; x1 ÞÞ2 ; ^ b ½h1 ðDx ð^x1 n where d- ¼ dx þ Dy Dz , and the linewidth function h ¼ f g, is the convolution of f with g. In the above, we emphasize the ^ b ; h1 , and h2 on the broadening d- and functional dependency of n and x1 . The broadening from the two the pulse parameters Dx different mechanisms contribute in the same way to the final signal. In our previous work we showed how the 90–180 SLSE should refocus broadening due EFG inhomogeneity [17]. That EFG and heteronuclear dipolar coupling contribute in the same way to the signal allows us to extend that conclusion to broadening due to heteronuclear coupling. The above solution, Eq. (56), is for a single crystal. To obtain the powder averaged signal the integrab would be made, with again tion on the direction cosine ^x B b x1 ¼ cB^x B. 2.3. Initial density matrix
2 ðiÞ ^ bðiÞ : c1 Ix1 jbihbj ^x1 n
ð47Þ
b
þ dxðiÞ where In Eqs. (38) and (39), we can replace Dx with Dx
¼x x xrf ; Dx
ð48Þ
x is the average NQR frequency, and dxðiÞ is the deviation from this x average by a particular nucleus i. For hard pulses, i.e. x1 dxðiÞ , the ^ bðiÞ of Eqs. (38) and (39) is defined by rotation axis n
As shown in Fig. 1, subtle differences in the timing of the sequence before the application of the repeat unit can have big impacts on the SLSE signal behavior. To derive an expression for the density matrix before the repeat units as they occur in our experiments, it is assumed that the sample is in thermal equilibrium and at the high-temperature limit. In this case the initial density matrix qð0Þ is taken as the reduced density matrix [9]
qðt < 0Þ ¼ xx Ix ^x3 : ^x3 Þ ^ 2 ¼ tp ðx1 ^x1 þ Dx h2 n ðiÞ ðiÞ ^ 1 ¼ s Dx þ dxðiÞ þ ðDyb Dzb Þ ^x3 : h1 n
ð49Þ ð50Þ
Significantly, the broadening mechanisms dx and ðDyb Dzb Þ do not appear separately in the above rotation, but are summed together. ðiÞ
The sample is excited by the same coil used to detect the signal. In this case the signal is proportional to
ð51Þ
ð57Þ
b cosðxrf t þ p=2Þ, which The rf pulse during excitation is B ¼ Be B differs in phase by p=2 compared to the refocusing pulse in Eq. (10). Similar to the derivation in Eq. (34), in the interaction representation of hH0 , the perturbing Hamiltonian of the excitation pulse is given by
e 1 ¼ DHQ þ eiH0 t Hrf ðtÞeiH0 t H ¼ DHQ þ x1e Ix2 ;
2.2. The observable
b ¼ TrfI B b qg; hI Bi
Z
ð58Þ ð59Þ
b and the secular approximation is used in where x1e ¼ c1 Be ^x B, the last line. The evolution of the density matrix from the excitation ^ 0 ¼ ^x3 , and pulse with length t e is given by Eq. (44), with n ^ ¼ ðx1e ^x2 þ Dx^x3 Þt e . The product x1 te is chosen in order to be hn
12
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
The coefficient in front of Ix1 is the c1 of Eq. (56). The excitation pulse is such that for Df ¼ 0; h p=2, while for increasing values of jDf j; h increases. For large Df , the second term in the square brackets compensates for what would be a loss of signal. This explains why the adjusted sequence produces the stronger signal in those cases since it creates a larger initial signal that can be spinlocked. 3. Experimental procedure
Fig. 1. The 90–90 SLSE signal on the left (red dots) due to a standard sequence has a stronger response as a function of off-resonance condition Df ¼ Dx=ð2pÞ than a nearly identical sequence on the right (black dots). The only difference is that the first free evolution of the left sequence has been adjusted to sR ¼ s te =2. As discussed below, this reduced free evolution ensures that the density matrix is suitable for pulse spin locking at large off-resonance conditions. The thick lines are the predicted signal strengths of the projection model Eq. (46), and agree quite well with the data. The thicker shaded lines correspond to the density matrix before application of the repeat unit, given in Eq. (62). Data is from the S1 sample of pchloroaniline, taken at xx with s0 ¼ 390 ls, tp ¼ 110 ls, and averaged over 63 ms. Both signals are normalized to the same value, and both sequences are symmetrical about Df ¼ 0. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
close to 90° for the majority of the crystallites contributing to the signal [6]. In order to isolate the echo train and reduce transient pulse effects, successive experiments in which Be is inverted are ~ ðt e Þ is effectively subtracted [18]. Therefore q
q~ ðte Þ ¼ xx Ix
