J . Phys. Chem. 1990, 94, 6879-6884
6879
Heterogeneous Spin-Lattice Relaxation Revealing the Activation Energy Distribution of Mobile Guests in Organic Glasses E. Rossler,* M. Taupitz, and H. M. Vieth Institut fur Atom- und Festkorperphysik. Freie Universitat Berlin, Arnimallee 14, D- 1000 Berlin 33, FRC (Received: April 19, 1989; In Final Form: February 16, 1990)
The dynamics of the mobile deuterated guest molecules hexamethylbenzene and benzene in the amorphous host systems polystyrene and phthalic acid di-n-butyl ester is investigated by studying 2H NMR. High-precision spin-lattice relaxation measurements have been performed in the temperature range from 40 to 300 K. Pronounced deviations from a monoexponential relaxation are observed, which are explained by a distribution of activation energies for the rotational reorientation of the guests. The deconvolution procedure applied to unravel details of the distribution function is more precise for spin-relaxation measurements as compared to many other methods, because the relevant pair of Laplace transformations is sensitive to a deviation from symmetry. We used a convolution of an exponential function with a log-Gauss function to describe the correlation time distribution on a logarithmic scale, which closely corresponds to the energy distribution. The maximum of this asymmetric energy distribution is found on the low-energy side. The fits reproduce well the time and temperature dependences of the observed relaxation curves; an independent line-shape analysis confirms the results.
Introduction
The glassy state of condensed matter is characterized by the absence of long-range order. The spatial correlation of the molecules or atoms extends only over a few nanometers; the glass is considered to be a frozen liquid. Topological disorder is found. Furthermore, it is a well-established fact that many physical properties of a glassy system are characterized by a distribution rather than by one single value.' I n particular, many relaxation experiments performed in disordered systems yield nonexponential decay functions resulting from a superposition of exponentials with a distribution of time constants. Often the symmetric Gauss distribution is used to interpret the observed data.2-4 In this concept, the measured relaxation function is given as the Laplace transform of the distribution function. However, only a few publications are found where a direct model-free evaluation of the distribution function from a given set of experimental data has been carried out. The main difficulty lies in the problem of performing the inverse Laplace transformation, that is, the back transformation from the observed time decay curves to the underlying distribution function. The numerical procedure is from general mathematical reasons a difficult problem, and in most cases, the extracted information has a pronounced degree of ambiguity, particularly when the signal-to-noise ratio is low. Often the same decay functions can be interpolated by using a variety of different distribution functions; hence, the evaluation of details of the distribution function is usually not On the other hand, only a few experiments are known where a distribution function is directly probed without being obscured by a Laplace transformation. In the frame of the free-volume theory,*v9a certain amount of free volume is considered to be frozen at the glass transition temperature, T . . In contrast to the liquid phase, no reorganization of the free volume is possible below this temperature. The following question arises: what kind of function describes the distribution of free volume in a glass. In particular, what parameters ( I ) Zallen. R. The Physics of Amorphous Solids; J. Wiley: New York,
of this distribution are probed by small molecules doped into the amorphous matrix, especially when they exhibit a characteristic reorientational motion, e.g., rotation of the whole molecule around a molecular axis, which depends on their local environment. It was shown that 2H N M R provides a promising tool to investigate motional nonuniformity in amorphous matrices, which corresponds to a distribution of motional correlation times.'*I2 Although solid-state N M R in diamagnetic organic compounds without intrinsic large-angle motion is usually hampered by long relaxation times, an indirect but more sensitive investigation of the glassy state becomes possible when an N M R active probe molecule with short spin-lattice relaxation is doped into the matrix. Molecules such as benzene or hexamethylbenzene are good candidates; they are known to exhibit anisotropic motion in their crystalline lattice. In many cases, the type of motion is conserved when the molecule is doped into a glassy matrix. Only the values of motional parameters such as correlation time and activation energy are changed.1° These changes of the dynamical parameters of the guest reorientation depend on the local environment; therefore, structural properties of an amorphous matrix can be investigated via analyzing the dynamics of such probe molecules. By studying (i) the nonexponential spin-lattice relaxation and (ii) the N M R line shape of the deuterated guest molecules in protonated organic glasses, two independent methods are given to probe the dynamical parameters. For 2H NMR, the relevant interaction in these molecules is given by the quadrupolar coupling with the intramolecular electric field gradient of the C-2H bonds. In contrast to 'H NMR, spin diffusion plays a minor r01e.l~ This condition allows the detection of site-specific relaxation in a disordered system. No averaging between subensembles with different relaxation times, e.g., between different sites in the matrix, is observed. This advantage is lost for ' H NMR.I4 In this publication, we want to present the first results on the activation energy distribution for the rotational motion of the guest molecules benzene and hexamethylbenzene, which are well-suited because of their high sensitivity and the simplicity of their motional process. As matrices, we used phthalic acid di-n-butyl ester and polystyrene. This choice offers the possibility to study differences
1983.
