Granite Self-Heating Math

Introduction

I came across the following statement in an article about the self-heating of the granite in an article about how radioactivity heats the interior of the Earth.

Radioactivity is present not only in the mantle, but in the rocks of Earth's crust. For example, Marone explains, a 1-kilogram block of granite on the surface emanates a tiny but measurable amount of heat (about as much as a .000000001 Watt [1.0E-9 W or 1 nW] light bulb) through radioactive decay.

I will do a quick calculation to verify that I understand where this number originates.

Background

I will assume that all the thermal power generated in a 1 kg granite piece comes from the radioactive decay of uranium and thorium that make up a small part of many granite deposits. It turns out that I have had to deal with the fact that many substances are slightly radioactive throughout my career. For example, memory chips have long been recognized as vulnerable to soft errors from alpha particles originating from trace radioactive materials in electronic packaging. I have spent much time implementing error correcting codes on memory systems in order to deal with single bit upsets.

Analysis

These calculations are similar in principle to those done in this blog post. My approach is simple:

• Gather information on uranium and thorium concentrations in granite (see Appendix C)
• Gather information on how uranium and thorium decay (see Appendices A and B, respectively)
• Perform routine radioactive decay calculations, assuming that all the alpha particle energy eventually ends up as heat.

Setup

Figure 1 illustrates the setup I went through for the calculation.

Figure 1: Setup for the Analysis of Granite's Self-Heating.

Calculations

Figure 2 shows the actual calculations.

Figure 2: Granite Self-Heating Calculations.

Conclusion

I encountered a statement in an article on the web that says that ~1 nW of power is generated in a 1 kg block of granite. I compute that the value is 0.52 nW, which is about 1 nW. So I have confirmed the number.

Another byproduct of radioactive decay in granite is radon. However, that is a topic for another blog post.

Appendix A: Uranium Isotope Characteristics

Figure 3 is a table from the Wikipedia entry for uranium.

Figure 3: Uranium Isotopes (Wikipedia).

Appendix B: Thorium Isotope Characteristics

Figure 4 is a table from the Wikipedia entry for thorium.

Figure 4: Thorium Isotopes (Wikipedia).

Appendix C: Uranium and Thorium Concentrations in Granite

I obtained my uranium and thorium concentrations in granite from the Wikipedia. Here is the quote with the pertinent text hightlighted.

Some granites contain around 10 to 20 parts per million of uranium. By contrast, more mafic rocks such as tonalite, gabbro or diorite have 1 to 5 PPM uranium, and limestones and sedimentary rocks usually have equally low amounts. Many large granite plutons are the sources for palaeochannel-hosted or roll front uranium ore deposits, where the uranium washes into the sediments from the granite uplands and associated, often highly radioactive, pegmatites. Granite could be considered a potential natural radiological hazard as, for instance, villages located over granite may be susceptible to higher doses of radiation than other communities.[11] Cellars and basements sunk into soils over granite can become a trap for radon gas, which is formed by the decay of uranium.[12] Radon gas poses significant health concerns, and is the number two cause of lung cancer in the US behind smoking.[13]

Thorium occurs in all granites as well.[14] Conway granite has been noted for its relatively high thorium concentration of 56 (±6) PPM.[15]