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Residual linewidths of NMR spectra of spin-12 systems under magic-angle spinning
JOURNAL
OF MAGNETIC
RESONANCE
w,90-99
( 1990)
Residual Linewidths of NMR Spectra of Spin-4 Systems under Magic-Angle Spinning E. BRUNNER,*
D. FREUDE,*
* Karl-Marx-Universiti and tAmes Laboratory,
B. C. GERSTEIN,~~$
Leipzig, Linntfstrasse U.S. Department Iowa State University,
AND H. PEEIEER*
5, Leipzig, German Democratic Republic. of Energy, and Department of Chemistry, Ames, Iowa 50010
Received January 26, 1990 The residual linewidths of NMR spectra of spin-f systems in solids under magic-angle spinning conditions are calculated in the rigid lattice limit for the general case where both home- and heteronuclear dipolar interactions aswell as shielding anisotropy may be present. The results are used to quantitatively interpret ‘H MAS NMR spectra of crystalline water in gypsum and of hydroxyl groups in zeolites. o 1990 Academic PPS, hc.
The recent decade has seen the development of high-resolution ( 1) . The present work considers sources of residual line broadening in solids in the case where secular portions of shielding anisotropy homonuclear dipolar interactions are present, under conditions of ning, but no strong radiofrequency pulse decoupling.
solid-state NMR for spin-4 nuclei and hetero- and magic-angle spin-
THEORY
The general approach taken in calculating residual linewidths under MAS for the types of systems considered in the present work is to calculate, from the envelope of the time decay function, the second moment under MAS. A Gaussian form is then assumed for the lineshape function of the residual line, from which the F?VHM of the line is calculated. In order to calculate the envelope of the time decay, coherent averaging theory is used to relate static Hamiltonians to Hamiltonians under MAS. The internal interactions considered in this work are homo- and heteronuclear dipolar interactions and the shielding anisotropy of the resonant spin, I: ti
= %I, + @Is + c%& + %&,.
[ll
All Hamiltonians are expressed in units of radians per second. To avoid consideration of additional line broadening which occurs if the quadrupolar interaction of the nonresonant spins (for S # 1) is not much lessthan their Zeeman energy (2)) a sufficiently high magnetic field (Bs 1, 7T for ‘H coupled to 27A1 via a bridging oxygen in the present work) is assumed. Moreover, since it can be shown theoretically (3) that the
$ Work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Chemical Science Division, under Contract W-740%Eng-82. 90 0022-2364190
$3.00
Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
RESIDUAL
LINEWIDTHS
UNDER
91
MAS
nonsecular portions of these interactions affect the residual linewidths of spin-3 nuclei in solids only if the linewidth achieved under spinning is 10 5 smaller than that of the static spectrum, a condition beyond present capabilities, only the secular portion of %’ will be retained. Hence
*I = g rfh $ ! rG3( 1 - 3 COS200){
3Zz,Zzj - IiIj1
[21
bJ
and similarly for X& by interchanging I and S, &Is = 5 YlTsh $ 2 rG3( 1 - 3 cos20j)Z~jS~~
A&*
.i k ‘% i 7 { (3 COS'Oj - 1) +
= YIB&*
~CS.b,Sin2BjCOS
[31
2$j} Zzj
[41
J
aCSA= &I,/ - (1/3)tr{2) TICSA
and &,I,
=
(bx’
-
‘$y’)/(‘bz’
-
bA)
. . . are the principal values of the shielding tensor, with ISzyl 2 16y’y’l 2 (6,yl.
In order to calculate residual linewidths under conditions of MAS, we consider the calculation of the second moment, M, of the static powder sample, M=
-(Tr{[z, =
MI
+
C Ljl}2/Tr{ MS
+
C Zxj)2)powder
[51
McsA,
where (3) ‘y;h’Z(Z
+ 1) 2 ri6 i,i
MIS = 6 ($‘r:r:h’S(S
[61
+ 1) ; ; r,;(” i K
[71
1 MCSA
=
j
(YIBo&sA)
‘(
[81
1 +;&A).