(2) Albery, W.J.; Bartlett, P. N.; Wilde, C. P.; Darwent, J. R. J . A m . Chem. Sor. 1985. 107, 1854. (3) Jankowiak, R.; Richert, R.; BBssler, H. J. Phys. Chem. 1985,89,4569. (4) Doba, T.; Ingold, K.; Siedebrand, W.; Wildmann, T. Chem. Phys. Lett. 1985. 115. 5 1.
(5) Lindsey, C. P.; Patterson, G . D. J. Chem. Phys. 1980, 73, 3348. McWriter, J. G.; Pike, E. R. J. Phys. 1978, A l l , 1729. Provencher, S. W. Comput. Phys. Commun. 1982, 27, 213. Fox, T. G.; Flory, P. J. J . Appl. Phys. 1950, 21, 581. Turnbull, D.; Cohen. M. H. J . Chem. Phys. 1961, 34, 120.
(6) (7) (8) (9)
0022-3654/90/2094-6879$02.50/0
(IO) Jansen-Glaw, B.; Rossler, E.; Taupitz, M.; Vieth, H.-M. J . Chem. Phys. 1989, 90, 6858. ( I I ) Rossler, E.; Taupitz. M.; Vieth, H.-M. Springer Proceedings in Physics; Springer: Berlin, 1989; Vol. 37. (12) Rossler, E.; Taupitz, M.; BBrner, K.; Schulz, M.; Vieth, H.-M. J . Chem. Phys. 1990, 92, 5847. (13) Schajor, W.; Pislewski, N.; Zimmermann, H., Haeberlen, U. Chem. Phys. Letf. 1980, 76, 409. (14) Muller-Warmuth, W.; Otte, W. J . Chem. Phys. 1980, 72, 1749.
0 1990 American Chemical Society
6880 The Journal of Physical Chemistry, Vol. 94, No. 17, I990 for high and low molecular weight glass-forming systems. In particular, we want to demonstrate that 2H spin-lattice relaxation probes a given distribution function of correlation times via two different Laplace transformations and that the precision of such relaxation measurements is high enough to reveal detailed features of the activation energy distribution.
Theoretical Background When mobile deuterated guest molecules are doped into a disordered matrix, their nuclear spins are coupled to individual local lattice environments. Spins with identical environments form subensembles, and the various subensembles are, in general, characterized by different relaxation times. The overall spinlattice relaxation results from the superposition of exponential decays with different TI values for each subensemble in the matrix and can be described by the distributionf( T I ) . The distribution function f ( TI) is related to the distribution p ( 7 ) of motional correlation time. 7 . Then, the normalized relaxation function is given by
@ ( r ) = ( M o - M ( t ) ) / M o = l m f ( T , )exp(-t/T,) dT, = 0
L m p ( 7 )e x p ( - t / T , ( ~ ) ) d7 ( 1 )
Rossler et al. The well-known problem that the shape of a distribution function is obscured after a Laplace transformation is performeds7 is less severe for our NMR investigations, because the distribution can be checked by two different inverse Laplace transformations when the two limiting cases can be reached experimentally. For the case that a thermally activated process determines the temperature dependence of the correlation time, T and the activation energy (E) are related by z
= In
(7/7")
= ( E - Eo)/RT
(4)
where Eo is some reference energy and 70 is the corresponding correlation time. Here, the distribution of the motional correlation times originates from a distribution of activation energies, g ( E ) , in the disordered system when 70 is assumed to be constant for all sites. This interpretation refers to a situation of isolated molecules hopping over site-specific potential barriers. I n this case, it is convenient to transform p ( 7 ) to a logarithmic scale: p ( 7 ) d 7 = G(z) dz, with G ( z ) = G(ln 7) = RTg(RTz). Although g ( E ) in the first approximation is assumed to be temperature independent, this transformation leads to a temperature dependence of the correlation time distribution, C(ln 7 ) . The width of the distribution becomes larger when the temperature is decreased. Equations 3a and 3b transform to
M(t) is the nuclear magnetization as function of time and Mo is
its equilibrium value. It was shown that the heuristic approach of Bloembergen et aI.l5 for isotropic motion in liquids, which connects relaxation times with correlation times, is also an appropriate description for many solidsi0 Tl-l = -Q 3a2 10
