Note on ultrasonic effects due to third-order elastic coefficients in icosahedrally symmetric systems

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Note on ultrasonic effects due to third-order elastic coefficients in icosahedrally symmetric systems

Shmuel Goshen, Joseph Birman

To cite this version:

Shmuel Goshen, Joseph Birman. Note on ultrasonic effects due to third-order elastic coefficients in icosahedrally symmetric systems. Journal de Physique I, EDP Sciences, 1994, 4 (7), pp.1077-1084.

�10.1051/jp1:1994185�. �jpa-00246965�

Classification

Physics

Absn.acts

43.25 61.50E 62.65

Note

_on

ultrasonic effects due to third.order elastic coefficients in icosahedrally symmetric _systems

Shmuel Y. Goshen

(1)

and

Joseph

L. Birman

(2)

(') Physics Dept.,

Nuclear Research Center, Negev, ^{P-O- Box} 9001, Beer-Sheva 84190, Israel (2)

Physics Dept.,

City

College

of the

City

University ^{of New York, New York,} NY10031,

U-S-A-

(Received 6

September

J993, ievised14 Maiih 1994, accepted 2/ Maic.h 1994)

Abstract. The identification of

alloy

systems ^{of Almn and} other compounds ^with macroscopic icosahedral symmetry ^{bas been} reported. ^In this work _we discuss several

macroscopic

tensor

properties

which may

distinguish

between icosahedral

point

_{group symmetry} I, from either fuit isotropic SO (3) symmetry, or cubic O symmetry.

Emphasis

is given to trie structure of trie third order elastic coefficient tensor and the non-linear elastic coefficients which _anse

in I-symmetry systems.

Corresponding

sound velocities _are obtained these _can permit

experimental

measure-

ments using ultrasonic

techniques.

One of the very

interesting

recent

developments

_m condensed matter

physics

has been the

discovery

and

synthesis

of _a

variety

of cristalline systems ^whose symmetry ^is

non-crystallogra- phic.

That is neither the

point

_{group symmetry nor} the space group symmetry are among the

32,

_or 230

(respectively)

_groups of 3-dimensional

crystallography.

The first of these _new discovenes _{to our}

knowledge

_was the

synthesis

of _an Al-Mn

alloy

with

stoichiometry approximately Al~mn by

Shechtman et ai.

il ]. They

carned _out electron- diffraction studies which demonstrated that the pattem ^{of scattered electrons}

possessed

icosahedral symmetry. ^In

addition, algorithms

have been

developed

for

mdexing

of the

locations of observed diffraction spots, consistent with the

predictions

of

quasipenodic

translational order.

Quite recently, large perfect samples

of

quasicrystals

have been available owmg ^to new

developments

in material

preparation.

This has

permitted

the first observation of

dynamical

X- ray diffraction effects _as earlier

predicted [2],

_{on a}

quasicrystal

of _« f-c-c- _» Al-Pd-Mn,

including

the observation of the Borrmann

effect,

_{or «} anomalous transmission at the

Bragg angle

_»

[3]. Samples

_on which the expenments were

performed

_were of centimeter size. With such

samples available,

it can be

possible

to determine novel

macroscopic

_properties of these

icosahedral materais.

In this note we restnct attention to some of the novel

physical,

measurable macroscopic

consequences of icosahedral

point

_symmetry. To

phrase

the

problem

_more

exactly

we ask what type ^{and order of tensonal} response coefficient _can be used to charactenze the presence

JOURNAL DE PHYSIQUE -T 4 N' 7 JULY J994 ~9

1078 JOURNAL DE PHYSIQUE I N° 7

of icosahedral symmetry, ^and

distinguish

from

isotropic (complete

SO

(3)),

_or cubic

(point

group

O) symmetries

^? We suppose a

single crystal

of suitable _size is

available,

and _can be

aligned

so that suitable

polarized

measurements can be made.

As _a first step we recall

[4]

that the components ^{of the}

physical

_response functions for

crystal

systems can be classified _as cartesian tensors, the indices of which possess some

permutational symmetries.

These _{tensors are} reducible _{into a sum} of irreducible components. ^Thus in an

isotropic

SO

(3)

system ^the response functions transform _{as a sum} of

sphencal harmonics,

and the irreducible representations are

conventionally

denoted

D~~'.

