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Why should we care about anisotropy?

– Glaciology: ice discharge and sea level

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Why should we care about anisotropy?

– Ice cube: optical scatter length

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Anisotropy types

Intrinsic

Basically single material

• crystal orientation, crystal preferred orientation (CPO)

• crystal shapes, shape preferred orientation (SPO)

Composite

Two or more materials with different properties

• Isotropic or anisotropic in isolation

• E.g. stack of layers

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RHEINISCH- Postadresse: D-52056 Aachen, Germany WESTF¨ALISCHE Institut: J¨agerstraße 17-19, D-52066 Aachen TECHNISCHE Telefon: ++49 241 80 96900 HOCHSCHULE Telefax: ++49 241 80 92184 AACHEN

INSTITUT F ¨UR KRISTALLOGRAPHIE Prof. Dr. Gernot Heger, Dr. Michael Merz

GRUNDZ ÜGE DER KRISTALLOGRAPHIE Das Prinzip der stereographischen Projektion Die stereographische Projektion ermöglicht die übersichtliche zweidimensionale Darstel- lung von Richtungen und Flächenorientierungen. Die Orientierung einer Kristallfläche wird durch die Richtung der Flächennormalen repräsentiert. Der Schnittpunkt der im Mittel- punkt einer Kugel (auch Lagen- oder Projektionskugel genannt) angetragenen Flächennor- malen mit der Kugel ist derFlächenpol(Abb. 1, 2). Jeder Flächenpol wird - mit S bzw. N als Augpunkt (= Projektionspunkt) - in die Projektionsebene projiziert: Stereographische Projektion der Flächenpole (Abb. 2, 3).

Abb. 1Ein kubischer Kri- stall im Zentrum einer Kugel.

Die Flächennormalen des Kri- stalls schneiden die Kugelo- berfläche in Flächenpolen, die auf Großkreisen liegen.

Abb. 2Bei der stereographischen Projektion werden die Flächenpo- le der Nordhalbkugel unter Verwen- dung des Südpols als Projektions- punkt auf die Äquatorebene projiziert.

Abb. 3 Stereographische Projektion der oberen H¨alfte des Kristalls in Abb. 1. Die Fl¨achenpole sind indiziert.

Eigenschaften der stereographischen Projektion:

a) Kreise auf der Projektionskugel werden auf Kreise in der Projektionsebene projiziert (Kreistreue).

b) Der Winkel✏, unter dem sich zwei Kreise auf der Kugel schneiden, bleibt bei der stereographischen Projektion erhalten (Winkeltreue) (Abb. 4). (✏ist der Winkel zwischen den Tan- genten an die Kreise in deren Schnittpunkt.)

Abb. 4Der Winkel✏zwischen den Ebe- nen zweier Zonenkreise ist der Winkel zwischen den entsprechenden Zonenpolen ( ).

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Intrinsic anisotropy - CPO evolution

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Fabric statistical parameters

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S: length of mean c-axes vector N: c-axes number

S3: maximum

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Random Parallelism

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s₁"s₂"s₃"1/3 Strong single maximum:

0#s1"s2#1/6; 2/3#s3#1 Girdle:

0#s1#s2"1/3#s3#2/3 Unimodal:

Girdle:

Strength Shape

e.g.

Fabric statistical parameters

! " Degree of Orientation, R

! " Spherical aperture, !s

! " Eigenvalues, s1,s₂,s₃

! " Woodcocks k value

S: length of mean c-axes vector N: c-axes number

S3: maximum

S1: minimum S2: >S1 & <S3

Random Parallelism

0 100%

0°

90°

Random:

s1"s2"s3"1/3 Strong single maximum:

0#s1"s2#1/6; 2/3#s3#1 Girdle:

0#s1#s2"1/3#s3#2/3 Unimodal:

Girdle:

Strength Shape

e.g.

8"6:/;4$%"$%43=%)'*+

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

>6;$25M%P R""#%6-"%

-25";%H25%#43"2P -36C4$"%

5"-27;$5/-$627

! Z2^%RM74C6-%

;6$";

b2C"%#2;6$627 ,-"%R6W6R"% #2;6$627

F$$#IQQ^^^=6-"47R -36C4$"=706=:/=R:Q5

";"45-F

Intrinsic anisotropy -

CPO evolution with depth

.4C#%."7$/5M

bM"%& 9M5R

LV,1 L,!1,,

D25$FLV,1 c2;$2:

abGZ DaaG b2C"%.

b2C"%<

(6)

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

>6;$25M%P R""#%6-"%

-25";%H25%#43"2P -36C4$"%

5"-27;$5/-$627

! Z2^%RM74C6-%

;6$";

Intrinsic anisotropy - CPO evolution with depth

<65;$%#FM;6-433M%C2$6W4$"R%R""#%6-"%-25"I%

a4;$LV,1 q 34/7-F"R%67$2%$F"%345J";$%6-"%;$5"4C%67%L5""7347R

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

Intrinsic anisotropy -

Shape preferred orientation (texture)

:2)+0298+2/3) /. 50#23

"$/35#+2/3 -20"'+2/3)

