In principle you are right Sihan Zhang .
But unfortunately there is a 4 index system used for the hexagonal directions and planes.
These 4 indices of a plane have to be converted to the 3 index description (Miller index) of the plane family.
Formulae for conversion can be found in the book of Cullity.
Hangchen Xu , see for example: around page 45 of :
https://glass.rutgers.edu/sites/default/files/uploads/virtual/dir.cullity/Cullity-ch2.pdf
Best regards
G.M.
Sihan Zhang 它们的定义不一样吧，米勒指数是晶体学中的概念，是以晶胞基矢为单位矢量的，而晶面指数是固体物理中的概念，是以原胞基矢为单位矢量的，对于同一个晶面，这两个指数不一定相同呀。
Gerhard Martens Thank you very much for your answer, but what confuses me is that there are some differences in the definition of these two indices, the Miller exponent is a concept in crystallography, which is measured in units of the unit sag, while the crystal plane index is a concept in solid state physics, which is measured in units of the original cell sagittal and for the same crystal plane, these two indices are not necessarily the same, in the textbook, there is a formula for how to calculate the plane spacing of the crystal plane index, but when I use this formula to calculate the plane spacing of the Miller index, I find that some of the crystal planes will be missed, For example, for an FCC, the plane spacing of a plane with a Miller index of (110) is different from that of a direct geometric judgment
Hangchen Xu 你这个定义应该是国内教材特色的咬字眼。你可以多参考一些外国教材，晶面指数和米勒指数是等价的。常规的晶面指数是根据基础布拉维格子定义的，很少会从原胞出发。
Dear Hangchen Xu ,
thanks for feedback...
I did not really got your point yet.
You wrote above: "some of the crystal planes will be missing".
There are no crystal planes missing; but for example for the case of the FCC lattice the 110 planes are still there and the interplanar distance d110 can be calculated via
dhkl = a * (h2 + k2 + l2)-1/2
but there is no diffraction peak (110).
This is due to the 'selection rules of x-ray diffraction'*).
The structure factor governs the ON and OFF switching of the peaks.
*) see for example:
https://groups.mrl.illinois.edu/chiang/czoschke/diffraction-selection-rules.html
https://en.wikipedia.org/wiki/Structure_factor
(Scroll a bit down to: 'Definition of Fhkl' )
Best regards
G.M.
Sihan Zhang 确实，在材料学里这两一般不做区分，并且一般都是用晶胞基矢为单位矢量的，方便理解，只不过在计算晶面间距时，似乎单纯的用米勒指数套入晶面间距计算公式时会存在一些问题，因此需要对这两个概念进行区分。我找到一篇比较好的文献可以说明这个问题。
Dear Gerhard Martens
Sorry, maybe my description is a bit of a bother, but what you mentioned is an important phenomenon that we need to be concerned about when it comes to XRD, but what baffles me is some of the problems that arise when using the Miller index to calculate the plane spacing using the plane spacing formula. I searched the literature and described the problem well, but it still seems a bit complicated to solve the problem
Hangchen Xu 所以我才说这个东西是国内的人咬字眼硬造出来的为了考试的概念，还是那句话，你去看国外的教材，miller index和face index就是一个东西，唯一有不一样的地方就在于之前G. Marten提到的六方和三方晶系有的时候用四个坐标。包括你提到这个问题我也去查了我们学校的固体物理课件（我本科化学同时修过物院的课），也说晶面指数和米勒指数是一个东西。我的建议是，如果是为了考试的话，你们怎么讲就怎么讲。但是对于真实的研究，以及英语文献的通用说法，这俩没有区别，除非在一些领域特殊强调了是原胞。而且我看了你那个文献，参考意义不大，那里的式子9我没懂他什么意思。代单胞体积和单胞倒格矢也是成立的。只要前后统一就好。他那个意思好像是只有原胞才能定义倒格矢是吧，什么老学究。
顺带一提，国内发的中文文献，真别当真。多看看外国教材。