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Magnetic relaxation in conducting electrolytes
Advances
in Molecular
Relaxation
Processes,
I (1975)
@ Elsevier Scientific Publishing Company, Amsterdam
MAGNETIC
RELAXATION
l-l 1 - Printed in The Netherlands
IN CONDUCTING
ELECTROLYTES
P. W. ATKINS AND M. J. CLUGSTON Physical
Chemistry
Laboratory,
(Received 7 November
South Parks Road, Oxford
(Ct. Britain)
1974)
ABSTRACT
The effect of an electric field on the relaxation of a spin-& nucleus by a paramagnetic ion in fluid solution is calculated. The model relates to a spin-echo experiment, and depends on inter-ionic magnetic dipole-dipole relaxation. It is shown that under optimum conditions, using electric fields of strength 6 10 MV m-r, the value of the nuclear T, may be changed by about 50 %.
INTRODUCTION
The question we pose in this paper is how an electric field can influence the relaxation behaviour of ions in solution. If the relaxation is affected we have a way of studying the relative motion of ions in electrolytes. The system we concentrate on is one ion with a single nuclear spin of + (e.g. F-) which is to be observed by NMR, and another ion with an electron spin (e.g., a lanthanon or a transition metal). The latter ion dominates the relaxation of the nuclear spin because of its dipolar interaction, and the field modifies the translational correlation time of the relative separation of the species. We concentrate on pulsed NMR experiments, and in particular on the determination of the nuclear spin-spin relaxation time by spin-echo techniques. It follows that we have to investigate the effect of self-diffusion and flow on the echo observed from negative ions, and the effect of mutual flow on the transverse relaxation time. The natural starting point is the Smoluchowski-Debye equation for the conditional probability distribution in a liquid. For the effect on the spinecho we follow the work of Stejskal and Tanner [l, 21. It will be seen that with an appropriate choice of parameters, the transverse relaxation time dominates the effect of self-diffusion in attenuating the signal amplitude. The drift velocity causes a phase shift. The remaining step in the argument is the calculation of T2 on the
2 basis of Redfield
theory.
This involves
the calculation
of a correlation
function
with an electric field applied at some orientation to the magnetic field. We select two particular orientations, parallel and perpendicular, and discuss the effect on the spin-echo. Electric field effects in magnetic resonance have been observed in a wide variety of situations. In solids a nucleus or paramagnetic ion with a spin greater than 3 occupying a crystal site lacking inversion symmetry may respond to an electric field [3,4]. The effect can be incorporated in an effective hamiltonian by modifying the zero-field splitting, the g-tensor, and the quadrupole and hyperfine coupling constants. The applied electric field has a direct effect on the ion, and also secondary effects through the polarization and displacement of the local lattice. Non-linear response sets in above about 50 MV m-i (and E-resonance may be observed) [5]. Partial alignment of molecular dipoles may be expected in liquids, and anisotropic interactions will no longer be averaged to zero. First experimental evidence for this was provided by Buckingham and McLauchlan [6] but their results were doubted until the situation was resolved by Hilbers and MacLean [7]. Two effects are observed: direct dipolar interactions augment the electron-coupled spin-spin coupling constants, and quadrupole splittings can be seen. In a typical experiment the electric and magnetic fields are parallel and the former has a strength of about 5 MV m-l. The calculation of the magnitude of the effect depends on the determination of the alignment, the ensemble average of the second-rank legendre polynomial P,(cos 0), and some knowledge of the local electric field. Hilbers, MacLean, and Mandel [8] made detailed calculations of the alignment correct to secondorder in the field. Their results, on neglect of the molecular polarizability, agree with those of Deutch and Waugh [9] who show the importance of 2- and 3-body correlations in the determination of the magnitude of the effect. Virtually all results on liquids relate to non-electrolyte solutions. The problems of working with high field strengths in conducting electrolytes do not need emphasis, but Packer has indicated that he might be able to measure drift rates using his pulsed field-gradient spin-echo spectrometer [lo, II]. Effects have also been observed in liquid crystals [12] (by rotation of the optic-axis) and in gases [13] (via the Stark effect).
