So first thing we'll do is talk about a procedure that we're going to follow to go through this.

所以，我們首先要做的就是談談我們要遵循的程序。

So first thing we're going to do is we're going to pretend that the A1, A2, Z, three-axis parallel pipette within the hexagonal unit cell is actually X, Y, Z orthogonal.

是以，我們要做的第一件事就是假裝六邊形單元格中的 A1、A2、Z 三軸平行移液管實際上是 X、Y、Z 正交的。

We're going to sort of imagine that that's where our direction exists, and then we're going to proceed from there and determine the three-index notation for that vector.

我們可以想象一下，我們的方向就存在於這裡，然後從這裡出發，確定該向量的三索引符號。

Now we're going to call this sort of a temporary three-index system because we're not going to stop there.

現在，我們把它稱為臨時三索引系統，因為我們不會就此止步。

We could.

我們可以

In fact, a while ago, a few decades ago, people used to do that.

事實上，不久前，也就是幾十年前，人們經常這樣做。

But the convention nowadays is to go to four-axis system to avoid any confusion with cubic systems.

但現在的慣例是採用四軸系統，以避免與立方系統混淆。

So we'll take this temporary three-axis system then and then convert it to a four-axis system and then just enclose it.

是以，我們將使用這個臨時的三軸系統，然後將其轉換為四軸系統，並將其包圍起來。

So it's sort of that straightforward.

所以就是這麼簡單。

So let's go ahead and look at an example problem now.

現在，讓我們來看一個例題。

So this first example problem, we've got a vector that originates off on the left-hand side of the base of the hexagonal unit cell and then travels up.

是以，在第一個例題中，我們得到了一個向量，它從六邊形單元底面的左側出發，然後向上移動。

And so what we have to do, the first step, and this is really I think the hard step, is we have to be able to picture that direction within the A1, A2, Z three-axis system.

是以，我們必須做的第一步，也是我認為最難的一步，就是在 A1、A2、Z 三軸系統中描繪出這個方向。

Our conventional origin in the middle of the unit cell there, in the middle of the base of the unit cell, is usually a place that we're comfortable visualizing this.

我們傳統的原點位於組織、部門晶胞的中間，也就是組織、部門晶胞底部的中間，這通常是我們可以將其形象化的地方。

So what we could do is we could actually just translate that vector, translate it over, so that it now originates at our conventional origin, right in the very center of the basal plane.

是以，我們可以做的是，將矢量平移，使其位於我們的傳統原點，也就是基底面的正中心。

And if we do that, we'll see the direction that the vector travels up in the Z direction and directly over the negative Y, sorry, the negative A3 axis.

如果我們這樣做，就會看到矢量在 Z 軸方向上的移動方向，並直接越過負 Y 軸，對不起，負 A3 軸。

And it travels by one step over in the horizontal direction, that is the A lattice parameter, and then during that distance, that travel, it travels up a rise of the lattice parameter.

它在水準方向上移動了一個臺階，也就是 A 晶格參數，然後在這一距離、這一移動過程中，它的晶格參數上升了一個臺階。

So if we look at that vector in the three-axis system now, it would be the vector that originates at the conventional origin and then travels one over in the X, one over in the Y, and one up in the Z.

是以，如果我們現在在三軸系統中查看該矢量，它將是以傳統原點為起點，然後在 X 軸、Y 軸和 Z 軸各向上移動一圈的矢量。

So it's in fact one of those cube diagonals, and specifically, it's the 1, 1, 1 direction in a cubic system.

是以，它實際上是立方體對角線之一，具體來說，它是立方體系統中的 1、1、1 方向。

So U prime, V prime, and W prime are 1, 1, and 1.

所以 U 質數、V 質數和 W 質數分別是 1、1 和 1。

So then all we need to do is convert that to the three-axis system.

是以，我們只需將其轉換為三軸系統即可。

And actually, another thing we could do, instead of shifting the vector over, another thing we could do is we could just define the origin of the vector as it originally sat as our 1, 1, A2 axis originating from that origin.

實際上，我們還可以做另一件事，那就是將矢量的原點定義為我們的 1、1、A2 軸，從原點出發，而不是將矢量移過來。

It's entirely equivalent.

這是完全等同的。

I either define a new origin or translate the vector over.

我要麼定義一個新的原點，要麼將矢量平移過去。

They're the same thing.

它們是一回事。

OK.

好的。

So anyway, we've got U prime, V prime, W prime, 1, 1, and 1.

總之，我們已經有了 U 質數、V 質數、W 質數、1、1 和 1。

So therefore, U is equal to 1 third times 2 times 1 minus 1.

是以，U 等於 1/3 乘 2 乘 1 減 1。

V is, again, 1 third times 2 times 1 minus 1.

V 又是 1/3 乘以 2 再乘以 1 減 1。

And T, the third, this new index, is, remember, it's equal to the sum of U and V made negative.