[+ 1
(OT)2
+ 1 + 47 4(w7)2
]
J 0 p ( 7 ) exp(-tClr) d7 = L ( p ( 7 ) )
@(f)= J m p ( 7 ) exp(-tC2/7) d7 = L ( P ( r ) ) 0
07
>
(3b)
1
C, and C2 are corresponding constants derived from eq 2 for the limiting cases. Accordingly, the relationship between the relaxation function, @ ( t ) ,and the correlation time distribution, p ( 7 ) , as given by eqs 3a and 3b is greatly simplified; now, two different Laplace transformations, L, describe the relaxation. For 07 > 1 by the Laplace transformation of the rate distribution P ( r ) , where r = 1 /7; P(r) is related to p ( 7 ) by P(r) = p ( 1 / r ) / ? . The different constants, C, and C, rescale the time axis only. Equation 3b is mathematically identical with expressions known from other methods for analyzing disordered media, e.g., from dielectric relaxation,I6from photon correlation spectroscopy," or from NMR applied to supercooled liquids when the system is in the limit of fast e x ~ h a n g e . ' ~ , ' ~ ~~
~~~~
( 1 5 ) Bloembergen, N.; Purcell, E.; Pound, R. Phys. Reo. 1948, 73, 679.
(16) Bottcher, C. J . F.; Borderwijk, P. Theory of EIectric Polarization; Elsevier Scientific Publishing Company: Amsterdam, 1978; Vol. 11. (17) Patterson, G.D. Ado. Polym. Sci. 1983, 48, 125. ( I 8) Noack, F. NMR-Basic Principles and Progress; Springer: Berlin, 1971; Vol. 3. (19) Dries, Th.; Fujara, F.: Kiebel, M.; Rossler, E.: Sillescu. H. J . Chem. Phys. 1988, 88,21 39.
The two different integrals result in similar relaxation functions for the two limiting cases, when the distribution function is characterized by the symmetry relation G ( z ) = G(ln
7)
= G(-(In
7))
= G(-z)
(6)
The only difference is a change in the time scale; hence, for a given symmetric distribution function, G(z), one expects for @ ( t ) no qualitative change of the relaxation function. Referring to eqs 3a and 3b, the symmetry relation is equivalent with the statement that the correlation time distribution, p ( 7 ) , and the rate distribution, P ( r ) , are basically the same. As a consequence, any asymmetry of the distribution function will cause different relaxation behavior for the two limiting regimes in a NMR experiment. It can be shown that many phenomenologically applied distribution functions that give similar decay behavior for the case of eq 5b exhibit pronouncedly distinguishable decay functions when eq 5a is applied. Usually in a N M R experiment, the condition of eq Sa and eq Sb is established by proper choice of the temperature. In the intermediate range ( U T I ) , eq 1 in connection with eq 2 has to be used to describe the relaxation function. The sensitivity of the spin-lattice relaxation with respect to such a break in symmetry offers the change to reveal more details of a distribution function in disordered systems. The proper distribution function g ( E ) has to describe both cases for the relaxation functions at high and low temperatures, respectively.
Experimental Section The glass-forming systems polystyrene (PS) and phthalic acid di-n-butyl (PDB) ester were commerical grade and used without further purification. The deuterated hexamethylbenzene (HMB) was synthesized with a degree of deuteration higher than 95% (the authors thank Herbert Zimmermann, Max-Planck Institut fur Medizinische Forschung, Heidelberg, for supplying deuterated hexamethylbenzene) and sublimated several times. The deuterated benzene was commerical grade with a degree of deuteration higher than 99%. The matrices were degassed by using a freeze-and-pump method for PDB. For polystyrene, the sample was pumped for several days at a temperature of 400 K. The mixing of the components was achieved by solving HMB or benzene in the glass-forming
Heterogeneous Spin-Lattice Relaxation
The Journal of Physical Chemistry, Vol. 94, No. 17. 1990 6881 sixfold
+ 2 9% HMB In PS
rotati011
5 p% benzene
c2 L
10
*,O
I
!
