Muus, L. T. & Atkins, P. W.. (Plenum Press, New York), pp. 341-381. 10. Wilkinson, J. H. (1965) The Algebraic Eigenvalue Problem. (Clarendon Press, Oxford) ...
Proc. Nat. Acad. Sci. USA Vol. 71, No. 5, pp. 1925-1929, May 1974
Investigation of Very Slowly Tumbling Spin Labels by Nonlinear Spin Response Techniques: Theory and Experiment for Stationary Electron Electron Double Resonance (isotropic rotational diffusion/free diffusion/adiabatic rapid passage/passage electron double resonance)
MURRAY D. SMIGELt, LARRY R. DALTONT, JAMES S. HYDE§, AND LAURAINE A. DALTONJ Departments of t Molecular Biology and $ Chemistry, Vanderbilt University, Nashville, Tennessee 37235; and § Varian Associates,
Instrument Division, Palo Alto, California 94303
Communicated by Dudley Herschbach, February 21, 1974 The investigation of very slowly tumbling ABSTRACT spin labels by nonlinear electron spin response techniques is discussed. Such techniques permit characterization of rotational processes with correlation times from 10-3 to 10 -7 sec even though the linear spin response (ESR) technique is insensitive to motion in this region. Nonlinear techniques fall into two categories: (a) Techniques (referred to as passage techniques) in which the distribution of saturation throughout the spin system is determined both by the applied magnetic field modulation of the resonance condition and by the modulation of the resonance frequency induced by the molecular motion. The time dependence of this distribution produces phase and amplitude changes in the observed signals. (b) Techniques that measure the integral of the distributioni function of the time required for saturated spin packets to move between pumped and observed portions of the spectrum [stationary and pulsed electron electron double resonance (ELDOR) techniques]. Quantitative analysis of passage ESR and stationary ELDOR techniques can be accomplished employing a density matrix treatment thai explicitly includes the interaction of the spins with applied radiation and modulation fields. The effect of molecular motion inducing a random modulation of the anisotropic spin interactions can be calculated by describing the motion by the diffusion equation appropriate to the motional model assumed. For infinitesimal steps the eigenfunctions of the diffusion operator are known analytically, while for random motion of arbitrary step size they are determined by diagonalizing the transition matrix appropriate for the step model used. The present communication reports investigation of the rotational diffusion of the spin label probes 2,2,6,6-tetramethyl-4-piperidinol-1oxyl and 17f3-hydroxy-4',4'-dimethylspiro-[5a-androstane3,2'-oxazolidin]-3'-oxyl in sec-butylbenzene. Experimental spectra are compared with computer simulations of spectra carried out for isotropic Brownian (limit of infinitesimal step size) and free diffusion (arbitrary step size) models.
INTRODUCTION
Studies of electron spin resonance (ESR) lineshapes of spinlabeled biological molecules, as pioneered by H. M. McConnell (1), have become an important tool for determining both the motion of the spin label and the dielectric properties of its micro environment (2). However, these experiments are limited to correlation times between 3 X 10-7 and 10-10 see (to Abbreviations: ESR, electron spin resonance; ELDOR, electron electron double resonance; TANOL, 2,2,6,6-tetramethyl4-piperidinol-l-oxyl; HDA, 17jB-hydroxy-4',4'-dimethylspiro[5a-androstane-3,2 '-oxazolidin] -3 '-oxyl. 1925
rotational frequencies of the order of the magnetic anisotropy). Many important biological molecules exhibit much slower (10-1-10-7 see) rotational times, and an alternative technique is required for their investigation. Recently an alternative has been demonstrated through the investigation of the nonlinear response of a spin system to an intense radiation field (3, 4). The response of a spin system to an intense radiation field is critically dependent upon energy transfer between the spin system and the lattice and between resonant and nonresonant portions of the spin system (5). This latter effect,- known as spectral diffusion, has usually beef associated with Heisenberg spin exchange; however, molecular motion which modulates an anisotropic magnetic interaction can also induce spectral diffusion. Indeed, for a dilute solution of slowly tumbling paramagnets, molecular reorientation is the dominant spectral diffusion mechanism. Two classes of nonlinear spin response techniques 'have been developed to map molecular motions. We will briefly discuss the first of these, adiabatic rapid passage, and then analyze in more detail the second, stationary electron-electron double resonance (ELDOR) (6). Rapid passage ESR experiments involve modulating the applied magnetic field (hence, the eigenvalue equation) and detecting either the first harmonic of the dispersion signal with a phase-sensitive detector 900 outof-phase with respect to the modulation (3) or the second harmonic of the absorption signal with the phase-sensitive detector 90° out-of-phase (3, 7). Passage phenomena can also be detected employing an ELDOR configuration (3) and passage ESR and ELDOR spectra are sensitive to molecular motion characterized by correlation times in the range of 1010-7 sec, where an ordinary ESR experiment gives a spectrum virtually identical to that of a rigid powder. Passage experiments are relatively easy to carry out, but extraction of details of the molecular motion from the spectra is nontrivial. Unlike ordinary ESR spectra, the interaction of the spin system with both the applied radiation and modulation fields must be taken into explicit account. A complete density matrix treatment is required for which computation is carried out employing the stochastic Liouville method (8), where the density matrix equation of motion due to the random Hamiltonian JC(t) is written as A(at) = -i[3C(%t), p(lt)] - Po(Q)-'rF[Po(Q)p(Qt)] [11 Po(Q,t) is the equilibrium probability of finding the "orientation" Ql at the particular state at time t and rF is a time-in-
Chemistry: Smigel et al.