In the evaluation of the second moment of the spinning sample, X, given by Eq. [I] in the most general case, becomes time dependent. Under these conditions the envelope of the time decay after a 90; pulse, with phase detection along x in the rotating frame, is given by dMAS(t) = (L(t))
= tr{ u(l) C
Lju-'(t)
C
Zxj}/tr{
C
ZZj>
[91
with [lOI
92
BRUNNER
ET
AL.
where T denotes the Dyson time-ordering operator (4). Since s(t) is periodic and CyCliCwith period tc = 2n/ wa, with wR being the angular rotational frequency, average Hamiltonian theory (5) may be applied to evaluate the effect of sample rotation at times corresponding to multiples of the rotational periods. At times n tc, the propagator, in the absence of irreversible relaxation, is u(n&)
= exp{-i%MC>
=exp(-int,[%(“)+
&(‘)+*
- .I}
Ull
and tc
SF(O) = (l/t,)
11la1
dt’rE”( t’) s0 t’
tc
SF(‘) = (-i/2t,)
dt’
%(f’)l.
dt”[S(t’),
s0
s0
11lb1
Furthermore, if tip is defined as the difference between the time-dependent Hamiltonian governing the time evolution of the system, Y?(t), and the constant Hamiltonian &definedinEq. [ll], S(t)
= tip + 2%
[I21
then the time dependence of the density operator is idp(t)/dt
The usual transformations, &, with
= [ti,
p] = [SP + &, p(t)].
iI31
first to the frame of tip and then to the frame of $@ = u-‘&u P
P,
yield p(t) = upUp(0)ir’u,’
1141
with U= Texp-[i~dt’&(t’)).
[I51
The time decay function is (JMAS(t) = tr{exp(-i&t)
C ZXjexp(i&t)u,‘(t)
C ZXjUP(t)}/tr{(C
Zxj)*).
[16]
Since the total internal Hamiltonian associated with spinning of the sample is periodic and cyclic with period tc = 2n/ wR, so will be sp. Therefore at times II t, , from time zero, measured as the time immediately after a 90; pulse places the magnetization of the ensemble along x in the rotating frame, up( nt,, 0) = 1. The envelope of the time decay function (i.e., the values of the decay observed at cycle times nt,, where up = 1) is described by 4MAS(t) = tr{exp(-i&t)
2
Z,jeXp(L%t)
2 Zxj}/tr((
2 Zxj)*}.
[I71
RESIDUAL
The second moment evaluated to be
LINEWIDTHS
UNDER
93
MAS
of the central line is described by this decay function and is
MMAS = (-tr(f.s,
C
~xj12}/tr{(C
[I81
Lj)2})powder.
At periods n tc = 27rn/wR after the 90; pulse, the first term in the Magnus expansion, 2(O), Eq. [ 1 la], vanishes. Hence, for sufficiently high spinning speeds, the leading term in the residual linewidth is given by the second term, Eq. [ 11b] : ,g g $(I).
[I91
It then follows that kfMAS
=
(l/&K&I
+
K&f,,&
+
K&fSShfiS
+
&b&k&A).
[201
Other products of the static second moments, Eqs. [ 6]-[ 81, do not contribute to MMAS . Mss is defined analogously to Eq. [ 18 ] by interchanging I and S. The coefficients K,-K4 depend upon geometrical parameters. For the pure dipolar [22a] interactions, the values of k-,-K3 are given by K, = (2 rG6)-' $ Alrk i,J
K2
=
N
Ns
i
2 k
(2
PM
I,.i,k
r$-’
NI
Ns
2 i,j
c
[21bl
Aij/c k
[21cl
[21dl aok = (7 + 26
- 33 cos4ai)/60
COS2(Yi
a$
+
2
j
rii r- ( COS3CYi tk
COS (Yi)
WeI .
The plus sign holds for K1, and the minus sign for Kz and K3. The meanings of the symbols are further illustrated in Fig. 1. For the present work, we consider K4 only in the case of axial symmetry for the shielding tensor. This assumption is justified on the basis of known hydroxyl tensors being axial, or almost axial, with anisotropies near 15 ppm. With VCsA= 0, K4 = (2 r;6)-’
i,j
s r;6Cij
[22al
i,i
.[33( 3,. ~i)( ~ii'~j)
+ 7( 2;' Zj)] + 7[ fo(3i X
Zj)]'}.
Wbl
94
BRUNNER
ET AL. P 0
t. JP t pk
a.3
j
:
i?. Jk
0
k
FIG. 1. Definition of the angle ai, and of the distance between three interacting nuclei, i, j, and k.
Here, Z0 and Zi are the unit vectors along the internuclear distance between nuclei i and j, and along the axis of symmetry of the nuclear shielding tensor of nucleus i, respectively (cf. Fig. 2). Special cases. 1. Homonuclear dipolar interaction: K2 = Kj = K4 = 0. 1.1. Two spins (zj): rik = rjk = cc and hence Auk = 0 or K, = 0 or MMAS = 0. 1.2. Three spins at the corners of an equilateral triangle: rik = rij =
rjk
= r;
ai = 60”.
