Among all random chat sites, Chat Alternative provides the most immediate way to the huge community of guys and girls dating online and looking to get acquainted. You stay completely anonymous in the chat unless you would like to share some secrets with your new partner.
Earth radiometric dating Chat Alternative, you try your luck in finding a very special earth radiometric dating - for a secret marrakech morrocco escorts, or a new friend - for a long lasting friendship, or even a love of your life - for a lifetime.
It goes at zero cost for everyone addicted to random chatting, and the chat can bring you a priceless joy of new relationship.
He did not publish these results, which was fortunate because they were flawed by measurement errors and poor estimates of the half-life of radium. Boltwood refined his work and finally published the results in Boltwood's paper pointed out that samples taken from comparable layers of strata had similar lead-to-uranium ratios, and that samples from older layers had a higher proportion of lead, except where there was evidence that lead had leached out of the sample.
His studies were flawed by the fact that the decay series of thorium was not understood, which led to incorrect results for samples that contained both uranium and thorium.
However, his calculations were far more accurate than any that had been performed to that time. Refinements in the technique would later give ages for Boltwood's 26 samples of million to 2.
Although Boltwood published his paper in a prominent geological journal, the geological community had little interest in radioactivity. Rutherford remained mildly curious about the issue of the age of Earth but did little work on it. Robert Strutt tinkered with Rutherford's helium method until and then ceased. However, Strutt's student Arthur Holmes became interested in radiometric dating and continued to work on it after everyone else had given up.
Holmes focused on lead dating, because he regarded the helium method as unpromising. He performed measurements on rock samples and concluded in that the oldest a sample from Ceylon was about 1.
For example, he assumed that the samples had contained only uranium and no lead when they were formed. More important research was published in It showed that elements generally exist in multiple variants with different masses, or " isotopes ". In the s, isotopes would be shown to have nuclei with differing numbers of the neutral particles known as " neutrons ".
In that same year, other research was published establishing the rules for radioactive decay, allowing more precise identification of decay series.
Many geologists felt these new discoveries made radiometric dating so complicated as to be worthless. His work was generally ignored until the s, though in Joseph Barrell , a professor of geology at Yale, redrew geological history as it was understood at the time to conform to Holmes's findings in radiometric dating. Barrell's research determined that the layers of strata had not all been laid down at the same rate, and so current rates of geological change could not be used to provide accurate timelines of the history of Earth.
Holmes' persistence finally began to pay off in , when the speakers at the yearly meeting of the British Association for the Advancement of Science came to a rough consensus that Earth was a few billion years old, and that radiometric dating was credible.
Holmes published The Age of the Earth, an Introduction to Geological Ideas in in which he presented a range of 1. No great push to embrace radiometric dating followed, however, and the die-hards in the geological community stubbornly resisted. They had never cared for attempts by physicists to intrude in their domain, and had successfully ignored them so far. Holmes, being one of the few people on Earth who was trained in radiometric dating techniques, was a committee member, and in fact wrote most of the final report.
Thus, Arthur Holmes' report concluded that radioactive dating was the only reliable means of pinning down geological time scales. Questions of bias were deflected by the great and exacting detail of the report. It described the methods used, the care with which measurements were made, and their error bars and limitations.
Radiometric dating continues to be the predominant way scientists date geologic timescales. Techniques for radioactive dating have been tested and fine-tuned on an ongoing basis since the s. Forty or so different dating techniques have been utilized to date, working on a wide variety of materials.
Dates for the same sample using these different techniques are in very close agreement on the age of the material. Possible contamination problems do exist, but they have been studied and dealt with by careful investigation, leading to sample preparation procedures being minimized to limit the chance of contamination.
An age of 4. The quoted age of Earth is derived, in part, from the Canyon Diablo meteorite for several important reasons and is built upon a modern understanding of cosmochemistry built up over decades of research. Most geological samples from Earth are unable to give a direct date of the formation of Earth from the solar nebula because Earth has undergone differentiation into the core, mantle, and crust, and this has then undergone a long history of mixing and unmixing of these sample reservoirs by plate tectonics , weathering and hydrothermal circulation.
All of these processes may adversely affect isotopic dating mechanisms because the sample cannot always be assumed to have remained as a closed system, by which it is meant that either the parent or daughter nuclide a species of atom characterised by the number of neutrons and protons an atom contains or an intermediate daughter nuclide may have been partially removed from the sample, which will skew the resulting isotopic date.
