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Multiplet relaxation studies in anisotropic media
Volume 54, number 3
.
MULTIPLET RELAXATION
CHEMICAL PHYSICS LETTERS
15 March 1978
STUDIES iN ANISOTROPIC MED;A
Lawrence G. WERBELOW and David M. GRANT Department of Chemistry, University of Utafz. Scrlt Lake Czty. Utah 841 I2. USA Received 28 November 1977
Equations descrlbinp the magnetization recovery rate of the three identical spms one-half system arc presented. The equations are tailored for nuclear spin relaxation studies performed in anisotropic flurds The ramifications of these expressions are briefly discussed.
1. Introduction
The detailed analysis of complete sets of coupled magnetization mode variabies provides the basis for a very powerful methodology [l] currently being exploited by a number of researchers in the field of nuclear magnetic reIaxation spectroscopy. These studies have indicated that one obtains maximum information from polarization relaxation experiments only if the time evolution of both virtual spin magnetizations (asymmetry and symmetry modes) and real spin magnetizations (total intensity modes) are compb tely characterized and the associated evolution matrix determined. The longitudinal magnetization for a spin system composed of n dlstinguishable spins one-half is fully described by n real spin magnetizations, 2”-l -n asymmetric virtual spin magnetizations and 2”-’ -1 symmetric virtual spin magnetizations 121. A symmetric mode representing the total spin population does not couple in with either set of magnetization variabIes_ A judicious choice of spin excitation and the incorporation of symmetry considerations often can reduce the compIexity of the probIem to a reasonably tractable form. In interesting systems containing one or more subsets of indistmguishable spins, the orientationally dependent spin-lattice couplings fail to differentiate between the population kinetics of irreducibte multiplicities sharing identical total spin. Consequently, there is a concomitant contraction of the dlmenslonality of the‘ kinetic expressions for such spin systems. For example. in the grouping of three identical spins one-half, the doubly degenerate doublet states evolve in time as a single variable thus condensing the linearly coupled magnetization modes from seven to five in number. Contrary to expectations, this contraction can compIicate the relaxation analysis. Since sets of indistinguishable spins exhibit no observable internal scalar couplings and are characterized by degenerate spectral features. the observable magnetization modes are also reduced in number. In these cases, many of the virtual spin magnetizations correspond to nonmeasurables yet must be retained to obtain a correct anaIysis of one’s relaxation data since they still couple into the measurables. For example, the three identical-spins system is characterized by one totd! magnetization and by four unobservable virtual spin magnetizations whereas the comparable system comprised of three nonidentical spins is characterized by three real spin magnetizations and four virtual spin magnetizations. In this latter case, ail seven variables are observables. The lack of measurables in systems composed of groupings of identical spins is a fundamental problem of conceptual and numerical concern. Studies of spin population kinetics in anisotropic media provides an intriguing approach capable of rendering measurable these hidden virtual spin magnetizations unique to systems composed of 571
Volume 54. number 3
CHEMICAL
15 March 1978
PHYSICS LETTERS
indistinguishable spins one-half [3] _ It is the intention of this note to demonstrate how such studies interject a complicating feature which removes the complete equivalence of irreducible multiplicities and hence expands the conventional kinetic dimensionahty. These features should prove important in future analyses-
2. Theory
Quantitative descriptions of relaxation phenomena in the magnetic resonance experiment generally utilize the Redfield-Bloch density operator formalism_ The pivotal feature of this method is the ability to obtain a solution to the matrix differential equation, -(d,‘dt)
