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Two-dimensional NMR separation of inhomogeneous from from homogeneous lineshapes in the glassy state. 87Rb in a proton glass DRADP
JOURNAL
OF MAGNETIC
RESONANCE
92, 3 12-3 19 ( 199 1)
Two-Dimensional NMR Separationof Inhomogeneous from HomogeneousLineshapesin the Glassy State. s7Rbin a Proton Glass DRADP J. DOLINSEK J. Stefan Institute, E. Karderj University of Ljubljana, Jamova 39, 61 I I I Ljubljana,
Yugoslavia
Received June 2 1, 1990; revised October 25, 1990 In systems with a glassy type of disorder NMR absorption lines are strongly inhomogeneously broadened, thus completely masking the shape of a homogeneous line. A twodimensional NMR technique is capable of showing in one frequency domain the inhomogeneous frequency distribution characteristic of a glassy disorder, which is static on the NMR timescale, whereas in the second frequency domain the undistorted homogeneous lineshape appears. The inhomogeneous lineshape is a convolution of the frequency distribution functionf( v), which is known theoretically for different types of glasses,and the homogeneous lineshape. The 2D technique thus makes it possible to obtain the experimental f(v) via a deconvolution of the inhomogeneous lineshape with the homogeneous one. 0 1991 Academic Press. Inc.
Strongly disordered materials like glasses are characterized by the fact that, on a small spatial scale (a few unit cells), the system of molecules looks ordered as in a crystal, whereas on a large scale there is no long-range order. This is a consequence of the fact that some local atomic property shows a distortion from its value in a perfect lattice. The distortion is randomly distributed from site to site according to some probability distribution function. In a NMR experiment this random modulation implies that the NMR resonance frequency of a given type of nucleus becomes space dependent since it reflects the disorder of the lattice via the magnetic and electrostatic couplings to the surrounding ions. We thus have in glasses an inhomogeneous distribution of resonance frequenciesf( u) for each physically nonequivalent site in the unit cell instead of a single homogeneous resonance line as in translationally invariant single crystals. The inhomogeneous broadening of the resonance lines has a different origin for different types of glasses. In magnetic spin glasses the randomly distributed local magnetic fields are responsible, whereas with insulating glasses (e.g., electric dipole, quadrupole, and orientational glasses) the randomly distributed local electric field gradients interact with nuclei of nonzero electric quadrupole moment. Because of the spatial disorder, electric field gradient (EFG) tensor components are distributed in both orientation and magnitude. The resulting inhomogeneity of the electric quadrupole interaction is essentially static in nature and reflects quenched random disorder in the lattice. A study of the inhomogeneous frequency distribution f(u) provides useful information with respect to static disorder in these glassy systems (I, 2). There is, 0022-236419 I $3.00 Copyright 0 1991 by Academic PT.%+Inc. All rights of reproduction in any form reserved.
312
SEPARATION
OF INHOMOGENEOUS
313
LINESHAPES
however, an important question concerning homogeneous lineshapes in such systems since the homogeneous lines are, in fact, distributed with the frequency distribution function f( v). The observed inhomogeneous NMR lineshape F(V) is a convolution of the frequency distribution functionjj V) with the homogeneous lineshape thus making it necessary to make a deconvolution of F( u) with the homogeneous lineshape to obtain pure f( v). In view of that there arises the question of how to experimentally observe the homogeneous lineshape which is inhomogeneously broadened by the static smearing of the electric quadrupole interaction. A two-dimensional NMR inhomogeneous vs homogeneous lineshape separation technique provides an answer to this problem. This 2D technique is, in essence, a homonuclear “separation of interactions” experiment. It is discussed here for the case of inhomogeneous quadrupolar interactions with special emphasis on the central transition ( 1 --t - 1) of nuclei with noninteger spins. It is shown that the effect of the inhomogeneous quadrupole interaction, since it is static, can be refocused with an appropriate pulse sequence to give a zero effect on the spectrum in the o,-frequency domain, whereas it is allowed to act fully in the 02frequency domain. A fictitious-spin- f formalism is appropriate to describe the central transition f --, - 1 when the quadrupole interaction is sufficiently strong that only the central transition is irradiated by the RF pulse excitation spectrum, whereas the satellites remain unaffected. A comparison with experiment is made by observing “Rb 2D inhomogenous vs homogeneous lineshape resolved spectra in a proton glass Rbi-,( ND4)xD2P04 with x = 0.44 (DRADP-44) at an orientation c 11Ho. It is shown that the homogeneous lineshape is Lorentzian while the broad, highly asymmetric, inhomogeneous lineshape is well described by a theory of Blinc et al. (1, 2). FORMAL
