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Studies of superlattice structure of opals as compared to other rocks by use of positron annihilation, x-ray powder diffraction and scanning electron microscopy methods.

ABSTRACT

Positron lifetime and Doppler broadening spectroscopy on opals show a presence of positronium formation. The positron annihilation signal observed in the opals is affected by the phase of diagenesis (maturity) and the degree of dehydration of opals. The short-lived positron sources produced by gamma activation methods and the standard long-lived source and novel analytical methods were applied to PA measurements and analysis. Scanning electron microscopy and powder X-ray diffraction methods are also used as the complementary techniques to identify the diagenesis stage and the degree of crystallinity/amorphicity of the opal samples. The results of our investigations on opals are presented in this article.

Keywords: positron annihilation, opals, diagenesis, positronium, Doppler broadening, lifetime, rocks, sandstones, limestones, carbonates positron sources, DBS, PAL, PA

MOTIVATION FOR THE RESEARCH

Our preliminary research on sedimentary rocks found relationships between rocks saturation in hydrocarbons and their positron annihilation (PA) signal (Urban and Quarles, 1995). Thus, the idea for the construction of a hydrocarbon sensor for a well logging application, which will distinguish between rocks saturated in hydrocarbons versus dry was proposed (Quarles, 1994). The PA studies on sandstone and carbonate rocks (Urban-Klaehn and Quarles, 1999) did not show any significant positronium formation so the presence of a positronium atom and its relatively long lifetime in opals was unexpected and needed an explanation.

A positronium atom is a hydrogen-like system that consists of a positron (electron anti-particle) and an electron. Positronium atoms form two states: the singlet state, called para-positronium and the triplet state called ortho-positronium (o-Ps). The singlet state, with total spin equal zero (when spins of the electron and positron have opposite signs), forms 25% of the time and decays into two gamma rays with a lifetime of 125 ps. The triplet state, with a total spin of one, if undisturbed, decays into three gamma rays with the lifetime 142 ns in vacuum. In condensed media, the lifetime of the o-Ps is greatly reduced to the range of nanoseconds by the pick-off process in which the positron from the positronium atom annihilates in the interaction with an electron from the surrounding medium.

Positron lifetime in the condensed media never exceeds 0.5 ns. Due to its positive charge, the presence of positron in the medium causes the enhancement of electrons nearby (so called enhancement effect) and speeds up annihilation process.

Since a positronium atom is neutral, its lifetime is greatly enhanced compared to the positron lifetime in the medium. Also, its interaction with media is weakened. Positronium is usually detected in non-conducting materials. Its intensity is enhanced by a presence of defects, voids, vacancies, vacancy clusters or micropores where it can be trapped in the regions with lower electron density. The lifetime of positronium depends on the size of the cavity or micropore where positroniu, is localized. The o-Ps pick-off annihilation lifetime is related to the size of the cavities or pores in material structure (Jean, 1990).

[FIGURE 1a OMITTED]

[FIGURE 1b OMITTED]

OPAL--PROPERTIES

All non-opal rocks and opal samples used in this research were obtained from Chevron Petroleum Co., Spencer Opal Mine in Idaho and the Department of Geosciences at Idaho State University.

Opal is a hydrated silica, SiO2 * nH20, with a water content of 3-10%. Opals are not stable over geologic time. They undergo a diagenesis process that changes their structure and composition over time. However, this process can be accelerated by the increased pressure and temperature. Opal, when first formed, contains large amounts of water and is amorphous (opal-A). Over time, opal loses water and changes its structure from amorphous to micro-quartz by the following sequence:

[ILLUSTRATION OMITTED]

Since opal-A is amorphous, it does not show the diffraction pattern in the X-ray range that is typical for crystalline substances. But, it may exhibit a short-range, domain-type, ordered structure that can be seen by use of the Scanning Electron Microscopy (SEM), especially if it is of organic origin (siliceous ooze). Opal of organic origin is usually an agglomerate of the siliceous shells of radiolaria or diatoms (called diatomites after fossilation)--each with the unique size and pattern of the internal pore structure (Figure1a). Opal-A of inorganic origin is formed from deposits accumulated near hot springs and geysers.