THE MODEL
When the time-scale of interest is sufficiently long the motion of ions in solution is governed by a Langevin equation with a Lorentz force term. The cyclotron motion is negligible if ]zJeB/[ << 1 (C is the friction constant), and for all reasonable parameters this is guaranteed. (For example, for B - 1.5T and 5‘characteristic of water the ratio is about lo- ‘.) It follows that only linear drift need be investigated. There is also an extra drift caused by the magnetic field gradient
3
acting on the magnetic dipole moment [14] but we disregard this small effect. We choose a positive ion with electron spin S (this could be a lanthanon or a transition metal ion), the nuclear spin-4 species is either charged (such as F-) or neutral (thesolvent). Weconcentrateon the former. Weassume that the nuclearspin relaxationisdominated bytheinteractionwith theelectronspinofthecation,andthat the direct dipolar interaction dominates any short-range scalar interaction [15, 161. A typical ionic mobility is 5 x lo-’ m2 s -I V-l, and a field strength just unlikely to cause dielectric breakdown is 10 MV m-r. It follows that oppositely charged ions will have a relative velocity of about 1 m s-l under these conditions. On an NMR time-scale this means that ions will have separated by about 10 nm, which is large on a molecular scale. The dipolar relaxation is a strong function of distance (rm6) and so it is likely that the electric field will generate a significant effect. Such large fields as we envisage cannot be maintained for long, and so we concentrate on pulsed techniques. Experiments related to those we have in mind have been discussed by Stejskal [2] who considered the use of pulsed magnetic-field gradients (PMFG) on a Hahn sequence to study anisotropic, restricted diffusion (for example, water in vermiculite crystals [ 171). Packer et al. have studied slow coherent motion (for example, flow) using three modified Carr-Purcell sequences either with or without PMFG [IO, 11, 181. As our anticipated values of T, are much shorter than theirs we need not consider these techniques, and a Hahn sequence will be adequate. Packer et al. [l I] have also reported the only magnetic resonance experiment on conducting ions: they studied the proton NMR of Me,NBr in D,O for low electric field strengths (about 1 kV m-l). As we have already indicated, we have to study self-diffusion and Sow in the spin-echo of negative ions. Then we must study the effect of mutual flow on the transverse relaxation time. This means that we must establish an expression for the probability of finding a positive ion (and hence an electron spin) at a distance Yfrom a nucleus at time t having applied the electric field at t = 0. The existence of an ionic atmosphere and concomitant asymmetry andelectrophoreticeffects is unlikely to be of severe difficulty because we envisage working in the first Wien effect region [19]. If the electric field effects can be followed into low field strength situations, these complications would have to be considered. This model assumes that the ions are free. At the other extreme the role of contact ion-pairs must be taken into account. An asymmetric ion-pair will be oriented in an electric field, and for a nucleus relaxing via an intramolecular quadrupole interaction, the signal from bound ions will be split by analogy with the work of Hilbers and MacLean [7]. PARTICLE
DISTRIBUTION
AND
ECHO PROPERTIES
We begin at a Fokker-Planck
equation for the n-particle distribution func-
tion, but regard the problem as a superposition of 2-particle interactions. We neglect all inter-ionic interactions and the implications of the Oseen tensor [20 1. The external electric field is static and homogeneous. It follows that the probability distribution satisfies ap(r, r)ldt =ig*Di Vi ’ (vi-PFi)P(r,
t,
(34
where P(Y, t) is the 2-body distribution function, Di is the diffusion coefficient of species i, Pi is the external force on i, and /? = I/kT. With P = P+ -r_ and the field, E, in the direction 4 this equation reduces to i3P(r, t)/dt = DV’P(r, t)-cS(r,
t)/@
(3.2)
with z+ = IzilDiej3, ci=uiE, c = c+ +c_, and D = D+ +D_. The solution of this equation by Fourier-Laplace transformation if initially an ion is at r,, at t = 0 gives the conditional probability for it being at r at t: P(r; t; ro> = (~Jn)+fw
(-ytlr’-ro12}
(3.3)
where r’ = r - ctk for an electric field along i, and yt = 1/4Dt. We a use modified Bloch-Torrey-Stejskal equation [2] for the precessing magnetization m, (P, t) in order to find the effect of diffusion and flow on the spinecho am+(r, t)/Jt = -iq,m+(r,
* G(t)m+(r,
t)-iy,v
+D_
t)-co2m+(r,
V2m+(r, t)-u(t)
t)
* Vm+(r, t)
(3.4)
The magnetic field B = B$ + r*G(t)$ consists of the homogeneous part, and the PMFG component r*G(t)$. The gradient G is uniform in space. In eqn. (3. 4) o, is the nuclear Larmor frequency, 7” its magnetogyric ratio, and l/o, = T2, the transverse relaxation time. u(t) is the drift velocity. On writing m+(~, t) = $(r, t) exp { - (ion + 02)t} the last equation transforms into %Q(r, t)/at = - i 7” r - G(t)$(r,
t)+D_V2$(r,
t)-u(t)
* V$(r, t)
(34
From this it follows that the principal echo at t = 22 is given by [I, 21 $(2*) = B(O)exp (-rfD_
[/:‘dtF(t)
*F(t)-4F(r)*r’dfF(t)
+ 4zF(z) * F(T)]) exp ( - iy,[H(2z)
where t F(t) =
H(t)
=
s
‘dt’G(t’) 0
‘dt’s(t’). G(t’) s0
- 2H(z)] + i(2+ -n)}
(3.6)
5
s(t) =
s
‘dr’v(t’) 0
with B(0) some initial value and 4 the relative phase angle of the 42 and n pulses in the Hahn sequence (7r/2 - r - rt - z - echo). This expression shows that diffusion attenuates the echo amplitude, while flow shifts the echo phase so long as the electric and magnetic fields are not perpendicular. Table 1 summarises the implication of the equation. With typical values of the experimental parameters (g N 0.1 T m-r, 6 N 5 x 10P4s) T, dominates the diffusional attenuation.
TABLE
1
THE VALUES
OF a AND
b IN THE EXPRESSION
REFER TO A PMFG EXPERIMENT
[I]
echo = B(0) exp{-2t/T,--y,2D_a}
AND
FOR THE PRINCIPAL
THE PARAMETERS
ECHO.