THEORY
OF
QUADRUPOLE ECHOS OF NONINTEGER
FOR THE SPINS
f +
-f
TRANSITION
Since the inhomogeneity of the quadrupole interaction is essentially static in nature, it is possible to apply a transient spin-echo method to refocus the damped free precession of nuclear magnetization caused by the spatial distribution of EFG tensors throughout the sample. Because the symmetry of the quadrupole Hamiltonian is different from that describing the Zeeman interaction in an inhomogeneous magnetic field, the usual classical explanation of the damped free precession of magnetic moments and their refocusing cannot be used and a quantum description is appropriate. A first-order perturbation treatment of quadrupole-perturbed Zeeman energy levels (3) shows that, for spin quantum number I, several allowed and forbidden echoes are formed in time. The central transition 4 + - 1 is unaffected by the first-order perturbation and is thus relatively insensitive to the inhomogeneous quadrupole interaction. However, as the strength of the quadrupole interaction increases, second-order effects influence the central transition. In this case, the satellite transitions are usually substantially displaced so that the excitation spectrum of an RF pulse extends only over the central transition. The problem is then reduced to a two-level system which can be treated by the fictitiousspin- ) formalism as explained in Ref. (3). In this formalism one deals with a system which has two degrees of freedom where
314
J. DOLINSEK
all the relevant physical observables can be represented by 2 X 2 matrices. Each operator Q can be written as
ill
Q = $qo+q*s,
where q. = tr { Q } inside the two-level manifold, and s = 4 0 represents the fictitious spin operator, with a,, uY, u, the usual Pauli matrices. The values of q are determined from the matrix elements of Q. The Hamiltonian can be thus written as S = $Eo - y’h(H’-
s),
121
where H’ and y’ are the fictitious magnetic field and the fictitious gyromagnetic ratio. The Hamiltonian in the rotating frame is a sum of a quadrupolar term and a term describing an RF field: 2@ = 2,
+ SRF = e-i~fz~Qeiootlz
- y hH, Z,.
131
Here w. = yHo and a! = x, y represents the direction of the RF field. By comparing the matrix elements of #a, inside the two-level manifold and the matrix element of Eq. [ 21, we write SaF in the fictitious-spin- f form ti cF = -rh(Z
+ $)Hg,,
[41
where s, = f ua. The quadrupolar term is transformed into the fictitious-spin formalism in the following way. We take &o to represent the full quadrupolar Hamiltonian in a general frame of reference,
.q =4z(2ep_ 1) Ko(3Zf
-
Z2)
+
v+l(z-zz
+
IX1
+ V-,(ZfZz + z,z+) + v+,(z-y
where VO = V,,, V&r = V, f iv,, of&o are ( WI&QI
(-1/2(*o] (1/21&o] (-1/21&o]
+ v-,(z+)2],
[5]
V+z = 1 (V,, - V,) + iv,... The matrix elements K/2
Wal
- l/2)
= E-,,2
[6bl
- l/2)
= e-i”0’(1/2]rEOo]
- l/2)
[6cl
= eiwo’(-1/2]tio]
l/2).
[6dl
W)
l/2)
=
Equations [6c] and [6d] are equal to zero since the matrix elements of So are zero. However to include the effect of the inhomogeneous quadrupole interaction on the central transition one must go to second-order perturbation theory for the quadrupole-perturbed Zeeman levels. There one has to correct the wavefunctions as well as the energy levels. Then Eqs. [ 6c] and [ 6d] may eventually become different from zero but will remain small compared to the diagonal terms, Eqs. [6a] and [ 6b]. Thus we can write, to a good approximation, (1/21&l
l/2)
= E1,2 = E;;; + E$
[W
LINESHAPES
315
- l/2)
= E-1/2 = ELI),, + EL:‘,,
[7bl
- l/2)
= (-l/21&(4
SEPARATION
OF INHOMOGENEOUS
(-1/2]S?,] (1/2I&QI
l/2)
= 0.
[7cl Ei” and Ej2’ represent first- and second-order quadrupole correction of the ith Zeeman level. Now we compare Eqs. [7a] - [ 7c] with the matrix elements of Eq. [ 21 and we get H; = H; = 0
WI
and Ew - y’H;
112
=
h
EL:‘,,
=--
@2’ h
.
After subtracting the constant part we get the following quadrupole in the fictitious-spin formalism: 9;
Hamiltonian
zz @2’s 2.