1926
Proc. Nat. Acad. Sci. USA 71 (1974)
dependent Markoff operator. In the high field approximation the spin Hamiltonian can be expressed in the form
3CA(Q.t)
3C14(2)
= 3CO +
+ e(t) + 3CR
[2]
where the time-independent Hamiltonian
3Co
=
gh -,eHoSz
-
E k
YnjIz, Ho - Ye Ek akSzlk
[3]
including electron Zeeman, nuclear Zeeman, and electrronnuclear hyperfine interactions determines the zero-orrder energy levels and transition frequencies. The anisotrc)pic electron Zeeman and hyperfine interactions that are rrandomly modulated by molecular motion are contained in tne term
3C3C1(Q) (&2) =
j A,
L,mm', ,k
Wm') (O2)F IAL'm'A k jA.k 59-~mwmQ n
[41]
The magnetic anisotropy, FAtk , is expressed in moleciulefixed coordinates while A (Lkm' is the nuclear and elect ron spin operator quantized in laboratory-fixed axes determi ned by the dc magnetic field direction. The D_ Lm (2) incl ude the transformation between these two coordinate syste ms. For axial symmetry 5C,(Q2) = 0D2 Sz[F + T'Ij] + ( D 2I I+SZ -D2 _IS)T [5] where F = 2 [g1 3 T
=
27 >X6
-
T' =
g1I]h-'#3Ho
[All-AI]
-V\(8/3)
T
The term e(t) describes the interaction of the spins with all the time-dependent applied fields. e(t) = do[S+e -W0t + S-eiwot] + d8S,[eiw't + e-icwt] +
Ek d8k1I~k[e
'
+
etw1] [6]
where do = '/2 yeho, d, = '/2 yeH, and dsk' = 1/2 YnkHs. ye is electron gyromagnetic ratio, 'Ynk is the gyromagnetic ratio of the kth nuclear species, ho is the amplitude of the microwave field of frequency wo and H. is the amplitude of the modulation field of frequency w.,. The commutator of the Hamiltonian 3CR which describes the coupling of the spin system to the lattice is approximated by -i[JCR,p] = -rR(p - pO)
=
-rnx
[7]
where p0 is the equilibrium spin density matrix. We express the off-diagonal matrix elements of Eq. 7 in terms of rotationally invariant spin-spin relaxation times and the diagonal elements in terms of rotationally invariant spin-lattice relaxation times. Eq. 1 can be solved for two different classes of diffusion operators (9). In the first, which we shall label the Brownian diffusion approach, ru is written as a rotational diffusion operator (for infinitesimally small steps); for example, for isotropic Brownian diffusion -rP = DVy2 where VQ2 is the angular portion of the Laplacian and D is the diffusion coefficient. For such an approach we take as the steady-
FIG. 1. Computer simulated rapid passage and ESR spectra where parameters are adjusted for best fit of experimental spectra of 1 mM 2,2,6,6-tetramethyl-4-piperidinol-1-oxyl (TANOL) in sec-butylbenzene at the following temperatures and modulation conditions: (A) T = -97.50C, c, = 27r X 103 Hz; (B) = 2v Xshown 104 Hz;represent co(, Spectra (C) andmicrowave (D) T = - 1140C, TC, == -106.50C, 27r X 105 Hz. absorption (ordinate) at constant frequency plotted versus magnetic field (abscissa) which increases to the right. Spectra A, B, and C show the first harmonic of the dispersion out-ofphase with the field modulation. Calculations show the absorption signal at the second harmonic of the modulation and out-of-phase with the modulation to exhibit a corresponding sensitivity to motion and theory further predicts that the in-phase first harmonic of the dispersion and out-of-phase first harmonic of the absorption to exhibit a reduced sensitivity to molecular and applied modulation effects. Spectrum D shows the simulated ESR spectrum, the in-phase first harmonic of the absorption signal, and demonstrates the configuration of the stationary ELDOR experiment. In all ELDOR experiments the observer monitors the spin packets indicated by the upward-pointing arrow. The horizontal arrow indicates the positioning and direction of scan of the microwave pump. state solution to Eq. 1
x(g,t) = E Z(Q,wo i rw8)ei(coorcs)t r
=A
r m
Cm(cwo i rws)Gm(Q)ei(wrw.)t, [8]
