Poisson's binomial

7/23/2019 Poisson's binomial

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LECTURE 15

X-Ray Difraction

Strcuture o Materials



Why should we be interested?





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LATTICE AA!" A#$ %A&AI" LATTICE"

Crystalline materials differ from amorphous materials in that in the former there is order in the arran'ement of the molecular contents whereas in the latter there is no order or at best a tendency for a short(ran'e order)

The packing of atoms, molecules or ions within a crystal occurs in a symmetrical manner and furthermore this symmetrical arran'ement is repetiti*e) A most important common characteristic that crystals may share is the manner in which repetition occurs) This will be e+pressed in a common lattice array)

A lattice array is constructed from the arran'ement of atomic material within the crystal as follows,



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A 2-dimensional Lattice -ic. any position within the 2 dimensional lattice in /i') 1a and note the arran'ement about this point) The chosen position can be indicated by a point a lattice point) In *iew of the repetiti*e arran'ement there will be a 2 dimensional array of identical positions and if these are also mar.ed by a point a 2(dimensional lattice will result if the points are 3oined)



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In a real 3-dimensional crystal lattice the same ideas apply.

When crystal structures are represented by lattices it transpires that all crystals

brea. down into one of fourteen three dimensional lattice arran'ements) Braais demonstrated mathematically that there are only fourteen ways in which repetiti*e symmetry can occur and the fourteen lattices representin' the ways in which repetition can occur are referred to as the Braais lattices)

!"IT C#LL

A unit cell can be any unit of a lattice array which when repeated in all directions and always maintaining the same orientation in space 'enerates the lattice array)

There is no uni4ue way of choosin' a unit cell) /or e+ample each of the cells A to $ in /i') 2 are 56)

7owe*er the cell fa*oured by crystallo'raphers is the one of smallest *olume that displays all of the symmetry of the lattice) Thus cells C and A are the preferred unit cells for the lattices

of /i's) 2 and 8 respecti*ely)

A B

C $

A

B

%ig. & %ig. 2

%ig. 2 %ig. 3



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When these unit cells are combined with possible 9centerin': there are 1; different %ra*ais lattices)

In 'eneral si+ parameters are re4uired to define the shape and si<e of a unit cellthese bein' three cell ed'e len'ths con*entionally defined as a b and cand three an'les con*entionally defined as α β and γ ) In the strict mathematicalsense a b and c are *ectors since they specify both len'th and direction)α is the an'le between b and c β is the an'le between a and c γ is the an'le

between a and b) The unit cell should be ri'ht handed) Chec. the cell abo*e withyour ri'ht hand

!"IT C#LL T'(#) and T*# )#+#" C')TAL )')T#)

Cubic a = b = c) α = β = γ = >0)Tetra'onal a = b ≠ c) α = β = γ = >0)

5rthorhombic a ≠ b ≠ c) α = β = γ = >0 )@onoclinic a ≠ b ≠ c) α = γ = >0 β ≠ >0)Triclinic a ≠ b ≠ c)) α ≠ β ≠ γ ≠ >0)hombohedral a = b = c) α = β = γ ≠ >0 )or Tri'onal7e+a'onal a = b ≠ c) α = β = >0 γ = 120)

5rthorhombica

c b



%ra*ais Lattices



Cubic Tetra'onal and 5rthorhombic



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Cuic

Tetragonal

/rthorhomic

onoclinic

Triclinic

(rimitie Cell

Body

Centred

Cell

%ace

Centred

Cell

#nd %ace Centred

Cell

(

I %

C

%ig. 3

Trigonal

*e0agonal



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&. E*ery crystal system has a primiti*e %ra*ais lattice)

2. The distribution of lattice points in a cell must be such as to maintain the total symmetry of the crystal system) Thus the cubic system cannot ha*e a C(type cell)

3. The fact that a unit cell meets the symmetry re4uirements of a crystal system does not 'uarantee its inclusion within the crystal system) This could result if the lattice it 'enerated could be e4ually well represented by a unit cell type which is already included within the crystal system) The C(type cell for the tetra'onal system see /i') ; pro*ides a 'ood e+ample)