Therefore K, = 35 / 1536 = 0.023. The line narrowing factor F is defined as F = [M/MM*S]
112
and therefore in this case F = 6.6~4 rM11.
FIG. 2. Zi, 3, and i5g are unit vectors. The first two are along the axes of symmetry of the shielding tensors of the interacting nuclei i and j, respectively, and Ltii is along the internuclear radius vector.
RESIDUAL
LINEWIDTHS
UNDER
95
MAS
Assuming a Gaussian lineshape for the static sample the FWHM MAS becomes
linewidth
under
AU MASz Ao2/ 15.6~~. 2. Heteronuclear dipolar interaction, and/or anisotropy of the chemical shift:
K, = K2 = K3 = K4 = 0 or
3. Homonuclear
plus heteronuclear dipolar interaction:
K4 = 0. 3.1. Two resonant and one nonresonant nuclei at the corners of an equilateral triangle:
K, = K3 = K4 = 0 K2 = 251288= 0.087 from which it follows that the line-narrowing
factor is
F = 3.4[(M,I + MIs)/MIIM&'~wR. 3.2. One resonant (I = 1) and two nonresonant (S = 1) nuclei at the corners of an equilateral triangle:
K, = K2 = Kg = 0 K3 = 251288 F = 3.4wR/rMss. 4. Homonuclear
dipolar interaction plus shielding anisotropy:
K2 = KS = 0;
K, (cf. Section 1).
K4 depends upon the relative orientation of the axes of symmetry of the shielding tensors and the internuclear vector. Some values of C, are collected in Table 1. TABLE
1
a,
0"
60”
90”
120”
0" 60” 90” 120”
0 0.033 0.020 0.033
0.033 0.087 0.023 0
0.020 0.023 0 0.023
0.033 0 0.023 0.087
Note. The angles nj and (Y; are defined
in Fig. 2.
96
BRUNNER
ET
AL.
As a general result, it can be seen (Eq. [ 22b]). that in all cases where the axes of symmetry of the shielding tensors in the pair o&interacting spins are oriented in the same direction ( P; = Zj), then K4 vanishes. EXPERIMENTAL
Crystalline water in nearly anhydrous CaS04. Gypsum, CaS04. 2Hz0, has long been used as the classical case of an ensemble of isolated spin-i systems. Gypsum itself, however, has sufficient many-body proton-proton interactions such that there is always a residual broadening from the collection of waters of crystallization surrounding the proton pair under consideration. As has been seen for an ensemble of isolated pairs of spin-4 nuclei, KI = K2 = K3 = 0. To calculate K4 it is assumed that the axis of symmetry of the shielding tensor is along the -OH bond. The H-O-H angle in the free water molecule (6) is 104.5” yielding K4 = 0.108. In crystalline water, where the angle is slightly enlarged ( 7), with the interproton distance increased from 0.152 to 0.154 nm, and assuming a constant -0-H distance of 0.096 nm, the increase in H-O-H angle is calculated to be 2”, yielding K4 = 0.106. Using a value (8) of 12.2 ppm for 6cSA, one calculates McsA = 1.06 X 10’ ss2, yielding MMAS = 2.16 X 1017 s-‘/w& Assumption of a Gaussian lineshape yields a line of FWHM AMMOS= (,-2j@‘*Q
In 2)“2 = 27.7 X lo6 s-~/Y~.
One would therefore need a spinning rate of about 18.5 kHz to achieve a linewidth of 1.5 kHz as observed in Fig. 3. However, it is known ( 7) that in gypsum at room temperature the protons of the water molecules are exchanging at a rate corresponding to a mean correlation time of 10 I.LS.In this case, the shielding tensors of both protons take the same average alignment, and K4 vanishes. Therefore a spinning rate of only 3.5 kHz, used in the present experiments, is sufficient to achieve the required narrowing. In the present system, K2 and K3 are zero, so the linewidth is that associated with dipolar broadening due to residual water molecules in the system. Similar experimental results have been published by Yesinowski et al. (9). Bridging OH groups in zeolites. Bridging OH groups in zeolites and related catalysts play a decisive role in catalytic reactions since they act as Broensted acid sites (proton donators). Moreover, it has been shown ( 10) that the isotropic chemical shift, &, of these groups is a quantitative measure of their catalytic activity, i.e., their acid strength. It is therefore of practical interest to enhance the resolution of their ‘H MAS NMR spectra. The structure of these groups is shown in Fig. 4. Using a proton shielding anisotropy of that for crystalline water it may be seen that the contribution of this broadening to the total linewidth is less than 20%. Therefore in the following, the anisotropy of the chemical shift is neglected so that in addition to K1 = K3 = 0, K4 is also zero. Assuming a statistical (powder average) distribution for (Y, it is necessary to use not K2 but (K2) in order to calculate MMAS, where ( ) denotes the average
RESIDUAL
LO
30
20
LINEWIDTHS
10
0 v
-
UNDER
-10
97
MAS
-20
-30
-Lo
v,/KHz
FIG. 3. ‘H MAS NMR spectrum of a powder sample of partially rehydrated CaSO., crystallites measured at room temperature and a resonance frequency of 220 MHz. The spinning rate was 4.5 kHz.