[91
AL?(*) can be found, for instance, in (4) to be &@2) = E;;; - E3,
=$
a2P2(
l/2),
0
where a = e2qQ/h and P2( l/2) = 31(VE)L2/eq12 - 6](VE’),,/eq12. The (V E)‘,i are linear combinations of EFG tensor elements. The total Hamiltonian can be now written as A?* = s@; + S’&
= AE’*‘s, - yh(Z + f)Hps,.
The signal induced in a coil by the precessing magnetization s(t) = Tr(p(t)Z+
1,
is proportional
[lOI to
[Ill
where p(t) is the density matrix describing the motion of the spin system in the rotating frame. When the system is in thermal equilibrium we have pq
a
h.
[I21
We assume that, during an RF pulse, .X& is much stronger than &G. We can relate the density matrix immediately after the pulse PI = (90”), to its equilibrium value as p(o) = ei(*12)~ysze-i(~12)~y= sxe At time have
t
= 7 we apply a second pulse P2 = (19)~.Just before the second pulse we p(7-) = e-~.+*hr/hS&+Fj~lhe
Immediately
[131
[141
after the second pulse Eq. [ 141 transforms into p(7+~
=
eiss,e-i~pb’/hSXei~bT/he-isSX,
[151
316
J. DOLINSEK
where0=yHI(Z+$).Att>7weget p(t) = e
-&$-7)/h
e i@s, epii,y/hSx.*
elJYb~jhe-rss,ei~fo(f-,)/~,
.
[I61
We now compute the trace in Eq. [ 1 I] within the + 1 manifold. The signal becomes a~(~)
=
1 eiA0(2)(f-27) 2
(-1/21&0”“( +
l/2)(-1/21e-‘@xI +iAw(*)l(-1/21~iB~~ 2
l/2) I - l/2)(
1/21e-i8Sxl l/2).
[17]
Here Awc2) = AEc2’/ ti. The matrix elements in Eq. [ 17 ] are the well-known Wigner rotation matrices O&( (Y,p, 7) and are tabulated in Ref. (5). By evaluating them we iset
1181 The first term represents the quadrupolar echo, since the phase factor at t = 27 equals unity and the signal is independent of the smeared quadrupole interaction. The echo is maximum for 0 = ?r, where the second term in Eq. [ 181 also vanishes. The shape of the echo is obtained by convoluting S(t) with the distribution function g(a) of quadrupole coupling constants a = e2qQ/h: E(t) =
mS(t)g(aW. s--co
1191
EXPERIMENT
A NMR technique used for separation of homogeneous and inhomogeneous lineshapes is the aforementioned 2D homonuclear separation of interactions method (6). The preparation period consists of a single 90” pulse. In the middle of the evolution period we apply a 180” pulse shifted in phase by 90”. The refocusing effect of this pulse is to eliminate, in the evolution period, the action of all parts of the total Hamiltonian which are linear in the Z, spin variable (chemical shifts, heteronuclear dipolar interaction, inhomogeneous external magnetic field) and the action of a much larger inhomogeneous quadrupole interaction. Thus the relevant Hamiltonian in the evolution period is only a homonuclear dipolar interaction. The detection period begins at the top of the echo. In that period the spin system is subjected to the full Hamiltonian. For the central transition of “Rb in DRADP-44, 256 incrementations of the evolution time tl were used. The Cyclops ( 7) phase cycle, together with superimposed phase alternation of the second pulse, was used to get rid of experimental artifacts. After a 2D complex Fourier transformation, a 2D spectrum appears. In the w2 domain, one finds the inhomogeneously broadened lineshape, whereas the o1 domain shows the homogeneous lineshape. The experiment was performed in a magnetic field of 6.3 T. The total acquisition time was about three hours and a magnitude spectrum was computed. DISCUSSION
In Fig. la a contour plot of the 87Rb 2D spectrum in DRADP-44 at orientation c 1)Ho and T = 140 K is shown. The projections on both frequency axes are also shown,
SEPARATION
OF INHOMOGENEOUS
;:II
317
LINESHAPES
IERTZ
-6000
1
0
-50000
HERTZ w2/2rI
FIG. 1. (a) A contour plot of a 2D spectrum of *‘Rb in DRADP-44 at c 11 Ha, vg(“Rb) = 88.34 MHz, T = 140 K. The w2 domain shows the inhomogeneously broadened lineshape, whereas in the wi domain one has the homogeneous lineshape. Projections on both frequency axes are also shown. (b) A three-dimensional plot of the spectrum shown in (a).