In the suitable conditions during diagenesis process, opal-A can transform to the ordered network structure called the "superlattice" structure. (Figure1b). The superlattice structure consists of silica grains that have regular layers and uniform size arranged in a hexagonal or cubic structure. The size and geometry of the superlattice structure along with the presence of any fluid filling opal's pores determines the refraction index of light passing and reflecting from opal therefore resulting with a play of colors (Sanders, 1964). The mechanism of color creation in precious opals is unique and the effect of enhancing color spectra by incorporating other materials, such as quartz, on to opals is used by the jewelry industry. For example, opals from the Spencer mine in Idaho are usually very thin and easily broken. To enhance their optical properties and protect them from damage, "triplets" are prepared with layers of quartz. Non-precious (potch) opals do not exhibit the play of colors because their internal structure is not uniform due to grains and pores of uneven sizes.

With aging, opals increase in density, lose water and become more crystalline. The crystallization process of opals goes through stages: first, open-volume crystals of tridymites and crystoballites are formed which are later replaced by more compact quartz.

POWDER X-RAY DIFFRACTION (XrD) AND SCANNING ELECTRON MISCROSCOPY (SEM)

SEM and XrD techniques were applied as complementary methods to Positron Annihilation spectroscopy. SEM was used to characterize the opals internal structure and its internal order. XrD helped to identify the type of present crystals and gave a rough idea about the amorphicity/crystallinity ratio which is very important in determining the stage of opal diagenesis.

SEM

Scanning Electron Microscope (SEM), model Leo 1430VP, was used to see the surface structure of etched in HF for 30 s samples. The magnification up to 20,000X allowed the resolution of structure details down to about 100 nm size or less. The SEM was coupled with a pure Si detector used in Energy Dispersive X-ray Analysis (EDX) system, model Oxford 7353 for elemental composition analysis. Some SEM images of opals from different diagenesis stages are presented in Figure 1.

XrD

Powder X-ray diffraction (XrD) analysis was carried out at the Idaho State University Laboratory for Environmental Geochemistry. The opal samples were ground to a fine powder with a ceramic mortar and pestle. Approximately one gram was mounted in a standard back filled aluminum holder. A copper X-ray tube was used in a Philips source (35 kV and 20 mA) with a Ni filter to produce monochromatic Cu [K.sub.[alpha]] radiation ([lambda][K.sub.[alpha]] = 1.5405A). Each crystal phase exhibits a unique XrD pattern, corresponding to distances between adjacent crystal lattice planes (called d-spacing) and follows Bragg's Law. The XrD techniques showed the presence of quartz, crystoballite and tridymites as well as the relative amorphicity/crystallinity ratio. This helped to determine opal's age and stage in diagenesis process.

Almost all samples showed the broad maximum in the range from 15-30[degrees] 2[theta]], centered about 0.42 nm due to the lack of long-range order which is characteristic of non-crystalline opal samples. Quantitative determination of the order/disorder rate and a quantitative comparison between different samples was difficult due to variations in powder particle sizes, presence of poorly crystalline states and also a preferred orientation that occurs in some minerals. According to XrD investigations precious and non-precious (potch) opals were characterized by lack on any type of crystallinity.

POSITRON SOURCES

Sources

Short-lived radioactive positron sources are inexpensive, easy to handle, and safe compared to long-lived standard sources. But, they are used rarely in positron research since very few laboratories have facilities to produce them. The short-lived positron radioactive sources were produced by ([gamma],n) reaction on pure foils, in the Idaho Accelerator Center (IAC) using

Bremsstrahlung from a 20 MeV electron pulsed LINAC incident upon a thick tungsten converter.

Different sources were used for two different applied PA techniques since these techniques have dissimilar requirements.

Positron Sources for Doppler Broadening Spectroscopy

These sources are strong positron emitters but with relatively weak gamma emission so that that the background around 511 keV (annihilation radiation) is relatively low.

Long-lived positron sources' characteristics:

[sup.68] Ge/Ga: [T.sub.1/2] = 271 days, [E.sub.[beta]max] = 1,899 keV, [I.sub.[gamma]max] <0.1%

[sup.22]Na: [T.sub.1/2] = 2.6 years, [E.sub.[beta]max] = 540 keV [E.sub.[gamma]] = 1,275 keV, [I.sub.[gamma]max] = 99.94 %

Short-lived positron sources' characteristics:

[sup.64]Cu ([T.sub.1/2] = 12.7 hours, [E.sub.[beta]max] = 579 keV, [I.sub.[gamma]max] <0.5 %)

Copper ([sup.64]Cu) was produced through the reactions:

[sup.65]Cu ([gamma], n) [sup.64]Cu, abundance of [sup.65]Cu = 30.8 %

[sup.63]Cu (n, [gamma]) [sup.64]Cu, abundance of [sup.63]Cu = 69.2 %

Positron Sources for Positron Annihilation Lifetime Spectroscopy

These sources require a gamma signal emitted in coincidence with a positron. Long-lived sources' characteristics:

[sup.22]Na ([T.sub.1/2] = 2.6 years, [E.sub.[beta]max] = 540 keV, [E.sub.[gamma]start] = 1,275 keV, [I.sub.[gamma]max] = 99.94 %)

Short-lived sources' characteristics:

[sup.57]Ni ([T.sub.1/2] = 35.6 hours, [E.sub.[beta]max], = 864 keV, [E.sub.[gamma]start] = 1,377.6 keV [I.sub.[gamma]] = 81.7 %)

Nickel ([sup.57]Ni) was produced n the reaction:

[sup.58] ([gamma], n) [sup.57]Ni, abundance of [sup.58]Ni = 68.3 %

The PA spectra obtained by short and long-lived sources were compared. Both types of sources gave the same results in the range of error. The short-lived sources had higher source contribution effect than the long-lived sources. The source contribution effect could be lowered, but there was a trade off. In order to have lower source contribution effect, thinner sources were used. The activation time were extended from tens of minutes to a couple of hours, increasing the costs significantly.

APPLIED POSITRON ANNIHILATION TECHNIQUES

Doppler Broadening Spectroscopy

The Doppler Broadening Technique measures a broadening of the 511 keV annihilation line due to a Doppler shift caused by the motion of the annihilating electron-positron pair. The high purity germanium detectors with a resolution 1.2 keV at 569 keV were used for Doppler Broadening (DB) Spectroscopy measurements. The signal was amplified and recorded by an ADC/MCA system.

Two parameters are commonly used for the interpretation of the DB spectrum: 1) S-parameter (shape-parameter) is defined as the ratio of the area under a fixed central part of the 511 keV peak to the total peak area and 2) W-parameter (wing parameter) is computed as the ratio of photons in the wing (edge) parts of the spectra to the total number of detected photons. For proper characterization of rocks, a "SW-parameter" was introduced, to represent the intermediate momentum between S and W-parameters ranges. The S-parameter is sensitive to low momentum annihilation processes. The W-parameter is sensitive to the annihilation events with higher momentum electrons. It is assumed that the positrons interact with electrons when thermalized (Jean, Schrader, 1988), so that the momentum of the annihilating pair reflects the momentum of the bound electron. Both, electrons and positrons are fermions, but there is usually only one positron present in a given time in the sample, while the electrons occupy all energy levels up to the Fermi level in a solid. According to the Pauli principle, identical Fermi particles cannot occupy the same energy levels; hence, one and only positron present in the sample in a given time has significantly lower momentum than the electrons. In this research, the energy range for the S-parameter covers the annihilation events with the valence electrons up to the energy 5 eV. The W-parameter the range is 14-30 eV.

The division of the DB spectrum into the central and core parts works well for metals such as copper or nickel. These metals have a higher effective Z than elements that are rock-constituents, such as SiO2 and CaCO3; therefore, their contribution of core electron momentum stretches to higher momentum values than in the rocks, in order to characterize the rocks properly, it is necessary to take into account lower momentum range than in metals (Urban and Quarles, 1997). As a result, the SW-parameter range characterizes rocks better than the W-parameter range which works well for metals. The shape of the 511 keV Doppler peak and its division into S, SW and W-

[FIGURE 2 OMITTED]

Positron Annihilation Lifetime Spectroscopy

In positron annihilation lifetime (PAL) spectroscopy, the time interval between the emission of a positron into the sample's interior and its subsequent annihilation in measured. Just as the energy resolution is critical in DB measurements, good timing is critical in PAL spectroscopy. We applied so-called fast-fast coincidence setup with two scintillators (either fast plastics or barium fluoride) linked with two CFDDs (Constant Fraction Differential Discriminator). One CFDD worked as a "start" registering the prompt gamma signal emitted in coincidence with the positron during a radioactive decay. The second detector was a "stop" registering one of two 511 keV gamma signals created as a result of the annihilation event. Time--to-Amplitude Converter (TAC) output for each coincident event was recorded by the ADC/MCA system and transmitted to the computer. J.Kansy's LT-9 software program was used for lifetime analysis (Kansy, 1996). Timing resolution of the system was 300-500 ps. The lifetime spectrum is the convolution of the instrumental resolution and one or more lifetimes due to different physical effects. The spectra are usually deconvoluted into three lifetimes: the shortest one due to the annihilation in the bulk; the intermediate due to positron trapping; and the longest with the value higher than 1 ns due to o-Ps pick off (Figure 3.).