To mitigate this effect it is usual to date several minerals in the same sample, to provide an isochron. Alternatively, more than one dating system may be used on a sample to check the date. Some meteorites are furthermore considered to represent the primitive material from which the accreting solar disk was formed.
Nevertheless, ancient Archaean lead ores of galena have been used to date the formation of Earth as these represent the earliest formed lead-only minerals on the planet and record the earliest homogeneous lead-lead isotope systems on the planet.
These have returned age dates of 4. Statistics for several meteorites that have undergone isochron dating are as follows: The Canyon Diablo meteorite was used because it is both large and representative of a particularly rare type of meteorite that contains sulfide minerals particularly troilite , FeS , metallic nickel - iron alloys, plus silicate minerals.
This is important because the presence of the three mineral phases allows investigation of isotopic dates using samples that provide a great separation in concentrations between parent and daughter nuclides. This is particularly true of uranium and lead. Lead is strongly chalcophilic and is found in the sulfide at a much greater concentration than in the silicate, versus uranium.
Because of this segregation in the parent and daughter nuclides during the formation of the meteorite, this allowed a much more precise date of the formation of the solar disk and hence the planets than ever before.
The age determined from the Canyon Diablo meteorite has been confirmed by hundreds of other age determinations, from both terrestrial samples and other meteorites. This is interpreted as the duration of formation of the solar nebula and its collapse into the solar disk to form the Sun and the planets. This 50 million year time span allows for accretion of the planets from the original solar dust and meteorites.
The moon, as another extraterrestrial body that has not undergone plate tectonics and that has no atmosphere, provides quite precise age dates from the samples returned from the Apollo missions. Rocks returned from the Moon have been dated at a maximum of 4. Martian meteorites that have landed upon Earth have also been dated to around 4.
Lunar samples, since they have not been disturbed by weathering, plate tectonics or material moved by organisms, can also provide dating by direct electron microscope examination of cosmic ray tracks. The accumulation of dislocations generated by high energy cosmic ray particle impacts provides another confirmation of the isotopic dates.
Cosmic ray dating is only useful on material that has not been melted, since melting erases the crystalline structure of the material, and wipes away the tracks left by the particles.
Altogether, the concordance of age dates of both the earliest terrestrial lead reservoirs and all other reservoirs within the Solar System found to date are used to support the fact that Earth and the rest of the Solar System formed at around 4. From Wikipedia, the free encyclopedia.
Scientific dating of the age of the Earth. Human timeline and Nature timeline. This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. October Learn how and when to remove this template message. Age of the Solar System. Archived from the original on 23 December Special Publications, Geological Society of London. Speculations about the age of the earth and primitive mantle characteristics".
Earth and Planetary Science Letters. Archived from the original on Meteorites and the Age of the Solar System". The Age of Everything. University of Chicago Press. The disintegration products of uranium". American Journal of Science. For the abstract, see: The Outcrop, Geology Alumni Newsletter. Archived PDF from the original on Australian Journal of Earth Sciences. Oxford University Press US. Archived from the original on 24 November Geological Society, London, Special Publications.
A missed opportunity in geodynamics". Lord Kelvin and the Age of the Earth. Journal of Geophysical Research. Meet your microbial mom". Archived from the original on June 29, The Rb-Sr method is based on the radioactivity of 87 Rb, which undergoes simple beta decay to 87 Sr with a half-life of Rubidium is a major constituent of very few minerals, but the chemistry of rubidium is similar to that of potassium and sodium, both of which do form many common minerals, and so rubidium occurs as a trace element in most rocks.
Because of the very long half-life of 87 Rb, Rb-Sr dating is used mostly on rocks older than about 50 to million years. This method is very useful on rocks with complex histories because the daughter product, strontium, does not escape from minerals nearly so easily as does argon. As a result, a sample can obey the closed-system requirements for Rb-Sr dating over a wider range of geologic conditions than can a sample for K-Ar dating.
Unlike argon, which escapes easily and entirely from most molten rocks, strontium is present as a trace element in most minerals when they form. For this reason, simple Rb-Sr ages can be calculated only for those minerals that are high in rubidium and contain a negligible amount of initial strontium.