the steady-state density matrix in the doubly rotating frame. Gm(Q) represent a complete orthogonal set; for isotropic diffusion rFGm(.Q) = L(L + 1)DGm(Q). ESR passage spectra can be calculated by evaluating the off-diagonal and diagonal matrix elements of Eq. 1 using as a basis set the eigenfunctions of 3Co. A typical passage response such as the dispersion signal at the first harmonic of the modulation frequency and out-ofphase with the modulation is given by ^j [ImCo(Wo + CO) ImCO(WO - (08) ]j. The imaginary part of the Co(wo ± W8)xj for the various transitions is obtained by solving the set of coupled equations where the coupling arises from the 3C,(Q) and e(t) terms of the Hamiltonian. The conditions of passage ELDOR experiments require abandonment of time-independent Z coefficients and the actual integration of the differential equation. Also, the evaluation of the matrix elements of Eq. 1 must be over a basis set which simultaneously diagonalizes JCo and e(t). The second method for treating the molecular motion in solving Eq. 1 is to artificially quantize the continuous variable Q into a finite number of jump sites. For a given step diffusion model the distribution of populations at the various sites at different times can be expressed as a state equation involving
Proc. Nat. Acad. Sci. USA 71
Nonlinear Spin Response Techniques
(1974)
the initial distribution, the final distribution, and a timedependent transition operator. Explicitly P(Q,t) is the column vector of probabilities at each of the discrete orientations
P(lt1) = S(flfj; t1 - to)P(&to)
[91
where S(%,22j; t1 - to) = [S(,Q{Q) ](t1-6O)/T with i. being the time required for one step. Time differentiation of this operator allows the expression of the state equations as a set of differential equations. The operator W(Qf,f7j) so derived becomes ra of Eq. 1 realized over a finite angular quadrature. The eigenfunctions of this matrix are analogous to the eigenfunctions of V02 realized over the same quadrature. In terms of the general differential operator W(f11,%j) the equation for the column vector of the density over spin states, o(Qt), becomes
(Q)= -i[(5C(<), e($2,t)] + [Po]-1 [WI(Qi2,)][Po]e(,0t) [10] where W(Qfl,f7j) is a transition probability matrix whose elements give the transition probability between the ith and jth discrete values of Q and Po is a diagonal matrix of the equilibrium density distribution. For isotropic Brownian diffusion W(42f,0j) is tridiagonal. Such descriptions of the density matrix equation of motion permit quantitative reproduction of observed passage spectra as shown in Fig. 1; however, the computational expense and effort involved motivated a search for nonlinear techniques which can be analyzed more easily, i.e., are more sensitive to the details of the motion and less sensitive to the details of the spin Hamiltonian. Conceptually, stationary ELDOR appeared to be such a technique. In the ELDOR experiment, one portion of the spectrum is saturated with an intense "pumping" microwave field, and another portion of the spectrum is monitored with a weaker "observing" microwave field. An ELDOR display consists of a plot of the fractional reduction in the observed ESR signal height versus the frequency difference between the pumping and observing microwave sources. A paramagnetic molecule in a strong magnetic field may be viewed as having its unpaired electronic and nuclear spins quantized along the dc magnetic field direction. The frequency at which an applied microwave field resonates the electronic spin depends upon the sum of the dc magnetic field felt at the spin plus the magnetic field contributed by nearby nuclear spins. Since this latter hyperfine interaction is mainly dipolar in nature, the resonance frequency depends upon the angle between the molecular axis and the axis of the dc field. Thus molecules with different orientations Ql contribute to different portions of the ESR spectrum. The most direct method to study molecular reorientationinduced spectral diffusion is to saturate a group of spin packets at one frequency (one group of orientations) and monitor the transfer of saturation to a second frequency with a low-level (nonsaturating) microwave field.
