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Given the information on swords here:

When you need to place a great deal of strength into a thin blade, steel remains the best all around choice. Titanium blades will be simultaneously weaker and softer than similar steel blades.

And the material (LDHEA) described here:

Summarized here from the Applications section of the Wikipedia article on Scandium, as:

An alloy called Al20Li20Mg10Sc20Ti30 has been shown to be as strong as titanium, light as aluminium, and hard as ceramic.

In combination with other materials, how could LDHEA be used to make a good shortsword?

Perhaps steel for the flat and LDHEA for the edge and point?

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As a HEMA practitioner and a physicist, I can tell you that different swords need different properties and it is unlikely one material will be the end-all for a sword material. Depending on the sword, you may actually want different parts of the blade to have different properties. Rapiers, for instance, are much less rigid than hand-and-a-half swords (or "longsword").

As I said earlier, different properties are desired for different parts of the blade. For instance, a high hardness on your edge is great, because you can deliver more cuts before it dulls. However, high hardness in your flat does not really help you, because it usually means brittleness, and your blade will come into contact with opponents' blades and potentially break or shatter, leaving you more-or-less defenseless.

Lighter blades are not always better. Sure, a lighter blade can ideally be moved faster, but the human body can only move so fast and benefits only so much from it. Cutting swords, however, need a certain degree of heft to them to enable their cuts. Trying to cut someone with a rapier, for instance, is usually laughable because it simply lacks the same momentum as a tulwar, katana, longsword, or any other sword. (Rapiers have a tendency to simply bounce off in such attempts to cut.) Also, some weight to your sword makes it harder to deflect, which is another problem rapiers classically encounter when attempting to cut.

It seems that the ideal blade, without regard to form, is made of a layering of both hard and soft metals, as seen in wootz or damascus steel. Many blacksmiths, when manipulating material properties for a sword, attempt to get a hard edge and soft flat. This material seems promising, but we need to know more about it.

How brittle is it? Dr. Carl Koch, the senior author for that paper, says "we think it’s tougher – less brittle – than ceramics." That bodes well for it as a sword material. At the very least, it would make a good material for the edge and points of swords. As noted earlier, the flexibility of this material may eliminate it for certain swords or recommend it for others.

A lighter material lets your sword be larger, which means you could (in theory) have swords that were formerly limited to anime and video games. You also run into the problem of simply larger size not being what you want. For instance, a two-handed sword is not used in the same way as a Messer.

In review, it seems like this material is promising, and could be good, but hardness and strength-to-weight are not the only properties which make a good sword material. Toughness (brittleness) is an important factor which we do not know.

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