Jan Trommel
MOLECULAR MOTIONS AND
COLLISIONS IN ORGANIC
FREE RADICAL SOLUTIONS
AS STUDIED BY DYNAMIC
NUCLEAR POLARIZATION
MOLECULAR MOTIONS AND COLLISIONS IN ORGANIC FREE RADICAL
SOLUTIONS AS STUDIED BY DYNAMIC NUCLEAR POLARIZATION
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE
TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE
HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR
MAGNIFICUS PROF. IR. L. HUISMAN, VOOR EEN
COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN
DEKANEN TE VERDEDIGEN OP WOENSDAG
8 MAART 1978 TE IU.OO UUR
DOOR
JAN TROMMEL
natuurkundig ingenieur
geboren te Stad aan het Haringvliet
DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR
PROF. DR. IR. J. SMIDT
Aan Ria, Huib en Sung-Min
Aan mijn ouders
Ter herinnering aan
mijn schoonmoeder
k
Molecular motions and collisions in organic free radical solutions as studied by Dynamic Nuclear Polarization.
Contents
Chapter 1 Introduction
Chapter 2 The theory of DNP in liquids lU 2.1 The basic equation for DNP lU 2.2 The calculation of the coupling factor 20
2.3 Dipolar coupling 2k
2.3.1 Modulation mechanisms for dipolar coupling 2k
2.3.2 Translational diffusion 26 2.3.3 The validity of the spectral densities SU
2.3.it The temperature dependence of the correlation
time 39 2.3.5 Mixed translational and rotational diffusion h^
2.1* Mixed scalar and dipolar coupling U6
2.i*.1 The coupling factor k^
2.1*.2 Models for scalar coupling kS
2.4.3 The pulse diffusion model 52 2.5 The three-spin effect 58
Chapter 3 Instrumentation and experimental aspects 3.1 Introduction
3.2 The NMR equipment
3.3 The equipment for ESR saturation 3.3.1 Microwave apparatus for DNP experiments 3.3.2 The X-band cavity
3.3.3 The K-band cavity 3.3.1* The Q-band probe
3.1* The noise decoupling equipment 3.5 Comparison with other systems 3.6 Radicals used 3.7 Sample preparation 61 61 62 68 68 70 72 75 80 82 83 85
1*.1 General introduction 86 1*.2 Experimental results for protons 89
1*.2.1 Introduction 89 1*.2.2 Experimental results for protons in benzene 89
1*.2.3 Experimental results for the OH -protons in
tri-isopropyl phosphite 90 lt.3 Experimental results for carbon nuclei 92
1*.3.1 Introduction 92 1*.3.2 Experimental results for carbon nuclei in
benzene 92 1*.3.3 Experimental results for carbon nuclei in
carbon tetrachloride 95 l+.l* Experimental results for phosphorus nuclei in
tri-isopropyl phosphite 98
Chapter 5 Interpretation and discussion 100
5.1 Introduction 100 5.2 Interpretation of the results for protons 100
5.2.1 Determination of the relevant parameters for
t h e protons in benzene 100 5.2.2 Determination of the relevant parameters for
t h e CH_-protons in tri-isopropyl phosphite IQl*
5.2.3 Comparison with literature data 105 5.3 Interpretation and discussion of the results
for carbon nuclei 107 5.3.1 Interpretation in terms of the pulse diffusion
m o d e l for carbon nuclei in benzene 107 5.3.2 Interpretation in terms of the pulse diffusion
m o d e l for carbon nuclei in carbon tetrachloride 117
5.3.3 Discussion 120 5.3.!*• Molecular orbital studies 128
5.1* Interpretation and discussion of the results
for phosphorus nuclei 133 5.1*. 1 Interpretation in terms of the pulse diffusion
6
5.1*.2 Comparison with literature d a t a 136
5.5 Conclusive remarks I38
References lltO
Concise list of symbols and abbreviations l46
Summary 150
7
Chapter 1 Introduction
Different names are used to denote the phenomenon to be discussed in this thesis. It is called the Overhauser effect, after the author who first predicted it; it is called nuclear-electron double resonance (NEDOR), in analogy with electron-nuclear double resonance (ENDOR); and it is called dynamic nuclear polarization (DNP). We shall use the last notation throughout this thesis.
DNP is a double magnetic resonance technique, which requires the presence of at least two non-identical interacting spin systems, for instance a nuclear spin system and a system of unpaired electron spins.
In 1953, Overhauser predicted that saturating the ESR of the conducting electrons in a metal should result in increased NMR signals. This theoretical prediction was verified experimentally by
2 .
Carver and Slichter in the same year. They found an enhancement of about a 100-fold for lithium nuclei in a sample consisting of small lithium particles dispersed in oil. Overhauser's prediction was based explicitly on the assumption of Fermi statistics for the
3 conducting electrons. However, it was pointed out by Bloch ,
1* 5 . . . .
Korringa and Overhauser that the assumption of Fermi statistics was by no means necessary and consequently a DNP effect could be expected in other paramagnetic substances as well. This idea was worked out by Abragam who showed that considerable enhancements are to be expected in, for instance, diamagnetic crystals containing paramagnetic impurities and in organic free radical solutions. From now on we shall confine ourselves to the latter case.
The magnitude and sign of the NMR enhancements in liquids depend upon the spin lattice relaxation processes in the solution, and these depend sensitively upon the time dependent interactions between the nuclei and unpaired electrons. The time dependence of the interactions is governed by the molecular motions present in the solution. It is possible to distinguish two types of interaction (otherwise denoted by coupling). First of all we have to consider dipole-dipole interactions between the nuclei and unpaired free radical electrons. Dipolar coupling is classical in nature and is
8
found therefore in every system of interacting spins. Time dependent dipolar coupling generally leads to negative enhancements for nuclei with a positive magnetogyric ratio. Secondly, we can be concerned with a dynamic scalar coupling, which occurs when unpaired electron spin density is found at nuclei in solvent molecules. This unpaired electron density is transmitted from the free radical to the nuclei by mechanisms similar to those giving rise to the nuclear hyperfine structure in normal ESR spectra. Scalar coupling is quantum
mechanical in nature and leads to positive enhancements, again for nuclei with a positive magnetogyric ratio. The strength of the scalar coupling differs greatly from system to system and is determined to a large extent by stereochemical factors and
properties of the chemical bond. In many cases we shall be concerned with a mixture of dipolar and scalar coupling. Sign and magnitude of the enhancement depend then upon the relative amounts of scalar and dipolar coupling. To conclude, we remark that DNP offers a very sensitive method to prove the existence of intermolecular scalar coupling.
Now we shall give a survey of important results achieved by means of DNP in liquids, mainly with the aim to illustrate the type of information obtainable from DNP measurements. We shall briefly
1 1Q 31 7 .
discuss the DNP of H, ^F, P and 'Li nuclei. The reader is 7
referred to the literature for a more extensive review of the subject.
After Overhauser's discovery and the pioneering work of 7 Abragam many DNP experiments concerning protons have been performed
(protons were selected first because of good sensitivity). As a most important result it was obtained that a pure dipolar interaction between protons and unpaired electrons applies to the majority of experimental results , regardless of the free radical or solvent molecule used. In addition, one was able to establish that the dipolar interaction is mainly time dependent by the random relative translational motion of the spin-carrying molecules. This time dependence can be characterized by a correlation time, which sets a time scale to the random motions. Many correlation times have been
9
determined by means of proton DNP in radical containing solutions. In non-viscous liquids the correlation times appear to be of the order 10~ - 10~ s.