In the presence of G point group

symmetry

^{the bases of each}

representation

D~~~

splits

into irreducible

representations

of G

D~~~

= D~~ e D~"~ e

where D~'~,

D~'~

are irreducible

representations

^{of G. Then}

by

_measunng the components associated with different irreducible

representations #1',

effects which _are

specific

_to the lower («

spht »)

symmetry can be identified. The number of

physical

parameters ^{is determined}

by

the number of times that the

identity representation

_appears.

Applymg

this to _{our case,} m table I _we grue the standard character table of the icosahedral group ^I and the

compatibility

(or

splittings)

table of the representations

D'~~

_{under I. From this} table, and the earlier work

by Ripamonti [5]

_{we can} find _a vanety ^of «

higher

order

» tensonal

properties

which may be used _to differentiate between

O,

I, and SO

(3).

These _are given in

table II.

Table I.

a)

Chaiac.tel table

defining

the

iii educible representations

of

the icosaheclial point groiip I.

b) Compatibility

table

foi-

ieduction

of

D~~~

of

SO

(3),

iestricted to point gioiip I.

a)

E 15

C~

20 C~ 12 C~ 12

Cl

A

~ ~ ~

,à

+ _,

,<'5

' ~

2 ^~ 2

~ ~ j ~

,,1- ,/j

+

~ ~

2 ^~ 2

G 4 0

H 5 0 0

b) Do

=A

Dj =F,

D~

=

H D~

=F~+G

D~

^=G+H

D~

=Fi+F~+H

D~ =A+Fj+G+H

An interesting ^and

practical

_{case is} of nonhnear elastic _wave

propagation

^{where the third} order elastic

(TOE)

coefficients _are used. For the icosahedral

symmetry

^there are four

Table II.

Highei

oider tensoi

piopeities (see Ref [2]).

Physical

_property, ^{Number of} parameters

Physical

_property

representing

relation

between _: Isotr. I

Symmetric

tensor and 3 4 6 56 Third order elastic

square of

symmetric

coefficients

tensor

subject

also to Maxwell relation

Symmetric

tensor and 4 5 9 126 Second order

photo-

square of

symmetric

elastic coefficients

tensor

Second rank

symmetric

2 3 6 90 Second order resis-

tensor and _a

totally

_tance second order

symmetric

fourth rank

magnetothermal

_con-

tensor

ductivity

Second rank

general

2 3 7 135 Second order magneto-

tensor and a

totally

electric power

symmetric

fourth rank tensor

Second rank

symmetric

5 6 12 216

Piezomagneto-

tensor and a fourth resistance

rank _tensor

symmetric

inside the first

pair

and the second

pair

parameters, while for the

isotropic

_case there _are three. For the second order elastic coefficients of the linear elastic _wave

propagation

both

symmetries,

the isotropic ^and icosahedral have

only

two parameters. ^{The 2 +4} = 6 parameters ^{of materais of} icosahedral symmetry can be

jnvestigated by

ultrasonics. Five of the parameters are the _{same as} t'or

isotropic

symmetry.

How

significantly

the sixth parameter differs from zero will grue the

special

properties ^related

to the icosahedral symmetry.

By

these

investigations

it _is

impossible

to decide about the _existence of _a center of symmetry because

they

_are based _on second order _tensors. Nonlinear effects in elastic _wave

propagation

may anse from several different _causes. We shall discuss and calculate the

following

two

cases. First, the

amplitude

of the elastic _wave may be

sufficiently large

so that finite _strams

anse. An

initially

sinusoidal

longitudinal

_wave of _a given

frequency

distorts as it propagates,

and energy is transferred from the fundamental to the harmonics. The _amount of energy

transferred

depends

_on the TOE coefficients and is _a way of

experimentally determmmg

them.

Second, a materai which in _its undeformed state behaves _{m a} lmear fashion may behave _{in a}

nonhnear fashion when mfinitesimal ultrasonic _{waves are}

propagated, provided

that _a

sufficient amount of externat _{static stress is}

imposed. Usually

this _{is in} the form of uniaxial _or

hydrostatic

compression. The formulas for the velocities for

longitudinal

and _transverse

1080 JOURNAL DE PHYSIQUE ^I N° 7

modes, in the different directions under different extemal _stresses

permits

^the

experimental

determination of TOE coefficients. We _use the well-established notation of Green

[6].

