It's a rather interesting question. I am not pretty sure, but you could try to take a look at porous materials, e.g. zeolites or MOFs. 

From minerals substances SiO2 has one of the largest volume growth in the transformation of crystalline in amorphous. Density of beta-quartz is 2.65 g/cm3 and of quartz glass  2.2 g/cm3.

With calcium carbonates, the formation of a crystalline phase from an amorphous state is normally guided by a dehydration step. Ikaite (CaCO3*6H2O, stable at low T) is more hydrated than its predecessor (presumably~CaCO3*1H2O), therefore the density decreases slightly upon crystallization. It must be said though that this is most likely a dissolution-reprecipitation mechanism, and therefore both compounds have little in common.

Dear Eric,

I'm sorry that I will not go into any particular examples. It seems crystalline state is the most efficient way (on certain structural level) to fill the space. It usually can be described in terms of "long-range order" or "close packing"... That's why the density of crystalline phase will be maximal. Disordered or "short-range" ordered amorphous state can't reach it because it isn't "packed" in optimal way (like the clothes in my closet :)).

Concerning real materials, hypothetically, there is a case when the "density of amorphous state" may be higher. It is when you have an impurity which is "heavier" (I mean with greater Z number) than your amorphous matrix. But just hypothetically.

Thanks for your question!

Andrei

Dear Erik Carton,

I have got the impression that your question lacks clear and sharp distinction of the cases to be envisaged.

1. The case of a compound  of given composition that can be present as a crystal and in an amorphous state at the same temperature

2. The case of a glass as an undercooled liquid (e.g. window plane glasses) does not have its counterpart in a crystalline form.

3. Many polymers are semi-crystalline, so the comparison between crystalline and amorphous in these polymers depends on the respective volume fractions.

4. At the melting temperature of e.g. a pure metal, solid and liquid are in equilibrium. Solidification only proceeds at a certain undercooling as compared to the equilibrium melting/solidification temperature. So, it seems to me very difficult to expect that at the equilibrium temperature solid would sink in the liquid of the same composition.

5.I think that the free volume should be taken into account when considering the relative density.

6. I note that only one example of an amorphous compound having a higher relative density as compared to its crystalline counterpart is sufficient for a negative answer to your question.