over (Y. Then it can be shown that for the interval rHA1
G
THH
G
l@+HAl,
which may reasonably well approximate the real distribution (&)
in such systems,
= 0.052 I!I 0.004.
For the homonuclear contribution of the dipolar interaction to the second moment, the value MI1 = 39 X lo6 se2 will be used. This has been determined experimentally
FIG. 4. Interaction between the proton of a bridging OH group H(‘) and a neighboring proton H’*). The silicon, aluminum, oxygen, and H(l) atoms are coplanar. The H “‘-Al distance (10, II) is 0.238 nm.
98
BRUNNER
ET AL.
(11) for a zeolite with Si/Al = 2.6. According to the distance rnAi = 0.238 nm, which has also been determined experimentally by NMR measurements (11, 12), the heteronuclear dipolar contribution to the second moment is (cf. Eq. [ 71 with S = 2) M*s = 500 x lo6 s-* leading to &fMAS N = lOI s-‘/C&. Assuming a Gaussian lineshape, the FWHM linewidth is calculated to be AuMAS = (7r-*MMAS2 In 2)“*
= 1.9 X lo6 s-*/vR.
A spinning rate of 3 kHz therefore yields a linewidth of 630 Hz, which is in agreement with the experimental results. The proton signals associated with the two types of bridging hydroxyl groups in HY zeolites, which are commonly denoted as lines (b) and (c), corresponding to the so-called HF and LF stretching vibration bands (IO), are separated by about 1 ppm, or 300 Hz at a resonance frequency of 300 MHz. A conventional lineshape analysis ( Linesim / MSL 300 ) of the spectrum shown in Fig. 5A yields a linewidth of about 600-700 Hz for these two lines. According to the above theory, this linewidth should decrease with decreasing proton content, i.e., increasing proton-proton distance ( rnn in Fig. 4). In fact Fig. 5B shows for a partially deuterated sample, apart from the signals (a) and (d) due to silanol groups and residual ammonium ions, the two lines (b) and (c) well separated.
(CA(b)
6Ippm
10
5
0
FIG. 5. ‘H MAS NMR spectra of a H-Y zeolite with a silicon-to-aluminum ratio of 2.6 measured at room temperature and at a resonance frequency of 300 MHz. The spinning rate was 3 kHz. (A) Nondeutemted sample; (B) partially deuterated sample.
RESIDUAL
LINEWIDTHS
UNDER
MAS
99
REFERENCES 1. 2. 3. 4.
B. J. E. B.
5. B. 6. 7. 8. 9.
L. C. R. J. 10. D. Il. D. 12. R.
C. GERSTEIN, Anal. Chem. 55,78lA, 899A (1983). BOHM, D. FENZKE, AND H. PFEIFER, J. Magn. Reson. 55, 197 ( 1983 ). BRUNNER, Ph.D. thesis, Karl-Marx-Universitit, Leipzig, DDR, 1989. C. GERSTEIN AND C. R. DYEOWSKI, “Transient Techniques in NMR of Solids,” Chap. 4, Academic Press, San Diego, 1985. C. GERSTEIN AND C. R. DYBOWSKI, “Transient Techniques in NMR of Solids,” Chap. 5, Sect. II, Academic Press, San Diego, 1985. PAULING, “The Nature of the Chemical Bond,” Cornell Univ. Press, Ithaca, New York, 1967. L. MCKNETT, C. R. DYBOWSKI, AND R. W. VAUGHAN, J. Chem. Phys. 63,4578 (1975). DITCHFIELD, J. Chem. Phys. 65,3 123 ( 1976). P. YESINOWSKI, H. ECKERT, AND G. R. ROSSMAN, J: Am. Chem. Sot. 110, 1367 ( 1988). FREUDE, M. HUNGER, AND H. PFEIFER, Z. Phys. Chemie (NF) 152, 171 ( 1987). FREUDE, J. KLINOWSKI, AND H. HAMDAN, Chem. Phys. Left. 149,355 ( 1988). L. STEVENSON, J. Catal. 21, 113 ( 1971).
