Acoustic phonon anisotropy: Phonon focusing

Computer Physics Communications 28 (1982) 103—107 North-Holland Publishing Company

103

ACOUSTIC PHONON ANISOTROPY: PHONON FOCUSING G.A. NORTHROP Physics Department and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Received 29 December 1981

PROGRAM SUMMARY Title ofprogram: ACOUSTIC PHONON ANISOTROPY Catalogue number: AAOJ Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer: DEC LSI 11-23

Nature ofthe physical problem Calculation of the phase velocity, polarization vector, group velocity, and phonon-focusing enhancement factor for non-dispersive acoustic plane-waves of any wave-vector. Method ofsolution The group velocity and the phonon-focusing enhancement are calculated from the derivatives of the characteristic equation of the Christoffel equations.

Operating system: RT-11 Version 3.0 Programming language: FORTRAN IV version 2.5

Restrictions on complexity The material must be described by a symmetric real 6-vector elastic constant matrix.

High speed storage required: 13 Kwords No. of bits in a word: 16

Typical running lime 0.3 5.

Peripheral used: line printer

References [I] H.J. Mans, J. Acoust. Soc. Am. 50 (1971) 812.

No. of cards in program and test deck: 1073

[2] G.A. Northrop and J.P. Wolfe, Phys. Rev. B22 (1980) 6196. [3] M.J.P. Musgrave, Crystal Acoustic (Holden-Day, San Francisco, 1970) p. 84.

Keywords: solid state, acoustics, phonons, elastic waves, anisotropy, phonon focusing, heat-pulse

OO1O-4655/0000—0000/$02.75 © 1982 North-Holland

104

GA. Northrop

/

Acoustic phonon anisotropy: photon focusing

LONG WRITE-UP

I. Introduction

The group velocity is the gradient of frequency wave-vector space. For non-dispersive media this reduces to

in

This paper describes a program for calculating certain properties of non-dispersive acoustic phonons in an anisotropic solid. This computation was originally applied to measurements of anisotropic phonon flux in heat pulse experiments, an effect known as phonon-focusing [1]. In the process of obtaining this anistropy, other useful values are calculated. These include the phase velocity, the group velocity and the phonon polarization. An elastic constant matrix, material density, mode and wave-vector direction are required as input, The program is a single subroutine which calculates the requested values for a set of input parameters each time it is called. It is intended to be linked with the user’s program to obtain the desired phonon characteristics. This subroutine is supplied in two versions: GCHAN, which will work for any real symmetric elastic constant matrix, and PCHAN, which assumes cubic symmetry and accepts only C 11, C12 and C~as input,

2. Calculation This calculation is based upon the plane wave solution to the linear elastic equation of motion for an infinite, continuous, but anisotropic medium [2]. Using a Cartesian coordinate system with subscripts i, j, 1, m = 1, 2, 3 (x,y, z) and assuming summation over repeated subscripts, this yields the Christoffel equations, (D, 2&,,)e, 0. (1) 1_v In eq. (1) D, = (1/p)Cjjimfljflm, n is the normalized wave-vector, C,J/m is the 3-vector elastic constant matrix, and p is the material density. The characteristic equation det(D,,_v2~,,) 0 (2) yields three solutions, or eigenvalues, v~.The mode, a, is labeled by a 0, 1, 2 and each mode has an eigenvector ~.

v = av + I

n,.

—

(3)

The direction of the group velocity or wave-normal may be defined by the two angles of a spherical coordinate system, which 0 meausred from the + z axis and ~ measured from the + x axis toward the +y axis, in the x—y plane. The group velocity 4)~)is a function of the direction of direction (0~~ the input wave-normal, (Ok,4’k). This mapping may be expressed as cos

= =

f(cos °k’4) g(cos 9k’ 4)k).

(4)

The phonon flux enhancement factor is the ratio of an incremental solid angle in wave-vector space to that in group velocity space [3]. Thus, the enhancement is the inverse of the Jacobian of the transform in eq. (4): J—=detl d~2~ d~k

/ II

I

aj a~O5e~,

aj I

84)k

0g

8g

a cos °k

84)k

I

(5)

/

Computation of J’~and J requires first and second derivatives of v with respect to the wave-vector components n,. This is accomplished by twice implicitly differentiating eq. (2).

3. Program organization The program deck consists of PCHAN and GCHAN, each of which may be used independently. If only cubic symmetry is required it is recommended that PCHAN be used, as it is somewhat faster and more accurate. For any other symmetry GCHAN must be used. They are functionally identical, with the exception of the form of the input elastic constant matrix.

GA. Northrop

/ Acoustic phonon anisorropy: photon focusing

105

Table I Input/output variables and their representations Variable

FORTRAN

I/O

9k

THETA PHI TYP IFLAG THETAV PHIV YACOB

I I I I 0 0 0

wave normal direction

CII, C12, C44 C(I, J) RHO JM VP VGR VG(I) PFI, PF2, PF3

I I I I 0 0 0 0

elastic constants for PCHAN elastic constants for GCHAN material density mode phase velocity group velocity group velocity components polarization vector

4’k — —

J C 11, C12, C~ C’1] p a v

V

The subroutine is called in the form CALL PCHAN (THETA, PHI, TYP, IFLAG, THETAV, PHIV, YACOB) and includes the followinglabeled common blocks: COMMON/PHYS/CI I, C12, C44, RHO, JM (PCHAN) COMMON/PHYS/C(6, 6), RHO, JM

sets degrees or radians sets degree of completion group velocity direction Jacobian

program. Setting IFLAG = 1 will produce solutions for only the phase velocity and the polarization vector. To obtain the group velocity and associated angles requires setting IFLAG 2. Setting IFLAG 3 completes the entire calculation. This feature = allows a considerable increase in execution speed for some applications. The only place in the calculation where units occur is in equations of the form