x1e te h
sin h^x1 þ
Dx t e ð1 cos hÞ^x2 ; h
ð60Þ
which means that for certain values of h which occur at large offresonance conditions the Ix ^x1 component of the effective density matrix that produces the locked signal is small, while the Ix ^x2 component that does not produce signal is large. The time between pulses in the echo train is 2s, or if measured from the center of the pulses 2s0 . However, there is some ambiguity in how far the excitation pulse should be separated from the first refocusing pulse. In practice, the midpoint of the excitation pulse to the midpoint of the first refocusing pulse is also set to s0 . This means that the time between the excitation pulse and the refocusing pulse must be reduced to sR ¼ s te =2. For delta function pulses, or for long values of s, the difference between s and sR is inconsequential. For short enough values of s, however, this adjustment provides advantages for large Df , as shown in Fig. 1. To incorporate this sR into the theory above only requires a small adjustment to the initial density matrix. This is done by performing a negative rotation after the excitation pulse using the operator
q~ ðt0 Þ eiDxte Ix3 =2 q~ ðte ÞeiDxte Ix3 =2
ð61Þ
0
^ ¼ Dxt e ^x3 =2, and recognizing that only Using again Eq. (44), with h n the Ix1 component gets locked by series of repeat units, the relevant expression for the density matrix before the first repeat unit is
q~ ðt0 Þ ¼ xx
x1e te h
sin h cos h0 þ
Experiments were performed on three 100 g powder samples of 98% purity p-chloroaniline from Sigma–Aldrich. These samples were never removed from their glass jars. All experiments were performed by placing the jars into a home built coil 8 cm in diameter and 20 cm long, located inside a shielded box. The coil diameter was just large enough to hold the jars, and the coil was long enough to ensure a uniform field over the samples, which were only 4 cm tall. The coil was part of a tuned circuit with a quality factor of 30, consistent across all experiments, and was used both to excite the sample and detect the signal. All experiments were carried out at room temperature using a Tecmag-based spectrometer (Tecmag, Houston, TX), and an rf input power of 1000 W or less. The experiments consisted of performing pulse sequences on each sample at various off-resonance conditions. A typical trial consisted of 16 scans at a given off-resonance condition in order to provide a useful signal to noise ratio (SNR). Before a trial it was necessary to measure the exact NQR frequency with an FID to ensure that the correct off-resonance condition was being used for the SLSE. This measurement was accurate to within 12 Hz. For the actual off-resonance condition experiments, two standard SLSE sequences were used: the 90–90 and the 90–180. Both sequences begin with an optimal pulse, the so-called 90 pulse, where h0 ¼ cI B1 te ¼ 119 because this produces the strongest signal for a powder [6]. For the 90–90, the refocusing pulses are also 90° pulses, which our previous work revealed refocus homonuclear dipolar coupling for the majority of the nuclei. For the 90–180, h1 ¼ cN B1 tp ¼ 2 119 , and our previous work addressed how this sequence largely refocuses EFG broadening [17]. Three samples of p-chloroaniline were used. Initially all three samples had the same FWHM linewidth as the first sample, S1. In order to create samples with a range in broadening due to EFG inhomogeneity the second and third samples, S2 and S3, were melted by submerging their jars in water warmed above 74 °C. Once the samples had entirely melted they were allowed to recrystallize. For S2, the heat source of the water was simply turned off and sample and water were allowed to cool slowly. This only increased the linewidth slightly from 117 Hz to 124 Hz, as shown in Table 1. For S3, after melting, the jar was placed in a bath of room temperature water and gently rocked as the sample recrystallized. Interestingly, an audible cracking noise was observed as the sample solidified. This process greatly increased the linewidth of the sample to from 117 Hz to 190 Hz, however, over the course of two weeks the linewidth then fell to 160 Hz, where it remained for the duration of data collection. For this reason we distinguish between S3, the sample immediately after recrystallization, and S3 , the sample after the linewidth stopped dropping.