In such minerals, the calculated age is insensitive to the initial strontium amount and composition. For most rocks, however, initial strontium is present in significant amounts, so dating is done by the isochron method, which completely eliminates the problem of initial strontium. In the Rb-Sr isochron method, several three or more minerals from the same rock, or several cogenetic rocks with different rubidium and strontium contents, are analyzed and the data plotted on an isochron diagram Figure 2.
The 87 Rb and 87 Sr contents are normalized to the amount of 86 Sr, which is not a radiogenic daughter product. The intercept of this line with the ordinate represents the isotopic composition of the initial strontium. From then on, as each atom of 87 Rb decays to 87 Sr, the points will follow the paths 3 shown by the arrows. The intercept of the line on the ordinate gives the isotopic composition of the initial strontium present when the rock formed.
Note that the intercepts of lines a-b-c and a'-b'-c' are identical, so the initial strontium isotopic composition can be determined from this intercept regardless of the age of the rock. Note that the Rb-Sr isochron method requires no knowledge or assumptions about either the isotopic composition or the amount of the initial daughter isotope — in fact, these are learned from the method. The rocks or minerals must have remained systems closed to rubidium and strontium since their formation; if this condition is not true, then the data will not plot on an isochron.
Also, if either the initial isotopic composition of strontium is not uniform or the samples analyzed are not cogenetic, then the data will not fall on a straight line. As the reader can easily see, the Rb-Sr isochron method is elegantly self-checking.
If the requirements of the method have been violated, the data clearly show it. An example of a Rb-Sr isochron is shown in Figure 3 , which includes analyses of five separate phases from the meteorite Juvinas 3.
The data form an isochron indicating an age for Juvinas of 4. This meteorite has also been dated by the Sm-Nd isochron method, which works like the Rb-Sr isochron method, at 4. The U-Pb method relies on the decays of U and U. These two parent isotopes undergo series decay involving several intermediate radioactive daughter isotopes before the stable daughter product, lead Table 1 , is reached.
If necessary, a correction can be made for the initial lead in these systems using Pb as an index. If these three age calculations agree, then the age represents the true age of the rock. Lead, however, is a volatile element, and so lead loss is commonly a problem. As a result, simple U-Pb ages are often discordant. The U-Pb concordia-discordia method circumvents the problem of lead loss in discordant systems and provides an internal check on reliability. This method involves the U and U decays and is used in such minerals as zircon, a common accessory mineral in igneous rocks, that contains uranium but no or negligible initial lead.
This latter requirement can be checked, if necessary, by checking for the presence of Pb, which would indicate the presence and amount of initial lead. The location of the point on concordia depends only on the age of the sample.
If at some later date say, 2. At any time after the episodic lead loss say, 1. This chord is called discordia. If we now consider what would happen to several different samples, say different zircons, from the same rock, each of which lost differing amounts of lead during the episode, we find that at any time after the lead loss, say today, all of the points for these samples will lie on discordia.
The upper intercept of discordia with concordia gives the original age of the rock, or 3. There are several hypotheses for the interpretation of the lower intercept, but the most common interpretation is that it indicates the age of the event that caused the lead loss, or 1 billion years in Figure 4. Note that this method is not only self-checking, but it also works on open systems. What about uranium loss?
Uranium is so refractory that its loss does not seem to be a problem. If uranium were lost, however, the concordia-discordia plot would indicate that also. The U-Pb concordia-discordia method is one of the most powerful and reliable dating methods available.
It is especially resistant to heating and metamorphic events and thus is extremely useful in rocks with complex histories. Quite often this method is used in conjunction with the K-Ar and the Rb-Sr isochron methods to unravel the history of metamorphic rocks, because each of these methods responds differently to metamorphism and heating.
For example, the U-Pb discordia age might give the age of initial formation of the rock, whereas the K-Ar method, which is especially sensitive to argon loss by heating, might give the age of the latest heating event.
An example of a U-Pb discordia age is shown in Figure 5. This example shows an age of 3. The K-Ar ages on rocks and minerals from this area in southwestern Minnesota also record this 1. This argument is specious and akin to concluding that all wristwatches do not work because you happen to find one that does not keep accurate time.
Like any complex procedure, radiometric dating does not work all the time under all circumstances. Each technique works only under a particular set of geologic conditions and occasionally a method is inadvertently misapplied.