There are a few exceptions to the rule of only dipolar coupling for protons. For instance, scalar coupling was found to be of importance for water protons in aqueous solutions of three
9
different organic nitroxide radicals . The scalar coupling appeared to be due to hydrogen bridge formation between the free radicals and water molecules. Detailed results were obtained for the equilibrium of complex formation, for the mean life time of the H-bond, and for the other motional processes in the solution. (The above-mentioned results were achieved by means of a combination of different methods; namely DNP, spin-lattice relaxation times and proton contact shifts.) To conclude, we remark that scalar coupling in case of protons often indicates the presence of a relatively strong complex formation.
7
Fluorine nuclei have been studied frequently by means of DNP. The NMR enhancements appear to depend upon competition between the dipolar and scalar part of the nucleus-electron interaction.
19
This was obtained as a general result for F nuclei. The chemical 19
environment of the F nucleus and the structure of the radical are of great importance with respect to the magnitude of the
enhancement . Aliphatic fluorocarbons always display less scalar coupling than aromatic fluorocarbons in similar solutions. Radicals with a poorly shielded unpaired electron give rise to a relatively strong scalar coupling which is in contrast to radicals with a sterically well-shielded unpaired electron. The scalar coupling for 19
F nuclei in aromatic compounds could be related to weak complexation tendencies between radicaJ-S and aromatic solvent
19 . molecules. Scalar coupling between free radicals and F nuclei has been studied very carefully by Miiller-Warmuth and co-workers . They found that the scalar frequency spectrum, corresponding to the dynamic scalar coupling, must be described by more than one correlation time in most cases. The different scalar correlation times for the same system are closely connected with the different collision attitudes of the interacting molecules in that system.
10
Many DNP experiments concerning phosphorus nuclei have been 7 . . . .
performed . A mixture of scalar and dipolar coupling is usually 19
found, which compares with F nuclei. The enhancement factors are . . 31 . extremely sensitive to the chemical environment of the P nuclei. Very large positive enhancements are found for instance in case of
31
tri-alkyl phosphites, which contain poorly shielded P nuclei. On the contrary, the shielded phosphorus nuclei in tri-alkyl phosphates give rise to a rather moderate degree of scalar coupling resulting in weak negative enhancements. This striking difference in DNP behaviour is closely connected with the presence of a lone pair of 3s electrons in tri-alkyl phosphite molecules. The lone pair belongs to the phosphorus atom and is absent in case of tri-alkyl phosphate compounds. Scalar frequency spectra for phosphorus nuclei often
12 indicate the presence of two or more correlation times
It is not our intention to give a complete survey of the DNP results in liquids. However, recent experiments, which were carried
13 . .
out by Potenza et al. , are certainly worth-while to be mentioned. The DNP of Li ions, dissolved in methanol, was investigated with several radical anions and with one radical cation. All systems
7 . . . .
showed negative Li NMR enhancements, at first sight indicating a relatively weak scalar interaction. However, a careful analysis of
7 . . . the Li DNP results, obtained with the radical anions and
supplemented by Li spin lattice relaxation time measurements for the Li ions, proved that the observed weak scalar interaction was not caused by a small dynamic scalar coupling constant but by a long scalar correlation time leading to a small scalar spectral density at the frequency of interest. One was able to interpret the
experimental data by assuming a strong complexation of Li ions with radical anions. The corresponding scalar correlation times, being equal to the mean life time of the complexes, were found to be of the
-8 -9 .
order 10 - 10 s, which is very long compared with other scalar correlation times due to complexation of free radicals with solvent molecules.
On the contrary, the measurements obtained with the radical cation could be explained easily without assuming any complexation of Li ions.
To conclude, we remark that the above-mentioned experiments 7 •
on Li nuclei clearly demonstrate the influence of coulombic forces on DNP data for charged molecules.
The original aim of the research reported in the present thesis was concerned exclusively with the DNP of carbon-thirteen nuclei. This primary interest was based upon the following
considerations. Carbon-thirteen NMR is a very useful spectroscopic method for the structure elucidation of molecules in liquids.
13
However, the natural abundance of C nuclei is about ^%, which is a great disadvantage from the viewpoint of sensitivity. In addition, there still are other factors giving rise to a poor signal-to-noise ratio, for instance a comparatively low resonance frequency.
Consequently, we may state in general that the signal-to-noise ratio 13
for C NMR signals is extremely bad m comparison with that for protons. The DNP method might be therefore, at least in principle, a valuable tool in enhancing these weak NMR signals. Moreover,
favourable results could be expected according to some literature ll* . . .
data . Tentative measurements with a variety of free radicals and solvent molecules, which will be described in one of the next
13 chapters, gave however very disappointing results. The C
enhancements were sometimes positive, sometimes negative, but never very large. The reasons for this disappointing behaviour were difficult to understand at first sight, mainly because of an insufficient knowledge about the occurrence of scalar coupling. It
13
was decided therefore to study the mechanism of C DNP very thoroughly. Such a study requires DNP measurements at different magnetic fields because DNP effects always display a more or less strong field dependence. Useful information about molecular motions and interactions in liquids can be obtained from this field
dependence of DNP results. As a consequence, a great effort in constructing DNP apparatus had to be made, because this type of apparatus is not commercially available. Three DNP spectrometers were built, ranging from medium to high magnetic field. However, a complete explanation of the mechanism underlying DNP also requires measurements at a rather weak magnetic field. Such DNP measurements
12
are difficult to perform and very time consuming, particularly
1 3 .
with respect to C nuclei. Fortunately, in the course of this investigation a succesful co-operation was achieved with Prof. Miiller-Warmuth and his co-worker Dr. Vilhjalmsson, both from the University of Munster. Prof. Miiller-Warmuth's group has the
disposal of extensive low field DNP apparatus which is particularly suitable for the detection of weak NMR signals. The difficult low field DNP experiments were performed therefore by Dr. Vilhjalmsson. Combining our low and high field DNP data resulted in a rather
13 . . detailed understanding of the DNP behaviour of C nuclei in liquids. The main topic of the present thesis concerns the
13
presentation and interpretation of these C DNP data.
As we already remarked two dynamic coupling mechanisms are of importance with respect to the explanation of DNP measurements. In chapter 2 we shall discuss the models for the dipolar and scalar coupling. The modulation of the dipolar coupling is described in terms of a mixture of translational and rotational diffusion. The dynamic scalar coupling will be treated by means of the pulse diffusion model. Chapter 3 contains a discussion of the
instrumentation. Three DNP spectrometers, respectively operating at X-, K- and Q-band frequency, will be considered. Attention is also paid to noise decoupling apparatus because many measurements were performed using this decoupling technique. Experimental DNP results
1 13 31 . .
for H, C and P nuclei, obtained with a variety of solvents and free radicals, are presented in chapter 1*. The majority of the
13
experimental results is concerned with C nuclei, as we already
31 31 remarked. P DNP measurements have been performed, because P DNP
is interesting with respect to the occurrence of a strong sceilar coupling, whilst H DNP data appear to be required for a complete
13 31
interpretation of the C and P measurements. The interpretation of the results is given in chapter 5. Several correlation times and other quantities, characterizing the motional behaviour of
molecules in organic free radical solutions, are determined from 13 31
the proton measurements. The C and P data appear to provide rather detailed information about different collision attitudes
between the colliding free radicals and solvent molecules. Lastly, 13
it will be shown that high-field C enhancements are often small because of a nearly perfect balancing of scalar and dipolar coupling in many radical containing solutions.