x = @x _
(1)
The elements of the magnetization transport matrix, IQare appropriately weighted spectral density sums and differences that can be calcuIated by standard prescriptions. Members of the thermal deviation population/phase vector, x. relevant for the treatment of the longitudinal relaxation experiment correspond to thermally normalized diagonal elements (populations) and other elements of the density operator connecting states degenerate in energy (multiplicity phase relations). It is generally accepted that these eigenvector phase reIations do not couple into the population variables and can be ignored so Iong as the density operator is defined in a set of basis functions which diagonalize the Zeeman hamiltonian. It can be rigorously shown that this statement is true if all spectral densities are real- However, spin systems embedded in spatially anisotropic environments are generally characterized by cross-correlation spectral densities with both real and imaginary componentsWhen a magnetization mode treatment is applied to the three identical spin one-half system subject to dipolar and random-field-type (e.g. spin rotation) interactions, the following magnetization mode evolution equation is obtained, ‘a
-(d/W
Vl
“v2 aV3 a”4
= _(dfdr)&=
ae(s-l&
l-
=
+I +2
(2)
.+3
The elements of the evolution submatrices a’e defined below, ‘25 (W) + ar, (2w) + 281 (w) 1OK, (w)-SK2
(2U)
2K1(0)
4L,
8& (2w)
(w)
-5
+
8L2 (2~)
Q,(w)
ar, ia) + 2X, (a) +6 - (,)+4/r (0) /”
al--=
-2K,
(W) +4K, (2W) -Jr
2L, (u) + 4L, (20)
-$Q,
572
(0)
(a)-& -* (a)
-L,(w)-_:Ql(w)
’ (3)
(0)
214 (o)--L2(2~ll -$QrW
-2L2
(2~)
- ;
Q1 (~1
gx+
2 [Jl (w&K1 (w) * 2 I& (2w)--K2 (2w)l +9 rj1 (w) -& (w)l
Volume 54, number 3
CHEMICALPHYSICS LE-L-PERS
-5Jt (W) + 7Kt (w) f ar, (2oJ)
+ 4K, (20) + 4jlb-)
SY=
--$X-J,(o)+&&)
+ 2& (0)
x + 5 [J1(0) - K, (aI1 + 2 [J2 (2~) + K2 (2u)l + 2 [j1(0) - & (~11
-3J, (WI - 9Kl Cd -12K, (20) - 6kl (a)
3 [L1 (w) - 4L2 + 6Q, (w)
15 March 1978
Cb)l
-3 [L, (Cd) f XL,
(ad)1
-2 [2Ll (cd) + L2 OWN
+x+2[J&J)
-K+)I
i- 2 [J2 (2~) - Kz (2w)l + $ [jI 63) - Al b)l
+2Q,tw)
(4)
_
The zero frequency contributions to the above equations are written in the abbreviated form, X E 3 [JO(O)--&-,(O)] + 2JliO (0) - &) WI The magnetization mode transformation matrix, 6, is defined in a manner such that a maximum number of virtual spin magnetizations (Qj+ 1, ‘vi) correspond to measurables of the system. a~l corresponds to the total magnetization of the system and is the only observable in isotropic media. In anisotropic environments, the three identical spins one-half spectrum is split into a I:211 triplet. The “vz mode corresponds to the intensity of the outer two components minus the intensity of the central component whereas Sal corresponds to the intensity difference between the outer two components. The two modes a~3 and s~2 correspond to irreducible quartet/doublet intensity and population asymmetries of the central component and are not observable even in anisotropic fluids_ “v4 and s~3 correspond to doublet state phase variables and are unique to anisotropic fluid studies. The various spectral density terms appearing in eqs. (3j and (4) can be written in the general form,
([Y,” (@(O))
(B; (0)
- (Y; (nii))]
Bk_, (t)> cos(nrt&,t)
dt
[Yp
.
(aik (f)) - ( qn
(fiik))])
COS(~I~~~~~~)
dt
,
(5)
(6)
The superscripts, i, j, and k index the individual spin particles. The autocorrelation spectral densities (J$) result when spins j and k are taken to be identical. The cross-correlation spectral densities (i f fc) may be complex with a real part, K, (I&), and a pure imaginary part, Ln (!$J The results presented in eq. (3) are compatible with previous discussions of isotropic fluids [4,5] and the investigations of Runnels [6] and Hubbard [7,8] on one-dimensional fluids. For isotroprc fluids, L(Q) vanishes, ( Y$ (L)) vanishes, and the n dependence occurring in the various nonvanishing spectral densities can be suppressed. Furthermore,J(/) and K(k) can be expressed in a relatively tractable closed form. All of these simplifications are absent in the amsotropic phase.