cZ
0
.6-
ILi.
z 0 Ic.)
0.4
F A
0.2 B C D 40 30 MHz FIG. 2. The stationary ELDOR responses obtained for 1 mM TANOL in sec-butylbenzene at temperatures of (A) - 106.50C, (B) -114oC, (C) - 120'C, and (D) -130'C are shown. Macroscopic viscosities for these temperatures can be obtained from
10
20
(Vp-Vo)
an extrapolation of the data of Barlow et al. (11) and from these viscosities correlation times can be estimated, yielding (A) -q = 110 P, x2 = 2 X 10 sec; (B) q = 1100 P, T2 = 2 X 10-6 sec; (C) q = 11000 P, T2 2 X 104sec; (D) n = 3 X 106 P. r2 = 5 X 10-2 sec. Frequency swept ELDOR traces were recorded as continuous scans between 0 and 45 MHz with the spectra being independent of scan direction. Field modulation of 14 Gauss amplitude at a frequency of 270 Hz was employed to gain signal enhancement and baseline stability. The strength of the pump microwave field was 0.32 Gauss while that of the observer was 0.063 Gauss.
An effort was made to obtain stationary ELDOR signals without field modulation, as the computation of signals under such conditions is less difficult than the computation of passage spectra. Unfortunately, field modulation and accompanying phase-sensitive detection were, in the present study, found to be necessary for adequate signal to noise and baseline stability. Thus we were forced to employ Eqs. 1 and 2, where e(t) is modified with the added term d,[S+e- 'Pt + S-e"tt]
o 0.6 I-
0 U-
04 0
A
0
cr
0. 2
B
C D E
EXPERIMENTAL RESULTS AND DISCUSSION
The instrumentation, the preparation of samples, and general measurement procedures have been discussed elsewhere (3, 6). To simplify matters, all ELDOR experiments to be discussed here were performed by positioning the monitor microwave field at the high field turning point of the ESR spectrum (see Fig. ID). The pump was then positioned at variable (, - aO)/2Tw = Av up to +45 MHz above the ohserver. Experimental spectra are shown in Figs. 2 and 3.
1927
10
20
(Vp--o)
30
40
MHZ
FIG. 3. The stationary ELDOR responses obtained for 1 mM 17ft-hydroxy4',4'-dimethylspiro [5a-androstane- 3,2'- oxazolidin]3'-oxyl (HDA) in sec-butylbenzene at temperatures of (A) -97.50C, (B) -106.50C, (C) -1140C, (D) -120'C, and (E) -130'C are shown. The spectra were obtained employing the same instrument settings as given in Fig. 2.
~
Chemistry: Smigel et al.
1928
Proc. Nat. Acad. Sci. USA 71 (1974)
0.6
=
CD
0I..40
'C%!
0.4
= CD C-)
-4~ A
B
----
c0 .1-
10
--A
20
30
step i - 1 all possible orientations after a step of fixed angular size, Ostep, lie on a conical surface with axis Oi-1 and half angle Ostep. For free diffusion the molecular axes after the ith step will be uniformly distributed over this conical surface. We compute the polar coordinates of the molecules after the ith step and transform the uniform distribution over the cone to the desired distribution function in 6. We find: A~o1~~,)=1
r
D
40
FIG. 4. The stationary ELDOR response computed for correlation times of (A) 3 X 10-6 sec, (B) 6 X 10-6 sec, (C) 1.2 X 10-5 sec, and (D) 3 X 10-5 sec. Calculations were performed employing a free diffusion model with a 0.03 radian step size. Other parameters employed in the computation include a modulation amplitude of 14 Gauss and frequency of 270 Hz, and a microwave pump field strength of 0.32 Gauss. Magnetic g-tensor and hyperfine tensors were obtained from ESRI and double resonance measurements. Electron spin-lattice and spin-spin relaxation times were obtained by A\I. Huisjen and J. S. Hyde by pulsed microwave techniques.
