2079
POWDER ENDOR LINESHAPES
Powder ENDOR Line Shapes: Nuclear Relaxation Induced by Motion of Nearby Electron Spins
by Daniel S. Leniart,* James 5. Hyde, and Jacques C. Vedrinel Varian Associates, Analytical Instrument Division, Palo Alto, California 9@708
(Received M a y 20,1971)
Publication costs assisted by Varian Associates
The line shape and intensity of electron-nuclear double resonance (ENDOR) spectra in powders is determined, in part, by the specific form of the mechanism(s) that produce nuclear spin-lattice relaxation. In all cases investigated, nuclear relaxation is assumed to stem from modulation of the electron-nuclear dipolar (END) interaction. The initial model considered assumes that the unpaired electron is trapped in a cage and undergoes a series of jumps between at least two equally probable sites. The dimensions of the cage are such that any fluctuations in angle are severely restricted, and this jumping motion results only in a modulation of the interparticle distance between the trapped H atom and surrounding nuclei outside the cage. It is shown that the dominant relaxation component of the surrounding nuclei is proportional to sin40 and leads t o a doubly peaked ENDOR spectrum. The second model assumes that the trapped electron resides in a cage whose walls are at infinite potential, and the system is treated quantum mechanically as a particle in a three-dimensionalbox. The thermal lattice motions of the cage modulate the dipolar interaction in some random manner to produce a spectral density which differsfrom that of the jump model by the appearance of a Boltamann factor describing the particle populations and a frequency term corresponding to the quantized transitions induced between the energy levels of the box. It is shown that for particle transition frequencies, wvvt, which are less than or equal to spin transition frequencies,we f wn, the quantum mechanical spectral densities reduce to those of the semiclassical jump model. As the temperature is reduced below 20°K the ENDOR line shape expected from the "box" model is shown to resemble that of a matrix ENDOR signal. At very low temperatures a change in both the ENDOR line shapes and intensities is predicted.
I. Introduction We have been concerned for some time in this laboratory with the development of methods for performing and analyzing electron-nuclear double resonance experiments on unordered solids, so-called powder ENDOR.2-7 The motivation is improved epr resolution from syst)ems where sufficiently large single crystals cannot be prepared. It is nevertheless assumed that the local order is high such that the sample can be visualized as a large number of crystallites isotropically distributed in orientation. The companion paper to this one describes an ENDOR experiment on Y-type zeolites 7-irradiated at 77OK.* Hydrogen atoms are trapped and the ENDOR experiment is performed on protons in the vicinity of the trapped atoms. The present theoretical analysis was developed during the course of this experiment. It is this type of sample which is envisioned throughout, although we have attempted to generalize the analysis. If a double resonance experiment is t o be performed on a system having a nuclear longitudinal relaxation time, TI,,that can be made comparable to the electronic longitudinal relaxation time, 2'1,) then the intensity and line shape of the stationary ENDOR signal are dependent on the magnitude and mechanism of this nuclear relaxation. I n general, the overall line shape of a powder ENDOR signal may be written as
g(v
- vo(r,0,+))r2sin O d4dOdr (1)
where g ( v - vo(r,6,4) is the spin-packet distribution function-which is commonly assumed to be either Gaussian or Lorentzian-centered at a resonant frequency vo(r,O,+) that may be orientation dependent, Tln--l is proportional to the number of nuclear spins relaxing per unit time, and k is a constant which affects the amplitude, but not the shape, of the entire spectrum. Equation 1 implies that 2'1, is orientation independent. Hyde,'la et al., initially investigated this problem from the standpoint of ENDOR and evaluated eq 1 using a (1) Varian Associates Postdoctoral Fellow, 1970. (2) (a) J. S. Hyde, G. H. Rist, and and L. E. G. Eriksson, J . Phys. Chem., 72, 4269 (1969); (b) A. L. Kwiram, Bull. Amer. Phys. Soc., 88, 4763 (1968). (3) G. H. Rist and J. S. Hyde, J . Chem. Phys., 50, 4532 (1969). (4) L. E. G. Eriksson, J. S. Hyde, and A. Ehrenberg, Biochim. Biophys. Acta, 192, 211 (1969). ( 5 ) G. H. Rist, J. S. Hyde, and T. Vanngard, Proc. Nat. Acad. Sci. U.S., 67, 79 (1970). (6) G. H. Rist and J. 6.Hyde, J . Chem. Phys., 52, 4633 (1970). (7) J. 8. Hyde, T. Astlind, L. E. G. Eriksson, and A. Ehrenberg, Rev. Sci. Instrum., 41, 1598 (1970). (8) J. C. Vedrine, J. S. Hyde, and D. S. Leniart, J . Phys. Chem., 76, 2087 (1972).