1U
Chapter 2 The theory of DNP in liquids
2.1 The basic equation for DNP
The essential features of DNP can be seen most easily by considering only nucleus-electron pairs. At first sight this may appear a drastic oversimplification of the real situation , because rather dilute free radical solutions are used in practice. (The number of free radical molecules is, roughly speaking, three orders of magnitude smaller than the number of solvent molecules.) The rapid random diffusion of the molecules ensures however that the solvent molecules are near a free radical many times during the nuclear relaxation time. Therefore any nucleus in solution may be considered to be in continuous interaction with an unpaired free radical electron.
The spin Hamiltonian for a nucleus-electron pair, placed in an external magnetic field H- in the z direction, is given by
- Y s H « 0 - Yl^^z«0 -^ «Sl(*^ (2.1)
where S refers to the electron spin quantum number and I to the
z z nuclear spin quantum number. The quantities Yo ^^^ Y T ^^^ ^^^
o 1
magnetogyric ratios of the electron and the nucleus. The ratio Y O / Y J appears to be of utmost importance with respect to the obtainable enhancements. This ratio equals -660 for protons, -2650
13 31 for C nuclei and -161*0 for P nuclei.
In (2.1) the first two terms represent a simple four level system if we confine ourselves to nuclei with spin 1/2. The energy level diagram is given in fig. 2.1.1 where we have denoted the possible states by |+->, |++>, | — > and |-+>. The corresponding level populations will be given by N , N , H and N . In the foregoing notation the plus or minus sign by convention represents the aligning of the spin with or against the magnetic field, whilst the first sign refers to the electron spin S and the second sign to the nuclear spin I.
nucleus-15
electron interactions, gives rise to the phenomenon of DNP. (in principle, the spin Hamiltonian still contains many more terms, which describe the other interactions of the electrons and the nuclei with their respective environments. However, these terms have been omitted, because they are largely uninteresting in an understanding of DNP.) As a consequence of the presence of H„_(t) and its stochastic dependence upon time, which we already discussed in the introductory chapter and which we will discuss in greater detail further on, relaxation transitions are induced between the energy levels of the nucleus-electron system. The allowed
transitions, denoted by the transition probabilities W , W , W' and W , are also shown in fig. 2.1.1. It will appear that the phenomenon of DNP is particularly connected with the coupled relaxation
transitions W and W in which case the electron spin and the nuclear spin change their orientation simultaneously.
w:
w;
Fig. 2.1.1 Relaxation transitions in the nucleus-electron system.
According to Solomon , we may now derive a quantitative expression for the enhancement factor by considering the rate of change of the four level populations. From fig. 2.1.1, it follows very easily that
dN,
16
Of course, similar equations apply for the remaining level populations. The constants can be obtained by considering the system at thermal equilibrium where dN /dt=0, etc.
It appears to be useful to introduce a quantity I , proportional to the NMR signal strength and a quantity S , proportional to the ESR signal strength. These two quantities are defined by
J^^z = (N+++N_+) - (N^_+N__) (2.3)
kS^ = (N_^_+N_^_^) - (N__+N_^) (2.1*)
where k is a constant. We can now calculate the rate of change of I by combining the expressions for the rate of change of the individual level populations and the eqns. (2.3) and (2.1*), by which we obtain
at
By considering the system at thermal equilibrium where dl /dt=0, we find for the constant C:
C = ( W Q + 2 W ^ + W 2 ) I Q + ( W 2 - W Q ) S P (2.6)
I_ and S- are the thermal equilibrium values of I and S . By 0 0 ^ z z "^ inserting the constant in (2.5) and rearranging, we have
dT
- r
= - tV2«l*W2'(iz-^o) - (V"o)(^z-^o) (2.7)
atIn order to obtain the basic equation for the description of DNP experiments we still have to make a few remarks. Firstly, in a DNP experiment one tries to achieve in principle a completely
saturated ESR signal. However, the actual degree of saturation always depends on the type of free radical used and several instrumental aspects. It is necessary therefore to introduce a quantity which is a measure for the degree of saturation. This quantity, usually denoted by s and referred to as the saturation factor, is defined by
1 - ^ (2.8) ^0
It will be clear that s=0 when there is no saturation and that s=1 in case of complete saturation. Secondly, in the treatment above we considered only relaxation transitions due to the nucleus-electron
interactions. However, it is evident that in the absence of free radicals the nuclei also undergo relaxation transitions, which can be taken into account most readily by introducing an additional
transition probability W between the levels 1 and 2 and between the levels 3 and 1* of the system depicted in fig. 2.1.1. Consequently, in the eqns. (2.5) - (2.7) W, must be replaced then by W.+W...
Further, we introduce the so-called leakage factor f, defined ty W +2W +W — ^—^~ ( 2 . 9 ) W Q + 2 W ^ + 2 W J + W 2 Thus f measures t h e f r a c t i o n of t h e t o t a l n u c l e a r r e l a x a t i o n r a t e due t o n u c l e u s - e l e c t r o n i n t e r a c t i o n s ; when n u c l e u s - e l e c t r o n i n t e r a c t i o n s are n e g l i g i b l e , f=0 and when t h e y a r e dominant, f = 1 . The l e a k a g e f a c t o r i s o f t e n expressed in a d i f f e r e n t way by means of r e l a x a t i o n t i m e s . From ( 2 . 9 ) , we can deduce in a simple way t h a t
' ' l
^ ( 2 . 1 0 ) 1.0
18
r a d i c a l c o n t a i n i n g s o l u t i o n and T . t h a t in the p u r e s o l v e n t . By
1 ,u
inserting the saturation factor and the influence of leakage in (2,7), and considering the system in a steady state so that dT /dt=0, we obtain
z
^ = 1 + pfs - 0 ( 2 . n ; 0 0
where p i s t h e s o - c a l l e d coupling f a c t o r , given by
W -W
p = ^-2— ( 2 . 1 2 )
WQ-H2W^+W2
Lastly, we introduce the enhancement factor A=(l -I )/I and remark that the factor S /I is equal to Y O / Y T > ^y which we obtain
0 U 0 i
^s
A = pfs - ^ (2.13) ''l
Equation (2.13) is the basic equation for the description of DNP experiments.
The treatment above clesirly indicates that the coupling factor p is the most revealing term about the nucleus-electron system. The coupling factor appears to be a sensitive test of the form eind time dependence of the nucleus-electron interaction H- (t) which determines the various transition probabilities W-, W. and W_. Consequently, in the next sections much attention will be paid to the calculation of these transition probabilities. Especially the transitions W^ and W„ will be considered in detail with respect to their magnetic field dependence, thereby providing information about molecular diffusion and collisional processes in the solution.
It is useful to make a few remarks concerning the magnitude and sign of the coupling factor, and thus the enhancement factor A. It can be shown very simply (see section 2.2) that W is the only
allowed transition in case of a purely scalar nucleus-electron interaction, giving p=-1 according to eqn. (2.12). Assuming f=1 and s=1, the maximum enhancement to be expected is then equal to +66O
13
for protons and +265O for C nuclei according to (2.13) and the respective values for the magnetogyric ratios. However, these large enhancements are rarely realized in practice,
In case of dipolar coupling, the situation is somewhat more complicated because of the fact that the transitions W^^, W, and W^ are all allowed. Under optimum conditions (extreme narrowing which is always realized at a sufficiently low magnetic field strength) the ratios of W :W :Wp are 2:3:12, and thus p has a value of +1/2. Hence the maximum observable enhancement for pure dipolar coupling
13 is 1/2 Yg/Yj, which is equal to -330 for protons and -1325 for C nuclei. In practice, both scalar and dipolar coupling may be
present and the value for the observed enhancement will range from + 1 / 2 Y g / Y j t O - Y g / Y j .