PDB/ 2 2% benzene in PDBI
A
it1
** 0.01
,.
0
" " '
1
' . ' ' ' ' , ' ' ' ' ,
2
3
4
" " " " "
"
6
5
7
8
9
t/s Figure 1. Spin-lattice relaxation of 2.9% HMB-d18in polystyrene, high temperatures; solid line, log-Gauss fit.
0
I
10
100
200
300
400
T/K
Figure 3. Width parameter 0 for the used distributions of correlation times as a function of temperature; dashed line, log-Gauss fit; solid line, convolution of the exponential and the log-Gauss distribution.
Using eq 1, we assume for a first approach a Gaussian distribution for the activation energy distribution, g ( E ) , with a width
145 K
SE. 01
This leads to a log-Gauss distribution of correlation times 0 01 . _
0
0 2
0 4
0 6
0 8
10
12
14
16
1.8
2 0
t/s
Figure 2. Spin-lattice relaxation of 5.2% benzene-d, in phthalic acid di-n-butyl ester, high temperatures; solid line, log-Gauss fit.
liquid and in the case of polystyrene by additional tempering of the annealed N M R tube for 1 week at 430 K. The guest concentrations of the samples are given in mass percentage. This corresponds for HMB to 5 mg and for benzene to 25 mg in one sample. The NMR experiments were performed on a Bruker CXP 300 spectrometer operating at a Larmor frequency of 46.07 MHz for deuterons. The temperature was controlled by using a home-built N M R probe inserted into an Oxford flow cryostat. The temperature stability was better than 0.1 K. The length of the 90-deg pulse was between 2.5 and 3.3 p s when using a coil with 5-mm diameter and 25-mm length. The spin-lattice relaxation was measured by using a saturation pulse sequence which was followed after a variable time delay by a solid echo sequence consisting of two a / 2 pulses with 90-deg phase difference. The first pulse of the echo sequence was phase-cycled to reduce artefacts from the second pulse. For the solid echo, a pulse delay of 50 p s was used. Accumulations of 30 scans at high temperatures and 12 scans at low temperatures were carried out.
Results We measured the spin-lattice relaxation of deuterated hexamethylbenzene in protonated polystyrene and of deuterated benzene in protonated phthalic acid di-n-butyl ester in a concentration range of 2-5 mass%. The relaxation behavior was mapped out down to an amplitude of about 0.02 of the normalized saturation recovery function, @ ( t ) . In Figure 1 and Figure 2, the spin-lattice relaxation is shown for different temperatures, for the case of 2.9% HMB in PS and 5.2% benzene in PDB, respectively. The relaxation behavior for temperatures above the relaxation minimum is plotted, only. We find strong deviations from exponential decays. The temperature dependence is small, and the degree of nonexponentiallity increases as the temperature is lowered. For the case of HMB in PS,temperatures up to about 360 K are accessible. For benzene in PDB, temperatures up to about 170 K are measurable. This reflects the differences of the glass transition temperature for the two mixed systems, that is, the temperature where the glassy matrix starts to soften and the heterogeneity in the spin-lattice relaxation disappears due to fast exchanging molecules and to fluctuations of the local environments.