ARE DEFINED
THE LAST THREE COLUMNS THERE
exp{iy,,c_b+i(2+-n)} b
b
b
b
b
Start of application of electric field
0
t
tl
II
t,+s
End of application of electric field
2t
2t
tl+A+8
r,+A
t,+A
t3 (end-start)
2t
T
A+6
A
A-6
t4 (t-start)
5
0
t--t1
t-t,
t-tl--d
a
%q(3t3t-t42) Steady gradient Pulsed gradient (9 >> go
: A >> 6)
g2Ab’
g4AS gJ(x-tl-
;)
gpA6
- q
go4(t~2/2-t~2)
RELAXATION TIMES IN THE PRESENCE OF AN ELECTRIC FIELD
We now use Redfield theory to calculate electric field strength. The hamiltonian structure ff = Ho+Hs, H, = -w,I,-qS,
Ho = H,+H,
of o2 the dependence of the problem is
Hs*HL=O
on the
(4.1) (4.2)
dipoleHs is the spin hamiltonian, HL the lattice, and HsL the electron-nucleus dipole hamiltonian. The equation of motion for I+ may be dealt with in the normal way of iterating in an interaction picture, approximating the second-order term, and ensemble averaging. The linear term disappears even in the presence of an electric field. In order to show this it is sufficient to show that P(r, t) remains
6
isotropic
if P(r,, 0) is isotropic. If P(r,, 0) is isotropic the angular integration in dr,P(re, 0) P(r; t; r,,)may be carried out, which leaves P(r, t) indeof any angles. Thus P(r, t) is isotropic, and the relaxation of I+ is governed
P(r, t) = J
pendent
by a correlation function of the form < ZZsL*ZZ&(r)*Z+ (t) > , where H&(r) is the evolution under the influence of Ho. The relaxation time that this leads to is 02
~~,“~‘(~0/471)~(4$0(0)+
=
$0(0,-0,)+3~1(0”)+6~1(~,) +6$2(c~+~,)I
(4.3)
where j$(o)
= 2J
dr T,(r) exp {(iw
1/r&r}
-
(4.4)
0 r,(t)
=
~C~~‘(~,~r~~C~~‘*(~~)~
(i-p
(4.5)
and ~1is the magnetic dipole moment in Bohr magnetons. When w, appears in an argument zA = t2, when o, does not appear r1 = rr : z1 and r2 are the electronspin relaxation times. The central problem is the evaluation of the correlation function. Using the conditional probability given by the k-space solution of eqn. (3.2), via multipole tensor expansion with an electric field at an angle Sz to the z-direction we have P(r,; t ;
=
1
X
x
ro)
1/2n2) C ?-‘1-‘2(2f1 + 1)(2E, + 1)(21+ 1) urn>
(
1,
im
-tnl
m2 f2)(:
c~:‘(o)c~‘(~)*c~‘(s)*( :
-
1)m’
:2)
dk k2j,,(kro) j,,(kcz) j,(kr,) exp (- k2Dz)
I
(4.6)
which is to be used in
r,(z) =
s
dr, dr, P(r, , O)P(r,; z ;
“C~‘(z)C~‘(O)*r~
ro)r;
3
(4.7)
0) is equal to some isotropic, homogeneous concentration IZ.The integrations implied by these expressions are straightforward if the r-integrations are confined to a region outside some distance of closest approach, d [21]. When the electric and magnetic fields are parallel we find
P(r,,
Z;‘(r) = (4rrn/d3)T(-$‘(2f+l)
(;
‘,
;)(;
;
A)
dk k- ‘Jf( kd) j,( kcz) exp ( - k2Dz)
x s
When the field is perpendicular only the term I = 0 contributes. The spectral density corresponding to this correlation function
is
(4.8)
in which x = kd and (4.10) with zd
a
k
(x) = x2+ zd/zA+ iwr,
(Td =
d2/D)
b(x) = xc/d and where the coefficients
pFC) take the values
/?P’ = 415 By’ = -8/7
/I’,” = -4/7
p:“’ = 8/7
pb”’ = 72135
D(4) r = -48/35
fly’ = 12/35
The r-integral in eqn. (4.10) is the Hankel-Gegenbauer generalization of the integral of Lipschitz [21]. So long as ju+ I > b it is known that the integral may be expanded in terms of hypergeometric functions: IYP)(.x) = (;):(fj2P+i(&)2u+i(~~) _
,F,(y+f,
P+l;
2P+3;
-b2/a2,) The restriction implies c* 2 D/z, when o = 0, and c2 5 WD when wf not known whether the expansion is valid outside this region. Only the p = 0 contribution can be summed explicitly: @‘(x)
= (l/b)
in powers of (b/a,),
0. It is
(4.12)
arctan (b/a,).
The others may be expanded
(4.11)
and we truncate
at second-order: (4.13a)
(4.13b)
(4.14a)
(4.14b) with
u2 = 2Wtd yo = 11,
v= = td/rl y1 = 9,
y2 = 3.
These integrals may be evaluated by contour integration. We concentrate on their values when w = 0 (so that u = 0), and when o+ 0 but U’ B v’, as is true in all reasonable cases (because o,rA B 1):
P2’(0, 0) =
P’(u,O)
.s i
[(3~~-15)+(4~~+14v~+27u~+30u+15)e-~”] >
= -
(4.15)
-$ (
(3(2~=+10)+[(2t14+14113+41u=-30)c0sU 1 +(2u4-41u=
- 60~ - 30) sin u]e-“}
For small values of v, Z(O) (0, v) N z/15 and Z’=’ (0, u) - 7c/48u; this suggests that at v = 0.1 the ratio I(‘) (O,u)/Z(‘) (0, u) has th e value 3.12 in comparison with the accurate value of 2.95. This is virtually the end of the calculation. Equations (4.13) give the spectral densities in the absence of the field (c = 0) and when it is present (c = c+ + c_, C+ = u,E). Substitution of the appropriate spectral densities, as calculated from the Z(u, v), into eqn. (4.3) enables one to calculate the nuclear spin relaxation time in the absence, or in the presence of the electric field. The Z(u, o) are simple, explicit functions expressed wholly in terms of the dynamical characteristics of the medium and the electron-spin relaxation times, which are assumed to be independent of the electric field strength. The functions Z(‘) ( 0 , u) and Z(O) (a, 0) are plotted in Fig. 1 and the ratio ZC2)(0, v)/Z(‘) (0, v) in Fig. 2.
Ob-
0.4o,tO0.1
1 0.3
Fig. 1. The functions
1 1 0.5 01
I PO u orv
1(0)(0, u) and I(@(u, 0) normalized
5.0 7.0 100 to n/15.
Fig. 2. The ratio P(O,
o)/P(O,
v).