%ig. 1

C - cell

( - Cell

/our simple points on crystal lattices,

1. If you repeat 8) within the orthorhombic system you will find that the primiti*e cell you

'enerate will not ha*e >0 an'les) This is not orthorhombic and thus orthorhombic C is included in the crystal system)



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%ig. 1

C - cell

( - Cell

A simplified *iew down c(a+is can beused to illustrate points 8 and ;

5rthorhombica b c a = b = γ = >0

ab

An'le not >0B smaller cellnot orthorhombic

Tetra'onal

a = b c a = b = γ = >0

ab

"maller cell is Tetra'onal -



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)'#T' (/I"T /!( )'#T' A"$ )(AC# /!( )'#T'

-oint 'roup theory is not dealt with here) What follows is 3ust a summary) -oint 'roup symmetry defines the symmetry of an isolated ob3ect or

'roup of ob3ects whereas space 'roup symmetry further defines the systematic fashion in which an ob3ect or 'roup of ob3ects is repeated in space to 'enerate an infinite periodic array in 8$)

-oint 'roup symmetry is 4uantified in terms of symmetry elements e+istin' within the ob3ect or 'roup of ob3ects and their associated operations) /our symmetry elements are used to 4uantify point 'roup symmetry

)ymmetry #lement )ymmetry /peration

otation a+is n(fold otation@irror plane eflection

Centre of "ymmetry In*ersionotor(reflection a+is n(fold otation and reflection

or otor(in*ersion a+is n(fold otation and in*ersion



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!"IT C#LL !"IT C#LL

(ositions of 2-fold a0esand mirror planes

Centres ofsymmetry

a

a

%ig. 4

5a6 56

-oint roup and "pace roup "ymmetry

To 'enerate a 8$ lattice from an ob3ect it is necessary to add translational

symmetry to point 'roup symmetry) The two important space 'roup symmetry operations which mo*e ob3ects are 'lide planes and screw a+es) These operations combine translation and reflection and translation and rotation respecti*ely)

The penta'ons on the

left are related by simple translation) In Db the penta'on on the top left of the cell is related to the one in the centre by

translation a2 followed by either reflection or rotation) Centres of in*ersion in Db are mar.ed with tiny circles)





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The A%"E# pro'ram within 5scail can pro*ide %ar Chartsof the contents of the Cambrid'e $ata %ase C"$

The number of entries by crystal system Entries in the first 2D space 'roups



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C')TAL (LA"#) A"$ ILL# I"$IC#)

The use of crystal planes to describe the

structure of crystals 'oes bac. to the start of crystallo'raphy and crystal planes were used by %ra'' to e+plain diffraction as will be seen later) Crystal planes are defined by the intercepts they ma.e on the

crystal a+es of the unit cell) The in*erse of these fractions are the @iller Indices of the planes)

In a the intercepts are G G 1 and the @iller Indices are 2 2 1)

In c the intercepts on b and c are at infinity the in*erse of which is 0 and the plane is the 2 0 0) In d the plane cuts the ne'ati*e c a+is at (1 and thus is 1 1 (1) In crystallo'raphy (1 is often written H

and pronounced 9%ar 1:)



Characteristics of cubic lattices

)imple Body-centered %ace-centered

Lattice

pointscell1 2 ;

#umber ofnearest

nei'hbours J 12

nearest

nei'hbourdistance a √82 a a√2

-ac.in'fraction π = 0)D2; √8J π = 0)J0 √2 π = 0)F;0



7e+a'onalK @iller(%ra*ais indices



@etallic bondin'



elation between 7C- and /CC



%CC to 7C- transformation



-roduction and -roperties of (ays

• (rays were disco*eredin 1J>D by Mnt'en

• In*isible in nature• -enetrate much more

than *isible li'ht



(ray interaction with matter





/undamentals of $iffraction







$iffraction

• If a wa*e encounters a barrier that has an openin' of dimensionsimilar to the wa*elen'th the passin' wa*e will flare spread Nwill diffract N in the re'ion behind the barrier)