The Journal of Physical Chemistry, Vol. 76, N o . 16,1972
D. S. LENIART,J. S. HYDE,AND J. C. VEDRINE
2080
special set of conditions. They assumed that the unpaired electron spin, evenly distributed over a sphere of radius ro < 3.5 8, interacted with surrounding nuclei assumed to be in a uniform continuous distribution throughout the matrix. The nuclear resonance frequency was determined solely by the electron-nuclear dipolar interaction and a spin-packet line shape of Lorentzian character was employed. They postulated a specific form of the nuclear relaxation and proceeded to solve eq 1 using a variety of limits when performing the integration over dr; i e . , they calculated the line shape for various values of the dipolar interaction. Their results show that an unpaired electron which interacts with a continuous distribution of protons will give rise to an ENDOR signal whose position is unshifted from the free proton resonance frequency and whose intensity and line shape are dependent on the magnitude of the dipolar term. Their calculations are in agreement with the experimental results which consistently show a single resonance peak at the free proton frequency. This particular type of spectrum has been coined as the matrix ENDOR signal. I n the present work vie investigate the class of powder ENDOR line shapes produced by the interaction of an unpaired electron spin with a single nucleus. The spin Hamiltonian includes contributions from the nuclear Zeeman, isotropic hyperfine, and electron-nuclear dipolar interactions which may produce a variety of ENDOR line shapes depending on their magnitudes relative to each other. A second equally important factor contributing to the line shape is the form of the dominant mechanism that produces nuclear relaxation. A few of the more probable relaxation mechanisms are discussed, and their effects on the EKDOR lines are given explicitly in the form of computer simulations. The new ideas presented here are largely concerned with nuclear relaxation induced by the END interaction when (a) a trapped H atom undergoes random motion within the cage and (b) the trapped H atom is quantized within the cage in much the same manner as is the model of a particle in a box. These models are then related to the EKDOR line shapes that can be expected from powder samples.
11. Effect of Nuclear Relaxation on ENDOR Line Shape Case I. Tln-l Independent of e and 4. Initially we make the simplifying supposition that the nuclear relaxation probability is a constant which may be removed from the integrand of eq 1 and included in k . All rotationally invariant interactions which induce relaxation, such as a fluctuation in the hyperfine splitting whose form is written as a scalar product, XI = a ( Q l . 8 , may be grouped in this category. For an electron interacting with protons at specific distances ( i e . , shells of protons), the spectral line shape is then written as The Journal of Physical Chemistry, Vol. 7 6 , N o . 16,1972
f(v,r) =
e - (dri,O,+) - 2 a ~ ) ~ / ( T z - ' )sin ~ eded4
h i 2
(2)
where the resonant frequency w(ri,O,q5) is determined from the Hamiltonian X(ri,e,q5) describing the spin system of interest. When dealing with zeolites the appropriate Hamiltonian is simply X = Xz
+ XFC+ XD(ri,e,4)
(3)
where
XZ = gePeXoSz XFC
+ gnPnHolzi
= hYeaiIi.8 = hyeaiIzi8,
(44 (4b)
XD(ri,e,4) =
and ai = gePegn,PnI/2~i3*When investigating samples other than zeolites, one may wish to include additional interactions such as quadrupole, zero field, etc., but here we will restrict ourselves to the three interactions mentioned above and examine their effects on powder ENDOR of protons. If the hyperfine interaction is negligible (ai = 0), the ENDOR position will be determined by the Zeeman interaction only, and the ENDOR line shape by the E N D interaction. When monitoring a particular epr hyperfine line (e.g., m~ = - 1/2) with the microwave field and sweeping the radiofrequency through nuclear resonance, each nu'cleus will resonate at two possible frequencies depending upon the sign of m, (cf. eq 4c). If the sample is a powder, the intensity distribution obei ~ / is2 similar tained by integrating eq 2 over 0 to that given by Pake and other^.