There are, to be sure, inconsistencies, errors, and results that are poorly understood, but these are very few in comparison with the vast body of consistent and sensible results that clearly indicate that the methods do work and that the results, properly applied and carefully evaluated, can be trusted. A few examples will demonstrate that their criticisms are without merit. The creationist author J. He claims that these examples cast serious doubt on the validity of radiometric dating.
The use of radiometric dating in Geology involves a very selective acceptance of data. Discrepant dates, attributed to open systems, may instead be evidence against the validity of radiometric dating. However, close examination of his examples, a few of which are listed in Table 2 , shows that he misrepresents both the data and their meaning. The two ages from gulf coast localities Table 2 are from a report by Evernden and others These are K-Ar data obtained on glauconite, a potassium-bearing clay mineral that forms in some marine sediment.
Woodmorappe fails to mention, however, that these data were obtained as part of a controlled experiment to test, on samples of known age, the applicability of the K-Ar method to glauconite and to illite, another clay mineral. He also neglects to mention that most of the 89 K-Ar ages reported in their study agree very well with the expected ages. Evernden and others 43 found that these clay minerals are extremely susceptible to argon loss when heated even slightly, such as occurs when sedimentary rocks are deeply buried.
As a result, glauconite is used for dating only with extreme caution. The ages from the Coast Range batholith in Alaska Table 2 are referenced by Woodmorappe to a report by Lanphere and others Whereas Lanphere and his colleagues referred to these two K-Ar ages of and million years, the ages are actually from another report and were obtained from samples collected at two localities in Canada, not Alaska.
There is nothing wrong with these ages; they are consistent with the known geologic relations and represent the crystallization ages of the Canadian samples. The Liberian example Table 2 is from a report by Dalrymple and others These authors studied dikes of basalt that intruded Precambrian crystalline basement rocks and Mesozoic sedimentary rocks in western Liberia.
The dikes cutting the Precambrian basement gave K-Ar ages ranging from to million years Woodmorappe erroneously lists this higher age as million years , whereas those cutting the Mesozoic sedimentary rocks gave K-Ar ages of from to million years. Woodmorappe does not mention that the experiments in this study were designed such that the anomalous results were evident, the cause of the anomalous results was discovered, and the crystallization ages of the Liberian dikes were unambiguously determined.
The Liberian study is, in fact, an excellent example of how geochronologists design experiments so that the results can be checked and verified. The final example listed in Table 2 is a supposed 34 billion-year Rb-Sr isochron age on diabase of the Pahrump Group from Panamint Valley, California, and is referenced to a book by Faure and Powell Again, Woodmorappe badly misrepresents the facts. The data do not fall on any straight line and do not, therefore, form an isochron.
The original data are from a report by Wasserburg and others , who plotted the data as shown but did not draw a billion-year isochron on the diagram. As discussed above, one feature of the Rb-Sr isochron diagram is that, to a great extent, it is self-diagnostic. The scatter of the data in Figure 6 shows clearly that the sample has been an open system to 87 Sr and perhaps to other isotopes as well and that no meaningful Rb-Sr age can be calculated from these data.
This conclusion was clearly stated by both Wasserburg and others and by Faure and Powell There are two things wrong with this argument. First, the lead data that Kofahl and Segraves 77 cite, which come from a report by Oversby , are common lead measurements done primarily to obtain information on the genesis of the Reunion lavas and secondarily to estimate when the parent magma the lava was derived from was separated from primitive mantle material.
These data cannot be used to calculate the age of the lava flows and no knowledgeable scientist would attempt to do so. We can only speculate on where Kofahl and Segraves obtained their numbers. The data Morris 92 refers to were published by Evernden and others 44 , but include samples from different islands that formed at different times! The age of 3. The approximate age of , years was the mean of the results from four samples from the Island of Hawaii, which is much younger than Kauai.
Many of the rocks seem to have inherited Ar 40 from the magma from which the rocks were derived. Volcanic rocks erupted into the ocean definitely inherit Ar 40 and helium and thus when these are dated by the K 40 -Ar 40 clock, old ages are obtained for very recent flows. For example, lavas taken from the ocean bottom off the island [sic] of Hawaii on a submarine extension of the east rift zone of Kilauea volcano gave an age of 22 million years, but the actual flow happened less than years ago.