OF MAGNETIC
RESONANCE
w,90-99
( 1990)
Residual Linewidths of NMR Spectra of Spin-4 Systems under Magic-Angle Spinning E. BRUNNER,*
D. FREUDE,*
* Karl-Marx-Universiti and tAmes Laboratory,
B. C. GERSTEIN,~~$
Leipzig, Linntfstrasse U.S. Department Iowa State University,
AND H. PEEIEER*
5, Leipzig, German Democratic Republic. of Energy, and Department of Chemistry, Ames, Iowa 50010
Received January 26, 1990 The residual linewidths of NMR spectra of spin-f systems in solids under magic-angle spinning conditions are calculated in the rigid lattice limit for the general case where both home- and heteronuclear dipolar interactions aswell as shielding anisotropy may be present. The results are used to quantitatively interpret ‘H MAS NMR spectra of crystalline water in gypsum and of hydroxyl groups in zeolites. o 1990 Academic PPS, hc.
The recent decade has seen the development of high-resolution ( 1) . The present work considers sources of residual line broadening in solids in the case where secular portions of shielding anisotropy homonuclear dipolar interactions are present, under conditions of ning, but no strong radiofrequency pulse decoupling.
solid-state NMR for spin-4 nuclei and hetero- and magic-angle spin-
THEORY
The general approach taken in calculating residual linewidths under MAS for the types of systems considered in the present work is to calculate, from the envelope of the time decay function, the second moment under MAS. A Gaussian form is then assumed for the lineshape function of the residual line, from which the F?VHM of the line is calculated. In order to calculate the envelope of the time decay, coherent averaging theory is used to relate static Hamiltonians to Hamiltonians under MAS. The internal interactions considered in this work are homo- and heteronuclear dipolar interactions and the shielding anisotropy of the resonant spin, I: ti
= %I, + @Is + c%& + %&,.
[ll
All Hamiltonians are expressed in units of radians per second. To avoid consideration of additional line broadening which occurs if the quadrupolar interaction of the nonresonant spins (for S # 1) is not much lessthan their Zeeman energy (2)) a sufficiently high magnetic field (Bs 1, 7T for ‘H coupled to 27A1 via a bridging oxygen in the present work) is assumed. Moreover, since it can be shown theoretically (3) that the
$ Work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Chemical Science Division, under Contract W-740%Eng-82. 90 0022-2364190
$3.00
Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
RESIDUAL
LINEWIDTHS
UNDER
91
MAS
nonsecular portions of these interactions affect the residual linewidths of spin-3 nuclei in solids only if the linewidth achieved under spinning is 10 5 smaller than that of the static spectrum, a condition beyond present capabilities, only the secular portion of %’ will be retained. Hence
*I = g rfh $ ! rG3( 1 - 3 COS200){
3Zz,Zzj - IiIj1
[21
bJ
and similarly for X& by interchanging I and S, &Is = 5 YlTsh $ 2 rG3( 1 - 3 cos20j)Z~jS~~
A&*
.i k ‘% i 7 { (3 COS'Oj - 1) +
= YIB&*
~CS.b,Sin2BjCOS
[31
2$j} Zzj
[41
J
aCSA= &I,/ - (1/3)tr{2) TICSA
and &,I,
=
(bx’
-
‘$y’)/(‘bz’
-
bA)
. . . are the principal values of the shielding tensor, with ISzyl 2 16y’y’l 2 (6,yl.
In order to calculate residual linewidths under conditions of MAS, we consider the calculation of the second moment, M, of the static powder sample, M=
-(Tr{[z, =
MI
+
C Ljl}2/Tr{ MS
+
C Zxj)2)powder
[51
McsA,
where (3) ‘y;h’Z(Z
+ 1) 2 ri6 i,i
MIS = 6 ($‘r:r:h’S(S
[61
+ 1) ; ; r,;(” i K
[71
1 MCSA
=
j
(YIBo&sA)
‘(
[81
1 +;&A).
