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In my book, it's been mentioned that crystalline solids are anisotropic whereas amorphous solids are isotropic in nature. The reason for these has been explained (with a diagram) as:

Crystalline solids are anisotropic in nature, that is, some of their physical properties like electrical resistance or refractive index show different values when measured along different directions in the same crystals. This arises from different arrangement of particles in different directions. Since the arrangement of particles is different along different directions, the value of the same physical property is found to be different along each direction.

Amorphous solids on the other hand are isotropic in nature. It is because there is no long range order in them and arrangement is irregular along all directions. Therefore, value of any physical property would be same along any direction.

My question is: similar arrangement of particles in different directions in crystalline solids gives rise to different physical properties. Then, how can disarrangement in amorphous solids in all directions give rise to same physical properties?

In other words, why do crystalline solids have properties (like electric conductivity, thermal conductivity, mechanical strength, and refractive index) different in different directions, and the amorphous solids have these properties same in all directions?

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    $\begingroup$ Imagine that your amorphous solid is made up of many small crystals, each anisotropic in properties, but in a random arrangement both in position and orientation. $\endgroup$ – porphyrin Jul 8 '17 at 10:46
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    $\begingroup$ Because it's disarranged in all directions equally, "degree of disorder" is same in any direction. It's not disarranged in any direction more than in another. It's isotropic because it's uniformly disarranged in all orientations. $\endgroup$ – voldermot Jul 8 '17 at 12:54
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    $\begingroup$ @Jamil Ahmed Keep in mind that the mentioned anisotropy electrical resistance, refractive indeces, etc. affects them because they are (at minimum) vectorial properties. It will not affect properties of purely scalar in nature, like density, for example. $\endgroup$ – Buttonwood Jul 8 '17 at 12:59
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enter image description here

Assume that the spheres in the drawing are atoms.

In the second picture atoms are not orderly arranged. No matter where you go in crystal of a amorphous solid, the disarrangement will be same. Since in every direction, disarrangement is same, physical properties along every direction is also same. Therefore, amorphous solids are isotropic.

In the first picture, the arrangement is orderly. So, when you go in any direction, the arrangement will be different. It is also clear in the rectangles shown in the picture. Try doing it yourself on a piece of paper. Try finding any four different directions along which the arrangement is same.

Ps: Ignore my poor drawing skills.

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Read the following article by C.V. Raman which explains how structural anisotropy exists in amorphous solids and also gives rise to the phenomenon "Structural Birefringence". This is the first and the only article that speaks about birefringence and anisotropy in amorphous solids.

Structural birefringence in amorphous solids

Abstract:

The result reported in an earlier investigation with vitreous silica is now shown to be true also for other amorphous solids, including especially inorganic glasses; besides the well-known photo-elastic effect, another kind of birefringence may be observed differing from the former both in its origins and in its observable characters. This “structural birefringence” arises from anisotropy of structure present in the solid by reason of the circumstances of its formation. It is conspieuously seen with plate glass whose optical behaviour shows it to have a highly laminated structure, while in moulded glass, it exhibits itself as luminous streaks or sheets of variously curved forms.

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