Crystal Structure vector question

[Caution]

Homework Statement

Identify the [221] vector?

A
B
C
D
E

Homework Equations

The Attempt at a Solution

I reduce [221] to [1 1 0.5].BUT after getting the wrong answer, the feed back is as follows

"Feedback: [221]: has a 0.5 component along the x-axis; a 0.5 component along the y-axis and a 1 component along the z-axis.

Why is it[0.5 0.5 1]? And even so i can't seem to find the answer from the choices.. Pls help (I jus started this module.)

 

[collinsmark]

Hello Caution,

Welcome to Physics Forums!

Homework Statement

Identify the [221] vector?

A
B
C
D
E

Homework Equations

The Attempt at a Solution

I reduce [221] to [1 1 0.5].BUT after getting the wrong answer, the feed back is as follows

"Feedback: [221]: has a 0.5 component along the x-axis; a 0.5 component along the y-axis and a 1 component along the z-axis.

Why is it[0.5 0.5 1]? And even so i can't seem to find the answer from the choices.. Pls help (I jus started this module.)

The key idea is that you are really looking for the orientation of a plane. The plane in which you are looking for passes through three points. Those points are
(2,0,0)
(0,2,0)
(0,0,1)

In other words, the plane crosses the x-axis at 2, the y-axis at 2, and the z-axis at 1.

Now find a vector that is perpendicular (i.e., "normal") to the surface of that plane, and put its tail at the origin.

These articles might help:
http://en.wikipedia.org/wiki/Crystal_structure
http://en.wikipedia.org/wiki/Miller_index

 

[Caution]

Hello Caution,

Welcome to Physics Forums!



The key idea is that you are really looking for the orientation of a plane. The plane in which you are looking for passes through three points. Those points are
(2,0,0)
(0,2,0)
(0,0,1)

In other words, the plane crosses the x-axis at 2, the y-axis at 2, and the z-axis at 1.

Now find a vector that is perpendicular (i.e., "normal") to the surface of that plane, and put its tail at the origin.

why is planes involved?The examples in my textbook which I am looking at,under the section "Crystallographic directions" just use a method of simply reducing the vectors to its simplest form and draw it accordingly. Theres a section next called "Crystallographic Planes" but this question doesn't seem to imply anything about plane??Anyway thank you:D

 

[Caution]

According to wikipedia,

"the related notation [hkℓ] denotes the direction:

h \mathbf{a}_1 + k \mathbf{a}_2 + \ell \mathbf{a}_3 .(

That is, it uses the direct lattice basis instead of the reciprocal lattice.


Im just confused why are the solution using the reciprocal meant for planes?? Went to Youtube for a lecture and it does exactly the same thing reflected in the textbook,e.g simply reduce the vector to the lowest integer.Anyone can explain to me why is this question different from the usual crystal directions's method?

 

[collinsmark]

It's just a convention.

Keep in mind, the information that is being expressed, is the orientation of a plane. But it doesn't refer to the plane's position. You can move the plane around with respect to the origin; as long as you don't tilt or rotate the plane it will have the same direct vector indexes, the same Miller indexes, and the same vector associated with it.

For example, consider the x-y plane (where z = 0). That has the same orientation if the plane were raised up to z = 1. The Miller index representation for that is (001).

That plane also has another type of vector representation. The vector is orthogonal to the plane. (This vector representation is more common representation of plane/surface orientation in physics, outside of material science). Using our example of the x-y plane, or the plane where z = 1, the vector representation of the plane's orientation is $\hat a_z$. As you can see, the Miller Index representation matches up with the more traditional vector representation.

--------------

Let's look more closely at how one gets obtains the Miller indexes.

Any plane crosses the x-axis, the y-axis and the z-axis. In the special cases where the plane is parallel to any of these axes, it can be said that the plane crosses the axis at infinity.

Let's start with the previous example where z = 1. In this example, the plane is parallel to both the x-axis and the y-axis. So it is said that the plane crosses these axes at infinity. But it crosses the z-axis at 1. So the direct lattice vector representation is $[\infty \ \infty \ 1]$.

So the Miller indexes are:

$$\left( \frac{1}{\infty} \ \ \frac{1}{\infty} \ \ \frac{1}{1} \right) = (001)$$
So you ask, why reduce the values to integers? It's just a convention (in both representations). [221] is really the same thing (certainly the same orientation) as [1 1 1/2]. Similarly, (1/2 1/2 1) is the same orientation as (112).

 

[Caution]

It's just a convention.

Keep in mind, the information that is being expressed, is the orientation of a plane.
The information is [221], not(221),so how does it got to do with planes?Also, I understand that [221] is perpendicular to (221) plane, which has the intercepts 0.5 0.5 1. BUT the answer in the mcq clearly gives a [0.5 0.5 1] VECTOR, not a [221] VECTOR.(0.5 0.5 1) should be intercepts of the plane.Notice how () is used to refer to planes and [ ] is used to refer to vectors.Thats why I am confused why this qns is the only exception out the many qns I do that have a weird answer.

 

[collinsmark]

,so how does it got to do with planes?

I just assumed it was about the orientation of a plane, given the context of the original post. Perhaps I was wrong. I just figured it was since the orientation of planes is a large part of the study of crystal structure. You should be able to figure out the context though by checking with corresponding section of your textbook/coursework.

If the problem statement is in fact dealing with the orientation of a plane, there are multiple ways of representing the same orientation. And it is possible to convert from one vector representation to another. That was what I was originally getting at.

On the other hand, if the given answer is an anomaly compared to answers of other, similar questions, then maybe there is a mistake in the textbook/coursework. Perhaps you might consider asking your instructor.