Lastly, we shall discuss some aspects with respect to the experimental determination of the coupling factor. The enhancement factor A is the measurable quantity in a DNP experiment. However, for a determination of p we also need the values of f and s. From eqn. (2.10), it follows that the leakage factor can be determined very easily by measuring the nuclear relaxation time in the
presence of the radical and in its absence. Unfortunately, a direct determination of the saturation factor s is extremely difficult to achieve, because the required parameters (see below) are usually unknown. However, it is possible to show that the enhancement factor in case of complete saturation (s=l) can be determined
2
indirectly. With the help of the Bloch equations , it follows that the dependence of S on the strength H,_ of an oscillating magnetic field applied exactly at resonance is given by
^0
^^--TT
(2.11*)
^•^V's^is"2S
20
times of the ESR line in question. With the help of (2.8), (2,13) 2
and (2,ll*), and the fact that H _ is proportional to the applied microwave power P, we find
^"^ = ( f p ^ r ^ { 1 - ^ ^ P J T > (2.15)
'I " IS 2Swhere a is a constant. Consequently, a plot of reciprocal
enhancement A~ against reciprocal power P~ should give a straight line with intercept A~ =(fpY„/Yj)" where A^ is the value of the enhancement for complete saturation of the ESR line. In this way the coupling factor p can be obtained. The extrapolation method
3
above was already pointed out by Carver and Slichter and is frequently applied in DNP experiments.
2.2 The calculation of the coupling factor
In this section we shall outline the calculation of relaxation transition probabilities in order to be able to study the coupling factor p. The Hamiltonian for a nucleus-electron pair may formally be written as
H = HQ+H^(t) (2.16)
The operator H- represents the time-independent part of the Hamiltonian and has eigenstates denoted as |k>, |m>,,.. The stationary random operator H,(t), in our case representing the dynamic nucleus-electron interaction, is the relaxation inducing perturbation. In general H,(t) is different for each
nucleus-electron pair. The time average of H.(t) is assumed to be zero. We shall now consider the simple case that H (t) can be expressed as
where U i s an o p e r a t o r a c t i n g on t h e spin v a r i a b l e s of t h e system and F ( t ) a random f u n c t i o n . S t a r t i n g from a well-known r e s u l t i n
. 1 * time-dependent p e r t u r b a t i o n t h e o r y , we o b t a i n f o r t h e t r a n s i t i o n p r o b a b i l i t y W, between t h e s t a t e s | k> and |m> ^ | < k | u | m > | 2 j(u), ) ( 2 . 1 8 ) where JM = /g(T)e"^"'^dT ( 2 . 1 9 ) and g(T) = F ( t ) F ' ' ( t + T ) ( 2 . 2 0 )
The symbol u) denotes the distance in angular frequency units between the energy levels |k> and |m>. The bar in (2.20) indicates an ensemble average. The spectral density J (to) is the Fourier transform of the auto-correlation function g(T). The formulae above clearly illustrate the important facts about spin-lattice
relaxation. The allowance for a transition is determined by the matrix element of the perturbing Hamiltonian connecting the two states in question. The function F(t) is a lattice (thermal bath) parameter and has a random time dependence owing to rapid molecular motions, thereby providing a process for energy exchange between the spin system and the lattice. The effectiveness of this process at the appropriate frequency is determined by the spectral density, which in turn is closely connected with the correlation function. The correlation function is a very important concept in relaxation theories, because it contains the information about the physical nature of the dynamic processes which give rise to the random fluctuations of H..(t). Consequently, we shall deal with correlation functions in great detail in the sections 2.3.2 and 2.1*.3.
22
By writing the dipolar coupling in spherical polar coordinates and by expressing the magnetic moments in terms of raising and
lowering operators, it is possible to write the Hamiltonian for the nucleus-electron interaction as
3 ~
H„.^(t) = I U.F.(t) (2.21)
^^ j=-2 J J
where, as in (2.17), the F. are random functions of only the relative positions of the two spins of interest and the U. are operators acting only on the spin variables of the nucleus-electron system,
* ' t
with the convention F.=F . and U.=U .. The functions F. are given by J -J J -J J
„ /8Tr „1i€,. ,> -3
' 1 " ""^TJ ^2 (s>*'-r
(2.22)
F^ = /fpYf(e,<t.).r-3 ; F^ = A(t;
The functions Y^ are the normalized spherical harmonics of the second order . The angles 6 and (j) aire the polar coordinates of the interspin vector r in the laboratory frame. The quantity A(t) represents the intermolecular hyperfine coupling between the nucleus and the electron. The operators U. are given by
"o = °'^- f ^z^z * i (^+^- * ^-^)>
U, = o{l S^ + I S } ^ z + + z (2.23) 1 - - 3YgYT^^ U„ = lal.S. ; a - ^ ^ -2 = 2 "^+^+ ' « B ^S
= 'z^z ^ i (
^^- ^ ^-^+^
2, is time-dependent owing to random fluctuations in both the polar angles and the interspin distance. The scalar coupling, given by the term with j=3, may be time-dependent due to a
distance-dependence of A, However, there still exist other mechanisms leading to a time-dependent scalar interaction. We shall discuss these mechanisms in section 2,1*,2.
With the help of the eqns. (2.l8) and (2.21), it is possible to express the various transition probabilities as
1 3 - p
Wv„ = - ^ 2 |<k|U.|m>| J.(iD.) (2.2l*) km .^2 . _ 2 I 1 jl I 0 J
where we have introduced spectral densities J.(a)), and hence
J
correlation functions g.(T), according to (2.19) and (2.20) given by
+00 - i u . T
J.(w.) = ; g.(T)e ^ dT (2.19a)
and
g.(T) = F.(t)F*(t+T) (2.20a)
From the foregoing discussion, it will be clear that J.=J ..
J "J
In deriving (2.2l*), the tacit assumption has been made that the dipolar and scalar coupling are uncorrelated, which is certainly disputable, because both mechanisms may depend upon the interspin distance. However, there is only an appreciable scalar coupling when the nucleus and the electron are separated by about the closest distance of approach. The intermolecular scalar coupling decreases very rapidly with increasing interspin distemce. On the contrary, the dipolar coupling has a much less steep distance-dependence. It is allowed therefore to neglect the correlation between the dipolar and scalar coupling.
2l»
easily derived from (2.2l*), leading to the following expression for the coupling factor
p= 9J2(a)g-a),) - Jo(^s^<^i) - (8TrUp/YgYiti2)2 J^{^^^^^) ^^ ^
9J2(Wg-a)j) + Jp(Wg+a)j) + 18J^ (Uj)+(8TrviQ/YgYj1i^)^ J2(Ug+Uj)
where u and u denote the NMR and ESR resonance frequency. In 1 D
order to work out (2.25) further, we have to choose models for the molecular motions. This will be done in the next sections.
2.3 Dipolar coupling
2.3.1 Modulation mechanisms for dipolar coupling
In this section we shall briefly discuss the possible mechanisms for the random modulation of the dipolsir coupling, mainly with the aim to point out the source of the time-dependence in the various practical cases. The following mechanisms can be encountered in DNP experiments,
1) Relative translational diffusion,
The nucleeir and electron spins are in the centre of distinct molecules, which will be assumed spherical. It will be clear that the sole cause for time-dependence arises from
fluctuations in the interspin distance. We shall frequently be concerned with this mechanism of relative translational
diffusion. The much more general case of off-centre spins could not be solved completely up to now. We shall discuss this point further in section 2,3.3.
2) Mixed translational and rotational diffusion.