where the inhomogeneously broadened line appears in the w2 domain, whereas the homogeneous line is shown in the wI domain. At temperature T = 140 K the inhomogeneous line is strongly asymmetric and has a full width at half-height of 12 kHz, whereas the homogeneous linewidth is a Lorentzian with a full width at half-height of 700 Hz. In Fig. lb the same spectrum is shown in a three-dimensional plot. By lowering the temperature the inhomogeneous line continuously broadens, as shown in the temperature dependence plot of the o2 projections (Fig. 2a). At T = 33 K the full width of the o2 projection at half-height amounts to 25 kHz. The homogeneous linewidth, however, exhibits a different behavior. It increases slowly from its room temperature value of about 600 Hz and exhibits a maximum at the temperature of the minimum of spin-lattice relaxation time T, (at T = 85 K) (Fig. 3). With T = 73 K the homogeneous linewidth at half-height is 1900 Hz (Fig. 2b). At lower temperatures
318
J. DOLINSEK
& 50000
)( T=33 K 0 HERTZ
Wp12rI
-50000
10000
5000
0 HERTZ
-5000
w2/2rI
FIG. 2. Temperature dependence of the (a) w2 projection (inhomogeneous line) and (b) w, projection (homogeneous line) of the 2D spectrum of *‘Rb in DRADP-44 at c /)Ho. All lines are scaled to equal height.
the homogeneous width decreases again and reaches a value of 1000 Hz at 33 K. The same behavior was confirmed by measurement of the spin-spin relaxation time T2 using the Hahn-echo experiment. T2 is the inverse of the homogeneous linewidth (Fig. 3). Here T2 exhibits a minimum at a temperature approximately equal to the temperature of the T, minimum. Since the minimum value of T1 is very short (300 ps) it seems clear that the homogeneous linewidth maximum is a consequence of lifetime broadening. The fact that the homogeneous lineshape is more than 10 times narrower than the inhomogeneous one is a characteristic feature of glasses with static disorder. For insulating glasses, it was shown ( 1, 2, 8) that the inhomogeneous NMR lineshape F(V) can be written as a convolution of the frequency distribution function f( V) and the homogeneous lineshape L ( v ) :
The frequency distribution function f( V) has been computed theoretically (8). At the particular orientation c 1)Ho in DRADP, it was shown to be asymmetric since at this orientation the quadrupole-perturbed NMR frequency of a *‘Rb nucleus depends quadratically on the local order parameter p. This order parameter describes the polarization of the proton in a hydrogen bond representing an elementary electric dipole which orders randomly in the ferroelectric-antiferroelectric mixture DRADP. The monotonic broadening of the inhomogeneous line with decreasing temperature can be attributed to the appearance of an Edwards-Anderson-type order parameter (9,
SEPARATION
OF INHOMOGENEOUS
319
LINESHAPES
. 11; 72
I
[msl
It
.
10-2
I 5
f
11 WI.56
(WO.bb
02POb
v. ia7RbI =88.34MHz
I 10
I 15
I 20
I 25
I 30
I I I I 35 40 .'+5 50 lo3 IT IK-‘1
I
55
FIG. 3. Temperature dependence of the spin-lattice relaxation time T, and spin-spin relaxation time Tz of “Rb in DRADP-44 at c 11Ho.
10) describing the static disorder in the glassy state. In order to obtain the pure experimental frequency distribution function f( V) one must make a deconvolution of Rq. [ 201. This is possible since the 2D NMR homogeneous-inhomogeneous lineshape separation technique allows precise measurements of F( v) and L(v). REFERENCES 1. R. BLINC, D. C. AILION, B. GUNTHER, AND S. ZUMER, Phys. Rev. Lett. 57,2826 ( 1986). 2. R. BLINC, S. ZUMER, M. KOREN, AND D. C. AILION, Phys. Rev. B 37,7276 ( 1988). 3. A. ABRAGAM, “The Principles of Magnetic Resonance,” Oxford Univ. Press/Clarendon Press, London, 1960. 4. G. M. VOLKOFF, C’unad. .I. Phys. 31,820 (1953). 5. M. MEHRING, “High Resolution NMR Spectroscopy in Solids,” Springer, New York/Berlin, 1976. 6. R. R. ERNST, G. BODENHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic Resonance in One and Two Dimensions” Oxford Univ. Press, London/New York, 1987. 7. D. I. HOULT AND R. E. RICHARDS, Proc. R. Sot. London A 344, 311 ( 1975). 8. R. BLINC, J. DOLINSEK, R. PIRC, B. TADI~, B. ZALAR, R. KIND, AND 0. LIECHTI, Phys. Rev. Lat. 63, 2248(1989). 9. S.F.EDWARDSANDP.
W. ANDERSON, J. Phys.F5,965( 1975). 10. R. BLINC, J. DOLINSEK, V. H. SCHMIDT, AND D. C. AILION, Europhys. Lett. 6, 55 (1988).