G(ln
7)
=
exp(-P2z2)
The width parameter, energy distribution
with
0,is closely
z = In
(T/T,)
(8)
related to the width of the
p = RT/(~SE)'/~
(9)
Thus, a proportionality holds between @ and T. The width parameter and the logarithm of the mean correlation time in seconds, In (T,/s), are fitting parameters of the numerically solved integral of eq 1 in connection with eq 2 for describing the nonexponential spin-lattice relaxation. The corresponding fits for @ ( t ) are drawn in Figure 1 and Figure 2 as solid lines. For the fit procedure, we used for benzene a coupling constant C = 147 kHz corresponding to the 6-fold rotation of the molecule. For HMB, two kinds of motions have to be considered: first, the 6-fold rotation of the whole molecule that determines the relaxation at temperatures above 160 K and for which a coupling constant of 38.6 kHz was used; second, the methyl rotation that governs the relaxation at temperatures below 100 K, where we used 154.4 kHz as the coupling constant. All coupling constants were taken from the respective minimum values of T, in neat HMB and neat benzene. In an intermediate temperature range (100-160 K), no evaluation in this straightforward way can be performed for HMB, because the two motional processes are competing with comparable relaxation efficiency. Hence, for this temperature range, no fit data are given for HMB. In Figure 3, the width parameter is given as function of temperature as obtained from a least-squares fit. For all three systems, a proportionality holds between (3 and T a s claimed for an activated process. For benzene as a probe molecule, two guest concentrations were studied; no indication for different relaxation behavior could be seen. We found a width of the activation energy distribution (cf. eq 9) for the 6-fold rotation of HMB, SE= 2.3 kJ/mol and SE= 0.57 kJ/mol for its methyl group rotation; for benzene, we got 1.9 kJ/mol. The second fitting parameter of the log-Gauss distribution, the logarithm of the mean correlation time, log ( T , / s ) , is plotted in Figure 4. For all three motional processes, an Arrhenius law is found, with activation energies of 7.6 kJ/mol for the 6-fold rotation and 1.6 kJ/mol for the methyl group rotation of HMB and of 5.8 kJ/mol for the 6-fold rotation of benzene. Again, no difference is seen for the two systems of benzene in PDB with different concentrations. The activation energies can be compared to the corresponding values in the neat crystalline state of the probe molecules. For HMB, 22.0 kJ/mol is reported for the 6-fold rotation and 6.5 kJ/mol for the methyl group.'0 I n the case of neat benzene, a value of 18.4 kJ/mol is
6882
The Journal of Physical Chemistry, Vol. 94, No. 17, 1990 -12
k,
Rossler et al.
.
methyl group rotation
-
2 9% H M ~ ; S - ' - ]
-
+ 2 9% HMB It1 PS
sixfold rotation
A
5 2% benzene in PDB
0,022% benzene i t i
PDB
T
I
-.-
-
-
9
- -- _
3
I
001;
=,
"
20
'
"
40
\
t/s
"
60
~
70K 74 K
-
-
* - -*--
65K
-
-t
b
-
-
"
80
+ J
100
Figure 7. Long time behavior of the low-temperature relaxation of 2.2% benzene in PDB.
i / /~~ o - ~ K - '
Figure 4. Correlation times T,,, and 7, of the used fits as a function of reciprocal temperature;dashed line, log-Gauss fit; solid line, convolution of the exponential and the log-Gauss distribution A
74K 92K
4
138 K
0
20K 38K
* 48K A 70K 0 160 K
-220 K
- 6 001
0 01
0 1
10
10
.D
t/s 0011 ' 0 001
:A ,,'"
" " " "
0 1
0 01
"
"
'
t/s
0
""' 100
10
10
Low- and high-temperature relaxations of 2.9% HMB in polystyrene; solid line, log-Gauss fit.
Figure 8. Relaxation curves after subtracting the exponential long time behavior for 2.2% benzene in PDB; solid line, fit applying the convolution of the exponential and the log-Gauss distribution.
Figure 5
___ 65 K 70 K 74 K 92 K 123 K 163 K
straight line for intermediate times rather than by a bent curve as is the case for high temperatures. It is no longer possible to perform a reasonable log-Gauss fit. For the longest times studied, the slope of the curves falls off again. We want to emphasize that this change of structure of the relaxation takes place mainly below an amplitude of 0.1 for a([). This demonstrates the necessity of careful measurements in this regime. Particularly, the determination of M o has to be very accurate; any systematic error distorts very delicately the shape of @ ( t )at amplitudes of 0.1 and lower. At T = 65 K, 800-s delay times were used for the measurement of M,, and the experimental error of Mo is less than 1%.