• $iffraction limits 'eometircal optics N will cause the li'ht to spread

a a a

$iffractedwa*e

)0 λ8)0 λ 1)D λ

λ

"creen



5ri'in of (rays



$iffraction and %ra''Os law

• "tructural information can be obtained by

usin' (rays electrons or neutrons• The incident radiation must ha*e a

wa*elen'th between 0)1 and 1 )•

The interference of +(rays when scatteredfrom the atoms within a crystal lattice)



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$I%%ACTI/" A"$ T*# BA #<!ATI/"

@a+ *on Laue was the first to

su''est that crystals mi'ht diffract (rays and he also pro*ided the first e+planation for the diffraction obser*ed) 7owe*er it is the e+planation pro*ided by %ra'' that is simpler

and more popular)

incidenteam

reflectedeam

0

y

m n

d

o o

o o

o oA B

C $

a

incidenteam

reflectedeam

0

y

m n

d

o o

o o

o oA B

C $

d

a!"IT

C#LL

=

/ (

5a6

56

%ig. &&

52,7,76

5&,7,76

# %

!"IT

C#LL

In the %ra'' *iew crystal planes act a mirrors) Constructi*e interference is obser*ed when the path

difference between the two reflected beams in a = nλ. The path difference in a is 2my) "ince myd = sinθ 2my = 2dsinθ = nλ where d is the interplanar spacin')



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In a abo*e it is clear that the planes are the 100 set of planes) If the path difference is simply one wa*elen'th the %ra'' condition can be stated as

This is a first order reflection) If the path difference istwo wa*e len'ths the %ra'' condition becomes

and the reflection is a second order reflection)

%ra''Ps Law animated

λ θ =sin2 )0,0,1(d

λ θ 2sin2 )0,0,1( =d

http://www.eserc.stonybrook.edu/ProjectJava/Bragg/

http://www.eserc.stonybrook.edu/ProjectJava/Bragg/



• A monochromatic beam of (rays with coherentradiations incident on a crystal

• The atoms are arran'ed as sets of parallel planesand these atoms act as a reflection mirror for (rays



elation between plane spacin' andlattice constant

• /or a cubic crystal

Where h . l are called miller indices•

/or constructi*e interference the beams scattered on differentplanes need to be in phase i)e)fi'ure 1

The abo*e relation is called %ra''Os law)/or the first order diffraction i)e) n =1

λ = 2dh.lsinθ



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C,')TAL )')T#-and

!"IT C#LL $I-#")I/")

%!LL $ATA )#TC/LL#CTI/"

B,A+AI) LATTIC#

)(AC# 3,/!(

C/")T,!CT A"#L#CT,/" $#")IT'

-A(

L/CAT# AT/-(/)ITI/")

)T,!CT!,#,#%I"#-#"T

)#L#CT A )!ITABL#

C,')TAL

A

B

C

$

#

%

3

)/L+I" A C')TAL )T!CT!# B'

)I"L# C')TAL $I%%ACTI/" T#C*"I<!#)

".B. The crystal must e a single crystal.

%ra''Qs e4uation specifies that if a crystal is rotated within a monochromatic (ray beam such that e*ery concei*able orientation of the crystal relati*e to the beam is achie*ed each set of planes will ha*e had the opportunity to satisfy the %ra'' e4uation and will ha*e

'i*en rise to reflection) In order to sol*e a crystal structure it is necessary to record a lar'e number of reflections) This implies accurately measurin' their intensities and recordin' their directions with respect to crystal orientation and initial (ray beam direction)

@any e+perimental techni4ues ha*e been de*ised to achie*e this) The steps in*ol*ed in a crystal structure determination are summarised in the flow chart)

$i 5f A A $ $ C ll i "



