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I got some sand out of a river bed and moved a magnet through it. Some of the particles I picked up were definitely small crystals of magnetite, but a lot of others were small, round brown rocks that did not look like magnetite at all.

I thought the only thing commonly found in country rock that was magnetic was magnetite. Does this mean the small brown rocks have magnetite crystals somewhere inside of them?

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Broadly speaking you are correct on both counts: magnetite is the most magnetic mineral, and it can oxidize, first to the matastable phase maghemite, and ultimately to the fully oxidized haemetite. This process can yield grains with a zoned appearance, both in hand specimen and more clearly under the microscope, with brown on the outside and residual black magnetite in the core. Maghemite, as the name suggests, is also magnetic. There are several other moderately to strongly magnetic minerals, closely related to magnetite, and with a similar molecular structure. For example ilmenite, a very common mineral that sometimes occurs with magnetite, is the double oxide of iron and titanium. Tantalite / columbite, or manganotantalite $(Fe, Mn)(Ta, Nb)_2O_6$ is not as magnetic as magnetite but may still be magnetic enough to swing a compass needle. It looks identical to magnetite, but can have a slightly purplish brown sheen. Similarly, chromite, the double oxide of chromium and iron, can be very magnetic, but I don't recall ever seeing anything other than black chromite. Franklinite, which is reddish purple, is another mineral of the magnetite group, and is a complex oxide of zinc,iron and manganese. There are several other weakly magnetic minerals of the magnetite group, most of which are quite rare. As a rough rule of thumb, the more oxidized or the more iron is replaced with some other metal, the less magnetic it is.

PS, if you ever find metallic looking sand that is magnetic, stake a claim! Platinum is also weakly magnetic.

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