1. المحاضرة 4 التركيب البلوري-التماثل

6. Plane of symmetry
Axis of symmetry

8. Center of Symmetry
هي نقطة وهمية متوسطة في البلورة تتميز بأن أي وجهين أو حرفين أو زاويتين مجسمتين يتناظران عبرها. وقد توجدهذه العناصر مجتمعة في بلورة واحدة مثل بلورة الجبس .وقد يوجد بعض هذه العناصر مثل بلورة اكسينايت التي يوجد بها مركز تماثل فقط.وقد لايوجد أي من هذه العناصر كما في بلورة ثيوسلفات الكالسيوم المائي.

10. أن أهم مايميز الشبيكية البلورية هي عمليات التماثل التي بإجرائها يعود البناء البلوري لوضعها الأصلي وأهم عمليات التماثل هي عملية التماثل النقطية والتي يمكن تطبيقها عند نقطة معينة في الشبكية البلورية ومن أمثلتها:

11. أهم عمليات التماثل هي عمليات التماثل النقطية والتي يمكن تطبيقها عند نفطة معينة في الشبكية البلورية ومن أمثلتها:
1- عملية الانقلاب Inversion operation وتتم هذه العملية حول نقطة شبكية تسمى مركز الانقلاب بحيت يتحول المتجه T إلى T.
Inversion operator

12. - عملية الانعكاس Mirror Reflections

13. 3-عملية الدورانRotation
(60,90,120,180,360).ولاتوجد أي شبكية يمكنها أن تعود الى وضعها الاصلي باي عملية دوران اخرى ،فلا يوجد مثلاً محور تماثل خماسي أو سباعي التماثل للشبكية البلورية.

14. Then the operation of the 2-fold leaves two points
Symbol for 2-fold axis
=180
n=2
2-fold rotation axis
Then the operation of the 2-fold leaves two points
Symbol for 3-fold axis
=120
n=3
3-fold rotation axis

15. =90
n=4
4-fold rotation axis
=60
n=6
6-fold rotation axis

16. 4- عمليات دوران انقلابيةRoto-Inversion تتم بالدوران حول محور بزاوية 2π/n= أو مضاعفاتها.حيث nعدد صحيح يصف محور التماثل ويأخذ القيم المحددة1,2,3,4, 6ثم يتبعها انقلاب حول نقطة شبكية يمر بها المحورويعبر عن محاور الدوان الانقلابية بالارقام 1,2,3,4,6

17. Notation for representing left and right handed objects
To start with we use the notation as described below. (Occasionally deviating from this as well!).
Ultimately, we will turn to ‘International Tables of Crystallography’ symbols in b/w. +R R +L L

18. Translation
The translation symmetry operator (t) moves an point or an object by a displacement t or a distance t.
A periodic array of points or objects is said to posses translational symmetry.
Translational symmetry could be in 2D or 3D (or in general nD).
If we have translational symmetry in a pattern then instead of describing the entire pattern we can describe the ‘repeat unit’ and the translation vector(s). t    t        

19. Mirror and Inversion Inversion operator
With note on left and right handed objects
The left hand of a human being cannot be superimposed on the right hand by mere translations and rotations
The left hand is related to the right hand by a mirror symmetry operation (m)
The right hand is called the enantiomorphic form of the left hand
Another operator which takes objects to enantiomorphic forms is the inversion operator (i) (in the figure to the right below- between the two hands (in the mid-plane) at the centre is an inversion operator)` m
Inversion operator

20. Rotation Axis
Rotation axis rotates a general point (and hence entire space) around the axis by a certain angle
On repeated operation (rotation) the ‘starting’ point leaves a set of ‘identity-points*’ before coming into coincidence with itself.
As we are interested mainly with crystals, we are interested in those rotations axes which are compatible with translational symmetry → these are the (1), 2, 3, 4, 6 – fold axis.
If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where:
The rotations compatible with translational symmetry are  (1, 2, 3, 4, 6)
Click here for proof 
Crystals can only have 1, 2, 3, 4 or 6 fold symmetry
* explained in an upcoming slide