DISCUSSION
We propose observing the NMR relaxation of a negative ion with nuclear spin 5 (e.g. F-) by a Hahn spin-echo sequence, the experiment being done in the presence of a paramagnetic ion so that the nuclear relaxation is dominated by the translational modulation of the inter-ionic dipole-dipole interaction. The modulation of the translational motion of the ion is modified by the application ofpulses of electric field, and the spectral density is modified. This modification will affect the relaxation behaviour of the nucleus, and by observation of the effect we anticipate that it may be possible to draw conclusions about ionic motion in electrolyte solutions. The calculation is based on a Smoluchowski-Debye type of equation and Redfield relaxation. The most serious approximation in the theory is the neglect of the consequences of the existence of an ionic atmosphere. (The importance of second coordination sphere effects is discussed by Stengle and Langford [22].) The field strengths we envisage correspond to the first Wien effect region, and so the complications may be minimal. The assumption that the relaxation is adequately described by a Redfield approach suffers from its normal limitations. Furthermore the calculations of the spectral densities depended on an integral with a value that has only been proved for a range of values. The fact that it has not yet been proved over the whole range of values of interest is upsetting, but unavoidable. The situation is alleviated by the fact that the range of values not yet accommodated by existing proofs is very narrow, and unless the integral is singular in this narrow range, the smooth interpolation we have used will not be greatly, if at all, in error. Experimental problems are to be expected in three inter-related areas. First there is the balance concerning the concentration of ions that should be used. Decreased concentration gives a longer T, and so it cannot be reduced too far
10 before diffusional attenuation assumes a greater importance. But decreased concentration diminishes heating and deposition.The second major problem is of course the duration of the electric field pulse. Very large fields (- 10 MV m- ‘) are needed to observe significant effects, and they should be applied for about 1 ms. This is very demanding experimentally, and might be unattainable. The third problem is the effect of the transient magnetic field generated by the conducting ions, and especially its effect on the homogeneity of the applied magnetic field and its gradient. If these problems can be overcome, and if the assumptions of the calculation are tolerable, then we can predict two effects of the electric field on the spin-echo from the conducting ions. First, the echo phase has a cosine modulation. Second, the echo amplitude is modified because l/T, is modified. We now establish possible magnitudes of the effects. The larger the values of r1 the larger the value of I’*‘(O, U) ( we recall that when w, appears in the argument of f(w) then r1 is to be identified with r2, and when it does not appear it is to be identified with rl). When u”( = 2wr,) is non-zero the values of both J(‘)(u, U) and Z(*)(U, U) are diminished. It follows that when o,~,>>l, y(q) is much less important than B(O) both in the presence and absence of the electric field, and so we neglect it. We take as an example numerical values appropriate to F- in the presence of CrOxi - (for which the magnetic moment is 3.8 ,~,[23]). The reason for treating this system is that experimental values for the electric field-free relaxation of Fare known, and so we can check that part of the calculation. This system would be inappropriate in practice because the relative velocity of ions of the same charge is small. If ions of opposite charge are chosen strong second coordination sphere interaction may occur (in which case there may be a significant second Wien effect). As to the first problem, we are concerned here to show how the calculation proceeds rather than to analyze a particular system. As to the second problem, systems can be chosen, for example Cr(H,O)z+/PF;, which have negligible second coordination [22]. The calculated electric field-free values at 0.1 mol dmM3 concentration of CrOxi- are w2 = 87 s-r (for w,zd < 1) and 30 s-r (for W,T~ >> 1). These figures are based on the values of d = 600 pm and D = 2.0 x 10m9 m2 s-l [24]. The experimental value is approximately 25 s-l [22]. When a perpendicular electric field is applied the fractional change in a2, with r1 - 1 ps and the above parameters, is given by 6oL2/w2 = Kc*, with K = 0.7 s2 m-* and c, the sum of the ionic velocities (ci = +!Z). The maximum velocity of the ions is about 1 m s-’ (at 10 MV m-l), and so a fractional shift of relaxation time of no more than 0.7 may be anticipated. This means that T2 may be changed by about 50 %, or so, under optimal conditions. The magnitude of the change is increased by increasing d and decreasing D. It follows that optimum conditions should be obtained for large, highly charged ions. Examples of negative ions which might usefully replace F- are PF, Al Fz-. The latter should give at least a five-fold increase over the change calcu-
11 lated above, but the more complex
nature of the ion might introduce
complications
due to polarization distortion and intramolecular effects. We thank the Science Research Council for a research studentship, and the Warden and Fellows of Wadham College for a Senior Scholarship, both to M.J.C. We thank Dr G. T. Evans and Dr R. Freeman for helpful discussions.
REFERENCES I E. 0.