而 T，即第三個新指數，記住，它等於 U 和 V 的負數之和。

It's not U prime plus V prime made negative.

這不是 U 質加 V 質的負數。

So just make sure you don't make a mistake with that.

是以，請務必不要在這一點上犯錯。

And so we'll have that U is 1 third.

是以，我們的 U 值為 1/3。

V is 1 third.

V 為 1/3。

T, then, is negative 2 thirds.

那麼，T 是負三分之二。

And W is just W prime, which is 1.

而 W 只是 W 的質數，也就是 1。

So we're almost done.

所以我們就快完成了。

We've just got the 3 in the denominator there, a little pesky little 3.

分母中只有 3，一個討厭的小 3。

And so we're going to just multiply across by 3 to clear the fractions.

是以，我們只需用 3 乘以 3，就可以清除分數。

And we're going to end up with 1, 1, negative 2, and 1.

結果是 1、1、負 2 和 1。

Sorry, and 3. 1 times 3 is 3.

對不起，還有 3。1 乘以 3 等於 3。

So our enclosure, then, in square brackets, is going to be 1, 1, 2 bar, 3.

是以，我們的包圍圈在方括號中將是 1、1、2 吧、3。

All right, fantastic.

好吧，太棒了。

Let's look at another vector now.

現在讓我們看看另一個矢量。

And this one is a little bit more challenging, perhaps.

而這個問題可能更具挑戰性。

So what we have to do is we have to translate that into our A1, A2, Z, 3-axis system, or define a new origin around there.

是以，我們必須將其轉換為 A1、A2、Z 三軸系統，或者在該系統周圍定義一個新的原點。

I think in this case, it's easier to translate the vector rather than draw the axes, but you could do either.

我認為在這種情況下，平移矢量比繪製座標軸更容易，但兩種方法都可以。

Either one is the same.

兩者都一樣。

So if we translate it over, you can realize that, in fact, it resides entirely over the A2-axis.

是以，如果我們把它轉換過來，你就會發現，事實上，它完全位於 A2 軸上。

So that means if you translate it back into space, back along the positive A2-axis, it'll just travel exactly over the A2-axis with no component hanging off into the A1 direction.

這意味著，如果你把它轉換回空間，沿著正 A2-軸返回，它將正好在 A2-軸上移動，而不會有任何分量懸浮在 A1 方向上。

So when we translate it back by one A lattice parameter in the positive A2 direction, it now originates in the base, in the conventional origin, right in the very middle of the basal plane, and travels out.

是以，當我們在正 A2 方向上將它向後平移一個 A 格參數時，它現在就會從底面的傳統原點，也就是底面的正中間開始，然後向外移動。

But you'll notice that it travels two steps before it makes one rise in the Z direction.

但你會注意到，它在 Z 方向上走了兩步後才上升了一步。

Therefore, it's going to exit our unit cell at half the height.

是以，它將以一半的高度離開我們的單元格。

So now we've got this vector that's going out straight along the A2 direction, with a rise for that one step in the A2 direction, a rise of 1 half.

是以，現在我們得到的這個矢量沿著 A2 方向筆直延伸，在 A2 方向上的每一步都會上升，上升幅度為 1/2。

So that vector in a three-axis notation would be, well, it's got no component in X.

是以，用三軸符號表示的矢量是，它在 X 軸上沒有分量。

It's got 1 in Y and 1 half in Z.

Y區有一個，Z區有一個半。

So we've got 0, 1, and 1 half.

是以，我們有 0、1 和 1 個半數。

Multiply across by 2, we get 0, 2, 1.

橫乘 2，得到 0、2、1。

So that's the 0, 2, 1 vector.

這就是 0、2、1 向量。

OK.

好的。

So then we just have to convert from U prime, V prime, W prime to UVTW for an index system.

是以，我們只需將 U質數、V質數、W質數轉換為 UVTW，即可建立索引系統。

So let's do that.

那就這麼辦吧。

And if we want to convert to U, we've got 1 third times 2 times U prime, which is 0.

如果我們想換算成 U，就會得到 1/3 乘以 2 乘以 U 的質數，也就是 0。

So minus 1 times 1 third is negative 1 third.

所以負 1 乘以 1/3 就是負 1/3。

And then, sorry, minus 2, minus 2.

然後，對不起，減去 2，減去 2。

So it's negative 2 thirds.

所以是負三分之二。

And then we've got V being 1 third times 2 times 2 minus 0.

然後，V 是 1/3 乘以 2 再乘以 2 減 0。

So that's 4 thirds.

所以是三分之二。

And then we sum U and V and make it all negative.

然後將 U 和 V 相加，得到負數。

So we've got 4 thirds minus 2 thirds is 2 thirds made negative.

所以，4/3 減去 2/3，就是 2/3 的負數。

So that's negative 2 thirds.

所以是負三分之二。

And then W is just W prime, which is just 1.