Dxt e ð1 cos hÞ sin h0 Ix1 : h
ð62Þ
4. Results If broadening due to heteronuclear coupling behaves like EFG inhomogeneity, as argued above, then the experimental parameters under which homonuclear dipolar coupling appear for a substance with heteronuclear dipolar coupling should be the same as before [17]: perform a 90–180 SLSE, at jDf j ¼ 41s0 , and for s
13
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
Table 1 The NQR signals from numerous resonant FIDs were averaged together and fit to a Voigt function, Eq. (64). This was done on two samples of sodium nitrite (Narrow and Broad) and three samples of p-chloroaniline (S1, S2, S3, and S3 ) that varied only in their broadening due to EFG inhomogeneity. Individually fitting the lineshapes revealed the Gaussian width’s agreed within 10 Hz for the same substance and transition frequency. The Lorentzian components, however, varied with the full width at half maximum linewidths, showing that EFG inhomogeneity manifests itself as a Lorentzian line shape, while the homogenous broadening mechanisms are well represented by a Gaussian lineshape. The results in this table are from fitting corresponding lineshapes simultaneously to the same Gaussian width, with the expected rG calculated from the expected broadening due to homogenous dipolar coupling only. The results show p-chloroaniline has an additional source of Gaussian broadening, seemingly due to heteronuclear coupling. Sodium Nitrite
P-Chloroaniline
xx
FWHM ± 2 Hz cL 5 Hz Observed rG 5 Hz Expected Homonuclear rG Hz Observed/homonuclear (%)
xy
xz
N.
B.
N.
B.
N.
B.
S1
S2
S3
S3
178 51 48 44
364 163
154 37 46 39
289 123
122 25 39 44
173 61
117 27 36 17
124 32
187 73
161 57
109
118
89
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi greater than a minimum s, smin ¼ tp =aeff , with the procedure for calculating the dipolar parameter aeff given in Ref. [18]. For p-chloroanaline, aeff ¼ 2pð12 HzÞ [18], so the time conpexpected ffiffiffiffiffiffiffiffiffiffiffiffi stant due to homonuclear coupling T d ¼ 1:16 ð3=2Þ=aeff ¼ 18:8 ms for the powder sample [17], which was less than the observed T 1 of 57 ± 10 ms. To see if a homonuclear response in p-chloroaniline could be obtained, a series of 90–180 SLSEs were run both at Df ¼ 0 and Df ¼ 41s0 for various values of s. The echo trains were then fit to a function composed of the addition of a Gaussian and exponential functions:
"
1 t SðtÞ ¼ A pg exp 2 Td
s s0
2 #
xx
t þ ½1 pg exp T 2e
! ;
ð63Þ
where A is the initial amplitude at t ¼ 0; T 2e is the long term decay constant, pg is the percent of the signal due to a Gaussian decay, and T d is the width of the Gaussian decay associated with the strength of the homonuclear dipolar coupling. The addition of the ss0 term to the Gaussian component is because there is effectively no evolution due to the dipolar coupling during the finite pulse lengths [17]. The results shown in Fig. 2 are consistent with what we previously observed for sodium nitrite. As s0 increases the Gaussian contribution gets stronger and the value of T d converges to the expected value of 18.8 ms, regardless of the EFG component of the sample. This is shown for both S1 and S3, with the values for both samples converging to within 10% of the expected value. The values of pg are somewhat smaller than that observed for sodium
(a)
212
nitrite [17]. This is attributed to the smaller geff , which is less than half the value for sodium nitrite. In addition, for p-chloroaniline the homonuclear dipolar coupling represents a smaller fraction of the total linebroadening, particularly for S3, as shown in Table 1. The expected second moment due to heteronuclear coupling for p-chloroaniline was predicted to be approximately 1 kHz by Vega, but the linewidths of the samples were as low as 117 Hz, far less than could be expected from Vega’s prediction. This had been noted before for p-chloroaniline [32]. Vega’s expected broadening was then re-derived assuming the protons were free to move rapidly within a sphere about their location [25], since the original prediction was based on a purely static model. This motional averaging was found to decrease the expected broadening by orders of magnitude as the sphere of uncertainty increased in size. This suggests that motion is responsible for the narrower than expected line, but since the calculations were only qualitative no prediction of the second moment was made. The inability to confirm the presence of the heteronuclear coupling from the linewidth alone resulted in uncertainty in declaring that the substance had heteronuclear broadening. However, there are three broadening mechanisms that are important to the initial signal: EFG, homonuclear coupling, and heteronuclear coupling. In principle, one could measure the heteronuclear dipolar coupling after isolating the homonuclear and EFG contributions. In order to do that, the FID lineshapes from all of the samples were independently fit to a Voigt function:
(b)
Fig. 2. The fit parameters obtained from the 90–180 SLSE echo trains using Eq. (63) for various ss at Df ¼ 1=ð4s0 Þ for both S1 (thin black line, squares) and S3 (thick red lines, circles). In (a) the strength of the Gaussian contribution pg is low for values of s below the minimum s where the homonuclear response is predicted to be inhibited, and is consistently higher above that value. In (b) both samples show T d approach the expected value for a powder 18.8 ms, while also remaining distinct from T 2e .