!"#$%&'()'*+,-'./01,2&'+3'1"4%+5*%$4*$%&'10,,"6#',"4*$%&' 7899(:;'1'<&,*=-'06<'%&5$2*"6#'#%0"6'>+$6<0%?'6&*@+%A' +>*0"6&<'>?'"10#&'5&#1&6*0*"+6:' B&3*',"4*$%&'5=+@5' 5*%+6#'>204A'2"6&5'@="4='0%&'#%0"6'>+$6<0%"&5'%&C&02&<'>?' 5$>2"10*"+6'&*4="6#:'>+**+1-'./01,2&'+3'DEDF,"4*$%&' 78;9G:;'1'<&,*=-'3+%'BHI'1&05$%&1&6*5' 70$*+10*"4' 30>%"4' 0602?5&%-'06<'%&5$2*"6#'#%0"6F>+$6<0%?'6&*@+%A'+>*0"6&<'

>?'"10#&'5&#1&6*0*"+6' C"0'&<#&'3"6<"6#:'B&3*',"4*$%&'3025&F 4+2+$%'4+<&J'5&&'@=&&2'2&#&6<:

(11

G11

E2 #

E2#

1553.0 m depth

1056.0 m depth

,3:4%8"6:/;4$

anisotropy -

orientation (texture

E2 # E2 #

abGZ%6-"% -25"

8"6:/;4$%"$%43=%)'*+

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

Intrinsic anisotropy -

Shape preferred orientation (texture)

C6-52;$5/-$/543% C2R"3367J%

DEFFEGHHIJ

P W6;-2#34;$6-% R"H25C4$627%

2H%6-"

**P 4"/%-+$11&K$&('. (5.&/"**

Z325"7;% "$%43=%)'*O()'*+

orientation (texture)

L4MGN"5 O.ABP L4MGN"5 O.@>P L4MGN"5 O.@P L4MGN"5 O.>P

,-"%67%74$/5"%6;%4%F2$%C4$"5643 EQE

C

l%'=`

! aS$"7;6W"3M% 5"-5M;$4336B67J

Z325"7;% "$%43=%)'*O()'*+

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

Intrinsic

anisotropy - - Shape

preferred orientation

abGZ%6-"% -25"

;2)8#$()+0#+250#6*7(

:2)+0298+2/3)( /.(50#23("$/35#+2/3)

12345%!-6"7-"%8!(%9"5367

- -

,3:4%8"6:/;4$

8"6:/;4$%"$%43=%)'*+

(7)

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

Composite anisotropy -

Dusty layers – “cloudy bands“

G"$F2RI

c6;/43%;$54$6J54#FM% 367"%;-477"5

!-4$$"5%20T"-$I

• !"-27R%#F4;"% #45$6-3";

• A65%0/003";

• L5467% 02/7R456";

126m

127m 487m

488m 808m

809m 1282m

1283m 1491m

1492m 1691m

1692m

Ee1

abGZ%-25"

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

Composite anisotropy - Dusty layers – “cloudy bands“

G"$F2RI

c6;/43%;$54$6J54#FM% 367"%;-477"5

!-4$$"5%20T"-$I

• !"-27R%#F4;"% #45$6-3";

• A65%0/003";

• L5467% 02/7R456";

126m

127m 487m

488m 808m

809m 1282m

1283m 1491m

1492m 1691m

1692m

Ee1

abGZ%-25"

!W"7;;27 "$%43=%)''_

O-C

D25$FLV,1

@Q@AR@.,

@B>=R@.,

@ST=RC.,

A=?BR>.,.

“Dust“ distribution

a6-F3"5%"$%43=%)'+ !"#$%&'(%)'+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

Composite anisotropy - Dusty layers –

“cloudy bands“

abGZ%0"32^% )&`'P)&U'%C

**F+$.&'$"%21/&1 ;"'#1$&,$".21/&1**

!-4$$"5%20T"-$I

• !"-27R%#F4;"% #45$6-3";

• A65%0/003";

• L5467% 02/7R456";

(8)

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

Composite anisotropy -

Dusty layers – “cloudy bands“

c6;/43%;$54$6J54#FM% 367"%;-477"5

abGZ%-25"

2055m

2056m 2409m

2410m 2455m

2456m

Ee1

Z4M"5;%67-367"R% 47R%H23R"R

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$

Layer deformation

EF2C4;%Y3"67"5 **U/" +8""$ 51(< ,(0"1 ;UVW.*$.ENWF.+&$"**

,7#/$I

P0"R 52-:%J"2C"$5M P;/5H4-" E(%A--V4$"

PJ"2$"5C43F"4$ H3/S

$"7;63" -2C#5";;6W"

a6J"7W43/";%2H ;$5467 54$"%$"7;25 a6J"7W"-$25;m%R65"-$627; 2H

;$5467 54$"%$"7;25 -2 C #5" ;;6 W" $" 7; 63"

<32^%C2R"3 ^6$F 6;2$52#6- H32^ R";-56#$627r

8"6:/;4$%"$%43=%)'*+

Anisotropy in ice sheets - some types and causes

!"#$%&'(%)'*+(%,-"./0" 12345%!-6"7-"%8!(%9"5367 ,3:4%8"6:/;4$