2 3 4 5 6 7 8 9 10 II I2 13 14 I5 16 17 18 19 20 21 22 23 24
Stejskal and J. E. Tanner, J. Chem. Phys., 42 (1965) 288. E. 0. Stejskal, J. Chem. Phys., 43 (1965) 3597. N. Bloembergen, Science, 133 (1961) 1363. A. Kiel and W. B. Mims, Phys. Reo., B., 5 (1972) 803. M. F. Deigen, XVIth Colloque Amp&e, (1970) 49. A. D. Buckingham and K. A. McLauchlan, Proc. Chem. Sot., London, (1963) 144. C. W. Hilbers and C. MacLean, Ber. Bunsenges. Phys. Chem., 75 (1971) 277. C. W. Hilbers, C. MacLean and M. Mandel, Physica (The Hague), 51 (1971) 246. J. M. Deutch and J. S. Waugh, J. Chem. Phys.. 43 (1965) 2568 (erratum, idem, 44 (1966) 4366). K. J. Packer, Mol. Phys., 17 (1969) 355. K. J. Packer, C. Rees and D. J. Tomlinson, Aduan. Mol. Relaxation Processes, 3 (1972) 119. P. Diehl, C. L. Khetrapal, H. P. Kellerhals, U. Lienhard and W. Niederberger, J. Magn. Reson., 1 (1969) 527. A. Carrington, D. H. Levy and T. A. Miller, J. Chem. Phys., 47 (1967) 3801. H. C. Torrey, Phys. Reu., 104 (1956) 563. D. Kivelson, 1972, in Electron Spin Relaxation in Liquids (ed. L. T. Muus and P. W. Atkins) (Plenum), p. 229. P. W. Atkins and M. J. Clugston, Mol. Phys., 27 (1974) 1619. B. D. Boss and E. 0. Stejskal, J. Co/l. Interface Sci., 26 (1968) 271. K. J. Packer, C. Rees and D. J. Tomlinson, Mol. Phys., 18 (1970) 421. H. Falkenhagen, Electrolytes (OUP), (1934) p. 82. R. Zwanzig, Adu. Gem. Phys., 15 (1969) 325. G. N. Watson, Theory of Bessel Functions (C.U.P.), (1966) pp. 132, 385. T. R. Stengle and C. H. Langford, J. Phys. Chem., 69 (1965) 3299. B. N. Figgis and J. Lewis, Prog. Inorg. Chem., 6 (1964) 37. H. G. Hertz, G. Keller and H. Versmold, Ber. Bansenges. Phys. Chem., 73 (1969) 549.
in Molecular
Relaxation
Processes,
I (1975)
@ Elsevier Scientific Publishing Company, Amsterdam
MAGNETIC
RELAXATION
l-l 1 - Printed in The Netherlands
IN CONDUCTING
ELECTROLYTES
P. W. ATKINS AND M. J. CLUGSTON Physical
Chemistry
Laboratory,
(Received 7 November
South Parks Road, Oxford
(Ct. Britain)
1974)
ABSTRACT
The effect of an electric field on the relaxation of a spin-& nucleus by a paramagnetic ion in fluid solution is calculated. The model relates to a spin-echo experiment, and depends on inter-ionic magnetic dipole-dipole relaxation. It is shown that under optimum conditions, using electric fields of strength 6 10 MV m-r, the value of the nuclear T, may be changed by about 50 %.
INTRODUCTION
The question we pose in this paper is how an electric field can influence the relaxation behaviour of ions in solution. If the relaxation is affected we have a way of studying the relative motion of ions in electrolytes. The system we concentrate on is one ion with a single nuclear spin of + (e.g. F-) which is to be observed by NMR, and another ion with an electron spin (e.g., a lanthanon or a transition metal). The latter ion dominates the relaxation of the nuclear spin because of its dipolar interaction, and the field modifies the translational correlation time of the relative separation of the species. We concentrate on pulsed NMR experiments, and in particular on the determination of the nuclear spin-spin relaxation time by spin-echo techniques. It follows that we have to investigate the effect of self-diffusion and flow on the echo observed from negative ions, and the effect of mutual flow on the transverse relaxation time. The natural starting point is the Smoluchowski-Debye equation for the conditional probability distribution in a liquid. For the effect on the spinecho we follow the work of Stejskal and Tanner [l, 21. It will be seen that with an appropriate choice of parameters, the transverse relaxation time dominates the effect of self-diffusion in attenuating the signal amplitude. The drift velocity causes a phase shift. The remaining step in the argument is the calculation of T2 on the
2 basis of Redfield
theory.
This involves
the calculation
of a correlation
function
with an electric field applied at some orientation to the magnetic field. We select two particular orientations, parallel and perpendicular, and discuss the effect on the spin-echo. Electric field effects in magnetic resonance have been observed in a wide variety of situations. In solids a nucleus or paramagnetic ion with a spin greater than 3 occupying a crystal site lacking inversion symmetry may respond to an electric field [3,4]. The effect can be incorporated in an effective hamiltonian by modifying the zero-field splitting, the g-tensor, and the quadrupole and hyperfine coupling constants. The applied electric field has a direct effect on the ion, and also secondary effects through the polarization and displacement of the local lattice. Non-linear response sets in above about 50 MV m-i (and E-resonance may be observed) [5]. Partial alignment of molecular dipoles may be expected in liquids, and anisotropic interactions will no longer be averaged to zero. First experimental evidence for this was provided by Buckingham and McLauchlan [6] but their results were doubted until the situation was resolved by Hilbers and MacLean [7]. Two effects are observed: direct dipolar interactions augment the electron-coupled spin-spin coupling constants, and quadrupole splittings can be seen. In a typical experiment the electric and magnetic fields are parallel and the former has a strength of about 5 MV m-l. The calculation of the magnitude of the effect depends on the determination of the alignment, the ensemble average of the second-rank legendre polynomial P,(cos 0), and some knowledge of the local electric field. Hilbers, MacLean, and Mandel [8] made detailed calculations of the alignment correct to secondorder in the field. Their results, on neglect of the molecular polarizability, agree with those of Deutch and Waugh [9] who show the importance of 2- and 3-body correlations in the determination of the magnitude of the effect. Virtually all results on liquids relate to non-electrolyte solutions. The problems of working with high field strengths in conducting electrolytes do not need emphasis, but Packer has indicated that he might be able to measure drift rates using his pulsed field-gradient spin-echo spectrometer [lo, II]. Effects have also been observed in liquid crystals [12] (by rotation of the optic-axis) and in gases [13] (via the Stark effect).