In the evaluation of the second moment of the spinning sample, X, given by Eq. [I] in the most general case, becomes time dependent. Under these conditions the envelope of the time decay after a 90; pulse, with phase detection along x in the rotating frame, is given by dMAS(t) = (L(t))
= tr{ u(l) C
Lju-'(t)
C
Zxj}/tr{
C
ZZj>
[91
with [lOI
92
BRUNNER
ET
AL.
where T denotes the Dyson time-ordering operator (4). Since s(t) is periodic and CyCliCwith period tc = 2n/ wa, with wR being the angular rotational frequency, average Hamiltonian theory (5) may be applied to evaluate the effect of sample rotation at times corresponding to multiples of the rotational periods. At times n tc, the propagator, in the absence of irreversible relaxation, is u(n&)
= exp{-i%MC>
=exp(-int,[%(“)+
&(‘)+*
- .I}
Ull
and tc
SF(O) = (l/t,)
11la1
dt’rE”( t’) s0 t’
tc
SF(‘) = (-i/2t,)
dt’
%(f’)l.
dt”[S(t’),
s0
s0
11lb1
Furthermore, if tip is defined as the difference between the time-dependent Hamiltonian governing the time evolution of the system, Y?(t), and the constant Hamiltonian &definedinEq. [ll], S(t)
= tip + 2%
[I21
then the time dependence of the density operator is idp(t)/dt
The usual transformations, &, with
= [ti,
p] = [SP + &, p(t)].
iI31
first to the frame of tip and then to the frame of $@ = u-‘&u P
P,
yield p(t) = upUp(0)ir’u,’
1141
with U= Texp-[i~dt’&(t’)).
[I51
The time decay function is (JMAS(t) = tr{exp(-i&t)
C ZXjexp(i&t)u,‘(t)
C ZXjUP(t)}/tr{(C
Zxj)*).
[16]
Since the total internal Hamiltonian associated with spinning of the sample is periodic and cyclic with period tc = 2n/ wR, so will be sp. Therefore at times II t, , from time zero, measured as the time immediately after a 90; pulse places the magnetization of the ensemble along x in the rotating frame, up( nt,, 0) = 1. The envelope of the time decay function (i.e., the values of the decay observed at cycle times nt,, where up = 1) is described by 4MAS(t) = tr{exp(-i&t)
2
Z,jeXp(L%t)
2 Zxj}/tr((
2 Zxj)*}.
[I71
RESIDUAL
The second moment evaluated to be
LINEWIDTHS
UNDER
93
MAS
of the central line is described by this decay function and is
MMAS = (-tr(f.s,
C
~xj12}/tr{(C
[I81
Lj)2})powder.
At periods n tc = 27rn/wR after the 90; pulse, the first term in the Magnus expansion, 2(O), Eq. [ 1 la], vanishes. Hence, for sufficiently high spinning speeds, the leading term in the residual linewidth is given by the second term, Eq. [ 11b] : ,g g $(I).
[I91
It then follows that kfMAS
=
(l/&K&I
+
K&f,,&
+
K&fSShfiS
+
&b&k&A).
[201
Other products of the static second moments, Eqs. [ 6]-[ 81, do not contribute to MMAS . Mss is defined analogously to Eq. [ 18 ] by interchanging I and S. The coefficients K,-K4 depend upon geometrical parameters. For the pure dipolar [22a] interactions, the values of k-,-K3 are given by K, = (2 rG6)-' $ Alrk i,J
K2
=
N
Ns
i
2 k
(2
PM
I,.i,k
r$-’
NI
Ns
2 i,j
c
[21bl
Aij/c k
[21cl
[21dl aok = (7 + 26
- 33 cos4ai)/60
COS2(Yi
a$
+
2
j
rii r- ( COS3CYi tk
COS (Yi)
WeI .
The plus sign holds for K1, and the minus sign for Kz and K3. The meanings of the symbols are further illustrated in Fig. 1. For the present work, we consider K4 only in the case of axial symmetry for the shielding tensor. This assumption is justified on the basis of known hydroxyl tensors being axial, or almost axial, with anisotropies near 15 ppm. With VCsA= 0, K4 = (2 r;6)-’
i,j
s r;6Cij
[22al
i,i
.[33( 3,. ~i)( ~ii'~j)
+ 7( 2;' Zj)] + 7[ fo(3i X
Zj)]'}.
Wbl
94
BRUNNER
ET AL. P 0
t. JP t pk
a.3
j
:
i?. Jk
0
k
FIG. 1. Definition of the angle ai, and of the distance between three interacting nuclei, i, j, and k.
Here, Z0 and Zi are the unit vectors along the internuclear distance between nuclei i and j, and along the axis of symmetry of the nuclear shielding tensor of nucleus i, respectively (cf. Fig. 2). Special cases. 1. Homonuclear dipolar interaction: K2 = Kj = K4 = 0. 1.1. Two spins (zj): rik = rjk = cc and hence Auk = 0 or K, = 0 or MMAS = 0. 1.2. Three spins at the corners of an equilateral triangle: rik = rij =
rjk
= r;
ai = 60”.