We suppose the same mechanism as pointed out above, but we allow for the possibility of weak complex formation between the spin-carrying molecules. When the molecules with the I and S
25
spin are stuck together, the dipolar coupling is only modulated by fluctuations in the polaur angles, which means that we are now concerned with a random rotational diffusion of the
complexing molecules as a whole. When the molecules are apart, the dipolar coupling is again modulated by translational diffusion. It will be clear that we have to deal in this case with a mixture of rotational and translational diffusion to which we shall pay much attention in section 2.3,5»
3) Rotational diffusion,
The I and S spin are in the same molecule which applies to nuclei within a free radical molecule and in principle also to tight bound complexes. We only need to consider random rotational diffusion in these cases. (The S spin is assumed to be localized) However, the possibilities just-mentioned for the occurrence of pure rotational diffusion are of minor importance for the DNP experiments to be described in this thesis. A much more
interesting situation arises when the S spin refers to a nucleus, residing in the same molecule as the I spin. Saturating the NMR of the S spin may result in a polarization effect for the I spin. We shall treat this situation in somewhat more detail when we discuss the so-called three-spin effect.
The most important problem with respect to the motional models discussed above is formed by the calculation of the
corresponding correlation functions. The auto-correlation function g(T) for a stationary random function F(r) can be defined in two ways. Firstly, we can express g(T) by means of an ensemble average, leading to
+0O +00
g(T) = F(t)F*(t+T) = f I p ( ? ^ , ?Q, T ) F (?g) F* (?^) <i?Qd?.| (2.26)
where p(r-, r , T ; determines the probability of finding F(r-^) at time t etnd F(r-) at time t+T. Secondly, we can define g(T) by means
26
of a time average, which results in+T
g d ) = F(t)F*(t+T) = l i m ^ / F(t') F* (t'+r) df (2.27)
The equivalence of both definitions is guaranteed by the ergodic
Q
theorem . We shall use (2.26) with respect to the calculation of correlation functions in case of dipolsir coupling. The time average definition will be employed in case of scalar coupling because of reasons becoming clear later on,
2.3.2 Translational diffusion
In this section we shall outline the calculation of the spectral density in case of dipolar coupling modulated by relative translational diffusion of the spin-carrying molecules. We have to start then with the calculation of the corresponding correlation function, as was pointed out previously. This last problem mainly consists of finding an expression for the probability density p(r-, r., T ) , defined m the preceding section. However, instead of p(r , r^, T ) it IS useful to introduce a probability density
P(r , r., T ) which has a slightly different meaning. P(r-, r., T ; is the conditional probability density that the interspin vector r has the value r, at time t+T if it has the value r. at time t. For stationary random processes we have the obvious relation
p(?^, T Q , T ) = P(?^, TQ, T ) P(?Q) (2.28)
where p(r.) dr. is the probability that at time t the interspin vector lies in the interval r., r.+dr.. According to (2.26), the correlation function g(T) for one nucleus-electron pair is now given by
g(T) = // P(r^, r^, T ) F (f^) F* (r^) p (r^) dr^ dr^ (2.29)
where the function F (r) is given by one of the expressions in (2.22). Actually, p (r.) is an initial datum depending on the circumstances of the particular problem. If we assume a uniform distribution of the spins, and if N represents the total number of unpaired electrons (free radical molecules) in the sample and n the number of unpaired electrons per unit volume, p vr_; is simply equal to n /N . The correlation function g(T) for the interaction of one nucleus with N unpaired electrons is then given by
g(T) = n^ ;/ P(r^, ?o. T ) F (r^) F* (r^) dr^dr^ (2.30)
where we have assumed that the motions of the N electron spins
e ^
are uncorrelated. In order to be able to calculate g(T), we have . - • - » • .
to find an expression for P(r , r., T ) . We shall solve this problem by means of the theory of random flights about which a thorough treatment has been given by Chandrasekhar . (This theory is
especially applied here, because it can give a useful contribution to a somewhat controversial discussion.) The general problem of random flights can be stated as follows. We consider a particle ->d -»-d ->-d which undergoes a sequence of displacements r,, T „ , ....r., .,., the magnitude and direction of each displacement being independent of all preceding ones. The position r of the particle after N displacements is given by
I T^ (2.31)
i=1 "
Further, let
T.(x^, y^, zf) dxjdy^dzj = T.(?J) dr^ (i = 1 N) (2.32)
28
r. + d r . . We are now interested in the probability P„(r)dr that the position of the particle after N displacements lies between r and r + d r . This problem can b e solved completely by using a method originally devised by Markoff . Because of the fact that the derivation of this method is rather lengthy and very formal, we only ment
given by
only mention the final results. The probability density P,j(r) is
1 ^
Pj,(r) = - ^ / exp (-ip.r) A ^ (p) dp (2.33) 8Tr -"
where the function A,, (p) can b e obtained from A « ( P
N +«
A^(p) = n / T . (r^) exp (ip.?^) d ? ^ (2.31*) j = l -CD J J JO
According to ( 2 . 3 3 ) , the function A ^ (p) is the 3-dimensional Fourier transform of the probability density function P ( r ) . It represents the distribution of wave vectors present in the system. In order t o be able to calculate P (r) w e have to choose an
,-+d> . . .
expression for T . ( r . ) . Assiaming isotropic diffusion, we have to J J
deal with a spherically symmetric distribution of the displacements. ,->-d< .
The function T.(r.) is then given by
J J
T . ( r ^ ) = T ( | r ^ | 2 ) ( j = 1 N) (2.35)
and we o b t a i n f o r A^ (p)
Ajj (p) = { / e x p ( i p . r ) T ( r 2 ) d ? } ^ (2.36)
By using polar coordinates with the z axis in the direction of p , the integral in (2.36) can easily be transformed in
/ exp ( i p . r ) T ( r 2 ) dr = 1*TT / ^ ^ " ^ i P l ' ' ^ r 2 T ( r 2 ) d r ( 2 . 3 7 ) - D O 0 | p | r / • • • > by which we o b t a i n for A^ (p) 00 Ajj ( p ) = { 1*TT / (1 - • l | p | 2 r 2 + . . . )r2 T ( r 2 ) dr } " ( 2 . 3 8 ) 2
We i n t r o d u c e t h e mean square displacement r , d e f i n e d by
r 2 = UTT / T(r2) r dr ( 2 . 3 9 )
If we assume that the mean square displacement is very small (in fact infinitely small because of mathematical reasons), we finally obtain from (2.38) for the function K ^P)
Ajj (p) = exp ( - N | P I 2 r2/6) (2.1*0)
By s u b s t i t u t i n g (2.1*0) i n ( 2 . 3 3 ) , we can o b t a i n t h e f o l l o w i n g e x p r e s s i o n for t h e p r o b a b i l i t y d e n s i t y P„ ( r )
P„ (?) = i ^ e x p (-3|?i2/2Nr2) (2.1*1) " (2TrNr2/3)3/2
If we assume that N=nt where n is the number of displacements per unit time, and if we introduce the diffusion constant D, defined by
2
D = ^i2L (2.1*2)
30
P ( r , t ) = ^ 7 ^ exp ( - | r | 2 / l * D t ) (2.1*3)
(l*TTDt)3'2
Maybe i t i s u s e f u l t o n o t i c e t h a t (2.1*3) r e p r e s e n t s t h e s o l u t i o n of t h e well-known d i f f u s i o n equation for a free p a r t i c l e , given by
3P = D ( ^ + ^ + i ! ^ ) (2 1*1*)
dX dy dz
It is also well-known that (2.1*1*) is only valid under the assumption of small diffusion steps (displacements).