^^'^ If each spinpacket line shape is represented by a Gaussian distribution with a half-width of half-height, TZ-I, equal to some fraction of the magnitude of the dipolar interaction (e.g., O.lai), then the simulated line shape is given in Figure 1. Under certain conditions it is possible to obtain an EKDOR line shape from a hydrogen atom and only one proton that is (1) a single line centered at the free proton frequency, vi, produced by a dipolar interaction which is small compared to the spin-packet line width, ai < T2-l; (2) a doublet with peaks v =t ai when the single line above is split by a larger dipolar interaction, ai > Tz-'; and (3) a doublet with two distinct peaks at
<
> 1, the ENDOR line shape will have a peak and a shoulder both above and below the free proton frequency centered a t v f ~eai/2. However, the frequency separation of the ENDOR peaks for a positive splitting constant will be (yeai 2ai) MHz compared to a shoulder separation of (yeai 4 4 MHz. On the other hand, if ai is negative, the peak-to-peak separation of (yea; - 2ai) now becomes less than the shoulder to shoulder separation of (yeai 4ai) MHz. ENDOR of powders under these conditions thus provides an easy, convenient method for determining the sign of the hyperfine splitting con~ t a n t . If ~ the ratio (ai/Tz-') is roughly equal to unity,
+
+
the shoulders will disappear; however, the sign of ai can still be determined from the absolute value of the theoretical peak-to-peak splitting by plotting the theoretical spectrum employing both positive and negative values of ai and comparing each of these results with the experimental value of the peak-to-peak separation. It would be impossible to extract the sign of ai when ( ~ I / T ~ > 1, then one might expect addiThe Journal of Physical Chemistry, Vol. 76,No. 16,1,972
I
TRAPPING SITE
Figure 4. A physical model of the H-atom trapping site. The unpaired electron is assumed to jump between a t least two sites, A and €3. This motion changes both e and ri, but the amplitude of the angular fluctuations is small compared to fluctuations in the magnitude of ri.
tional peaks or shoulders to be found at v,
* yean/2 f
2a.21
(c) Quantum Mechanical Model. At this point it is appropriate to consider other models that may describe the mechanisms of spin-lattice relaxation and compare the resulting ENDOR line shape with that derived from the jump model. If the walls of the p cage are assumed to be at infinite potential, the trapped H atom is analogous to a particle in a three-dimensional box. The size of a sodalite cage determines the allowed energy levels and the boundary conditions at the walls define the nature of the eigenfunctions. The particle populations are obtained by taking a thermal Boltzmann distribution over the different allowed states. To produce spin relaxation the particle-in-a-box must be coupled to (20) Recall v Z occurs at 0 = 5 4 . 7 O whereas the pseudosecular term peaks a t 45'. However, this small difference is averaged out when both ms = f l / z and ms = - I/z are considered. (21) I n both cases the sign of the hyperfine splitting has been assumed t o be positive. If it is negative then replace the f preceding both terms by F.
POWDER ENDOR LINESHAPES A.
a
2085 c.
+
-(F(q))(F(q'))[~/l
Figure 5. ENDOR line shapes produced by all five components of the nuclear relaxation probability. The upper portion of A, B, and C shows a dashed line depicting the double resonance line shape related to W (f 2 ) a sin4 e, a dotted line depicting the contribution from the W (i1) 0: sin2 e cos2 0 terms, and a solid line showing the W ( 0 ) a (1 - 3 cos2 e ) 2 component. The lower protions of A, B, and C are superpositions of the three line shapes shown above. D is EI superposition of the lower three spectra of A, B, arid C. The data used to compute these line shapes is as follows: A, TP- 1 = 50 kHz, an = 0.0, ai = 20 kHz; B, Tz-' = 50 kHz, a n = 0.0, ai = 132 kHz; C, Tz-1 = 50 kHz, an = 0.0, ai = 225 kHe.