THE MODEL
When the time-scale of interest is sufficiently long the motion of ions in solution is governed by a Langevin equation with a Lorentz force term. The cyclotron motion is negligible if ]zJeB/[ << 1 (C is the friction constant), and for all reasonable parameters this is guaranteed. (For example, for B - 1.5T and 5‘characteristic of water the ratio is about lo- ‘.) It follows that only linear drift need be investigated. There is also an extra drift caused by the magnetic field gradient
3
acting on the magnetic dipole moment [14] but we disregard this small effect. We choose a positive ion with electron spin S (this could be a lanthanon or a transition metal ion), the nuclear spin-4 species is either charged (such as F-) or neutral (thesolvent). Weconcentrateon the former. Weassume that the nuclearspin relaxationisdominated bytheinteractionwith theelectronspinofthecation,andthat the direct dipolar interaction dominates any short-range scalar interaction [15, 161. A typical ionic mobility is 5 x lo-’ m2 s -I V-l, and a field strength just unlikely to cause dielectric breakdown is 10 MV m-r. It follows that oppositely charged ions will have a relative velocity of about 1 m s-l under these conditions. On an NMR time-scale this means that ions will have separated by about 10 nm, which is large on a molecular scale. The dipolar relaxation is a strong function of distance (rm6) and so it is likely that the electric field will generate a significant effect. Such large fields as we envisage cannot be maintained for long, and so we concentrate on pulsed techniques. Experiments related to those we have in mind have been discussed by Stejskal [2] who considered the use of pulsed magnetic-field gradients (PMFG) on a Hahn sequence to study anisotropic, restricted diffusion (for example, water in vermiculite crystals [ 171). Packer et al. have studied slow coherent motion (for example, flow) using three modified Carr-Purcell sequences either with or without PMFG [IO, 11, 181. As our anticipated values of T, are much shorter than theirs we need not consider these techniques, and a Hahn sequence will be adequate. Packer et al. [l I] have also reported the only magnetic resonance experiment on conducting ions: they studied the proton NMR of Me,NBr in D,O for low electric field strengths (about 1 kV m-l). As we have already indicated, we have to study self-diffusion and Sow in the spin-echo of negative ions. Then we must study the effect of mutual flow on the transverse relaxation time. This means that we must establish an expression for the probability of finding a positive ion (and hence an electron spin) at a distance Yfrom a nucleus at time t having applied the electric field at t = 0. The existence of an ionic atmosphere and concomitant asymmetry andelectrophoreticeffects is unlikely to be of severe difficulty because we envisage working in the first Wien effect region [19]. If the electric field effects can be followed into low field strength situations, these complications would have to be considered. This model assumes that the ions are free. At the other extreme the role of contact ion-pairs must be taken into account. An asymmetric ion-pair will be oriented in an electric field, and for a nucleus relaxing via an intramolecular quadrupole interaction, the signal from bound ions will be split by analogy with the work of Hilbers and MacLean [7]. PARTICLE
DISTRIBUTION
AND
ECHO PROPERTIES
We begin at a Fokker-Planck
equation for the n-particle distribution func-
tion, but regard the problem as a superposition of 2-particle interactions. We neglect all inter-ionic interactions and the implications of the Oseen tensor [20 1. The external electric field is static and homogeneous. It follows that the probability distribution satisfies ap(r, r)ldt =ig*Di Vi ’ (vi-PFi)P(r,
t,
(34
where P(Y, t) is the 2-body distribution function, Di is the diffusion coefficient of species i, Pi is the external force on i, and /? = I/kT. With P = P+ -r_ and the field, E, in the direction 4 this equation reduces to i3P(r, t)/dt = DV’P(r, t)-cS(r,
t)/@
(3.2)
with z+ = IzilDiej3, ci=uiE, c = c+ +c_, and D = D+ +D_. The solution of this equation by Fourier-Laplace transformation if initially an ion is at r,, at t = 0 gives the conditional probability for it being at r at t: P(r; t; ro> = (~Jn)+fw
(-ytlr’-ro12}
(3.3)
where r’ = r - ctk for an electric field along i, and yt = 1/4Dt. We a use modified Bloch-Torrey-Stejskal equation [2] for the precessing magnetization m, (P, t) in order to find the effect of diffusion and flow on the spinecho am+(r, t)/Jt = -iq,m+(r,
* G(t)m+(r,
t)-iy,v
+D_
t)-co2m+(r,
V2m+(r, t)-u(t)
t)
* Vm+(r, t)
(3.4)
The magnetic field B = B$ + r*G(t)$ consists of the homogeneous part, and the PMFG component r*G(t)$. The gradient G is uniform in space. In eqn. (3. 4) o, is the nuclear Larmor frequency, 7” its magnetogyric ratio, and l/o, = T2, the transverse relaxation time. u(t) is the drift velocity. On writing m+(~, t) = $(r, t) exp { - (ion + 02)t} the last equation transforms into %Q(r, t)/at = - i 7” r - G(t)$(r,
t)+D_V2$(r,
t)-u(t)
* V$(r, t)
(34
From this it follows that the principal echo at t = 22 is given by [I, 21 $(2*) = B(O)exp (-rfD_
[/:‘dtF(t)
*F(t)-4F(r)*r’dfF(t)
+ 4zF(z) * F(T)]) exp ( - iy,[H(2z)
where t F(t) =
H(t)
=
s
‘dt’G(t’) 0
‘dt’s(t’). G(t’) s0
- 2H(z)] + i(2+ -n)}
(3.6)
5
s(t) =
s
‘dr’v(t’) 0
with B(0) some initial value and 4 the relative phase angle of the 42 and n pulses in the Hahn sequence (7r/2 - r - rt - z - echo). This expression shows that diffusion attenuates the echo amplitude, while flow shifts the echo phase so long as the electric and magnetic fields are not perpendicular. Table 1 summarises the implication of the equation. With typical values of the experimental parameters (g N 0.1 T m-r, 6 N 5 x 10P4s) T, dominates the diffusional attenuation.
TABLE
1
THE VALUES
OF a AND
b IN THE EXPRESSION
REFER TO A PMFG EXPERIMENT
[I]
echo = B(0) exp{-2t/T,--y,2D_a}
AND
FOR THE PRINCIPAL
THE PARAMETERS
ECHO.