A few remarks with respect to the usefulness of eqn. (2,1*3) have to be made. In the foregoing we have shown that this equation is veilid under the condition of very small displacements. On account
12 .
of this condition Torrey states , in a paper concerning nuclear spin relaxation by translational diffusion, that it is unlikely that (2.1*3) corresponds to any physical reality. However, many DNP experiments have been interpreted by means of (2.1*3). Useful information (e.g. correlation times, distances of closest approach and diffusion constants) could be obtained in this way, which seems to justify the validity of (2.1*3). In addition, it is certainly worth-while to emphasize a particular result from the general theory of random flights, which up to now has always been overlooked to my best knowledge. In deriving (2.1*3) we did not use an exact
->-d
expression for the distribution of the displacements r-. We only ,-»-d> "^
assumed a spherically symmetric form for T.(r.). However, it appeeirs J J
to be instructive and it seems reasonable to choose a Gaussian distribution with spherical symmetry for the displacements in which case T.(r.) is given by
J J
^n = 5 - ^ 7 2 ^^P (-Sl^tl^/Sl.^) (2.1*5)
2
where 1. denotes the mean square displacement to be expected on the
"^ 2 jth occasion. In principle the quantity 1. may differ from one
displacement to another. The function A„ (p) can be calculated
9 N ^ exactly in this case just as the probability density P (r). If
2 2 -*• we replace r by 1. the result for P (r) is given by (2.1*1) without any restriction. For a Gaussian distribution of the displacements, eqn. (2.1*1) represents therefore an exact solution valid for einy value of N and for all possible magnitudes of the diffusion steps. Consequently, we can conclude that (2.1*3) presumably represents a useful expression for the sought probability function P(r).
Equation (2,1*3) gives the solution for a diffusing particle which starts from a fixed origin at zero time. However, we need
• • • • • / ^ • * \
the conditional probability function P(r-, r„, t) for a molecule diffusing relative to another molecule. Therefore, we form the product
P (r^, t) P (rg - T Q , t) dr^ dr^ (2.1*6)
where r and r are the position vectors of the spins I and S
J. o
relative to a common origin. At zero time the position of the S -> . . .
spin IS given by r., whilst the I spin is then assumed to be in the origin. We transform to relative centroidal coordinates by means of the substitutions
r^ = rj - rg ; R = r^ + r^ (2.1*7)
The sought expression for P (r,, r., t ) is found by integrating (2.1*6) over aJ.1 possible positions R. The result is given by
P (? , r , t) = ^ ^ 7 ^ exp {-1? -? |2/1*(D +D )t} (2.1*8) ^ ° ^ {l*TT(Dj+Dg)t}3/2 1 0 I S
where D and D„ are the self diffusion constants for the molecules 1 o
containing the I and S spins. A comparison of (2.1*8) with (2.1*3) yields immediately that the diffusion constant D for relative
32
diffusion is given by
D = Dj + Dg (2,1*9)
and not by D = 1/2(D + D„) as is sometimes met in the literature^^'^'''^^.
According to (2.22), (2,30) and (2.1*8), the correlation function of any one of the three functions F , F and F will be given by
, yJ (e ,* )Y^(9,*) -I?-? r/U(D+D„)T ^ _^ g.(T)=C.n^(l*n(D^+Dg)T}-3/2^; ^ ^ e ^ ° ^ ^ dr^d?,
^ 0 ^ 1
(2.50) where
^0 - IT^ ' ^1 " Tf • ^2 - T5^ (2-51)
In the integration (2.50), r. and r, can not go below a lower limit d which is the closest distance of approach between the I and S spins. The integral in (2.50) can be obtained after some
mathematics. The Fourier transform of the correlation function thus obtained is immediately calculable. The result is given by
2C.n •»
where J. , (u) is a 3/2 order Bessel function of the first kind. In (2.52) we have introduced the so-called correlation time T for relative translational diffusion. The quantity T is defined by
T^ =-7—2^ (2.53)
|(V^s)
It is useful to notice that the definition (2.53) is consistent with Kubo's general definition of correlation times in magnetic spin systems. This general definition is given by
00
It will be clear from (2.52) that it is allowed to express the spectral density as a function of U)T. as a unit. A closed form
• 13 17 expression for the integral in (2.52) is available * as are
7
computed curves . However, we are much more interested m a graphical representation of the coupling factor p as a function of UqT.. A formal expression for p in case of dipolar coupling modulated by translational diffusion can easily be obtained from (2.25) and
(2.52). This expression is given by
p D = i F ^ 1 (2-55)
" '^ 0.7 + 0.3Jp ((Oj)/ J* (U)g)
where w e have introduced the spectral density J ( u ) , defined by
J^ ((o) = J. ( u ) / C. (2.56)
Moreover, we assumed that a)«a)o. We have calculated p_ for protons
I D D
as a function of a)„T. with t h e h e l p of t h e computer. The r e s u l t i s
b t
given b y the continuous curve in fig. 2.3.2.1. The depicted curve is of utmost importance for the interpretation of DNP measurements in case of dipolar coupling modulated by relative translational diffusion. (Two curves have been plotted in fig. 2.3.2.1. The
continuous curve is based upon the diffusion model discussed in the foregoing. This m o d e l is referred to as the independent diffusion model. The force-free diffusion model, leading to the dashed curve for p , will be discussed in the next section.) The decreasing p value w i t h increasing W „ T . is easily understood from eqn. (2.55)«
3k 0 1 -0 -01 V-L
h
rT^"^^^^""^^
rP
r[-u
\ 1 1 1 1 1 1 1 11
independent diffusionforce-^s.
\ \
\ \
\ 10\
\
\ \
\ \
\ \
1 1 1 1 1 1 1 11 \ free diffusion\
< \ \ lOx\ \
\ \
\ \
\ \
\ \
\ \J
\ 1
— » " s tt \ \ 1 - , . • ,\1 1 01 100F i g . 2 . 3 . 2 . 1 The coupling f a c t o r p as a function of io„T. for t h e
o t
independent and force free diffusion model. The latter model will be discussed in the next section.
The spectral densities have their maximum value for small tii„T^. t
values (white spectrimi region). The spectral density J^ (oj_) t decreases with increasing field strength much faster than Jy, (a)_) because of the large difference between u„ and u,.. Consequently, the
t t b 1
ratio J (u ) / J_ (u ) strongly increases with increasing u_T, , D X D o o t explaining the functional behaviour of the coupling factor p.
2.3,3 The validity of the spectral densities
In calcuQ.ating the correlation function in case of translational diffusion, we applied for the required conditional probability P(r., r , T ) the solution of the well-known diffusion equation under the assumption of spherical molecules with the spins in the centre of these molecules. This approach can be attacked in
35
principle with regard to various aspects. Now, we shall successively discuss these points of criticism.
1) In the previous section the influence of the distance of closest approach between the two spins was simply taken into account by means of appropriate integration boundaries in (2.50), which only means that the volume of the two colliding molecules is excluded from the region where a solution for P (r,, r., T ) exists. This diffusion model is known as the independent diffusion model, as we already remarked. However, it was recently pointed out by Hwang and Freed and independently by Ayant et al, that the diffusion equation has to be solved under the reflecting wall boundary condition
( F ) ^ = 0 (2.57)
dr r=d
The validity of the condition (2,57) is based upon Em investigation of the one-dimensional random walk problem with a reflecting
barrier. The one-dimensional analogue of (2.57) is derived easily for this problem . The diffusion model which allows for (2.57) is referred to as the force-free diffusion model. Solving the diffusion equation under the above-mentioned condition leads to a modified expression for the spectral density J_^ (u) and hence to a modified expression for the coupling factor p. The reader is referred to the literature ' for these expressions. In fig. 2.3.3.1 we have plotted the spectral densities as a function of OJT. for both models. The larger value of J_ (03) for the force-free model at small o) is probably due to the reflecting-wall condition
(2.57), which would increase the time the interacting molecules remain in contact at r=d, where the dipolar interaction has its maximum. The different behaviour at large o) is less easy to understand froto a physical point of view. At large o) (uT.»l) the
t -2 spectral density J_ (o)) for the force-free model decreases with to"
for increasing 00 whilst for the independent diffusion model J_ (u) -3/2 . .