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$ia'ram 5f An Area $etector (ay $ata Collection "ystem)

The crystal is oscillated o*er R 2B while an ima'e is collected then rotated by the same amountand oscillated a'ain) The process is

repeated o*er a total ran'e of about 1J0B) Each ima'e is e+posed for R 100s) Thus if readout time is i'nored total data collection time is often R 8 hr) A typical ima'e shown to the left) A computer pro'ram is used to predict

the unit cell from se*eral ima'es)

The first crystallo'raphicdata collection systems usedphoto'raphic methods) Thesewere replaced by automateddiffractometers which measuredreflections one at a time) A typicaldata collection too. se*eral days)modern systems use area detectorswhich measure 100s at a time)



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$etermination of the Lattice type and "pace roup

7i'h symmetry can lead to reflections bein' systematically absent from the data set) Absent reflections ha*e no measurable intensity) There are two types of absences eneral Asences and )pecial Asences) The 'eneral absences determine the lattice typeK -rimiti*e ( has no 'eneral absences and no restrictions on h . or l) End Cantered C hS.=2nS1 are all absent)

/ace Cantered % only h . l all e*en or all odd are obser*ed) %ody Cantered I hS.Sl=2nS1 are all absent)

The special absences refer to specific sets of reflections and are used to detect the presence of 'lide planes and screw a+es) "ome "pace roups

are uni4uely determined by special absences but in many cases se*eral "pace roups will ha*e to be considered)

Computer pro'rams are able to lay out the data in tables with absencesindicated and possible "pace roups can be su''ested howe*er the choice of "pace roup will often need much thou'ht)



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The "eed for )ingle Crystals ( In order to carry out a detailed (ray structure determination it is essential to ha*e a crystal of the material in 4uestion) @any compounds cannot be crystallised and thus are not amenable to diffraction studies) There are also commercially important materials such as 'lasses and many ceramics which owe their uni4ue properties to their amorphous nature) %ein' amorphous no lon' ran'e order the structures of these materials cannot be in*esti'ated in detail by diffraction techni4ues)

Locating *ydrogen atoms ( 7ydro'en atoms ma.e e+tremely small contributions and for this reason (ray crystallo'raphy is not a 'ood techni4ue for accurately locatin' hydro'en atom positions) If the location of hydro'en atoms is of specific interest e.g. in the study of hydride structures and hydro'en bondin' interactions use has 'ot to be made of the much more e+pensi*e and less a*ailable techni4ue of neutron diffraction) The theory of neutron diffraction is *ery similar to that for (ray diffraction but an essential difference is that hydro'en atoms scatter neutrons as effecti*ely as many other atoms and for this reason they can be located with 'ood accuracy in the structure determination)

-roblems with (ray Crystallo'raphy

Lo> Temperature )tructure $etermination N When (ray data are collected at low temperature R(1D0 C thermal ellipsoids are smaller and better defined)

#)%) bond len'ths show *ery little *ariation with temperature)



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(/?$# @-A' $I%%ACT/#T'

A crystalline powder sample will diffract (rays but since the orientations of the indi*idual crystals are random the data set produced is a plot of intensity v.s.

diffraction an'le or %ra'' an'le θ)

7ere the sample is sittin' on a flat plate and the plate is turned about the centre of the diffractometer at half the rate throu'h which the counter

mo*es) This is the θ2θ or %ra'' scan method)

#otice the plot contains 2θ on the (a+is and (ray intensity on the y(a+is)



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!ses of @-ray (o>der $iffraction

In 'eneral powder diffraction data are unsuitable for sol*in' crystal structures) "ome ad*ances ha*e recently been made usin' the iet*eld method) 7owe*er

this is far from tri*ial and it wor.s best in relati*ely simple cases) It is *ery difficult to be sure that the unit cell is correct as the reflections o*erlap and are difficult to resol*e from one another)

Important adantages and uses of po>der diffraction

1) The need to 'row crystals is eliminated)