Therefore K, = 35 / 1536 = 0.023. The line narrowing factor F is defined as F = [M/MM*S]
112
and therefore in this case F = 6.6~4 rM11.
FIG. 2. Zi, 3, and i5g are unit vectors. The first two are along the axes of symmetry of the shielding tensors of the interacting nuclei i and j, respectively, and Ltii is along the internuclear radius vector.
RESIDUAL
LINEWIDTHS
UNDER
95
MAS
Assuming a Gaussian lineshape for the static sample the FWHM MAS becomes
linewidth
under
AU MASz Ao2/ 15.6~~. 2. Heteronuclear dipolar interaction, and/or anisotropy of the chemical shift:
K, = K2 = K3 = K4 = 0 or
3. Homonuclear
plus heteronuclear dipolar interaction:
K4 = 0. 3.1. Two resonant and one nonresonant nuclei at the corners of an equilateral triangle:
K, = K3 = K4 = 0 K2 = 251288= 0.087 from which it follows that the line-narrowing
factor is
F = 3.4[(M,I + MIs)/MIIM&'~wR. 3.2. One resonant (I = 1) and two nonresonant (S = 1) nuclei at the corners of an equilateral triangle:
K, = K2 = Kg = 0 K3 = 251288 F = 3.4wR/rMss. 4. Homonuclear
dipolar interaction plus shielding anisotropy:
K2 = KS = 0;
K, (cf. Section 1).
K4 depends upon the relative orientation of the axes of symmetry of the shielding tensors and the internuclear vector. Some values of C, are collected in Table 1. TABLE
1
a,
0"
60”
90”
120”
0" 60” 90” 120”
0 0.033 0.020 0.033
0.033 0.087 0.023 0
0.020 0.023 0 0.023
0.033 0 0.023 0.087
Note. The angles nj and (Y; are defined
in Fig. 2.
96
BRUNNER
ET
AL.
As a general result, it can be seen (Eq. [ 22b]). that in all cases where the axes of symmetry of the shielding tensors in the pair o&interacting spins are oriented in the same direction ( P; = Zj), then K4 vanishes. EXPERIMENTAL
Crystalline water in nearly anhydrous CaS04. Gypsum, CaS04. 2Hz0, has long been used as the classical case of an ensemble of isolated spin-i systems. Gypsum itself, however, has sufficient many-body proton-proton interactions such that there is always a residual broadening from the collection of waters of crystallization surrounding the proton pair under consideration. As has been seen for an ensemble of isolated pairs of spin-4 nuclei, KI = K2 = K3 = 0. To calculate K4 it is assumed that the axis of symmetry of the shielding tensor is along the -OH bond. The H-O-H angle in the free water molecule (6) is 104.5” yielding K4 = 0.108. In crystalline water, where the angle is slightly enlarged ( 7), with the interproton distance increased from 0.152 to 0.154 nm, and assuming a constant -0-H distance of 0.096 nm, the increase in H-O-H angle is calculated to be 2”, yielding K4 = 0.106. Using a value (8) of 12.2 ppm for 6cSA, one calculates McsA = 1.06 X 10’ ss2, yielding MMAS = 2.16 X 1017 s-‘/w& Assumption of a Gaussian lineshape yields a line of FWHM AMMOS= (,-2j@‘*Q
In 2)“2 = 27.7 X lo6 s-~/Y~.
One would therefore need a spinning rate of about 18.5 kHz to achieve a linewidth of 1.5 kHz as observed in Fig. 3. However, it is known ( 7) that in gypsum at room temperature the protons of the water molecules are exchanging at a rate corresponding to a mean correlation time of 10 I.LS.In this case, the shielding tensors of both protons take the same average alignment, and K4 vanishes. Therefore a spinning rate of only 3.5 kHz, used in the present experiments, is sufficient to achieve the required narrowing. In the present system, K2 and K3 are zero, so the linewidth is that associated with dipolar broadening due to residual water molecules in the system. Similar experimental results have been published by Yesinowski et al. (9). Bridging OH groups in zeolites. Bridging OH groups in zeolites and related catalysts play a decisive role in catalytic reactions since they act as Broensted acid sites (proton donators). Moreover, it has been shown ( 10) that the isotropic chemical shift, &, of these groups is a quantitative measure of their catalytic activity, i.e., their acid strength. It is therefore of practical interest to enhance the resolution of their ‘H MAS NMR spectra. The structure of these groups is shown in Fig. 4. Using a proton shielding anisotropy of that for crystalline water it may be seen that the contribution of this broadening to the total linewidth is less than 20%. Therefore in the following, the anisotropy of the chemical shift is neglected so that in addition to K1 = K3 = 0, K4 is also zero. Assuming a statistical (powder average) distribution for (Y, it is necessary to use not K2 but (K2) in order to calculate MMAS, where ( ) denotes the average
RESIDUAL
LO
30
20
LINEWIDTHS
10
0 v
-
UNDER
-10
97
MAS
-20
-30
-Lo
v,/KHz
FIG. 3. ‘H MAS NMR spectrum of a powder sample of partially rehydrated CaSO., crystallites measured at room temperature and a resonance frequency of 220 MHz. The spinning rate was 4.5 kHz.