36
Fig. 2.3.3.1 The spectral density J_^ (u) for the independent diffusion model (curve l) and the force-free diffusion model (curve 2 ) . The quantity c is a
1R constant. (From Hwang and Freed .)
coupling factor p as a function of u>„T. for the force-free model. o t
The result is shown in fig. 2.3.2.1 by means of the dashed curve. A comparison between the two models immediately leeirns that the difference in functional behaviour of p is only significant for large values of u_T. (a)„T. > 15).
o x o t
In the previous section we already mentioned that the application of the diffusion equation in obtaining an expression for P(r.,r.,t) was attacked heavily by Torrey. Torrey tried to achieve a better probability density function by means of a model which allows for larger diffusion steps.
12 The following diffusion model was proposed by Torrey . A molecule is "bound" in a trapped state for a time T - T', then it may jump into a thermally excited state for a time T ' before
f a l l i n g i n t o a n o t h e r t r a p p e d s t a t e . The time T ' i s small compared with T . I t i s assumed t h a t t h e motion i n t h e e x c i t e d s t a t e i s d e s c r i b e d by t h e d i f f u s i o n e q u a t i o n and t h a t t h i s motion does not c o n t r i b u t e t o t h e r e l a x a t i o n because of i t s extreme r a p i d i t y , implying a very s m a l l s p e c t r a l d e n s i t y . Torrey could show t h a t t h i s random f l i g h t d i f f u s i o n model l e a d s t o a function A (p) of t h e form
A (p) = ( 1 + &rp2 )-•' ( 2 . 5 8 )
The time T is the mean time between flights (displacements) and can be considered as the appropriate correlation time. It has to be noted that A (p) only depends on the magnitude of p which is a characteristic property of an isotropic random flight process.
12 Starting from (2.58) a suitable expression can be obtained for the spectral density J_ (u). The differences with the spectral density for the independent diffusion model can be seen very well by considering some limiting cases. Firstly, we consider the case
2 2
of a large jump distance which means that r » d . It has been . 20
shown by Kruger that under this condition Torrey's spectral 2 2 — 1
density is proportional to T (1 + u T ) . However, such a frequency spectrvun has never been observed for DNP experiments in liquids,
2 2 .
leading us to the conclusion that the case r » d is of minor importance. This is not so surprising, because it is not reasonable to expect jumps in a liquid much larger than a molecular diameter,
2 2
For small jump distances (r « d ) both models give identical results which is to be expected of course. Of particular interest for our applications is the behaviour for <*)eT.»1, because most of our experiments have been performed under this condition. It has been
21
shown by Held and Noack that in this case Torrey's model and the independent diffusion model give about identical results for medium
2 •v 2
jump distances (r ~ d ) . Only for u T « 1 there may occur more or
0 t
less significant differences between both models. We CEin conclude from the discussion above without any ambiguity that it is by no means necessary to apply Torrey's model for the interpretation of
38
our DNP experiments.3) In the previous section we assumed a uniform distribution of the spins, resulting in a simple expression for p (r„). However, it
22 . .
was pointed out by Seiden that this assumption may be wrong for short distances (r. < 5d) and hence may lead to erroneous results for the spectral densities. Consequently, we now introduce the following expression for p (r.)
vCr,)=ffCr,)
(2.59)
e
where the function f (r„) is used to describe the initial
equilibrium distribution of the spins. The function f (r.) is often denoted as the radial pair distribution function (rpdf). Considering
23
the molecules as hard spheres, it can be shown that the rpdf has a maximum (>1) near the core r=d. With increasing r- the rpdf
exhibits damped oscillations about unity and it is allowed to assume that f (r.) = 1 for distances r >5d. However, the behaviour of the rpdf sketched above is only of importance for a high density of spins. In DNP experiments we are concerned with dilute free radical solutions. There are only a few free radical molecules in a sphere with a radius of about 5d around a nucleus, which implies that we can safely use the simplest rpdf, namely f (r.) = 1 for r>d and f (r.) = 0 for r<d, m considering the influence of free radicals on the spin-lattice relaxation of nuclei.
1*) In the foregoing we always assumed spherical molecules with the spins in the centre. In many cases this will be a crude assumption and consequently, we shall have to estimate the influence of off-centre spins. This problem has already been considered in great
21*
detail by Hubbard , who pointed out that two effects are of importance. Firstly, the distance of closest approach of two spins will no longer be the sum of the molecular radii but something less, leading to an increased relaxation rate due to a stronger dipolar
39
interaction. Secondly, one must consider the effect of rotation of the spin-bearing molecules on the purely translational relaxation, resulting in a decreased relaxation rate due to a shorter
correlation time. The two effects clearly compete and the net result is small in many practical cases, which can be illustrated very well by mentioning a result, obtained for the protons in liquid ethane where the spins are strongly off-centred. Harmon and
25
Muller have applied Hubbard's theory to this molecule and obtained an off-centre correction for the relaxation rate of only 3 percent. Consequently, we can conclude that it is allowed to ignore the influence of off-centre spins if we do not claim an extreme accuracy.
It will be clear from the discussion above that the independent diffusion model leads to reasonable accurate spectral densities in radical containing solutions and that there is no need of applying various corrections and improvements which often are cumbersome and time consuming therefore. The applicability of the independent diffusion model at medium and high frequencies (w_T. > 1) is
o t
closely connected with the fact that high frequencies in the spectral density function correspond to short times, which in turn are due to small diffusion steps.
2.3.1* The temperature dependence of the correlation time
In section 2.3.2 we introduced the correlation time T. which is a very important quantity in NMR relaxation theories.
Unfortunately, it is difficult to give it a precise physical definition. It may be characterized somewhat loosely by saying that the relative motion of the two interacting spins is negligible for times smaller than T . In general the actual value of T. strongly depends on the behaviour of the Hamiltonian describing the
interaction between the two spins and on the molecular motions present in the solution. In non-viscous liquids the correlation times for dipolar coupling, modulated by translational diffusion,
uo
are of the order 10 to 10 s. It will appear that scalar correlation times, modulated by the same mechanism, can be much shorter.
We are interested in the temperature dependence of the correlation time, because most of our DNP experiments have been performed as a function of temperature. According to (2.53) it will be clear that the temperature dependence of T, is governed by the diffusion constants D.^ and D . For a pure solvent, consisting of
1 b
spherical molecules with radius r, the diffusion constant is given by
D = r ^ (2.60) 6Trnr
where k is Boltzmann's constant. T is the absolute temperature and ri is the viscosity. From (2.53) and (2.60), it follows directly that
Consequently, the temperature dependence of the correlation time is determined by the temperature dependence of r|/T.
Since in expressions for spectral densities (relaxation transition probabilities) the quantity a)„T, occurs as a unit, an
b t
increase in T has the same effect as an increase in to or an
t o increase in external field strength, which will appear to be of
great importance for the interpretation of our DNP experiments at different temperatures and at different field strengths.