the thermal lattice modes of motion and this coupling arises from the END interaction of the spin system (unpaired electron plus hydroxyl proton). The hydroxyl protons which form a part of the sodalite cage may be considered as fixed in the lattice and the dipolar interaction is modulated by those thermal modes of motion that affect the internuclear distance, rn1. I n this case the relaxing spin system gives up a quantum of energy which may induce a corresponding transition from some particle energy level. The state of the spin system plus particle is properly described by the density matrix pav,Bv,, where a,@ refer to spin states and v,v' refer to particle state^.^^-^^ Hence, for this model the dipolar interaction must be treated quantum mechanically as opposed to the semiclassical treatment used to describe the jump model. Following the work of Freed, one may solve for the time evolution of the density matrix in the interaction representation (14) where X D ( ~is) the dipolar term operating on both the spin and vibrational states of the system. Assuming that (1) p ( t ) := X(t)o(t) where X ( f ) and a(2) are the reduced density matrices depending on only vibrational and spin degrees of freedom, respectively, and that (2) the thermal motions of the lattice are random with a mean time interval T which is short enough so (3) ~ ( t ) is not appreciably changed and only (4) A ( t ) is affected (it?.,the thermal perturbations have no matrix elements between spin states), the resulting spectral density analogous t o eq 12 (and Table 11)may be written as p * ( t ) = -i[XD*(t),p*(t)]
(15)
The final assumption made to formally solve eq 14 is that the vibrational states are given by a Boltzmann distribution and the postvibrational distribution is independent of the distribution just before a vibration occurs. The basic nature of the quantum mechanical model is that the thermal motions represent a very strong particle perturbation which rapidly restores the vibrational states to equilibrium, while the dipolar ) , a much weaker perturbation that interaction, X D ( ~ is slowly tends to bring the spins to equilibrium. The spectral density of eq 15 reduces to the form of the semiclassical formulation similar to the relaxation probabilities given in Table I1 when the relaxing spin state induces no change in the vibrational configuration of the system. This process may be visualized in the following manner. The thermal vibrations cause a broadening of the H-atom eigenenergies. I n general, the natural infrared line widths obtained for the 0-H atoms comprising the @ cage are of the order of 10 cm-' a t room temperature corresponding to the T 0.5 X If we assume that the thermal molecular vibrations are also of this order, then each quantized particle level is broadened by nearly A w = 2 X 10I2 sec-l. The quantum of energy, waB, emitted by the relaxing spin system (-10" sec-l) is now well within the frequency spread (2 X 10l2sec-') of a particular Hatom eigenenergy [u = 3n2a2/8ma2 10I2sec-' ( n = l ) ] found in a cubical box of length 6.6 Thus the spin system may relax without changing the energy level of the particle and wvv = 0. The diagonal matrix elements become
-
-
(z
-
II sin2 (vqrq/Z) d r (16) q=z,y,z
and the nuclear transition probability for the H atom located outside the box at (xo,yo,xo)is given by L v
(22) J. H . Freed, J . Chem. Phys., 41, 7 (1964). (23) J. H. Freed, ibid., 45, 1251 (1966). (24) J. H . Freed, lecture presented at NATO Summer School on Electron-Spin Relaxation in Liquids, Spatind, Norway, Aug 1971. (25) (a) J. W. Ward, J . Catal., 9, 396 (1967): (b) J. B. Uytterhoeven, R. Schoonheydt, B. V. Liengme, and W. K. Hall, ibid., 13, 425 (1969); (0) J. W. Ward and R. C. Hansford, ibid., 13, 364 (1969); (d) J. W. Ward, Proceedings of the 2nd International Conference on Molecular Sieve Zeolites, Worcester, Mass., Sept 1970. (26) See note 17 of ref 22. The Journal of Physical Chemistry, Vol. 76, N o . 16,1978
2086
D. S. LENIART,J. S. HYDE,AND J. C. VEDRINE
Here B(v) is the Boltzmann distribution in vibrational states. An alternative situation is present if the energy levels we f of the box are separated by a frequency wVv, w n ; then the relaxing spin system may be accompanied by a corresponding particle transition. However, if (was wvvt) r is still less than unity (case i), the quantum mechanical, formulation of the relaxation probability will remain quite similar to that predicted from semiclassical theory
-
+
2Wn
[5Z(v)e-8:v/kTFVVtFV~vC B(v)F,, C B(v')FVw]7 Y
(18)
V'
wvVt
where the off-diagonal elements of F are given by
F,,I
=
( y sin (~~'nq/Z)Ir--~/ II sin (vqsq/Z))
(19)
P
and r U 3is defined in eq 16. Although no experimental ir data at 20°K could be found, the EXDOR experiments on zeolitess indicate that W , a r and the system may still be treated semiclassically, exhibiting ENDOR line shapes previously discussed in this section. Upon reducing the temperature (case ii) either the particle populations change or the eigenenergy broadw v V r ) r > 1. ening decreases, or both, such that ( w a p Nom W , is given by eq 15
+
2Wn = jqq'(~ag
+
(20)
wvvt)
and two situations may arise which affect the ENDOR line shape. First, if pure nuclear relaxation from t h e pseudosecular terms of the dipolar interaction is dominant, i.e. ( o a o r ~ wVv,)272