describing the interaction of the spins with the pump microwave field while do[S+e-iwot + S-eiwot] describes the interaction with the observing microwave field. The state-transition matrix elements in Eq. 9 are given for axially symmetric magnetic interactions and isotropic diffusion by 1
rn max
J
dO
JrOmax(OL )dT
rT A(Ollai-,)da
Co C-,>
[11]
Pi(O1)
OA
m
A
-0~B
r~~~~~~~~~~~~~~
10
20
FIG.
5.
=
sinoi/2. [14]
Since this distribution is unchanged upon further steps this is the equilibrium distribution. This, of course, corresponds to a uniform distribution over the surface of a sphere. For equally spaced angular regions of size 60 the inner integral of Eq. 11
[15]
hk(Oi) (D-
U60
can be expressed in terms of a difference of incomplete elliptic integrals (Sn-') of the first kind as: hk(Oi) =
sino i
2
[(a - 'Y)
r
xX Sn-
[ d
a) ]'/2
( Y ( -Y
/'2
cos(kb0i) cos(k60jf)
-
a-
1
\y Iy-cos[(ko-1)60 l/ I [16] -)G]}/2]} Sn-I[(aL -:
where
COS1max) (cOnOin COsollmax
1/2
2 COS9min
+
30
BcosGi + AsinOi ey= Cos~ ax = BcosO2 - AsinO
f3
=
cosOti
=
a = -1
D
>~
40
(Yp-YO) in MHz
The stationary ELDOR response computed employing
a free diffusion model with a are carried out for correlation
[13]
1
aY=
IV.
->
=
( Ain Oj) (OjOjl,)dOj_, 2
J
_1
C_')
0.2
=
a
TIc0c4 Co
i_)2]/21[12[2
BcosO
for all 0 i-l. The equilibrium distribution function can be found by noting that if the density function at step i - 1, Pi_,(Oj_1), is equal to sin(O,-1)/2 then
0.6
I--
-
0
n
where 0 is the angle between the principal axis of the magnetic anisotropy tensor and the dc magnetic field vector. A simple geometric argument allows determination of the transition function A(0jjIOj1) representing the probability that a molecule oriented along 0i- at step i - 1 in a random walk will be oriented at Oi at step i. For a molecule pointing along O-, at =
(cos1
where A = sin(Gstep) and B = cos(Ostep) when the denominator is real and A(OiOi-1) = 0 otherwise. This function is normalized so that
(Vp-Vo) in MHz
S(A,) =
sinO
[A2sin21_., -
0.15 radian step size. Calculations times of (A) 3 X 10-6 sec, (B) 6 X 10-6 sec, (C) 1.2 X 10-5 sec, and (D) 3 X 10-' sec. Other parameters are the same as given in Fig. 4.
We assume S(Q) can be expressed as S(Q) = MIXMII-' where X is a diagonal matrix of eigenvalues of S(Q) and of is the modal matrix whose columns are the eigenvectors of S(Q).
This implies that
S(Ott
-
to)
-
*
-(t10t)/rt M.