ARE DEFINED
THE LAST THREE COLUMNS THERE
exp{iy,,c_b+i(2+-n)} b
b
b
b
b
Start of application of electric field
0
t
tl
II
t,+s
End of application of electric field
2t
2t
tl+A+8
r,+A
t,+A
t3 (end-start)
2t
T
A+6
A
A-6
t4 (t-start)
5
0
t--t1
t-t,
t-tl--d
a
%q(3t3t-t42) Steady gradient Pulsed gradient (9 >> go
: A >> 6)
g2Ab’
g4AS gJ(x-tl-
;)
gpA6
- q
go4(t~2/2-t~2)
RELAXATION TIMES IN THE PRESENCE OF AN ELECTRIC FIELD
We now use Redfield theory to calculate electric field strength. The hamiltonian structure ff = Ho+Hs, H, = -w,I,-qS,
Ho = H,+H,
of o2 the dependence of the problem is
Hs*HL=O
on the
(4.1) (4.2)
dipoleHs is the spin hamiltonian, HL the lattice, and HsL the electron-nucleus dipole hamiltonian. The equation of motion for I+ may be dealt with in the normal way of iterating in an interaction picture, approximating the second-order term, and ensemble averaging. The linear term disappears even in the presence of an electric field. In order to show this it is sufficient to show that P(r, t) remains
6
isotropic
if P(r,, 0) is isotropic. If P(r,, 0) is isotropic the angular integration in dr,P(re, 0) P(r; t; r,,)may be carried out, which leaves P(r, t) indeof any angles. Thus P(r, t) is isotropic, and the relaxation of I+ is governed
P(r, t) = J
pendent
by a correlation function of the form < ZZsL*ZZ&(r)*Z+ (t) > , where H&(r) is the evolution under the influence of Ho. The relaxation time that this leads to is 02
~~,“~‘(~0/471)~(4$0(0)+
=
$0(0,-0,)+3~1(0”)+6~1(~,) +6$2(c~+~,)I
(4.3)
where j$(o)
= 2J
dr T,(r) exp {(iw
1/r&r}
-
(4.4)
0 r,(t)
=
~C~~‘(~,~r~~C~~‘*(~~)~
(i-p
(4.5)
and ~1is the magnetic dipole moment in Bohr magnetons. When w, appears in an argument zA = t2, when o, does not appear r1 = rr : z1 and r2 are the electronspin relaxation times. The central problem is the evaluation of the correlation function. Using the conditional probability given by the k-space solution of eqn. (3.2), via multipole tensor expansion with an electric field at an angle Sz to the z-direction we have P(r,; t ;
=
1
X
x
ro)
1/2n2) C ?-‘1-‘2(2f1 + 1)(2E, + 1)(21+ 1) urn>
(
1,
im
-tnl
m2 f2)(:
c~:‘(o)c~‘(~)*c~‘(s)*( :
-
1)m’
:2)
dk k2j,,(kro) j,,(kcz) j,(kr,) exp (- k2Dz)
I
(4.6)
which is to be used in
r,(z) =
s
dr, dr, P(r, , O)P(r,; z ;
“C~‘(z)C~‘(O)*r~
ro)r;
3
(4.7)
0) is equal to some isotropic, homogeneous concentration IZ.The integrations implied by these expressions are straightforward if the r-integrations are confined to a region outside some distance of closest approach, d [21]. When the electric and magnetic fields are parallel we find
P(r,,
Z;‘(r) = (4rrn/d3)T(-$‘(2f+l)
(;
‘,
;)(;
;
A)
dk k- ‘Jf( kd) j,( kcz) exp ( - k2Dz)
x s
When the field is perpendicular only the term I = 0 contributes. The spectral density corresponding to this correlation function
is
(4.8)
in which x = kd and (4.10) with zd
a
k
(x) = x2+ zd/zA+ iwr,
(Td =
d2/D)
b(x) = xc/d and where the coefficients
pFC) take the values
/?P’ = 415 By’ = -8/7
/I’,” = -4/7
p:“’ = 8/7
pb”’ = 72135
D(4) r = -48/35
fly’ = 12/35
The r-integral in eqn. (4.10) is the Hankel-Gegenbauer generalization of the integral of Lipschitz [21]. So long as ju+ I > b it is known that the integral may be expanded in terms of hypergeometric functions: IYP)(.x) = (;):(fj2P+i(&)2u+i(~~) _
,F,(y+f,
P+l;
2P+3;
-b2/a2,) The restriction implies c* 2 D/z, when o = 0, and c2 5 WD when wf not known whether the expansion is valid outside this region. Only the p = 0 contribution can be summed explicitly: @‘(x)
= (l/b)
in powers of (b/a,),
0. It is
(4.12)
arctan (b/a,).
The others may be expanded
(4.11)
and we truncate
at second-order: (4.13a)
(4.13b)
(4.14a)
(4.14b) with
u2 = 2Wtd yo = 11,
v= = td/rl y1 = 9,
y2 = 3.