OF MAGNETIC
RESONANCE
92, 3 12-3 19 ( 199 1)
Two-Dimensional NMR Separationof Inhomogeneous from HomogeneousLineshapesin the Glassy State. s7Rbin a Proton Glass DRADP J. DOLINSEK J. Stefan Institute, E. Karderj University of Ljubljana, Jamova 39, 61 I I I Ljubljana,
Yugoslavia
Received June 2 1, 1990; revised October 25, 1990 In systems with a glassy type of disorder NMR absorption lines are strongly inhomogeneously broadened, thus completely masking the shape of a homogeneous line. A twodimensional NMR technique is capable of showing in one frequency domain the inhomogeneous frequency distribution characteristic of a glassy disorder, which is static on the NMR timescale, whereas in the second frequency domain the undistorted homogeneous lineshape appears. The inhomogeneous lineshape is a convolution of the frequency distribution functionf( v), which is known theoretically for different types of glasses,and the homogeneous lineshape. The 2D technique thus makes it possible to obtain the experimental f(v) via a deconvolution of the inhomogeneous lineshape with the homogeneous one. 0 1991 Academic Press. Inc.
Strongly disordered materials like glasses are characterized by the fact that, on a small spatial scale (a few unit cells), the system of molecules looks ordered as in a crystal, whereas on a large scale there is no long-range order. This is a consequence of the fact that some local atomic property shows a distortion from its value in a perfect lattice. The distortion is randomly distributed from site to site according to some probability distribution function. In a NMR experiment this random modulation implies that the NMR resonance frequency of a given type of nucleus becomes space dependent since it reflects the disorder of the lattice via the magnetic and electrostatic couplings to the surrounding ions. We thus have in glasses an inhomogeneous distribution of resonance frequenciesf( u) for each physically nonequivalent site in the unit cell instead of a single homogeneous resonance line as in translationally invariant single crystals. The inhomogeneous broadening of the resonance lines has a different origin for different types of glasses. In magnetic spin glasses the randomly distributed local magnetic fields are responsible, whereas with insulating glasses (e.g., electric dipole, quadrupole, and orientational glasses) the randomly distributed local electric field gradients interact with nuclei of nonzero electric quadrupole moment. Because of the spatial disorder, electric field gradient (EFG) tensor components are distributed in both orientation and magnitude. The resulting inhomogeneity of the electric quadrupole interaction is essentially static in nature and reflects quenched random disorder in the lattice. A study of the inhomogeneous frequency distribution f(u) provides useful information with respect to static disorder in these glassy systems (I, 2). There is, 0022-236419 I $3.00 Copyright 0 1991 by Academic PT.%+Inc. All rights of reproduction in any form reserved.
312
SEPARATION
OF INHOMOGENEOUS
313
LINESHAPES
however, an important question concerning homogeneous lineshapes in such systems since the homogeneous lines are, in fact, distributed with the frequency distribution function f( v). The observed inhomogeneous NMR lineshape F(V) is a convolution of the frequency distribution functionjj V) with the homogeneous lineshape thus making it necessary to make a deconvolution of F( u) with the homogeneous lineshape to obtain pure f( v). In view of that there arises the question of how to experimentally observe the homogeneous lineshape which is inhomogeneously broadened by the static smearing of the electric quadrupole interaction. A two-dimensional NMR inhomogeneous vs homogeneous lineshape separation technique provides an answer to this problem. This 2D technique is, in essence, a homonuclear “separation of interactions” experiment. It is discussed here for the case of inhomogeneous quadrupolar interactions with special emphasis on the central transition ( 1 --t - 1) of nuclei with noninteger spins. It is shown that the effect of the inhomogeneous quadrupole interaction, since it is static, can be refocused with an appropriate pulse sequence to give a zero effect on the spectrum in the o,-frequency domain, whereas it is allowed to act fully in the 02frequency domain. A fictitious-spin- f formalism is appropriate to describe the central transition f --, - 1 when the quadrupole interaction is sufficiently strong that only the central transition is irradiated by the RF pulse excitation spectrum, whereas the satellites remain unaffected. A comparison with experiment is made by observing “Rb 2D inhomogenous vs homogeneous lineshape resolved spectra in a proton glass Rbi-,( ND4)xD2P04 with x = 0.44 (DRADP-44) at an orientation c 11Ho. It is shown that the homogeneous lineshape is Lorentzian while the broad, highly asymmetric, inhomogeneous lineshape is well described by a theory of Blinc et al. (1, 2). FORMAL