14
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
(a)
(b)
(c)
(d)
Fig. 3. The average 90–90 SLSE signals (black squares) obtained from the first 63 ms of echoes, to maximize SNR, from S2 for various values of s0 and for various off-resonance conditions (Df s0 ). Signals were normalized to the value at Df ¼ 0. The expected contribution to the lineshape due to EFG and heteronuclear broadening (dashed black line) is centered around Df ¼ 1=ð4s0 Þ along the bottom axis. For low values of s0 this distribution in Df s0 approaches a delta function, and the model that ignores line broadening behavior (thin red line) closely tracks the data. At longer s0 , incorporating the lineshape (thick black line) does a better job tracking the data. The linewidth data used here came from FIDs taken during a SLSE sequence, as discussed in the text. For S2 the linewidth narrowed from 124 Hz to 117 Hz; for S3 , from 161 Hz to 136 Hz.
VðmÞ /
Z
1
1
"
1 exp 2
x
rG
2 #
1 2
ðm xÞ þ c
2 L
dx;
ð64Þ
where rG is the second moment of the Gaussian component and cL is the width associated with the Lorentzian profile. The results of the fits, shown in Table 1, showed that the Gaussian component remained consistent between samples, within 10 Hz, while the Lorentzian components fluctuated on the scale of the observed changes in the linewidth. This suggests that EFG broadening has a Lorentzian behavior, since neither of the dipolar coupling broadening mechanisms should vary from sample to sample. As further confirmation the two samples of sodium nitrite from our previous work were similarly examined. The results from all of the experiments are shown in Table 1 and are consistent with the conclusion that EFG broadening has a Lorentzian contribution while the homogenous broadening mechanisms are primarily Gaussian. Since the Gaussian contribution for p-chloroaniline was so much greater than expected from homonuclear dipolar coupling alone, which was not observed for sodium nitrite, we attribute this additional broadening to heteronuclear dipolar coupling strength. For a 90–90 SLSE the dominant response should be due to offresonance condition behavior. Using the projection model of Eq. (46) the 90–90 was modeled for various values of s0 . This produced an expected signal that is independent of broadening mechanisms. To account for the expected broadening due to EFG and heteronuclear dipolar coupling, the known homonuclear response was
subtracted in quadrature from the Gaussian component of the observed lineshape for S2. This new distribution, h of Eq. (56), was then convolved with the projection model. Using the lineshape data obtained with standard FIDs, however, did not produce a model consistent with the data. Surprisingly it was found that by taking an FID during the SLSE sequence, i.e. obtaining an FID acquisition immediately after a repeat unit, that the linewidth was narrower than that observed from a single excitation pulse; while this effect is relatively small for S2, it is still significant. This was observed on both S2 and S3 but only after the second repeat unit; the linewidth after the first repeat unit was the same as a normal FID’s. Using the lineshape from this SLSE-FID, the projection model predicted the observed signal quite well, as shown in Fig. 3, confirming the projection model for the 90–90 SLSE.
5. Conclusion We have theoretically demonstrated that broadening due to heteronuclear coupling and EFG inhomogeneity have the same response to a pulse sequence. In particular they are both refocused under a 90–180 SLSE sequence. By examining p-chloroaniline we see that the homonuclear response, which was previously addressed [17], can still be observed, and at the same conditions observed in the simpler case, i.e. using a 90–180 SLSE, jDf j ¼ 41s0 , and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s P tp =aeff . In order to verify that heteronuclear coupling is present we found that the inhomogeneous broadening of the NQR line
M.W. Malone, K.L. Sauer / Journal of Magnetic Resonance 238 (2014) 8–15
