I've seen in some sources that the Earth's inner temperature is because of nuclear decay, namely uranium and such.
I find it hard to believe since volcanoes aren't know for spewing long half-life materials. I was wondering if this is true or not.
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Results published (the link is the the arXiv preprint, but the paper also appeared in Phys. Rev. B) by the Borexino collaboration in 2010 are highly consistent (at the three sigma level) with most of the geothermal power of the Earth being due to radiological decay. These results extend an earlier, lower precision result by the KamLAND collaboration (the KamLAND geo-neutrino paper appeared in Nature).
Breaking news as of August 2011: KamLAND has published a new paper on Geo-neutrinos in Nature Geoscience, where they compute the radiothermal power from U-238 and Th-232 at 20 [+8.8/-8.6] TW (out of about 44 TW total). Heating due to K-40 is unmeasured.
These measurements also put strict upper limits on the power of a theorized natural nuclear reactor deeper down, at the Earth's core (the data are now consistent with zero reactor power).
Both Borexino and KamLAND are large anti-neutrino detectors and are directly sensitive to the anti-neutrino emissions of radioactive beta decays such as those found in the uranium and thorium chains (but not potassium-40 due to limitation of the detector technology). From this data we can reconstruct the overall radioactive decay activity in the deep Earth, and compute the total power represented.
This is compared to estimates of the geothermal power obtained from temperature gradient measurements in deep bores.
To address the OP's concern about "volcanoes [...] spewing long half-life materials", the short answer is "they do", just not concentrations that are a concern.
You could, for instance look at the Wikipedia article on radiometric dating for survey of decay based dating. You'll note that many of the dating methods work on igneous rocks. Particularly potassium-argon dating of volcanic deposits has been used for dating hominid fossils and impressions in the Olduvai Gorge (Wikipedia credits Tattersall (1995), ISBN 0-19-506101-2).
Disclaimer: I worked on KamLAND for three years, but I am not named as an author on either of the papers cited herein.
I want to answer the second part of the question:
I find it hard to believe since volcanoes aren't know for spewing long half-life materials.
The calculations of heat generated by radioactive decay don't require unusual concentrations of radioactive material. The sheer quantity of rock that makes up the earth means that even small quantities decaying slowly can generate significant amounts of heat.
For example, in Donald L. Turcotte; Gerald Schubert (25 March 2002). Geodynamics. Cambridge University Press. ISBN 978-0-521-66624-4. Chapter 4-5 he uses the following table:
His heat calculations (correctly) use the concentrations found in undepleted mantle rocks, not the higher concentrations typical of crustal granite or shale. (Even the tholeiitic basalts of the ocean ridge volcanics are around 2x richer in long half-life elements than typical mantle estimates.)
Keep in mind that this heat doesn't have anywhere to go – it just stays trapped within the earth until temperatures rise sufficiently to generate the convection that drives plate tectonics. Even then, heat transfer to the surface is limited.
It's hard to get direct measurements of what's below the crust in the Earth's mantle, and scientists seem to be divided on mantle radioactivity's contribution to Earth's "internal heat budget".
Proponents of radiation-originated heat point to studies that take advantage of the relation between decay of Uranium and Thorium and the appearance of neutrinos in neutrino-measurement pools:
...the relative contributions from residual primordial heat and radiogenic decay remain uncertain. However, radiogenic decay can be estimated from the flux of geoneutrinos [that we measured in Japan and Italy]...
In a passage written to categorize the results from the scintillator-pool neutrino measurements, S.T. Dye of U of Hawaii writes,
The number of geochemical assumptions and predictions indicates the level at which geochemistry informs geo-neutrino results.
Dye makes three or four categories of results, ranging from most independent to the least. The first result from the geoneutrino measurements is precisely that: how many geoneutrinos are passing through the scintillator. The second is an estimate of the contributions to that rate of specific chemicals (uranium and thorium) which is "potentially biased". The third category is applying this to the earth's mantle, which has the limitation that "mantle fluxes do not uniquely define mesospheric distributions of uranium and thorium". And the last (and least independent) result is "A further result, which depends on the mantle distributions, the predicted lithospheric distributions, and the assumed chondritic spectral shape, assesses global radiogenic power from uranium and thorium." Dye goes on to write:
...more sophisticated analyses are needed for planning observations that independently confirm the geochemical prediction of non-uniform and differential uranium and thorium mass fractions.
Regarding the current knowledge of radiogenic heating, Šrámek, McDonough, Kite, Lekić, Sephen T. Dye, and Shijie Zhong write that "estimates of mantle radiogenic heat production vary by a factor of more than 20." They don't think this is always going to be the case, though, recommending that we place several new geoneutrino detectors in the ocean to get a better picture.
There seems to be wide agreement that radioactivity is a small contributor to heat down at the core. As these researchers wrote in 2003:
[The size of the radiogenic potassium heat source of our observations and analysis] corresponds to a present-day heat production of up to 0.2 TW, a small fraction compared to the estimated heat flux across the core-mantle boundary (∼ 6–15 TW, e.g. ).
5. J.C. Lassiter Constraints on the coupled thermal evolution of the Earth's core and mantle, the age of the inner core, and the origin of the 186Os/188Os [Osmium 186 / Osmium 188] “core signal” in plume-derived lavas Earth Planet. Sci. Lett., 250 (2006), pp. 306-317
When calculations are done, the Earth's temperature seems hotter than current mechanisms can be explained. One theory for the high temperature is that the Earth has gravitationally captured dark matter. This is a main theory for what is at the core of the gas giants.
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