The

Lagrangian

symmetric strain matrix _{y is}

given by Mumaghan [7]

_as

y =

(ÀJ à) (1)

where J is the Jacobian _{matrix :}

ôXj

J~t

w

(2)

ôat

Here a,

(1=1, 2, 3)

_are the _cartesian coordinates of _a point ^{in the unstrained}

solid, X,

(1 ₌ 1, 2,

3)

are the coordinates _at time _t of the point in the deformed solid, and à is the _unit

matrix.

The

engineering

stress tensor

(which

is not

symmetnc)

is defined

by

T=J~ ₍₃₎

Y

where

#

is the strain energy, and its form _is

specified by

the symmetry ^{of the solid. The} equations of motion in the

Lagrangian

coordinates for the deformations

u~ ^are

~Î

ôT~~

$

^{~ P° ~i}

^~~l

j J

where u, =

X,

_{a, (1}

= 1, 2, 3 ), _po the undeformed

density,

and ü _is

Î~

^~ _in the

Lagrangian

formulation.

Now

wnting

for the strain energy çS, when the _z axis has _a 5-fold symmetry, and the y-axis is

of _a ?-fold symmetry

~ ÀL

(Y22

_{Y33 Y23 Y32} + Y33 TII Y13 Y31 + TII Y22 Y12

Y21)

+

+ ~~ ~

~'~~

(Yjj

+ Y?? + Y33)~

3

2

m(yjj

+ y~~ ⁺

y33)(Y22

Y33 Y23 Y32 + Y33 Yl1 Y13 Y31 ⁺ Yj Y22 Y12 Y2j +

+

n(yj

_{y~2 Y33 Y11 Y23 Y32 Y21 Y12} Y31+ Y21 Y13 Y32 + Yii Y12 Y23 Y31 Yii

Y22)

+

~[YÎ3

₊

(TII

₊

YÎ2)

_Y33

YÎ3(Yll

₊

Y22)

₊ 2 _TII Y22 Y33

+ ~ Y13 Y31 TII ⁺ ~ Y23 Y32 Y22 ~ Y13 Y31 Y3~ ~ Y23 Y32 Y33

+

(Y13

+

Y31)(Yll

_{Y22)~ +}

~(Y22 Yll)(Y12

_{Y23 + Y32}

Y21)

~ _{Y12 Y21}

(Yl~

_{+ Y~l +} 4 Y12 Y23 Y31 ⁺ ~ Y13 Y~2 Y211

(5)

The five terms whose coefficients _are (2 _{M +}

),

_M, ^f ₊ 2 _{m, m} and _n,

respectively,

are

isotropic. ^The calculation of these terms is descnbed in Green

[6].

The last term, which is anisotropic ^but has

only

icosahedral symmetry, was calculated

by

Butler's tables. We shall also need the expression for _çS when the _{z axis} has _a 3-fold symmetry ^and _{the y} axis has _a 2-fold

symmetry. ^{Then this last} term becomes

~

ÎÎ °'~' /7

°'~~

°'~'

_°'~~

~°'~'

~ °'~~~ °'~~ ~ °'~~ °" _' ~ +

j _YÎ3(Yll

+

Y22)

+

)

TII Y22 Y33 + Y12

Y21(~

_Y22 ^~ _TII

Y31)

+ Yl~

Y31(~

_Y3~ ^~ _TII ~

Y22)

+ Y23

Y32(~

_Y33 Y22 ~ TII +

(Y12

_{Y23 Y31} + Yl~ Y32

Y21)

/j

+

'j _(Y13

₊

_{Y31)(5 YÎI}

~

YÎ2

^~ _Y11 Y22 ~ _Yl1 Y3~ + 8 y22 y~3

~ Y12 Y21 + 16 Y23 Y32

~/

^{Y13 Y31 +}

+'~~ (yj7

_{y71+ y17 y7j}

)(16

_vii 4 yjj 12

~~)j

(6)

9

First _case.