[FIGURE 3 OMITTED]

DOPPLER BROADENING SPECTROSCOPY--ANALYSIS OF RESULTS

According to our data, opals are characterized by the highest S-parameter and the lowest SW and W-parameter values when compared to other rocks. The DB-parameter values for sandstones and limestones are more alike. This is due to similar structures of sandstones and limestones characterized by low degree of microporosity thus the absence of positronium formation.

Table 1. contains the averaged S, SW and W-parameter values for all measured sandstones, carbonates (limestones) and opals. The variations among the different samples of the same rock, shown in the parentheses as one standard deviation error, were 2% or less in S-parameter values. However, the differences among the different rocks were over 10% in the S-parameter values.

The S, SW and W-values were calculated according to the Figure 2. The values of one standard deviation error from several measurements of the same rocks are showed it: the brackets

Doppler Broadening Spectroscopy is not commonly used to detect the effects of only positronium, since the central area of the peak is sensitive to both positron and positronium trapping. Special analytical methods in conjunction with the DBS measurements are applied to distinguish between positron and positronium annihilations. One of these methods is explaned in the following text.

Figure 2, showed earlier, depicts the original DB spectrum whereas the Figure 4., below, shows a derivative of the DB spectrum. Every derivative spectrum has a maximum that occurs at the inflection point of the original spectrum. When the curvature of the Gaussian changes, the center of the DB derivative spectrum is at the "momentum equaling zero." The DB spectrum is always broader than the nuclear line of [sup.207]Bi, therefore its derivative is shifted towards higher momentum. The Doppler spectrum is a convolution of the instrumental width (instrumental response function) with a real physical effect. The momentum of an annihilating electron and positron contributes to the broadening effect. According to Figure 4., the maximum for Boise sandstone is shifted towards higher momentum as compared to the opal-CT sample. The shift is due to a lack of p-Ps formation in sandstone as compared to opal-CT. The positronium annihilation contributes to less momentum than the positron annihilation with valence electrons which is the dominant effect in Boise sandstone.

[FIGURE 4 OMITTED]

LIFETIME SPECTROSCOPY (PAL)--ANALYSIS OF RESULTS

PAL spectra of opals were commonly deconvoluted into three or four lifetimes due to the presence of positronium. The spectra of sandstones and limestones were successfully deconvoluted into two or three lifetimes. Table 2 shows the values of lifetimes and their intensities (with standard deviation errors in parentheses) and their mean lifetime values. The mean lifetime value is defined as the sum of all the lifetimes and their intenstities. The mean lifetime value for opals is described by use of the following equation: <[tau]> = [I.sub.1] x [[tau].sub.1] + [I.sub.2] x [[tau].sub.2] + [I.sub.3] + [I.sub.4] x [[tau].sub.4]. The short lifetimes (up to 0.5 ns) caused by a self p-Ps annihilation in the bulk, and positron trapping are not presented here. The focus is on longer lifetimes that are caused by o-Ps pick-off annihilation.

Table 2. The Longest Lifetime Values ([tau] > 1 ns) and Their Intensities and the Mean Lifetime Values for Opals, Compared to Various Other Rocks. Standard Deviation in Brackets.

The following conclusions can be drawn from the PAL measurements on opals:

Opals exhibit significantly higher lifetimes and intensities than non-opals (see Table 2). The positronium is formed in the micro- and meso-porous matrix and superlattice. The positronium pick-off annihilation is also observed in open-volume crystals. The lifetime value around 1.7 ns, present in almost all opals, is a combination of two effects: the pick-off positronium annihilation in open crystals of cristobalites and tridymites (Chojcan, 1993), and the annihilation in water filled pores (Singru, 1976). The second process dominates in water-rich precious and potch opals that do not exhibit any crystallinity according to XrD analysis. Opal-CT and opal-A samples are characterized by longer lifetimes and intensities than any other examined opals. Both samples exhibit unusual appearance and features as compared to other opals that were solids with a pearly luster. Opal-A is a fine white powder while opal-CT has a lusterless and chalky appearance due to dehydration. The lifetime value of [tau] = 1.9 ns, in opal-A is due to its fine powderization. The negative positronium work function in Si[O.sub.2]] grains causes the positronium to be expelled to the interstitial sites outside of the grains (Gidley, 1976) and its subsequent pick-off annihilation. Opal-CT has a low density as compared to the other samples due to the lack of water. The longest lifetimes present in opal-CT and opal-A were caused by the trapping of positronium in the pores that were not-filled with water or any other fluid.