Slusher and Morris 92 advanced this argument in an attempt to show that the K-Ar method is unreliable, but the argument is a red herring. Two studies independently discovered that the glassy margins of submarine pillow basalts, so named because lava extruded under water forms globular shapes resembling pillows, trap 40 Ar dissolved in the melt before it can escape 36 , This effect is most serious in the rims of the pillows and increases in severity with water depth.
The excess 40 Ar content approaches zero toward pillow interiors, which cool more slowly and allow the 40 Ar to escape, and in water depths of less than about meters because of the lessening of hydrostatic pressure.
The purpose of these two studies was to determine, in a controlled experiment with samples of known age, the suitability of submarine pillow basalts for dating, because it was suspected that such samples might be unreliable.
Such studies are not unusual because each different type of mineral and rock has to be tested carefully before it can be used for any radiometric dating technique. In the case of the submarine pillow basalts, the results clearly indicated that these rocks are unsuitable for dating, and so they are not generally used for this purpose except in special circumstances and unless there is some independent way of verifying the results.
The citation for this statement is to a report by Turner Turner, however, made no such comment about excess argon in lunar rocks, and there are no data in his report on which such a conclusion could be based. The statement by Rofahl and Segraves 77 is simply unjustifiable. Volcanic rocks produced by lava flows which occurred in Hawaii in the years were dated by the potassium-argon method. Excess argon produced apparent ages ranging from million to 2.
Similar modern rocks formed in near Hualalai, Hawaii, were found to give potassium-argon ages ranging from million years to 3 billion years. Kofahl and Segraves 77 and Morris 92 cite a study by Funkhouser and Naughton 51 on xenolithic inclusions in the flow from Hualalai Volcano on the Island of Hawaii. The flow is unusual because it carries very abundant inclusions of rocks foreign to the lava.
These inclusions, called xenoliths meaning foreign rocks , consist primarily of olivine, a pale-green iron-magnesium silicate mineral. They come from deep within the mantle and were carried upward to the surface by the lava.
In the field, they look like large raisins in a pudding and even occur in beds piled one on top of the other, glued together by the lava. The study by Funkhouser and Naughton 51 was on the xenoliths, not on the lava. The xenoliths, which vary in composition and range in size from single mineral grains to rocks as big as basketballs, do, indeed, carry excess argon in large amounts. Quite simply, xenoliths are one of the types of rocks that cannot be dated by the K-Ar technique. Funkhouser and Naughton were able to determine that the excess gas resides primarily in fluid bubbles in the minerals of the xenoliths, where it cannot escape upon reaching the surface.
Studies such as the one by Funkhouser and Naughton are routinely done to ascertain which materials are suitable for dating and which are not, and to determine the cause of sometimes strange results.
They are part of a continuing effort to learn. Two extensive K-Ar studies on historical lava flows from around the world 31 , 79 showed that excess argon is not a serious problem for dating lava flows. In nearly every case, the measured K-Ar age was zero, as expected if excess argon is uncommon. An exception is the lava from the Hualalai flow, which is so badly contaminated by the xenoliths that it is impossible to obtain a completely inclusion-free sample. There is really no valid way of determining what the initial amounts of Sr 87 in rocks were.
As discussed above in the section on Rb-Sr dating the simplest form of Rb-Sr dating i. Such samples are rare, and so nearly all modern Rb-Sr dating is done by the isochron method. The beauty of the Rb-Sr isochron method is that knowledge of the initial Sr isotopic composition is not necessary — it is one of the results obtained.
A second advantage of the isochron method is that it contains internal checks on reliability. Look again at the isochron for the meteorite Juvinas Figure 3. The data are straightforward albeit technically complex measurements that fall on a straight line, indicating that the meteorite has obeyed the closed-system requirement. The decay constants used in the calculations were the same as those in use throughout the world in The age of 4. There is far too much Ar 40 in the earth for more than a small fraction of it to have been formed by radioactive decay of K This is true even if the earth were really 4.
In the atmosphere of the earth, Ar 40 constitutes This is around times the amount that would be generated by radioactive decay over the hypothetical 4. Certainly this is not produced by an influx from outer space. Thus it would seem that a large amount of Ar 40 was present in the beginning. Since geochronologists assume that errors due to presence of initial Ar 40 are small, their results are highly questionable. This statement contains several serious errors.