THEORY
OF
QUADRUPOLE ECHOS OF NONINTEGER
FOR THE SPINS
f +
-f
TRANSITION
Since the inhomogeneity of the quadrupole interaction is essentially static in nature, it is possible to apply a transient spin-echo method to refocus the damped free precession of nuclear magnetization caused by the spatial distribution of EFG tensors throughout the sample. Because the symmetry of the quadrupole Hamiltonian is different from that describing the Zeeman interaction in an inhomogeneous magnetic field, the usual classical explanation of the damped free precession of magnetic moments and their refocusing cannot be used and a quantum description is appropriate. A first-order perturbation treatment of quadrupole-perturbed Zeeman energy levels (3) shows that, for spin quantum number I, several allowed and forbidden echoes are formed in time. The central transition 4 + - 1 is unaffected by the first-order perturbation and is thus relatively insensitive to the inhomogeneous quadrupole interaction. However, as the strength of the quadrupole interaction increases, second-order effects influence the central transition. In this case, the satellite transitions are usually substantially displaced so that the excitation spectrum of an RF pulse extends only over the central transition. The problem is then reduced to a two-level system which can be treated by the fictitiousspin- ) formalism as explained in Ref. (3). In this formalism one deals with a system which has two degrees of freedom where
314
J. DOLINSEK
all the relevant physical observables can be represented by 2 X 2 matrices. Each operator Q can be written as
ill
Q = $qo+q*s,
where q. = tr { Q } inside the two-level manifold, and s = 4 0 represents the fictitious spin operator, with a,, uY, u, the usual Pauli matrices. The values of q are determined from the matrix elements of Q. The Hamiltonian can be thus written as S = $Eo - y’h(H’-
s),
121
where H’ and y’ are the fictitious magnetic field and the fictitious gyromagnetic ratio. The Hamiltonian in the rotating frame is a sum of a quadrupolar term and a term describing an RF field: 2@ = 2,
+ SRF = e-i~fz~Qeiootlz
- y hH, Z,.
131
Here w. = yHo and a! = x, y represents the direction of the RF field. By comparing the matrix elements of #a, inside the two-level manifold and the matrix element of Eq. [ 21, we write SaF in the fictitious-spin- f form ti cF = -rh(Z
+ $)Hg,,
[41
where s, = f ua. The quadrupolar term is transformed into the fictitious-spin formalism in the following way. We take &o to represent the full quadrupolar Hamiltonian in a general frame of reference,
.q =4z(2ep_ 1) Ko(3Zf
-
Z2)
+
v+l(z-zz
+
IX1
+ V-,(ZfZz + z,z+) + v+,(z-y
where VO = V,,, V&r = V, f iv,, of&o are ( WI&QI
(-1/2(*o] (1/21&o] (-1/21&o]
+ v-,(z+)2],
[5]
V+z = 1 (V,, - V,) + iv,... The matrix elements K/2
Wal
- l/2)
= E-,,2
[6bl
- l/2)
= e-i”0’(1/2]rEOo]
- l/2)
[6cl
= eiwo’(-1/2]tio]
l/2).
[6dl
W)
l/2)
=
Equations [6c] and [6d] are equal to zero since the matrix elements of So are zero. However to include the effect of the inhomogeneous quadrupole interaction on the central transition one must go to second-order perturbation theory for the quadrupole-perturbed Zeeman levels. There one has to correct the wavefunctions as well as the energy levels. Then Eqs. [ 6c] and [ 6d] may eventually become different from zero but will remain small compared to the diagonal terms, Eqs. [6a] and [ 6b]. Thus we can write, to a good approximation, (1/21&l
l/2)
= E1,2 = E;;; + E$
[W
LINESHAPES
315
- l/2)
= E-1/2 = ELI),, + EL:‘,,
[7bl
- l/2)
= (-l/21&(4
SEPARATION
OF INHOMOGENEOUS
(-1/2]S?,] (1/2I&QI
l/2)
= 0.
[7cl Ei” and Ej2’ represent first- and second-order quadrupole correction of the ith Zeeman level. Now we compare Eqs. [7a] - [ 7c] with the matrix elements of Eq. [ 21 and we get H; = H; = 0
WI
and Ew - y’H;
112
=
h
EL:‘,,
=--
@2’ h
.
After subtracting the constant part we get the following quadrupole in the fictitious-spin formalism: 9;
Hamiltonian
zz @2’s 2.
[91
AL?(*) can be found, for instance, in (4) to be &@2) = E;;; - E3,
=$
a2P2(
l/2),
0
where a = e2qQ/h and P2( l/2) = 31(VE)L2/eq12 - 6](VE’),,/eq12. The (V E)‘,i are linear combinations of EFG tensor elements. The total Hamiltonian can be now written as A?* = s@; + S’&
= AE’*‘s, - yh(Z + f)Hps,.