Harmonic distortion of _a finite

amplitude

harmonic wave. For a pure

Longitudinal

ultrasonic

wave we get

Po =

(2

_» + À

)(u,,u fia,,11,,,,) ^(7a)

p

is the

nonlineanty [6] parameter

:

P

=

3

+

~~Îi/ i

i~~~ ^(7b)

Here _:

K

=

3 _{g in} the direction of 5-fold axis K

=

0 in the direction of 2-fold _axis

(7c)

K

=

~

g in the direction of 3-fold _axis 9

The solution up to the second harmonic

[6]

r (a, t ₌

Ai

sin

(ka

wt + A7 _cos 2

(ka

_mi

(8a)

Ai

is the fundamental _wave

amplitude.

A~,

the

amplitude

of the

generated

second harmonic

A~ ₌

j _(A)

_{k2 ap}

_{). (8b)}

Therefore,

the

discontinuity

distance

[6]

_{is :}

~

p

2

po(~f)~

_ii~ ^~~~~

where _w

=

?

~f

and uo is the

amplitude

^{of the} wave at _a 0.

Second _case.

Measurement of the variation of ultrasonic _wave

velocity

with

applied

externat stress.

1082 JOURNAL DE PHYSIQUE I N° 7

Let _us define

[6]

_:

il _: the wave

velocity,

pjj the unstrained

density,

P _an

imposed

externat

hydrostatic

pressure,

T _an

imposed

externat uniaxial _stress in _a

specified

direction.

We get ^the

following

equations ^{for the}

velocity,

V, of the _{waves in} the directions of symmetry, ^with constraints indicated

Extemal

hydrostatic

_pressure

longitudinal

wave

along

ail symmetry directions.

poil~= _(2p

_+À)- ^~

_(7À +10p +6i+4m _+2g). (9)

3À +2 _p

II. Externat

hydro~tatic

_{pressure ;}

transverse wave

along

ail symmetry directions.

pov~=p-

^~

[6p

^{+3À +3m-}

_~nj (10)

(3À +2p) 2

III. Externat compressive stress

along

5-fold _{axis ;}

longitudinal

_wave ^I _stress.

p~V~=(2p+À)-~~~~~ ~(~~~(4À+10p+4m+6g)+~~~+À+2ij,

+-R _{R R}

(11)

IV. Externat compressive stress

along

2-fold _axis

longitudinal

_wave

along

5-fold _{axis 1 stress.}

Po~'~= ~2/l+À)~~~ l~~~12f-~/

_(m+À

_{+2JL)l~(2À +JL)1. ('2)}

V. Externat

compressive

stress

along

3-fold _axis

longitudinal

_wave ^I stress,

v~~?/~+~~-~~iili~jil~io~+~m~+~+2fH~'2~+~/~~'.

"~~

VI. Externat compressive ^stress

along

2-fold axis

longitudinal

wave

along

^3-fold axis 1stress.

T

[~(_~~(m+À+2Rl+@~~~~~~~Î

~~~~

p~~i'2=(2p+À)-~~~

+2p)

_R

VII. Externat

compressive

stress

along

5-fold axi~

longitudinal

wave

along

2-fold _{axis 1 stress,}

p~V~= _(2p

_+À)- ^~

[2i-~~ _(m+À ^+2p~+~~~ _~~~gj (15)

(3À +2p)

_{p R}

VIII. Externat

compressive

stress

along

2-fold axis

longitudinal

_wave ^I stress.

p~V~= _{(2p +À)-}

^~

[~

^~~

^(4À

^+10p

+4m)-~~

+À

+2ij. ₍₁₆₎

(3À +2p)

_p p

IX. Extemal

compressive

_stress

along

5-fold axis

transverse wave I stress with

particle displacements along

?-fold axis.

p~V~=p-

~

~

[m+4p+4À +£-~~~~~~~~j. (17)

À

+2p)

_p p

X. Externat

compressive

stress

along

2-fold axis

transverse wave

along

5-fold _axis 1 stress with

particle displacements

^I ~tress.

p~v2=

~_

T

[~~~

~ ~

nÀ

(3À+2p)

~~~~~/~)

^{~ ~~}

4p~

p

~' ~~~~

XI. Externat compressive ^stress

along

?-fold _{axis ;}

transverse wave

along

5-fold _{axis 1 stress} with

particle displacements

1 stres~.

p~

V~

= p

~

~

m

2 ~~ _{~ ~ ~~}

(

+2p) 2p

(19)

In conclusion _we have

presented

results which _can be used in the determination of the icosahedral component ^of macroscopic elastic response of _a

quasicrystal.