The absence of the lifetime values longer than 10 ns in precious opals suggests that the trapping in the larger pores (>100 nm) is not observed. The extended Tao-Eldrup model does not go beyond the pore size larger than 20 nm (Dull, 2001). Trapping probability for mesopores (size of 10 nm and more) is not well known. According to quantum physics principles, the bigger size of trap is associated with a shallower and broader potential well. Since little energy is needed for de-trapping from a shallow potential, these traps are less stable. The broad potential well may contain several temperature dependent excited levels where the chances for de-excitation are increased (Goworek, 2000). De Broglie' wavelength of thermalized positron and positronium is in the order of tens of nanometers. Therefore, the quantum effects stop playing a significant role with the growing size of the pore, and chances of positronium localization in meso-traps is less probable. The larger pores, if not isolated, are more likely to be filled with water or hydrocarbons shortening positron lifetime.

As a result the positron lifetime in many opal samples is due to its pick-off annihilation in water rather that due to the pick-off annihilation as a function of a pore size.

EFFECTS OF WATER AND DEHYDRATION IN LIFETIME MEASUREMENTS

Some of the opals were dried out to verify whether the lifetime value at around 1.7 ns was caused by the o-Ps annihilation in water. In addition, their lifetimes were measured before and after drying to test whether the intensity of the lifetime 1.7 ns would be affected.

The samples were placed in the oven and covered by graphite pellets to avoid oxidation. The heating process was done gradually with at 50[degrees]C per 1.5 hr to avoid damage caused by the temperature stress,. After heating to 425[degrees]C, the sample was kept at this temperature for 1.5 hr, then cooled down to 110[degrees]C overnight to avoid moisture. All samples were stored in dessicator after removing from the oven to avoid any moisture deposition.

Three opal samples (opal milky, opal ice-cream and precious opal) were weighted before and after dehydration. The measured weight loss was around 1-3% depending on the sample. All opals showed a decrease in the intensity of the lifetime 1.7 ns in a range between 10-30%, depending on the sample. The mean lifetime value increased to 30-45% due to the increase in the intensity of the long lifetime value (~ 8 ns). The long lifetime value is attributed to the trapping of the positronium in the dried samples contained more waterless micropores.

CONCLUSIONS

Several analytical techniques, PA spectroscopy, SEM, XrD, were used to study opals--their structure and diagenesis. The results of our studies show that opals' microstructure, especially its microporosity is responsible for a formation of positronium. But the dependence between the diagenesis of opals and the shape of the PA spectra is complex. For instance there is not a simple relationship between the diagenesis stage in opals and the shape of the positron lifetime spectrum since the lifetime spectrum is affected by the powderization of opals.

We were successful in showing the evidence how crystallinity and dehydration affect the value of the opals' lifetime. Both of these factors are at least partly related to its diagenesis stage.

Our studies showed that precious opal can not be used as a natural standard for lifetime measurements, therefore opal's play of color related to its superstructure size cannot be associated with the positronium lifetime. The positronium signal that was sensitive to the microporosity greater than 100 nm was not detected, since the lifetimes longer than 10 ns were not observed.

Positron annihilation techniques are rarely applied to complex materials like rocks and minerals in natural conditions. Studies presented here, as well as the earlier studies, (Quarles, 95, Urban, 99) showed that the PA techniques can be useful in understanding rocks structure, state and composition. The use of new analytical methods in DB spectroscopy allowed a distinction between positron and positronium annihilation processes.

Short-lived positron sources, [sup.64]Cu, for DB spectroscopy and [sup.57]Ni, for lifetime spectroscopy were successfully applied for positron annihilation studies. They produced comparable data with commonly used long-lived manufactured sources. Short-lived sources are easy to maintain, cheap and safe.