0 0 0 001
0 01
01
1 10
10
100
2 1000
t/s
Figure 6. Low- and high-temperature relaxations of 2.2% benzene in
PDB; solid line, log-Gauss fit. given in the literature.20 Accordingly, the mean activation energies are reduced by a factor of 3-4 for the studied motional processes in the amorphous matrices. An obvious interpretation is that the guest molecules are less densely packed in the amorphous matrix as compared to their own crystalline state and that the rotational barrier heights are flattened. So far, we have discussed only the relaxation for temperatures above the relaxation minimum; here a log-Gauss distribution provides a fair interpolation of the data. The situation changes completely when the relaxation is studied in the low-temperature regime, where the relaxation slows down with decreasing temperature. In Figure 5 and Figure 6, these qualitative changes are demonstrated for HMB and benzene in a double logarithmic plot of the relaxation function where now the whole investigated temperature range is covered. The nonexponential relaxation is monitored over more than 5 decades of time. For both systems, the observed changes of the relaxation at the lowest temperatures as compared to the high-temperature behavior are similar: as soon as the relaxation becomes slower with decreasing temperatures, the behavior is described in the double logarithmic plot by a ~
~~
(20) Van Steenwinkel. R Z Naturforsch. 1969. 24a, 1526
The relaxation minimum is found for benzene at about 100 K and for HMB at about 40 K. This big difference is due to the different motional processes that govern the relaxation at the minimum. For HMB, that is the methyl group rotation with its low mean activation energy, E,,,, while for benzene it is the 6-fold rotation with its much higher mean activation energy. As already discussed in detail, this change of relaxation behavior below the relaxation minimum is a strong indication for an asymmetry of the distribution function, G(ln 7). A properly chosen model function for the distribution has to fit both the high-temperature and the low-temperature behavior. For the following discussions, we restrict ourselves to the system benzene in PDB. Benzene with its rather simple motional processes is better suited to unravel more details of the distribution functions. In the case of HMB with its two motional processes, the temperature range where an accurate analysis is possible is also reduced by the line-shape changes due to the slowing down of the 6-fold rotation of the molecule. When changes in the line shape occur, eq 1 does not apply any longer, because an additional reduction factor that depends on the correlation time has to be taken into account.I0 Figure 7 shows a logarithmic plot of +(t) as a function of time on a linear scale for low temperatures of benzene in PDB. While the fast decay of the relaxation curves is not resolved in this representation, it is seen that the long time behaivor is determined in good approximation by an exponential decay with a temperature-independent time constant of T = 110 s for all shown relaxation curves. Its importance for the overall relaxation increases when the temperature is lowered. We postpone a detailed dis-
Heterogeneous Spin-Lattice Relaxation
The Journal of Physical Chemistry, Vol. 94, No. 17, 1990 6883
cussion of this topic to a subsequent publication. On a qualitative level, we explain this behavior by the onset of an averaging of the heterogeneous spin-lattice relaxation due to the small but not negligible spin diffusion. For the further discussion in this paper, we have in a first approach subtracted this exponential tail of the relaxation, leaving decay functions as shown in Figure 8. Incorporating spin diffusion in disordered systems to the relaxation theory will overcome this somewhat crude approach. The characteristic feature of the low-temperature relaxation in Figure 8 is straight lines for long times in the double logarithmic plot corresponding to a power law. This relaxation behavior is also observed in Figure 7 for intermediate times. Hence, the basic feature is not lost by subtracting the long time tail. The power law in the relaxation corresponds to a behavior of the correlation time distribution G(ln 7 ) a 7-0 for large 7 . This feature is compatible with several well-known distribution functions, e.g., the reverse Cole-Davidson, the Maxwell-Boltzmann, and the exponential distribution. At high-temperatures, Le., for UT > I , mainly the right side is probed corresponding to long correlation times. This situation is characteristic for the involved Laplace transformations; roughly speaking, only one-half of a distribution is probed in one transformation. Consequently, we used for describing the asymmetric distribution a convolution of the simple asymmetric exponential distribution and the symmetric log-Gauss distribution. The exponential distribution for z is given by G(z) =
6 exp(-pz) = p ( T / 7 , ) - ' G(z) = 0
for z
-12
-11
-10
-9
-7
-6
-5
-2
-1
0
.
2.9XHMB-d,ln polystyrene
(10)
T , characterizes the cut-off time of the asymmetric distribution. Such an exponential distribution was proposed first by Matsumoto and Higasi for the interpretation of dielectrical relaxation data.16 The convolution of a log-Gauss and such an exponential distribution function is given by
z = In
-3
-4
Figure 9. Upper part: three distribution functions: (a) log-Gauss distribution, (b) exponential distribution, and (c) convolution of both distributions. Lower part: temperature dependence of the measured distribution function.
for z 2 O