21. Then the operation of the 2-fold leaves two points
Symbol for 2-fold axis
=180
n=2
2-fold rotation axis
Then the operation of the 2-fold leaves two points
Symbol for 3-fold axis
=120
n=3
3-fold rotation axis

22. =90
n=4
4-fold rotation axis
=60
n=6
6-fold rotation axis

23. Identity Points/Objects
If we start with a general point, then the operation of symmetry operator(s) will leave a (finite) set of points. These symmetrically related set of points are called identity points.
An extension of the concept of Identity points is to use identity objects which can show left or right handedness.
Some examples are shown below.
4-fold leaves 4 identity points
Alternate diagram
4mm
Right Handed
4mm
Left Handed
Right Handed
4mm point group leaves 8 identity points: 4 left handed (orange circle) and 4 right handed (green circle)
Left Handed

24. Compound Symmetry Operators
Translation, mirror, inversion & rotation are simple symmetry operators which we had considered
Roto-inversion and Roto-reflection are compound symmetry operators which do not involve translation → both these take left handed objects to right handed form  For generating point groups (to be considered later) one of the two operators is sufficient and hence we will consider roto-inversion only in future.
Glide reflection and Screw are compound symmetry operators involving translation → Only Glide reflection takes left handed objects to right handed form
It is important to note in these operations the compound operator acts before leaving a identity-point (i.e. Roto-inversion is NOT rotation followed by a inversion).
In some cases these compound operators can be broken down into a combination of two operators. In a combination (unlike a compound) the individual operators express themselves fully – i.e. the first operator acts first and then the second acts on the result of the first operation.

25. Roto-inversion operations
A roto-inversion operator rotates a point/object and then inverts it (inversion operation) in one go.
A left handed object will be taken to its right handed form by the operation.
We will only consider 1,2,3,4,6 - fold rotations (crystallographic) as a part of the roto-inversion operation.
Roto-inversion operations
Compatible with translational symmetry

26. Screw Axis
A screw (axis) operator rotates a point/object and then moves it a fraction of the repeat distance in one go.
The faction which the screw axes move is called the pitch of the screw.
We will only consider (1, 2, 3, 4, 6) - fold rotations (crystallographic) as a part of the screw axes.
The screw axes to be considered are:  21  31 , 32  41 , 42 , 43  61 , 62 , 63 , 64 , 65
The normal and screw axis both give the same effect on the external symmetry of the crystal

27. The 32 axis produces a rotation of 120 along with a translation of 2/3.
The set of points generated are: (0,0) (120,2/3) (240,4/3=1/3) (360,6/3=2)…
This is equivalent to a left handed screw (LHS) of pitch 1/3

28. The 43 axis is a RHS with a pitch of 3/4
The effect of 43 axis can be thought of as a LHS with a pitch of 1/4

29. The 42 axis generates the following set of points: (0,0) (90,1/2) (180,2/2=1) (270,3/2=1/2) (360,4/2=2)
The grey arrowhead maps the (270,3/2) point to (270,1/2) → to keep points within unit cell

31. الشبكية الفراغية
ترتيب عدد لانهائي من النقط في الفراغ بحيث يكون لكل نقطة نفس الجيران فأنها تكون شبكية فراغية لكل نقطة فيها نفس عدد الجيران على نفس الابعاد والاتجاهات.أي أنها أصغر جزء مرتب من ذرات متماسكة تماثل تماماً البلورة الكبرى التي يتكون منها الجسم.
ويمكن تمثيل الشبكية البلورية بالمتجهات a,b,c

34. وقد صنف العالم برافييه الفرنسي كل احتمالات ترتيب الذرات داخل الشبكية البلورية فوجد إنها يمكن أن تمثل اربعة عشر احتمالا لطول a,b,c.وتبعاً للزوايا التي تميل بها ,, على الاحداثيات x,y,zعلى الترتيب.(أي اربعة عشرة طريقة يمكن بواسطتها ترتيب أي عدد من النقط بحيث تستوفي شروط تكوين الشبكية لكل طريقة من هذه الطرق تحدد وحدة الخلية عندما تتكرر هذه الوحدة في الاتجاهات الفراغية الثلاثه لنحصل على الجسم البلوري الكامل ).

54. And now for something completely different . . .