ACKNOWLEDGMENTS

The authors are grateful to Dr. Kathleen A. Campbell from Department of Geology at the University of Auckland in New Zealand and Dr. Chris Nelson from Physics Department of ISU for important discussions and providing the literature. They also thank the staff and students from Physics Department and the Idaho Accelerator Center for helping to carry on the experiments and for assistance during the measurements.

LITERATURE CITED

Chojcan, J. Sachanbinski, M., Acta Phys. Polon., 83, p.267, 1993.

Dull, T.L., Frieze, W.E., Gidley, D.W., Sun, J.N., Yee, A.F., J. Phys. Chem. B2001, 105, 4657.

Gidley, D.W., Marko, K.A., Rich, A., Phys. Rev. Lett., 36(8), p.395, 1976.

Goworek, T, J. Nucl. Radioch. Sciences, Vol. 1. No.1, 11, 2000.

Jean, Y.C., Microchem. J, 42,72, 1990.

Jean, Y.C., Schrader, D.M., Positron and Positronium Chemistry, Elsevier Scientific Pub, Amsterdam, 1988.

Kansy, J. Nucl. Instr.Meth. Phys. Rev.: A374, 235, 1996.

Quarles, C.A., Urban, J.M., Salaita, G., SPWLA 35-th Annual Logging Symposium, Tulsa vol. 1, paper X, 1994.

Sanders, J.V., Nature, Vol. 204, p.1151, 1964.

Singru, R.M., Lal, K.B., Tao, S.J., Atomic Data and Nuclear Data Tables, 17, 1976

Urban, J.M., Quarles, C.A., AIP Conf. Proc., CP475, AIP, 1-56396-825-8/99, p.345, 1999.

Urban, J.M., Quarles, C.A., Mat. Science Forum Vols. 255-257, p.623, 1997.

Urban-Klaehn, J.M., Quarles, C.A., J. Appl. Phys., VI.86, No.1, p.355, 1999.

Urban, J.M., Quarles, C.A., Acta Physica Polonica A, V88, p241, 1995.

Corresponding author's address:

Jagoda Urban-Klaehn

Idaho State University

Dept. Of Physics

Campus Box 8263

Pocatello, Idaho 83209

Tel: 208-282-5678

e-mail: klaehn@physics.isu.edu

Jagoda Urban-Klaehn, Alan W. Hunt, Mark Ford *, Lisa Lau **, C.A. Quarles ***, Randy Spaulding, Frank Harmon

Idaho State University (ISU), Physics Department, Pocatello, ID

* ISU, Department of Geosciences

** ISU, Department of Chemistry

*** Texas Christian University, Fort Worth, TX
Table 1. Averaged Doppler Broadening Values in Opals Compared
to Sandstones and Carbonates, with Error Bars in Brackets.

sample       S-parameter     SW-parameter    W-parameter

sandstones   0.519 (0.005)   0.456 (0.005)   0.027 (0.000)
carbonates   0.505 (0.002)   0.469 (0.002)   0.028 (0.000)
opals        0.561 (0.003)   0.417 (0.004)   0.023 (0.000)

 ISU--sample    [[tau].sub.3]     [I.sub.3]     [[tau].sub.4]

Opal-CT          1.7 (0.07)      3.4 (0.20)      35.4 (1.7)
Opal-A           1.9 (0.07)      4.7 (0.07)      13 (1.8)
Op. Ice-cream    1.7 (0.04)      9.7 (0.32)      8.9 (0.3)
Opal milky       1.7 (0.03)      10.8 (0.12)     6.8 (0.25)
Opal-VV          1.7 (0.04)      10.2 (0.11)     7.5 (0.3)
Opal precious    1.7 (0.05)      10.8 (0.17)     7.6 (0.40)
Rhyolite         2.8 (0.46)      0.5 (0.07)      --
Berea Sandst.    4.9 (l.40)      0.3 (0.05)      --

 ISU--sample      [I.sub.4]        <[tau]>

Opal-CT          1.9 (0.05)         1.131
Opal-A           0.7 (0.18)         0.746
Op. Ice-cream    2.7 (0.07)         0.860
Opal milky       2.2 (0.11)         0.772
Opal-VV          2.2 (0.10)         0.782
Opal precious    1.7 (0.11)         0.756
Rhyolite         --                 0.444
Berea Sandst.    --                 0.414
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