2) A powder diffraction pattern can be recorded *ery rapidly and the techni4ue is non(destructi*e)8) With special e4uipment *ery small samples may be used 1(2m');) A powder diffraction pattern may be used as a fin'erprint) It is often superior to an infrared spectrum in this respect)D) It can be used for the 4ualitati*e and often the 4uantitati*e determination of

the crystalline components of a powder mi+ture)) -owder diffractometry pro*ides an easy and fast method for the detection of crystal polymorphs) -owder patterns are pro*ided when a dru' is bein' re'istered with the /$A) -olymorphs are different crystal forms of the same substance)



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/or an ortho'onal system α = β = γ = >0B the relationship between interplanar spacin' d and the unit cell parameters is 'i*en by the

e+pression,

This is the e+pression for an orthorhombic crystal)

/or the tetra'onal system it reduces to

and for the cubic system it further reduces to

Calculations using @-ray po>der diffraction patterns.

2

2

2

2

2

2

2),,(

1

c

l

b

k

a

h

d l k h

++=

2

2

2

22

2),,(

)(1

c

l

a

k h

d l k h

++

=

2

222

2),,(

)(1

a

l k h

d l k h

++=

I t t C bi L tti T



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"ite #aS Cl(Central 0 1/ace 2 0Ed'e 0 12;Corner JJ 0Total ; ;

Important Cubic Lattice TypesTwo of the most important cubic lattice types are the #aCl type and theCsCl type)

"toichiometry formula from the nit Cell

In the CsCl structure both ions ha*e coordination numbers of J and the structureis a simple primiti*e one with no centrin')

/ormula Cs at centre = 1 J + 1JCl = 1 = CsCl

#aCl crystalli<es in the "pace roup %m-3m





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The %ra'' e4uation may be rearran'ed if n=1

from to

If the *alue of 1/(d h,k,l )2 in the cubic system e4uation abo*e is inserted into this form of the %ra'' e4uation you ha*e

"ince in any specific case a and λ are constant and if λ2;a2 = A

θ λ sin2d n = θ λ 2

2

2

sin4

=d

)(4

sin 2222

22 l k h

a

++= λ θ

)(sin 2222 l k h A ++=θ



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A"AL')I) /% @-A' (/?$# $I%%ACTI/" $ATA

$iffraction data ha*e been collected on a powder diffractometer for a series of compounds that crystallise in the cubic system)

In the practical course each person will calculate the unit cell parameter and density from an (ray powder diffraction pattern)

<uestion &

Aluminium powder 'i*es a diffraction pattern that yields the followin' ei'ht lar'est d(spacin's, 2)88J 2)02; 1);81 1)221 1)1> 1)012; 0)>2J>

and 0)>0DD U) Aluminium has a cubic close pac.ed structure and its atomic wei'ht is 2)>J and λ = 1)D;0D A ) Inde+ the diffraction data and calculate the density of aluminium)

The %ra'' e4uation can be used to obtain sinθ

The ccp lattice is an / type lattice and the only reflections obser*ed are those with all e*en or all odd indices)

Thus the only *alues of sin2θ in that are allowed

are 8A ;A JA 11A 12A1A and 1>A for the first ei'ht reflections)

)(sin 2222l k h A ++=θ

θ λ sin2d =d 2

sin λ

θ =



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dU "inθ "in2θ Calc) "in2θ h . I

2)88J 0)82>;D 0)10JD; 111

2)02; 0)8J0D 0)1;;J2 0)1;;F2 200

1);81 0)D8J2 0)2J>F2 0)2J>;; 220

1)221 0)80J; 0)8>F>D 0)8>F>J 811

1)1> 0)DJ>0 0);8;1; 0);8;1 222

1)012; 0)F0J2 0)DFJJ; 0)DFJJJ ;00

0)>2J> 0)J2>21 0)JFDJ 0)JF;2 881

0)>0DD 0)JD08 0)F28DJ 0)F280 ;20

Insert the *alues into a table and compute sinθ and sin2θ)"ince the lowest *alue of sin2θ is 8A and the ne+t is ;A the firstEntry in the Calc) sin2θ column is 0)10JD;8V; etc.