然後 W 就是質數 W，也就是 1。

So our indices are almost done.

是以，我們的指數幾乎已經完成。

Again, we've got to clear across to get rid of the 3 in the denominator.

同樣，我們要清除分母中的 3。

And so we're going to end up with 2 bar, 4, 2 bar, 3 once we've multiplied across by 3.

是以，在乘以 3 之後，我們將得到 2 小節、4 小節、2 小節、3 小節。

So that was fairly challenging.

是以，這相當具有挑戰性。

Let's look at one more.

讓我們再看一個。

And this one here, again, we've got to picture this somewhere within our A1, A2, Z 3-axis system somehow.

同樣，我們必須以某種方式在 A1、A2、Z 三軸系統中的某個位置將其描繪出來。

So we could translate it over, or we could just define the origin where it exists, where the origination of the vector resides right now.

是以，我們可以把它翻譯過來，或者直接在它存在的地方定義原點，也就是矢量的原點現在所在的地方。

And I think that's probably the easiest way to do it.

我認為這可能是最簡單的方法。

So the start of the vector, let's call that the origin.

是以，我們把矢量的起點稱為原點。

And we can draw in our A1, A2, Z parallelopiped around that.

我們可以在周圍繪製 A1、A2、Z 平行線。

Hopefully, you'll be able to see how that makes, or what that would look like in our XYZ orthogonal system.

希望你們能明白，在我們的 XYZ 正交系統中，這將會是怎樣的結果。

So in fact, it's traveling out along the positive A1 direction.

是以，事實上，它是沿著正 A1 方向傳播出去的。

It rises up a little bit.

它上升了一點。

It has a little component into positive y.

它有一點成分是正 y。

But it's going to come out of that front face, essentially, of the XYZ cube or system, OK?

但本質上，它將從 XYZ 立方體或系統的正面出來，明白嗎？

And so that's the first thing you need to picture is that.

是以，你首先需要想象的就是這一點。

It's a little difficult to see exactly where it's going to exit the unit cell.

要看清它從單元格的哪個地方流出有點困難。

Hopefully, if you sketch in your axes around the origin of the vector, it'll be a little more clear.

希望你能圍繞矢量的原點勾畫出你的座標軸，這樣會更清楚一些。

But you can see, if it actually traverses two steps along the A1 direction before it rises up 1 and Z, that means that you're going to have it exiting the front face at half the height.

但你可以看到，如果在上升 1 和 Z 之前，它實際上沿著 A1 方向走了兩步，這就意味著它將以一半的高度離開正面。

OK, so that means we've got this vector now that originates at the conventional origin of our cube and travels out towards us, towards you this way, I suppose, and rises up by half of the unit cell, half the height in the Z direction, before it exits that front face.

好了，這意味著我們現在有了一個矢量，它從立方體的傳統原點出發，向我們這邊，我猜是向你這邊，在離開正面之前，上升了組織、部門格的一半，也就是 Z 方向高度的一半。

So the point coordinates of that point where the vector exits the unit cell, starting at 0, 0, 0, would be 1 in the X, 1 half in the Y, and 1 half in the Z.

是以，從 0、0、0 開始，矢量離開組織、部門格的那一點的點座標在 X 軸上是 1，在 Y 軸上是 1 個半點，在 Z 軸上是 1 個半點。

So we've got a vector that you multiply across by 2.

是以，我們得到了一個向量，將其乘以 2。

The vector in 3-space, or in the 3-axis system, is 2, 1, 1.

三維空間或三軸系統中的矢量為 2、1、1。

So then we just have to convert 2, 1, 1 into our 4-axis system.

是以，我們只需將 2、1、1 轉換為我們的四軸系統。

So 1 third times 2 times 2 minus 1 is going to give us 4 minus 1, 3.

是以，1/3 乘以 2 再乘以 2 減 1 就得到 4 減 1，即 3。

That's actually 1.

實際上是 1。

That's actually 1, isn't it?

實際上是 1，不是嗎？

And then V is going to be 1 third times 2 times 1 minus 2.

然後 V 將是 1/3 乘以 2 再乘以 1 減 2。

So that's 0.

所以是 0。

OK, so that's 0.

好吧，那就是 0。

It is 0, in fact.

事實上是 0。

And then T is 1 plus 0, made negative.

然後 T 是 1 加 0，變成負數。

So it's negative 1.

所以是負 1。

And W, of course, is just W prime, which was 1.

當然，W 只是 W 的質數，也就是 1。

So our 4-axis notation now for that vector is just 1, 0, 1 bar, 1.

是以，我們現在對該矢量的四軸符號表示為 1、0、1 bar、1。

OK?

行嗎？

But that was hard.

但這很難。

It looks like an easy answer, but it was a challenging question.

這看起來是個簡單的答案，但卻是個具有挑戰性的問題。

I hope that helped.

希望對你有所幫助。

Thanks.

謝謝。