due to EFG has a Lorentzian shape, while the homogenous broadening has a Gaussian shape. Since the width of the Gaussian cannot be accounted for solely in terms of the homonuclear coupling width that we observed, heteronuclear coupling must be filling the remainder. We note that heteronuclear coupling strength is much smaller than expected, and we argue that this may be due to the motion of the light hydrogen nuclei in the crystalline environment. The ability to separately determine the contribution of these broadening mechanisms can be used to fingerprint samples, to track their manufacturing origin, and help optimize detection sequences depending on the relative size of these broadening mechanisms. References [1] U. Haeberlen, High Resolution NMR Spectroscopy in Solids: Selective Averaging, Academic Press, New York, 1976. [2] M. Mehring, High Resolution NMR Spectroscopy in Solids, Springer, Berlin, Heidelberg, New York, 1976. [3] E.D. Ostroff, J.S. Waugh, Multiple Spin Echoes and Spin Locking in Solids, Phys. Rev. Lett. 16 (1966) 24. [4] R.A. Marino, S.M. Klainer, Multiple spin echoes in pure quadrupole resonance, J. Chem. Phys. 67 (1977) 3388. [5] D.Ya. Osokin, Pulsed line narrowing in nitrogen-14 NQR, Phys. Stat. Sol. (b) 102 (1980) 681. [6] S. Vega, Theory of T1 relaxation measurements in pure nuclear quadrupole resonance for spins I = 1, J. Chem. Phys. 61 (1974) 1093. [7] O.S. Zueva, A.R. Kessel, Dynamic line narrowing of nuclear quadrupole resonance, J. Mol. Struct. 83 (1982) 383. [8] D.Ya. Osokin, Coherent multipulse sequences in nitrogen-14 NQR, J. Mol. Struct. 83 (1982) 243. [9] R.S. Cantor, J. Waugh, Pulsed spin locking in pure nuclear quadrupole resonance, J. Chem. Phys. 73 (1980) 1054. [10] B.N. Provotorov, A.K. Khitrin, J. Exp. Theor. Phys. Lett. 34 (1981) 157. [11] A. Gregorovic, T. Apih, Relaxation during spin-lock spin-echo pulse sequence in 14 N nuclear quadrupole resonance, J. Chem. Phys. 129 (2008) 214504. [12] M.M. Maricq, Quasistationary state and its decay to equilibrium in the pulsed spin locking of a nuclear quadrupole resonance, Phys. Rev. B 33 (1986) 4501.
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[13] G.B. Furman, G.E. Kibrik, A.Yu. Poljakov, I.G. Shaposhnikov, NQR multiple-pulse spin-locking as a method of study of molecular motions in solids, J. Mol. Struct. 192 (1989) 207. [14] G.B. Furman, I.G. Shaposhnikov, Multiple-pulse NQR dynamics of spin systems with strong heteronuclear coupling, J. Mol. Struct. 321 (1994) 239. [15] G.B. Furman, Nitrogen-14 NQR multiple-pulse spin-locking in the axial electric field gradient, Z. Nat. 49a (1994) 97. [16] M. Buess, S. Caulder, Factors affecting the NQR line width in nitramine explosives, Appl. Magn. Reson. 25 (2004) 383. [17] M.W. Malone, M. McGillvray, K.L. Sauer, Revealing dipolar coupling with NQR off-resonant pulsed spin locking, Phys. Rev. B 84 (2011) 214430. [18] K.L. Sauer, C.A. Klug, Spin dynamics in the pulsed spin locking of nuclear quadrupole resonance, Phys. Rev. B 74 (2006) 174410. [19] T. Gullion, D.B. Baker, M.S. Conradi, New, compensated Carr–Purcell sequences, J. Magn. Reson. 89 (1990) 479. [20] M.J. Lizak, T. Gullion, M.S. Conradi, Measurement of like-spin dipole couplings, J. Magn. Reson. 91 (1991) 254. [21] G.W. Leppelmeier, E.L. Hahn, Nuclear dipole field quenching of integer spins, Phys. Rev. 141 (1966) 724. [22] S. Vega, Adv. Magn. Reson. 6 (1973) 259. [23] R.E. Sheriff, D. Williams, Nuclear gyromagnetic ratios. III, Phys. Rev. 82 (1951) 651. [24] G.B. Carpenter, The crystal structure of sodium nitrite, Acta Crystal. 5 (1952) 132. [25] J. Trotter, S.H. Whitlow, T. Zobel, The crystal and molecular structure of pchloroaniline, J. Chem. Soc. A (1966) 353. [26] J.H. Palm, The crystal structure of p-chloroaniline, Acta Cryst. 21 (1966) 476. [27] A. Abragam, The Principles of Nuclear Magnetism, second ed., Clarendon, Oxford, 1961. [28] C.P. Slichter, Principles of Magnetic Resonance, third ed., Springer, New York, 1996. [29] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Wiley, New York, 1977. [30] S. Vega, Fictitious spin 1/2 operator formalism for multiple quantum NMR, J. Chem. Phys. 68 (1978) 5518. [31] K.L. Sauer, C.A. Klug, J.B. Miller, A.N. Garroway, Using quaternions to design composite pulses for spin-1 NQR, Appl. Magn. Reson. 25 (2004) 485. [32] R. Ambrosetti, A. Colligiani, P. Grigolini, F. Salvetti, Conformation of the NH2 group in a single crystal of p-chloroaniline as determined by combined Zeeman effects on the 35Cl and 14N nuclear quadrupole resonances, J. Chem. Phys. 60 (1974) 459.