The reflections ha*e now been inde+ed)



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/or the first reflection for which h2 S .2 S l2 = 8 sin2θ = 8A = 8 λ2 ;a2

a2 = 8λ2 ;sin2θ a = ;)0;>; U = ;)0;>; + 10(J cm)

Calculation of the density of aluminiuma8 = );08D U8 = );08D + 10(2; cm8)If the density of aluminium is ρ ') cm)(8 the mass of the unit cell is ρ + );08D + 10(2; ')The unit cell of aluminium contains ; atoms)The wei'ht of one aluminium atom is 2)>J)022 + 1028 = ;);J02; +

10(28

and the wei'ht of four atoms the content of the unit cell is 1F>)20> +10(2;)

ρ + );08D + 10(2; = 1F>)20> + 10(2;

p = 2)>JJ ')cm(8)

Calculation of a



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Answer each of the followin') 5n the basis that the structure is cubic and of either the #aCl or CsCl type1) Inde+ the first si+ reflections) &4 marks, 2) Calculate the unit cellparameter)4 marks, 8) Calculate the density of A'Cl) 4 marks

Assume the followin' atomic wei'hts, A' 10F)JJK Cl 8D);D8K and A*o'adroPs number is )022 + 1028

The (ray powder diffraction pattern of A'Cl obtained usin' radiation of wa*elen'th 1)D;U is shown below) The pea.s are labelled with 2 *alues



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"ince θ *alues are a*ailable sin2θ *alues can be calculated and inserted in a table)

1) for a face centred lattice 8A ;A JA 11A 12A and 1A 2) for a primiti*e lattice 1A 2A 8A ;A DA and A

2θ θ "in2

θ2F)J0 18)>0 0)0DFF

82)20 1)10 0)0F>

;)20 28)10 0)1D8>

D;)J0 2F);0 0)211J

DF);D 2J)F8 0)2810F);D 88)F8 0)80J8

The second option is not possible as the first 2 are not in the ratio of 1,2)To test the first option di*ide the first by 8 and multiply the result by ; J etc.

Calc) "in2

θ

0)0F>8

0)1D8>

0)211

0)280J

0)80FF

/rom "in2θ = Ah2 S .2 S l2 the possible *alues are,



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"ince sin2θ = λ2h2 S .2 S l2;a2

a2 = 1)D;2)1;0)80J8 usin' the lar'est most accurate 2θa2 = 80)F>2a = D)D;FX 1X = 10(J cm/ormula wt) of unit cell = ;A'Cl = DF8)2J;'This is the wei'ht of ; moles of A'Cl)

The wei'ht of ; molecules is DF8)2J; )02 + 1028$ensity = DF8)2J; )02 + 1028D)D;F + 10(J8 A is in X thus the answer should be multiplied by 1 10(2;

$ensity = D)DJ0 'cm8

$ensity of A'Cl



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(referred /rientation #ffects in @-ray (o>der $iffraction (atterns

It is possible to calculate the theoretical diffraction pattern if thecrystal structure is .nown)

Calculated pattern

5bser*ed pattern

There are no preferred orientation effects here as all reflections ha*e their

e+pected intensity)

#ifedipine



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%en<oic acid

Calculated pattern

5bser*ed pattern

There is clear preferred orientation here)The 002 is the flat face e+posed when theneedles lie down on a flat plate)

002

00;



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"ome points relatin' to preferred orientation effects)

• -referred orientation effects are often obser*ed for needles and plates)

• -referred orientation effects can be reduced by sample rotation etc)

•

7owe*er it is possible to obtain useful information when a stationary flat sample holder is used without rotation and when preferred orientation effects are at a ma+imum)

• If de*iations from a theoretical pattern are measured they may be used to monitor the morpholo'y shape of crystals in a production batch)



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Poisson's binomial