3. Discussion The experimental implications of eq. (2) are readily apparent. The importance of the phase variables is determined by the associated strength of coupling into the observables. The imaginary component of the dipolar crosscorrelation terms is on the order [9] of (Y: (SC!))(cos fl)K where p is the angle defined between the averaged molecular orientation and the methyl rotor axis. For solute species characterized by small ordering parameters, the 573
Volume 54, number 3
CHEMICAL
PHYSICS
LETTERS
15 March 1978
imaginary terms are expected to be relatively insignificant. However, for an exacting analysis, their presence-must he acknowiedged. For highly ordered systems, one must be extremely careful not to overlook these imaginary terms. A complete discussion of these problems and the most appropriate choice of magnetization mode variables wiU appear in a forthcoming treatment [IO] of partially oriented multispin systems complicated by spectral degeneracy.
Acknowledgement This work was sponsored by the National Institutes of Health under Grant GM 08521_
References [l] L.C. Werbelowand D.hf. Grant, Advan. Magn. Reson. 9 (1977) 189. [2] L.G. Werbeiow, 3. Chem. Educ., to be published. [3] J-M. Courtieu, CL. hlayne and D.M. Grant, 3. Chem. Phys. 66 (1977) 2669. [4] L.G. Werbelow and A-G. hfarshall, J. Map. Reson. 11 (1973) 299. ]S] P.S. Hubbard, J. Chem_ Phys. 51 (1969) 1647. [6] R.L. Runnels, Phys. Rev. 134 (1964) A28. [7] R-L_Hilt and P.S. Hubbard, Phys. Rev. 134 (1964) A392. [SJ P.S. Hubbard and C.S. Johnson, I. Chem. Phys. 63 (1975) 4933. [9] L-G_ Werbelow, unpublished results. [lo] L-G. WerbeIow, J-M. Courtieu, D.M. Grant and E-P. Black, Chem. Phys., to be published.
574
.
MULTIPLET RELAXATION
CHEMICAL PHYSICS LETTERS
15 March 1978
STUDIES iN ANISOTROPIC MED;A
Lawrence G. WERBELOW and David M. GRANT Department of Chemistry, University of Utafz. Scrlt Lake Czty. Utah 841 I2. USA Received 28 November 1977
Equations descrlbinp the magnetization recovery rate of the three identical spms one-half system arc presented. The equations are tailored for nuclear spin relaxation studies performed in anisotropic flurds The ramifications of these expressions are briefly discussed.
1. Introduction
The detailed analysis of complete sets of coupled magnetization mode variabies provides the basis for a very powerful methodology [l] currently being exploited by a number of researchers in the field of nuclear magnetic reIaxation spectroscopy. These studies have indicated that one obtains maximum information from polarization relaxation experiments only if the time evolution of both virtual spin magnetizations (asymmetry and symmetry modes) and real spin magnetizations (total intensity modes) are compb tely characterized and the associated evolution matrix determined. The longitudinal magnetization for a spin system composed of n dlstinguishable spins one-half is fully described by n real spin magnetizations, 2”-l -n asymmetric virtual spin magnetizations and 2”-’ -1 symmetric virtual spin magnetizations 121. A symmetric mode representing the total spin population does not couple in with either set of magnetization variabIes_ A judicious choice of spin excitation and the incorporation of symmetry considerations often can reduce the compIexity of the probIem to a reasonably tractable form. In interesting systems containing one or more subsets of indistmguishable spins, the orientationally dependent spin-lattice couplings fail to differentiate between the population kinetics of irreducibte multiplicities sharing identical total spin. Consequently, there is a concomitant contraction of the dlmenslonality of the‘ kinetic expressions for such spin systems. For example. in the grouping of three identical spins one-half, the doubly degenerate doublet states evolve in time as a single variable thus condensing