over (Y. Then it can be shown that for the interval rHA1
G
THH
G
l@+HAl,
which may reasonably well approximate the real distribution (&)
in such systems,
= 0.052 I!I 0.004.
For the homonuclear contribution of the dipolar interaction to the second moment, the value MI1 = 39 X lo6 se2 will be used. This has been determined experimentally
FIG. 4. Interaction between the proton of a bridging OH group H(‘) and a neighboring proton H’*). The silicon, aluminum, oxygen, and H(l) atoms are coplanar. The H “‘-Al distance (10, II) is 0.238 nm.
98
BRUNNER
ET AL.
(11) for a zeolite with Si/Al = 2.6. According to the distance rnAi = 0.238 nm, which has also been determined experimentally by NMR measurements (11, 12), the heteronuclear dipolar contribution to the second moment is (cf. Eq. [ 71 with S = 2) M*s = 500 x lo6 s-* leading to &fMAS N = lOI s-‘/C&. Assuming a Gaussian lineshape, the FWHM linewidth is calculated to be AuMAS = (7r-*MMAS2 In 2)“*
= 1.9 X lo6 s-*/vR.
A spinning rate of 3 kHz therefore yields a linewidth of 630 Hz, which is in agreement with the experimental results. The proton signals associated with the two types of bridging hydroxyl groups in HY zeolites, which are commonly denoted as lines (b) and (c), corresponding to the so-called HF and LF stretching vibration bands (IO), are separated by about 1 ppm, or 300 Hz at a resonance frequency of 300 MHz. A conventional lineshape analysis ( Linesim / MSL 300 ) of the spectrum shown in Fig. 5A yields a linewidth of about 600-700 Hz for these two lines. According to the above theory, this linewidth should decrease with decreasing proton content, i.e., increasing proton-proton distance ( rnn in Fig. 4). In fact Fig. 5B shows for a partially deuterated sample, apart from the signals (a) and (d) due to silanol groups and residual ammonium ions, the two lines (b) and (c) well separated.
(CA(b)
6Ippm
10
5
0
FIG. 5. ‘H MAS NMR spectra of a H-Y zeolite with a silicon-to-aluminum ratio of 2.6 measured at room temperature and at a resonance frequency of 300 MHz. The spinning rate was 3 kHz. (A) Nondeutemted sample; (B) partially deuterated sample.
RESIDUAL
LINEWIDTHS
UNDER
MAS
99
REFERENCES 1. 2. 3. 4.
B. J. E. B.
5. B. 6. 7. 8. 9.
L. C. R. J. 10. D. Il. D. 12. R.
C. GERSTEIN, Anal. Chem. 55,78lA, 899A (1983). BOHM, D. FENZKE, AND H. PFEIFER, J. Magn. Reson. 55, 197 ( 1983 ). BRUNNER, Ph.D. thesis, Karl-Marx-Universitit, Leipzig, DDR, 1989. C. GERSTEIN AND C. R. DYEOWSKI, “Transient Techniques in NMR of Solids,” Chap. 4, Academic Press, San Diego, 1985. C. GERSTEIN AND C. R. DYBOWSKI, “Transient Techniques in NMR of Solids,” Chap. 5, Sect. II, Academic Press, San Diego, 1985. PAULING, “The Nature of the Chemical Bond,” Cornell Univ. Press, Ithaca, New York, 1967. L. MCKNETT, C. R. DYBOWSKI, AND R. W. VAUGHAN, J. Chem. Phys. 63,4578 (1975). DITCHFIELD, J. Chem. Phys. 65,3 123 ( 1976). P. YESINOWSKI, H. ECKERT, AND G. R. ROSSMAN, J: Am. Chem. Sot. 110, 1367 ( 1988). FREUDE, M. HUNGER, AND H. PFEIFER, Z. Phys. Chemie (NF) 152, 171 ( 1987). FREUDE, J. KLINOWSKI, AND H. HAMDAN, Chem. Phys. Left. 149,355 ( 1988). L. STEVENSON, J. Catal. 21, 113 ( 1971).