The signal induced in a coil by the precessing magnetization s(t) = Tr(p(t)Z+
1,
is proportional
[lOI to
[Ill
where p(t) is the density matrix describing the motion of the spin system in the rotating frame. When the system is in thermal equilibrium we have pq
a
h.
[I21
We assume that, during an RF pulse, .X& is much stronger than &G. We can relate the density matrix immediately after the pulse PI = (90”), to its equilibrium value as p(o) = ei(*12)~ysze-i(~12)~y= sxe At time have
t
= 7 we apply a second pulse P2 = (19)~.Just before the second pulse we p(7-) = e-~.+*hr/hS&+Fj~lhe
Immediately
[131
[141
after the second pulse Eq. [ 141 transforms into p(7+~
=
eiss,e-i~pb’/hSXei~bT/he-isSX,
[151
316
J. DOLINSEK
where0=yHI(Z+$).Att>7weget p(t) = e
-&$-7)/h
e i@s, epii,y/hSx.*
elJYb~jhe-rss,ei~fo(f-,)/~,
.
[I61
We now compute the trace in Eq. [ 1 I] within the + 1 manifold. The signal becomes a~(~)
=
1 eiA0(2)(f-27) 2
(-1/21&0”“( +
l/2)(-1/21e-‘@xI +iAw(*)l(-1/21~iB~~ 2
l/2) I - l/2)(
1/21e-i8Sxl l/2).
[17]
Here Awc2) = AEc2’/ ti. The matrix elements in Eq. [ 17 ] are the well-known Wigner rotation matrices O&( (Y,p, 7) and are tabulated in Ref. (5). By evaluating them we iset
1181 The first term represents the quadrupolar echo, since the phase factor at t = 27 equals unity and the signal is independent of the smeared quadrupole interaction. The echo is maximum for 0 = ?r, where the second term in Eq. [ 181 also vanishes. The shape of the echo is obtained by convoluting S(t) with the distribution function g(a) of quadrupole coupling constants a = e2qQ/h: E(t) =
mS(t)g(aW. s--co
1191
EXPERIMENT
A NMR technique used for separation of homogeneous and inhomogeneous lineshapes is the aforementioned 2D homonuclear separation of interactions method (6). The preparation period consists of a single 90” pulse. In the middle of the evolution period we apply a 180” pulse shifted in phase by 90”. The refocusing effect of this pulse is to eliminate, in the evolution period, the action of all parts of the total Hamiltonian which are linear in the Z, spin variable (chemical shifts, heteronuclear dipolar interaction, inhomogeneous external magnetic field) and the action of a much larger inhomogeneous quadrupole interaction. Thus the relevant Hamiltonian in the evolution period is only a homonuclear dipolar interaction. The detection period begins at the top of the echo. In that period the spin system is subjected to the full Hamiltonian. For the central transition of “Rb in DRADP-44, 256 incrementations of the evolution time tl were used. The Cyclops ( 7) phase cycle, together with superimposed phase alternation of the second pulse, was used to get rid of experimental artifacts. After a 2D complex Fourier transformation, a 2D spectrum appears. In the w2 domain, one finds the inhomogeneously broadened lineshape, whereas the o1 domain shows the homogeneous lineshape. The experiment was performed in a magnetic field of 6.3 T. The total acquisition time was about three hours and a magnitude spectrum was computed. DISCUSSION
In Fig. la a contour plot of the 87Rb 2D spectrum in DRADP-44 at orientation c 1)Ho and T = 140 K is shown. The projections on both frequency axes are also shown,
SEPARATION
OF INHOMOGENEOUS
;:II
317
LINESHAPES
IERTZ
-6000
1
0
-50000
HERTZ w2/2rI
FIG. 1. (a) A contour plot of a 2D spectrum of *‘Rb in DRADP-44 at c 11 Ha, vg(“Rb) = 88.34 MHz, T = 140 K. The w2 domain shows the inhomogeneously broadened lineshape, whereas in the wi domain one has the homogeneous lineshape. Projections on both frequency axes are also shown. (b) A three-dimensional plot of the spectrum shown in (a).