the linearly coupled magnetization modes from seven to five in number. Contrary to expectations, this contraction can compIicate the relaxation analysis. Since sets of indistinguishable spins exhibit no observable internal scalar couplings and are characterized by degenerate spectral features. the observable magnetization modes are also reduced in number. In these cases, many of the virtual spin magnetizations correspond to nonmeasurables yet must be retained to obtain a correct anaIysis of one’s relaxation data since they still couple into the measurables. For example, the three identical-spins system is characterized by one totd! magnetization and by four unobservable virtual spin magnetizations whereas the comparable system comprised of three nonidentical spins is characterized by three real spin magnetizations and four virtual spin magnetizations. In this latter case, ail seven variables are observables. The lack of measurables in systems composed of groupings of identical spins is a fundamental problem of conceptual and numerical concern. Studies of spin population kinetics in anisotropic media provides an intriguing approach capable of rendering measurable these hidden virtual spin magnetizations unique to systems composed of 571
Volume 54. number 3
CHEMICAL
15 March 1978
PHYSICS LETTERS
indistinguishable spins one-half [3] _ It is the intention of this note to demonstrate how such studies interject a complicating feature which removes the complete equivalence of irreducible multiplicities and hence expands the conventional kinetic dimensionahty. These features should prove important in future analyses-
2. Theory
Quantitative descriptions of relaxation phenomena in the magnetic resonance experiment generally utilize the Redfield-Bloch density operator formalism_ The pivotal feature of this method is the ability to obtain a solution to the matrix differential equation, -(d,‘dt)
x = @x _
(1)
The elements of the magnetization transport matrix, IQare appropriately weighted spectral density sums and differences that can be calcuIated by standard prescriptions. Members of the thermal deviation population/phase vector, x. relevant for the treatment of the longitudinal relaxation experiment correspond to thermally normalized diagonal elements (populations) and other elements of the density operator connecting states degenerate in energy (multiplicity phase relations). It is generally accepted that these eigenvector phase reIations do not couple into the population variables and can be ignored so Iong as the density operator is defined in a set of basis functions which diagonalize the Zeeman hamiltonian. It can be rigorously shown that this statement is true if all spectral densities are real- However, spin systems embedded in spatially anisotropic environments are generally characterized by cross-correlation spectral densities with both real and imaginary componentsWhen a magnetization mode treatment is applied to the three identical spin one-half system subject to dipolar and random-field-type (e.g. spin rotation) interactions, the following magnetization mode evolution equation is obtained, ‘a
-(d/W
Vl
“v2 aV3 a”4
= _(dfdr)&=
ae(s-l&
l-
=
+I +2
(2)
.+3
The elements of the evolution submatrices a’e defined below, ‘25 (W) + ar, (2w) + 281 (w) 1OK, (w)-SK2
(2U)
2K1(0)
4L,
8& (2w)
(w)
-5
+
8L2 (2~)
Q,(w)
ar, ia) + 2X, (a) +6 - (,)+4/r (0) /”
al--=
-2K,
(W) +4K, (2W) -Jr
2L, (u) + 4L, (20)
-$Q,
572
(0)
(a)-& -* (a)
-L,(w)-_:Ql(w)
’ (3)
(0)
214 (o)--L2(2~ll -$QrW
-2L2
(2~)
- ;
Q1 (~1
gx+
2 [Jl (w&K1 (w) * 2 I& (2w)--K2 (2w)l +9 rj1 (w) -& (w)l
Volume 54, number 3
CHEMICALPHYSICS LE-L-PERS
-5Jt (W) + 7Kt (w) f ar, (2oJ)
+ 4K, (20) + 4jlb-)
SY=
--$X-J,(o)+&&)
+ 2& (0)
x + 5 [J1(0) - K, (aI1 + 2 [J2 (2~) + K2 (2u)l + 2 [j1(0) - & (~11
-3J, (WI - 9Kl Cd -12K, (20) - 6kl (a)
3 [L1 (w) - 4L2 + 6Q, (w)