In practice the temperature dependence of the correlation time is often assumed to obey the Arrhenius equation, expressed by
E./R T
\=\0' « (2.62)
The relation (2.62) is often fulfilled, at least in a certain ?6
temperature range . The activation energy E. and the
pre-exponential factor T - can be obtained by plotting InT against 1/T. According to (2.6l), the activation energy can tilso be
obtained by plotting ri/T against 1/T, because a relation similar to (2.62) holds for the temperature dependence of T\/T.
2.3.5 Mixed translational and rotational diffusion
At high temperatures and low frequencies the DNP of protons can often be explained by dipolar coupling modulated by pure translational diffusion. On the contrary, at high frequencies and low temperatures more or less serious deviations may occur, especially when aromatic molecules are present in the solution or when there is a good possibility for transient bond formation, for instance hydrogen bridges. It appears that the deviations can be explained by assuming a weak complex formation between the solvent molecules and the free radicals, in which case the dipolar coupling is modulated by rotational diffusion of the complex formed as was already pointed out in section 2.3.1.
Firstly, we shall now consider the case of pure rotational diffusion. Further on we shall be concerned with the much more important situation of mixed translational and rotational
diffusion. Because of the fact that dipolar coupling modulated by 27 pure rotational diffusion has been well-treated in the literature , we shall deal briefly with this case and in the main we shall
confine ourselves to summarizing the most important results. To start with, we need the conditional probability function P (fl, Q , t ) , which determines the probability that the axis of the two spins has the orientation fi at time t, when we know that it has the orientation n_ at time zero (the orientation fi is determined by the polar angles 6 and <ii, defining the nucleus-electron axis). Assuming spherical complexes, the function P (fi, fi-, t) is the solution of the following diffusion equation
i»2
9P °r
= ^ A P (2.63) ?t " 2 s
where A is the Laplacian operator on the surface of a sphere and a is the radius of the complex. D is the diffusion constant for rotation. An analyticEil expression for P can easily be obtained. Further, we can calculate the correlation function without difficulties. Fourier transformation then gives the necessary expressions for J , J. and J„ (see eqn, (2,25)). An equation for the coupling factor is easily found and is given by
Pl=i ^
(2.61*)
'^ 0,7 + 0,3 Jp ((Dj) / J^ (Wg)
The spectral density J (u) is introduced in the same way as in (2.56) and it appears that it can be written as
JI
(O))
=-^ \ - ^
(2.65)
2TTb 1 + 0) T r
where b is the nucleus-electron distance in the complex and T the correlation time for rotational diffusion. From the theory, it follows that T is given by
\=W-
(2-66)
r
The coupling factor for dipolar coupling modulated by pure rotationeil diffusion is shown in fig. 2.3.5.1 as a function of U _ T It can be seen that it differs considerably from the case of pure translational diffusion.
If weak complex formation is present in a sample we have to deal in fact with a system consisting of two different phases. Firstly, we have to consider nuclei in solvent molecules closely
_
-—"~"~-~^
^~-^^J^^\^
~^->. > v*^ \
p
^ C \
1 V X\ \
\ \
z
\ \
— \ \
\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ \
— ".'r-".^» \ \ 1 1 ' \ 1 \ 1 1 1 1 1 1 1 ooi"-01 1 10 100Fig. 2.3.5,1 The coupling factor as a function of oOgT^ (UgT^), The continuous curve applies to pure rotational diffusion. The dashed one describes the behaviour for independent translational diffusion.
associated with the unpaired electron spins of the free radical molecules. Secondly, because of the low free radical concentration, the majority of the solvent molecules will independently move in the liquid. We introduce therefore the probability W^ of finding a nucleus in the associated phase. The corresponding quantity for a nucleus in the non-associated phase will be given by W^, where W + W = 1. Assuming a fast exchange between the molecules in the
r t
two phases, we may write for the radical-induced relaxation rate 28
1/T of the nuclei, according to Beckert and Pfeifer 1 s
w w
JL = JL.^-L-
(2.67)
'^s ^ s ^is
where T.„ and T.„ are the longitudinal relaxation times of the IS IS
kk
nuclei in the associated and non-associated phase respectively. r t
(T and T,„ are only due to the presence of the free radicals.) lb lb
Since 1/T,„ is directly proportional to a sum of spectral 1 b
densities, it will be clear that we may write for the spectral density in case of a mixture of translational and rotational diffusion
Jp
M = JI M
+ w^j^
M
(2.68)
where we have assumed that W = 1, which is certainly allowed because of the low free radical concentration. It is obvious that the probability W is given by
n
W = m — (2.69) n^
where m is the average number of nuclei associated with a radical molecule and n is number of nuclei per unit volume. A short calculation with the help of (2.52), (2.56), (2.65) and (2.69) learns that J_ (co) can be expressed as
J^
M
=
1
^ { f.(uT.) +
n^timl)
} (2.70)
D 3 ^3 t t -^t "" ""
where we have introduced the reduced spectral densities f (uT.) and f (OJT ) , defined by
r r ' '
Jp, ( U ) Jp, (00)
f . ( a ) T . ) = ^ ; f ( u ) T ) = ^ (2.71)
* * 4 (0) -^ (0)
Moreover, we have made use of the fact that the integral in (2.52) equals the value 2/15 if u=0. Equation (2.70) was originally
29
derived by Kruger et al . It will be clear that the quantity R measures the relative influence of the dipolar coupling modulated
by rotational diffusion. It is given by
R=^2dL (2.72)
l*irb^njThe expression for R can be understood quite easily. It is obvious that R is proportional to the average co-ordination number m. The quantity d enters into the expression, because the spectral density due to the translational diffusion mechanism increases with
decreasing d, whilst the spectral density due to rotational
diffusion increases with decreasing b. The meaning of n., in (2.72) is clear from (2.69).
An expression for p in case of a mixture of translational and rotational diffusion is given by (2.55), but with J„ (u) replaced by the spectral density J_ (o)) from (2.70). The result can
29 be written as ^ P= o
^^t(Vt^""ft^r(yr)>
0.7{f,(Vt) ^ K T- ^r(Vr)> ^ 0-3{f^(aijT^) + R ^ ^^i'^jT^)} t T> (2.73)Equation (2.73) will be employed for the interpretation of the DNP results for protons, where it will appear that values for T + , R and T / T . can be obtained from the measurements. In fig. 2.3.5.2 the coupling factor is shown for some values of R and T / T . . Figure
r u
2.3.5.2 shows that the deviations from the case of pure translational diffusion are more or less negligible for small values of w^T.
(a)„T. < 1 ) . For medium and high frequencies (a)„T. > 1 ) , the influence
O t b t
of a small admixture of rotational diffusion can be considerably, especially when the value of R increases.
We conclude with a few remarks. Firstly, it will be clear from the discussion above that there is an upper limit for T since it
1*6
01 1 10 100
Fig. 2.3.5.2 The coupling factor for different contributions of rotational diffusion. I: R = 0.3 and T /T = 0.012; II: R = 0.3 and T / T ^ = 0.12; III: R = 1 and
r t '
T /TJ_ = 0.12. The dashed curve corresponds to r t
independent translational diffusion.
cannot be greater than the mean time during which a solvent molecule emd a free radical are joined. It appears that in our experiments T is always much smaller than this "sticking" time, which means that T gives indeed useful information about rotational motions in the solution.
Secondly, the eqns. (2.70) and (2.72) are valid for one single nucleus per complex. In practice, we are often concerned with solvent molecules which contain several equiveilent nuclei. Nevertheless, we shall apply (2.70) and (2.72) assuming that there exists an
appropriate average value for b (the nucleus-electron distance in the complex).