First, there is not more 40 Ar in the atmosphere than can be accounted for by radioactive decay of 40 K over 4. An amount of 40 Ar equivalent to all the 40 Ar now in the atmosphere could be generated in 4.
Current estimates of the composition of the Earth indicate that the crust contains about 1. The 40 Ar content of the atmosphere is well known and is 6. Thus, the Earth and the atmosphere now contain about equal amounts of 40 Ar, and the total could be generated if the Earth contained only ppm potassium and released half of its 40 Ar to the atmosphere.
Second, there have been sufficient tests to show that during their formation in the crust, igneous and metamorphic rocks nearly always release their entrapped 40 Ar, thus resetting the K-Ar clock.
In addition, scientists typically design their experiments so that anomalous results, such as might be caused by the rare case of initial 40 Ar, are readily apparent. The study of the Liberian diabase dikes, discussed above, is a good example of this practice. First, if it is assumed that there is a uniform distribution of Sr 87 in the rock, then it is assumed that there is also a uniform distribution of Rb It only requires that the Sr isotopic composition , i.
Even though the various minerals will incorporate different amounts of Sr as they cool and form, the Sr isotopic composition will be the same because natural processes do not significantly fractionate isotopes with so little mass difference as 87 Sr and 86 Sr. Second, Slusher has confused isotopes and elements. Rb and Sr are quite different elements and are incorporated into the various minerals in varying proportions according to the composition and structure of the minerals.
There is no way to correct for this natural isotopic variation since there is no way to determine it. This renders the Rb 87 -Sr 87 series useless as a clock. Slusher is wrong again. He has used an invalid analogy and come to an erroneous conclusion. Arndts and Overn 8 and Kramer and others 78 claim that Rb-Sr isochrons are the result of mixing, rather than of decay of 87 Rb over long periods:.
It is clear that mixing of pre-existent materials will yield a linear array of isotopic ratios. We need not assume that the isotopes, assumed to be daughter isotopes, were in fact produced in the rock by radioactive decay. Thus the assumption of immense ages has not been proven.
The straight lines, which seem to make radiometric dating meaningful, are easily assumed to be the result of simple mixing. This preliminary study of the recent evolutionary literature would suggest that there are many published Rb-Sr isochrons with allegedly measured ages of hundreds of millions of years which easily meet the criteria for mixing, and are therefore more cogently indicative of recent origin.
Kramer and others 78 and Arndts and Overn 8 have come to an incorrect conclusion because they have ignored several important facts about the geochemistry of Rb-Sr systems and the systematics of isochrons. First, the chemical properties of rubidium and strontium are quite different, and thus their behavior in minerals is dissimilar.
Both are trace elements and rarely form minerals of their own. It is chemically similar to potassium and tends to substitute for that element in minerals in which potassium is a major constituent, such as potassium feldspar and the micas muscovite and biotite.
It commonly substitutes for calcium in calcium minerals, such as the plagioclase feldspars. The chemical properties of rubidium and strontium are so dissimilar that minerals which readily accept rubidium into their crystal structure tend to exclude strontium and vice versa. Thus, rubidium and strontium in minerals tend to be inversely correlated; minerals high in rubidium are generally low in strontium and vice versa.
This relation, however, is a natural consequence of the chemical behavior of rubidium and strontium in minerals and of the decay of 87 Rb to 87 Sr over time, and has nothing to do with mixing.
Second, mixing is a mechanical process that is physically possible only in those rock systems where two or more components with different chemical and isotopic compositions are available for mixing.
Examples include the mingling of waters from two streams, the mixing of sediment from two different source rocks, and the contamination of lava from the mantle by interactions with the crustal rocks through which it travels to the surface. Mixing in such systems has been found 49 , 70 , but the Rb-Sr method is rarely used on these systems. The Rb-Sr isochron method is most commonly used on igneous rocks, which form by cooling from a liquid.
Mineral composition and the sequence of mineral formation are governed by chemical laws and do not involve mixing. In addition, a rock melt does not contain isotopic end members that can be mechanically mixed in different proportions into the various minerals as they form, nor could such end members be preserved if they were injected into a melt.
Fourth, if isochrons were the result of mixing, approximately half of them should have negative slopes. In fact, negative slopes are exceedingly rare and are confined to those types of systems, mentioned above, in which mechanical mixing is possible and evident.