15 March 1978
Cb)l
-3 [L, (Cd) f XL,
(ad)1
-2 [2Ll (cd) + L2 OWN
+x+2[J&J)
-K+)I
i- 2 [J2 (2~) - Kz (2w)l + $ [jI 63) - Al b)l
+2Q,tw)
(4)
_
The zero frequency contributions to the above equations are written in the abbreviated form, X E 3 [JO(O)--&-,(O)] + 2JliO (0) - &) WI The magnetization mode transformation matrix, 6, is defined in a manner such that a maximum number of virtual spin magnetizations (Qj+ 1, ‘vi) correspond to measurables of the system. a~l corresponds to the total magnetization of the system and is the only observable in isotropic media. In anisotropic environments, the three identical spins one-half spectrum is split into a I:211 triplet. The “vz mode corresponds to the intensity of the outer two components minus the intensity of the central component whereas Sal corresponds to the intensity difference between the outer two components. The two modes a~3 and s~2 correspond to irreducible quartet/doublet intensity and population asymmetries of the central component and are not observable even in anisotropic fluids_ “v4 and s~3 correspond to doublet state phase variables and are unique to anisotropic fluid studies. The various spectral density terms appearing in eqs. (3j and (4) can be written in the general form,
([Y,” (@(O))
(B; (0)
- (Y; (nii))]
Bk_, (t)> cos(nrt&,t)
dt
[Yp
.
(aik (f)) - ( qn
(fiik))])
COS(~I~~~~~~)
dt
,
(5)
(6)
The superscripts, i, j, and k index the individual spin particles. The autocorrelation spectral densities (J$) result when spins j and k are taken to be identical. The cross-correlation spectral densities (i f fc) may be complex with a real part, K, (I&), and a pure imaginary part, Ln (!$J The results presented in eq. (3) are compatible with previous discussions of isotropic fluids [4,5] and the investigations of Runnels [6] and Hubbard [7,8] on one-dimensional fluids. For isotroprc fluids, L(Q) vanishes, ( Y$ (L)) vanishes, and the n dependence occurring in the various nonvanishing spectral densities can be suppressed. Furthermore,J(/) and K(k) can be expressed in a relatively tractable closed form. All of these simplifications are absent in the amsotropic phase.
3. Discussion The experimental implications of eq. (2) are readily apparent. The importance of the phase variables is determined by the associated strength of coupling into the observables. The imaginary component of the dipolar crosscorrelation terms is on the order [9] of (Y: (SC!))(cos fl)K where p is the angle defined between the averaged molecular orientation and the methyl rotor axis. For solute species characterized by small ordering parameters, the 573
Volume 54, number 3
CHEMICAL
PHYSICS
LETTERS
15 March 1978
imaginary terms are expected to be relatively insignificant. However, for an exacting analysis, their presence-must he acknowiedged. For highly ordered systems, one must be extremely careful not to overlook these imaginary terms. A complete discussion of these problems and the most appropriate choice of magnetization mode variables wiU appear in a forthcoming treatment [IO] of partially oriented multispin systems complicated by spectral degeneracy.
Acknowledgement This work was sponsored by the National Institutes of Health under Grant GM 08521_
References [l] L.C. Werbelowand D.hf. Grant, Advan. Magn. Reson. 9 (1977) 189. [2] L.G. Werbeiow, 3. Chem. Educ., to be published. [3] J-M. Courtieu, CL. hlayne and D.M. Grant, 3. Chem. Phys. 66 (1977) 2669. [4] L.G. Werbelow and A-G. hfarshall, J. Map. Reson. 11 (1973) 299. ]S] P.S. Hubbard, J. Chem_ Phys. 51 (1969) 1647. [6] R.L. Runnels, Phys. Rev. 134 (1964) A28. [7] R-L_Hilt and P.S. Hubbard, Phys. Rev. 134 (1964) A392. [SJ P.S. Hubbard and C.S. Johnson, I. Chem. Phys. 63 (1975) 4933. [9] L-G_ Werbelow, unpublished results. [lo] L-G. WerbeIow, J-M. Courtieu, D.M. Grant and E-P. Black, Chem. Phys., to be published.
574