Ii would be of considerable interest _to have such measurements made _on the _new,

large-size

materais, in order to check these results, and aise with _{an atm} of

relating

to

microscopic

theories of

elasticity

of

quasicrystals recently proposed [9].

A recent

expenmental

review of the

physical

_properties of

quasicrystals (10] heightens

interest in the measurement of macroscopic properties ^{of these} systems. ^A

noteworthy

related

work

by

Amant, de Boissieu and Zarembowich

[1Ii

_measures elastic properties

ultrasonically

under

hydrostatic

_pressure. No anisotropy _was found and

they reported

measurement of

only

two

independent

linear combinations of the four TOE coefficients (see ^here

Eqs. (9, 10)

and Tab. I in

[1Ii).

Acknowledgments.

We thank Prof. F. Fumi, and _a referee for

pointing

Dut the relevant work of reference [5

].

One of _us (JLB) thanks Prof. A.Goldman for

sending

the

preprint, reference[3(,

before

publication,

aise Prof. R. Colleta for discussions of

possible Dynamical

Diffraction exper-

iments which _can be carned _{Dut on} the

large quasicrystals,

and Prof. A. Zarembowitch and

Dr. B. Perrin for discussions of their _recent work

[1Ii

The work was

supported

in part

by

a

PSC-FRAP-CUNY grant.

1084 JOURNAL DE PHYSIQUE ^{I N° 7}

References

il Schectman D., Blech I.. Grattas D. and Cahn J. W.,

Ffij,s.

Roi. Len 53 (1985~ 195 A review _is given in Gratias D.. Cfimenip Ffiys. 28 <1987) 219.

[2] Berenson R. and Birman J. L., Ffiy.i. Roi>. B 34 <1986~ 8926.

[3] Kycia ^S. W., Goldman A. I., Lograsso ^{T. A.,} Delaney D. W.. Black D., Sutton M., Dufresne E..

Bruning R., Rodencks B., Fhj~s. Rei>. B 48 <1993) 3544.

[4] Bhagavantam S., Crystal Structure and Physical Properties <Academic Pre~~. London-New York, 1966).

[5]

Ripamonti

C.. J. Ffi_i~5. Fiaiic.e 48 <1987) 493.

[6] Green R. E. Jr.. Ultrmonic

Investigation

of Mechanical

Properties

<Academic Pre~~. New York- London, 197~j wherein Table gives the relations between the different notations

Breazeale M. A. and Philip J., _{p. in} Fhi>sic.a/ Acoi<.çiic..ç <edited by W. P. Mason and R. N.

Thurston), Vol. XVII (Academic _Pres~, New York, 1984) _; Breazeale M. A. and Ford J.. J App/. Fhys. 36 <1965~ 3486.

[7] Murnaghan F. D., Finite Deformation of an Ela~tic Solid

(Wiley.

New York, 1951 Dover, New York, 1967~.

[8] Butler

Philip

Il., Point

Group Symmetry

Application~ <Plenum Press, New York-London, 1981) [91 ^Bak P., Ffiv.i Rei B 32 (1985) 5764 _:

also _~ee

Quasicrystal;,

Theory ^and Experiment (Vols. 1-3) M. V. Jaric Ed. <Acaclemic Press, 1989- 90).

loi Jeong H. C. and Stemhardt P. J., A New Approach _to Quasicrystal~ <prepnnt U. of Penn, 1994).

[1Ii Takeuchi S..

Phy~ical

Properties of Quasicrystal~. An Expenmental Review, _in

Quamcrystal~

anal

Imperfectly

Ordered

Crystals

(Tran~. Tech. Pub., 1994),

Proceedings

of _an International

Symposium

_in

Chengde,

China (August 1993).

See especially Amazit Y.. de Boissieu M.. Zarembowitch A.. Eurfipfi_1,sic.i Len. 20 (1992) 703 Amazit Y., Fischer M.. Perrin B., Zarembowitch A., de Boissieu M., ibid. 2511994) 441.