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When do we say a material is isotropic? When properties such as density, Young's modulus etc. are same in all directions. If these properties are direction-dependent then we can say that the material is anisotropic.

Now, when do we say a material is homogeneous? If I have steel with BCC crystal structure, when do we say that this is homogeneous and non-homogeneous? Can someone give specific examples to explain - especially what a non-homogeneous material would be?

Max
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    That was always the prelude to a problem. "Assume a homogeneous and isotropic medium". It is pretty simple. Homogeneous means there is the same stuff everywhere, like hydrogen gas or a block of copper. Isotropic means it has the same properties in all directions. Glass would be isotropic on a macro scale, a crystal would not. – C. Towne Springer Dec 13 '14 at 00:41
  • @C.TowneSpringer According to your comment, it doesn't make sense to say "homogeneous with respect to some property $f$". Should we speak of homogeneity only with respect to composition? – Anton Apr 28 '22 at 17:15

4 Answers4

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In short, to my understanding:

homogeneous : the property is not a function of position, i.e. it does not depend on $x$, $y$ or $z$.

isotropic: the property does not depend on a particular direction.

NB: you can have a homogenous property that is not isotropic, i.e. the refractive index of a birefringent material: it is a constant, but this constant has two different values along the two axes of the material.

A non-homogeneous material could be, say, the Earth itself: its density depends on whereabouts you are (which layer, crust, mantle etc.).

SuperCiocia
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    Also, isotropic is always homogeneous but the reverse is not true. And another way to say it all is that an isotropic property is invariant under translation and rotation. – tpg2114 Mar 31 '15 at 14:07
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    @tpg2114 False: isotropic but non homogeneous patterns are possible. The two properties are independent from each other. See here for example: http://www.astro.ucla.edu/~wright/cosmo_01.htm – valerio Jan 16 '18 at 12:01
  • @SuperCiocia How is it possible for a homogenous property to be not isotropic if it has the same value in every point? – Antonios Sarikas Jun 12 '20 at 11:09
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    See examples in Valerio’s answer. – SuperCiocia Jun 12 '20 at 15:42
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    @AntoniosSarikas I should have specified it, but a material can be isotropic with respect to 1 point or multiple points. If a material is isotropic with respect to multiple points, it will be also homogeneous. – valerio May 01 '22 at 09:53
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Homogeneity = translational invariance

A material is homogeneous with respect to the property $f$ (for example density) if

$$f(\mathbf r) = f (\mathbf r + \mathbf r')$$

i.e. property $f$ does not depend on the spatial position. If you measure property $f$ at point $\mathbf r$ or $\mathbf r+\mathbf r'$, you will find the same result.

Examples: most materials are homogeneous at a large enough scale, but they can reveal inhomogeneities if we look close enough. See the section about scale.

Isotropy = rotational invariance

A material is isotropic with respect to the property $f$ if

$$f(\mathbf r) = f (|\mathbf r|)$$

i.e. property $f$ does not depend on the direction of its argument. If you measure property $f$ along any direction in the material, you will find the same result.

Examples: fluids and amorphous solids are isotropic. Most crystals (with a few exceptions like the cubic crystal system) are not isotropic.

Scale dependence

Notice that both homogeneity and isotropy are scale-dependent quantities: they depend on the spatial scale where we choose to effectuate our measurements.

To give you a specific example, consider steel: steel is an iron-carbon alloy. At a large enough scale (let's say the mm scale), steel is homogeneous. However, if you look at it close enough ($\mu$m scale), this is what you see (source):

enter image description here

Definitely not homogeneous. Another example is granite:

enter image description here

Other examples of materials which are homogenous/isotropic on large scales but inhomogeneous/anisotropic on smaller scales, apart from alloys, are polycrystalline materials.

Also a normal simple cubic crystal (figure below), which is isotropic on large scales, is anisotropic on small scales. To see this, just think about standing in the center of the cube: how many atoms will you encounter if you move towards one of the faces? And how many if you move along one of the diagonals? The answer is different.

enter image description here

To conclude, I will just remark that homogeneity and isotropy are independent from each other. Below you can see an homogeneous but not isotropic pattern on the left and an isotropic but not homogeneous pattern on the right (source).

enter image description here

Edit: I should have specified this, but a material can be isotropic with respect to a point. The figure on the right is isotropic with respect to its center, however it is clearly not isotropic with respect to all points. If it was isotropic with respect to more than 1 point, it would also be homogeneous.

valerio
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  • You say that most crystals (except the cubic crystal system) are anisotropic, but the link you give states that the cubic crystal system is one of the most commonly found in nature. Anyway, my question is, how come the cubic crystal system is isotropic? If I use your mathematical definition I would get that it is isotropic only in the crystallic principal axis. But what about an arbitrary direction? If I measure the resistivity of say potassium in a non crystallographic direction, can I expect it to be the same as in the a-b plane or the c direction? – untreated_paramediensis_karnik Aug 28 '19 at 11:59
  • @valerio I can't understand the two concepts with your notation. If $$f(\mathbf{r}) = f(\mathbf{r} + \mathbf{r'})$$ then at every point the property $f$ has the same value. Shouldn't this imply that the material is isotropic? Also, your right image how it can be isotropic? There is a white space between the black lines, so the pattern doesn't look the same at all directions. – Anton Apr 28 '22 at 16:40
  • What you may imply with your notation is that it each point of the material the properties behave the same way (but may of course depend on direction, that is they can be anisotropic). With your notation although is not clear. For example, if we take your definition and ask if a system is homogeneous with respect to refractive index then it doesn't make sense to apply your first definition. Maybe for density it makes sense, but not for all properties. – Anton Apr 28 '22 at 16:58
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    @Anton Thanks for the comments. You are right about the last figure, I added a comment to better explain what I meant. And indeed if a material is isotropic with respect to multiple points, it is also homogeneous. Regarding the mathematical expressions reported, I acknowledge that they may not apply to all possible physical properties, but I wanted to give some "mathematical sense" of what homogeneity and isotropicity mean. – valerio May 01 '22 at 09:50
  • Hi @valerio, I seem to have 2 questions about your amazing answer. 1. if you got steel and check it on large scale, you say it's homogeneous. If so, in your notation, $f(r) = f(r+r')$, you should be using $r'$ to be in the $mm$ scale and not $µm$ scale, because if you do so on $µm$ scale, you will still get different results even on large scale. So I guess, if you use large scale, your $r'$ is always mm. Right ? – Giorgi Lagidze Sep 10 '23 at 19:35
  • So if we say material is isotropic around point $k$, then property must be the same in every direction - I get that, but is it enough to say it’s isotropic only for some specific $r$ - say $r=2$ and from our point $k$, $f(2) = f(|2|)$, but $f(2.1) \neq f(|2.1|)$. would this be enough to say material is isotropic around point $k$ ?
    • – Giorgi Lagidze Sep 10 '23 at 19:35