l
[17]
Since X is diagonal, X(t -to)/r- is simply the diagonal matrix whose nonzero elements are the eigenvalues taken to the indicated power. Rewriting the state Eq. 9 as a differential
Proc. Nat. Acad. Sci. USA 71
Nonlinear Spin Response Techniques
(1974)
SUMMARY
equation implies that
P(Qyt) = MAM-1 P(U,t)
1929
[18]
where A is a diagonal matrix in which the ith nonzero element is (1/Tr) lnX1. Eq. 18 is of the form needed for writing Eq. 1 with the identity of
r2 = W(Q iQ1) = MAM11A-1. We chose to find the eigenvalues of S(Q) by transforming to a Hessenberg form and using the QR algorithm of Franpis (10). The eigenvectors of S(Q) were found by inverse iteration as described by Wilkinson (10). The quadrature used was 50 equally spaced angular regions. Typical stationary absorption ELDOR signals at the first harmonic of c. and in phase with ca are shown in Figs. 4 and 5 for angular step sizes of 0.05 and 0.15 radians. The agreement between theory and experiment leaves little doubt as to the adequacy of the present theoretical approach, which is carried out without the employment of adjustable parameters. All parameters employed in the computation were measured by independent means. It is gratifying that the theory justifies our intuitive feeling that nonlinear techniques will be most sensitive to motional frequencies of the order of the spin-lattice relaxation rate. For nitroxide radicals this means that rotational correlation times between 10-7 and 10-4 see can be measured with high accuracy. Perhaps the most unique observation emanating from a detailed comparison of theoretical and experimental stationary ELDOR curves is that very slowly rotating small molecules such as those of the present study are best described by a step diffusion model with step size of approximately 0.15 radians. This observation is also supported by a detailed analysis of ESR passage spectra. Preliminary studies of maleimido-labeled hemoglobin and larger biological molecules indicate that the random motions of the larger molecules are better described by a small step size model (Brownian diffusion). It can also be noted that while microscopic and macroscopic viscosities (11) appear to be of the same magnitude at intermediate temperatures, there do exist some noticeable deviations at the highest viscosities, with the microscopic viscosities indicating that the spin labels continue to rotate after appreciable ordering of the solvent exists.
The stationary ELDOR technique is shown to represent a uniquely powerful method of investigating the rotational diffusion of paramagnetic probes in liquids and glasses. The experimental data can be quantitatively reproduced from model calculations which consider the motion in detail. Comparison of experimental and calculated spectra allows characterization of the motion in terms of both step size and correlation time. Deviations of the microscopic from macroscopic viscosity become accessible to measurement in the region of high viscosity. Acknowledgment. is made to the Donors of The Petroleum Research Fund, Administered bv the American Chemical Society, for partial support of this research. Partial support for this research was also provided by a grant from the Research Corporation. We acknowledge support for the construction of nonlinear spin response instrumentation from the Chemical Instrumentation Section, National Science Foundation. 1. Ohnishi, S. & McConnell, H. M. (1965) J. Amer. Chem. Soc. 87, 2293; Stone, T., Buckman, T., Nordio, P. & McConnell, H. M. (1965).Proc. Nat. Acad. Sci. USA 54, 1010-101?. 2. Smith, I. C. P. (1972) in Biological Applications of Electron Spin Resonance, eds. Swartz, H. M., Bolton, J. R. & Borg, D. C. (Wiley-Interscience, New York), pp. 483-539. 3. Hyde, J. S. & Dalton, L. (1972) Chem. Phys. Lett. 16, 56872; Dalton, L. R. & Dalton, L. A. (1973) Magn. Resonance Rev. 2, 361-397; see also Proceedings of the IVth International Biophysics Congress, Moscow, USSR, August, 1972. 4. Goldman, S. A., Bruno, G. V. & Freed, J. H. (1973) J. Chem. Phys. 59, 3071-3091. 5. 1)alton, L. R1., Kwiram, A. L. & Cowen, J. A. (1972) Chem. Phys. Lett. 17, 495-499; Dalton, L. 1R., Kwiram, A. L. & Cowen, J. A. (1972) Chem. Phys. Lett. 14, 77-81; Dalton, L. R. & Kwiram, A. L. (1972) J. Chem. Phys. 57, 1132-1145. 6. Hyde, J. S., Chien, J. C. & Freed, J. H. (1968) J. Chem. Phys. 48, 4211-4226. 7. Hyde, J. S. & Thomas, D. D. (1974) Ann. N.Y. Acad. Sci. 222, 680-692. 8. Kubo, R. (1969) J. Phys. Soc. Jap. Suppl. 26, 1-5; Kubo, R. (1969) in Stochastic Processes in Chemical Physics, ed. Shuler, K. E. (Wiley, New York), pp. 101-127; Freed, J. H., Bruno, G. V. & Polnaszek, C. F. (1971) J. Phys. Chem. 75, 3385-3399. 9. Gordon, R. G. & Messenger, T. (1972) in Electron Spin Relaxation in Liquids, eds. Muus, L. T. & Atkins, P. W. (Plenum Press, New York), pp. 341-381. 10. Wilkinson, J. H. (1965) The Algebraic Eigenvalue Problem (Clarendon Press, Oxford), 647 pp. 11. Barlow, A. J., Lamb, J. & Matheson, A. J. (1966) Proc. Roy. Soc. Ser. A 292, 322-342.