where the inhomogeneously broadened line appears in the w2 domain, whereas the homogeneous line is shown in the wI domain. At temperature T = 140 K the inhomogeneous line is strongly asymmetric and has a full width at half-height of 12 kHz, whereas the homogeneous linewidth is a Lorentzian with a full width at half-height of 700 Hz. In Fig. lb the same spectrum is shown in a three-dimensional plot. By lowering the temperature the inhomogeneous line continuously broadens, as shown in the temperature dependence plot of the o2 projections (Fig. 2a). At T = 33 K the full width of the o2 projection at half-height amounts to 25 kHz. The homogeneous linewidth, however, exhibits a different behavior. It increases slowly from its room temperature value of about 600 Hz and exhibits a maximum at the temperature of the minimum of spin-lattice relaxation time T, (at T = 85 K) (Fig. 3). With T = 73 K the homogeneous linewidth at half-height is 1900 Hz (Fig. 2b). At lower temperatures
318
J. DOLINSEK
& 50000
)( T=33 K 0 HERTZ
Wp12rI
-50000
10000
5000
0 HERTZ
-5000
w2/2rI
FIG. 2. Temperature dependence of the (a) w2 projection (inhomogeneous line) and (b) w, projection (homogeneous line) of the 2D spectrum of *‘Rb in DRADP-44 at c /)Ho. All lines are scaled to equal height.
the homogeneous width decreases again and reaches a value of 1000 Hz at 33 K. The same behavior was confirmed by measurement of the spin-spin relaxation time T2 using the Hahn-echo experiment. T2 is the inverse of the homogeneous linewidth (Fig. 3). Here T2 exhibits a minimum at a temperature approximately equal to the temperature of the T, minimum. Since the minimum value of T1 is very short (300 ps) it seems clear that the homogeneous linewidth maximum is a consequence of lifetime broadening. The fact that the homogeneous lineshape is more than 10 times narrower than the inhomogeneous one is a characteristic feature of glasses with static disorder. For insulating glasses, it was shown ( 1, 2, 8) that the inhomogeneous NMR lineshape F(V) can be written as a convolution of the frequency distribution function f( V) and the homogeneous lineshape L ( v ) :
The frequency distribution function f( V) has been computed theoretically (8). At the particular orientation c 1)Ho in DRADP, it was shown to be asymmetric since at this orientation the quadrupole-perturbed NMR frequency of a *‘Rb nucleus depends quadratically on the local order parameter p. This order parameter describes the polarization of the proton in a hydrogen bond representing an elementary electric dipole which orders randomly in the ferroelectric-antiferroelectric mixture DRADP. The monotonic broadening of the inhomogeneous line with decreasing temperature can be attributed to the appearance of an Edwards-Anderson-type order parameter (9,
SEPARATION
OF INHOMOGENEOUS
319
LINESHAPES
. 11; 72
I
[msl
It
.
10-2
I 5
f
11 WI.56
(WO.bb
02POb
v. ia7RbI =88.34MHz
I 10
I 15
I 20
I 25
I 30
I I I I 35 40 .'+5 50 lo3 IT IK-‘1
I
55
FIG. 3. Temperature dependence of the spin-lattice relaxation time T, and spin-spin relaxation time Tz of “Rb in DRADP-44 at c 11Ho.
10) describing the static disorder in the glassy state. In order to obtain the pure experimental frequency distribution function f( V) one must make a deconvolution of Rq. [ 201. This is possible since the 2D NMR homogeneous-inhomogeneous lineshape separation technique allows precise measurements of F( v) and L(v). REFERENCES 1. R. BLINC, D. C. AILION, B. GUNTHER, AND S. ZUMER, Phys. Rev. Lett. 57,2826 ( 1986). 2. R. BLINC, S. ZUMER, M. KOREN, AND D. C. AILION, Phys. Rev. B 37,7276 ( 1988). 3. A. ABRAGAM, “The Principles of Magnetic Resonance,” Oxford Univ. Press/Clarendon Press, London, 1960. 4. G. M. VOLKOFF, C’unad. .I. Phys. 31,820 (1953). 5. M. MEHRING, “High Resolution NMR Spectroscopy in Solids,” Springer, New York/Berlin, 1976. 6. R. R. ERNST, G. BODENHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic Resonance in One and Two Dimensions” Oxford Univ. Press, London/New York, 1987. 7. D. I. HOULT AND R. E. RICHARDS, Proc. R. Sot. London A 344, 311 ( 1975). 8. R. BLINC, J. DOLINSEK, R. PIRC, B. TADI~, B. ZALAR, R. KIND, AND 0. LIECHTI, Phys. Rev. Lat. 63, 2248(1989). 9. S.F.EDWARDSANDP.
W. ANDERSON, J. Phys.F5,965( 1975). 10. R. BLINC, J. DOLINSEK, V. H. SCHMIDT, AND D. C. AILION, Europhys. Lett. 6, 55 (1988).