Which of the three-dimensional primitive lattices has a unit cell where none of the internal angles is 90? (a) Orthorhombic, (b) hexagonal, (c) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

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Hello everyone. So in this video we want to identify the lattice that has an internal angle of 120° and also has equal on two dimensions. So let's go ahead and draw the lattices of each. So first of all I want to go ahead and kind of draw a key because we have so many angles and sides. I kind of want to just label them as being alpha, beta, gamma or a B and C. So let's say we have this angle here. I'll refer this to see this as A and this as B. And this angle to the left will be my beta. This angle to the right can be alpha and the bottom can be gamma. Okay, so this is going to be what I'll be referring to. So first number one we have to go. No, so that's just going to be kind of like a rectangle. Go ahead and connect those. Looks a little bit like this. So we can see that A. Is equal to B. But it's not equal to C. As for our angles are alpha is equal to beta, Is equal to gamma, which is about 90°. Next we have this Rumba hydro so we have like slanted rectangle. Go ahead and connect those. So we can see from this lattice we have that the A. Is equal to R. B. Which is equal to R. C. Same thing for the angles we have alpha equal to beta equal into gamma and that's not equal to 90 degrees. Next we have our chai clinic. So again it's kind of similar to the ramble hydro except it's a little bit longer. Let's go ahead actually. Pre draw that. So again we have slightly slanted rectangle shape. Go ahead and connect those. Although it's poorly drawn we can see that this a side is equal to R. B. Are not equal to R. B. Actually and not equal to R. C. So they're all different as for angles they're also all very different so they're not equal to each other at all. Our next one is going to be our Ortho ram bic. So we basically have rectangles and we connect them. Alright so for this one again all the sides are different so A is not equal to B. Which is not equal to C. As for the angles are alpha is equal to our beta equal to our gap and they're all equal to 90 degrees. Then we have are hexagonal. So that's similar to our tetra connell except it's a little bit more condensed again connecting here. Alright so from this we can see that the A. Is equal to B. But it's not equal to R. C. And the angles we have alpha equaling two beta which is at degrees but at gamma we have 120 degrees. Our last one will be our mono clinic and that one looks something like this here. Alright so from this you can say that our alpha is not equal to your beta which is not equal to let's go ahead actually redraw this a little bit. It might be a little bit confusing. So again, to sort of slanted brain tingles connecting that, They're all right, that's a lot better. Okay, so for size wise we have our we have eight Not going to be which is not going to see. And then for the angles we have alpha equaling two gamma, which is at 90°, but for our beta that is not at 90°. And so the only lattice that matches our description, our requirements of one having a internal angle of 120 as you can see will be this right here, which is going to be our hexagonal. And that means that Option five is going to be our final answer for this problem. Thank you all so much for watching.

Which of the three-dimensional primitive lattices has a unit cell where none of the internal angles is 90? (a) Orthorhombic, (b) hexagonal, (c) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

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Key Concepts

Primitive Lattices

Unit Cell Angles

Crystal Systems