These integrals may be evaluated by contour integration. We concentrate on their values when w = 0 (so that u = 0), and when o+ 0 but U’ B v’, as is true in all reasonable cases (because o,rA B 1):
P2’(0, 0) =
P’(u,O)
.s i
[(3~~-15)+(4~~+14v~+27u~+30u+15)e-~”] >
= -
(4.15)
-$ (
(3(2~=+10)+[(2t14+14113+41u=-30)c0sU 1 +(2u4-41u=
- 60~ - 30) sin u]e-“}
For small values of v, Z(O) (0, v) N z/15 and Z’=’ (0, u) - 7c/48u; this suggests that at v = 0.1 the ratio I(‘) (O,u)/Z(‘) (0, u) has th e value 3.12 in comparison with the accurate value of 2.95. This is virtually the end of the calculation. Equations (4.13) give the spectral densities in the absence of the field (c = 0) and when it is present (c = c+ + c_, C+ = u,E). Substitution of the appropriate spectral densities, as calculated from the Z(u, v), into eqn. (4.3) enables one to calculate the nuclear spin relaxation time in the absence, or in the presence of the electric field. The Z(u, o) are simple, explicit functions expressed wholly in terms of the dynamical characteristics of the medium and the electron-spin relaxation times, which are assumed to be independent of the electric field strength. The functions Z(‘) ( 0 , u) and Z(O) (a, 0) are plotted in Fig. 1 and the ratio ZC2)(0, v)/Z(‘) (0, v) in Fig. 2.
Ob-
0.4o,tO0.1
1 0.3
Fig. 1. The functions
1 1 0.5 01
I PO u orv
1(0)(0, u) and I(@(u, 0) normalized
5.0 7.0 100 to n/15.
Fig. 2. The ratio P(O,
o)/P(O,
v).
DISCUSSION
We propose observing the NMR relaxation of a negative ion with nuclear spin 5 (e.g. F-) by a Hahn spin-echo sequence, the experiment being done in the presence of a paramagnetic ion so that the nuclear relaxation is dominated by the translational modulation of the inter-ionic dipole-dipole interaction. The modulation of the translational motion of the ion is modified by the application ofpulses of electric field, and the spectral density is modified. This modification will affect the relaxation behaviour of the nucleus, and by observation of the effect we anticipate that it may be possible to draw conclusions about ionic motion in electrolyte solutions. The calculation is based on a Smoluchowski-Debye type of equation and Redfield relaxation. The most serious approximation in the theory is the neglect of the consequences of the existence of an ionic atmosphere. (The importance of second coordination sphere effects is discussed by Stengle and Langford [22].) The field strengths we envisage correspond to the first Wien effect region, and so the complications may be minimal. The assumption that the relaxation is adequately described by a Redfield approach suffers from its normal limitations. Furthermore the calculations of the spectral densities depended on an integral with a value that has only been proved for a range of values. The fact that it has not yet been proved over the whole range of values of interest is upsetting, but unavoidable. The situation is alleviated by the fact that the range of values not yet accommodated by existing proofs is very narrow, and unless the integral is singular in this narrow range, the smooth interpolation we have used will not be greatly, if at all, in error. Experimental problems are to be expected in three inter-related areas. First there is the balance concerning the concentration of ions that should be used. Decreased concentration gives a longer T, and so it cannot be reduced too far
10 before diffusional attenuation assumes a greater importance. But decreased concentration diminishes heating and deposition.The second major problem is of course the duration of the electric field pulse. Very large fields (- 10 MV m- ‘) are needed to observe significant effects, and they should be applied for about 1 ms. This is very demanding experimentally, and might be unattainable. The third problem is the effect of the transient magnetic field generated by the conducting ions, and especially its effect on the homogeneity of the applied magnetic field and its gradient. If these problems can be overcome, and if the assumptions of the calculation are tolerable, then we can predict two effects of the electric field on the spin-echo from the conducting ions. First, the echo phase has a cosine modulation. Second, the echo amplitude is modified because l/T, is modified. We now establish possible magnitudes of the effects. The larger the values of r1 the larger the value of I’*‘(O, U) ( we recall that when w, appears in the argument of f(w) then r1 is to be identified with r2, and when it does not appear it is to be identified with rl). When u”( = 2wr,) is non-zero the values of both J(‘)(u, U) and Z(*)(U, U) are diminished. It follows that when o,~,>>l, y(q) is much less important than B(O) both in the presence and absence of the electric field, and so we neglect it. We take as an example numerical values appropriate to F- in the presence of CrOxi - (for which the magnetic moment is 3.8 ,~,[23]). The reason for treating this system is that experimental values for the electric field-free relaxation of Fare known, and so we can check that part of the calculation. This system would be inappropriate in practice because the relative velocity of ions of the same charge is small. If ions of opposite charge are chosen strong second coordination sphere interaction may occur (in which case there may be a significant second Wien effect). As to the first problem, we are concerned here to show how the calculation proceeds rather than to analyze a particular system. As to the second problem, systems can be chosen, for example Cr(H,O)z+/PF;, which have negligible second coordination [22]. The calculated electric field-free values at 0.1 mol dmM3 concentration of CrOxi- are w2 = 87 s-r (for w,zd < 1) and 30 s-r (for W,T~ >> 1). These figures are based on the values of d = 600 pm and D = 2.0 x 10m9 m2 s-l [24]. The experimental value is approximately 25 s-l [22]. When a perpendicular electric field is applied the fractional change in a2, with r1 - 1 ps and the above parameters, is given by 6oL2/w2 = Kc*, with K = 0.7 s2 m-* and c, the sum of the ionic velocities (ci = +!Z). The maximum velocity of the ions is about 1 m s-’ (at 10 MV m-l), and so a fractional shift of relaxation time of no more than 0.7 may be anticipated. This means that T2 may be changed by about 50 %, or so, under optimal conditions. The magnitude of the change is increased by increasing d and decreasing D. It follows that optimum conditions should be obtained for large, highly charged ions. Examples of negative ions which might usefully replace F- are PF, Al Fz-. The latter should give at least a five-fold increase over the change calcu-
11 lated above, but the more complex
nature of the ion might introduce
complications
due to polarization distortion and intramolecular effects. We thank the Science Research Council for a research studentship, and the Warden and Fellows of Wadham College for a Senior Scholarship, both to M.J.C. We thank Dr G. T. Evans and Dr R. Freeman for helpful discussions.
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