~/

_________

(GCHAN)

velocity—

elastic constant density

(6)

COMMON/VEL/VP, VGR, VG(3) COMMON/POLVEC/PF I, PF2, PF3 GCOS, COMMON/YACDET/FCOS, FPHI, GPHI All of the input/output variables and their representations from section 2 are listed in table 1. Two of these, TYP and IFLAG require extra explanation. The parameter TYP allows the input of THETA and PHI and the output of THETAV and PHIV to be either in radians or degrees. Setting TYP = I HD indicates degrees and TYP = 1 HR indicates radians. The default is radians. The parameter IFLAG allows the subroutine to attain various stages of completion before returning to the main

6 the elastic constants mustinvolves be expressed in Since the characteristic equation units a reasonable absolute value, terms so as of to order yielding v avoid floating point overfiow/underfiow errors. A good example would be C’s in units of lO’° dyne/cm2, p in units of g/cm3 and v in units of io~cm/s.

4. Test programs and accuracy The programs PTEST and GTEST, when linked with PCHAN and GCHAN, will output all values for several directions and all modes in Ge and sapphire. Degeneracies in the eigenvalues occur

106

GA. Northrop / Acoustic phonon anisotropy: photon focusing

along certain directions, causing inaccuracy in the differentiation process. For PCHAN a 30 bit word size provides sufficient accuracy, except within 10 of a degeneracy where a 60 bit word is required. For GCHAN a 60 bit work is suggested for all directions.

ing support for and discussion of this work. I acknowledge the support of the National Science Foundation under the Materials Research Laboratory Grant DMR-80-20250.

References Acknowledgements I wish to thank S. Kirch for help in debugging GCHAN and Professor J.P. Wolfe for his continu-

[II G.A. Northrop and J.P. Wolfe, Phys. Rev. B22 (1980) 6196. [2] M.J.P. Musgrave, Crystal Acoustics (Holden-Day, San Francisco, 1970) p. 84. [3] H.J. Mans, J. Acoust. Soc. Am. 50 (1971) 812.

G.A. Northrop

ri I... 0..

C-~ Li.. c,~

— 0.. 0..

1100 01.30 ‘5110

r-~’~

000 000 000 000 0

000

000

00-~

000 000

Or-..-4 ‘-4N..-~

99

r-.O’)N 1111)0 ‘SIN

,4’O ,-1,-40 r..r-.o 000 N.r-.O

00

001

000

0110 0)030

0N0

r1’So 0.00 Coo

r~10rN 11030 ‘0)1

,-~—0 ‘-4,-40 r—.N0 000 000

000

000

000

c.c’-o

Coo 000

I’.~ON) 0 0.

0.. 0’

000000 000000 000000, 000000 0090

0) 0 U C >-

0000003

000000 000000 000000 000000 900000

0. 4-. 0 0

000000

0. .0 I— 03 0

~-

0000300 “3

1~’00 0.~-JO 100)0

o~~o

,—11N,-4

‘-40,-s

,-4-41’)

011)0

101.1-0

W3C1N.

1.101’) 0—,-40 MC’ILl

03,-sC-I ,-4L3 ‘SO’S

C~.M ‘SOr-. ‘SOIl)

L1M1’)

L1r’)N)

Oc-Ir’)

0-101’) 110-I. 00)1’) d)’SN. 00.11

‘-40.11 0-NO 110110 0.0)0. lION)

—s,-~N) 1411-IN (30-Fl ‘54301. ~r..o

100-IF’)

OF’)

~ .0 E

~ ~

O

Z ~

~

O 40 4-(1) Z 03 O U — (1) C .J

w

0303 C~— 034’ U~ 0.. Li 00 03 — Cc 0)0

oo-sooo 000000 000000 000000 01101000 03 0-0-000 ‘S’SFl ‘-4 0’—,OOOO’S 0000001’) 000000 0000000 011”011000

3: )

0 0 •

— 0 0.

— 0. ~

<0 4— 1.41 03 4—

110-0.

C’iCJL.) 1—4 0-’S03 C

‘SO,0003 O’S’S 00311

loor-.

N.00 ‘SLl~~ 9010

000

000

000

r-~ ‘5-010

C-ION ~-4NL3

lf)’~.03

03110 ‘-4111’)

M’S0) ~0’003 1•F1.0 01103

000

CS

000

0—0 ,-4C4-.O3 ~‘)~O—

1f3—Ifi OL3,-4

0300. 0-.~0C?.

001-lId 1-1141’S 0-0-0(3-43— -CON)

000000 000000 000000 000000 0000011’S

0010

0r~o

0100

0000111.

000 000 000 000 4

0011 0(1-0 00-0 00-0 00-0

0-0 001-0 0O-0 ooo 00-0

0-00000 000000 000000 000000 0000’5tl

000

0’S143 ‘S’S’S

10’S,,’) ‘S’S’S

0000,,’)

,-400 1410110 00-0 14300 ‘500

0.0.0 00111(3 01100. 011011—s INN~,

000 000 000 000 000

000000 000000 000000 000000 C-IC-Io’SOO

0.00 Fl 0

0-00. M’s.-~

000 0.0-0-

CI 4— C 3:

1100000 010-1 ‘0 —

000 000 000 000 000

000 000 000 000 000

000 000 000 000 000

103 C ~ 0)

000000 000000 000000 000000 00(1000

0~(1

0~s01i

0

0 U

000000 ,-s-s0-

000 000 000 000 000

000 000 000 000 000

000 900 000 000 000

10) C

000

0141141 ‘S’S’S

10014) ‘S’S’S

= I— — 0

000 000 000 000 000

000 000 000 000 000

000 000 000 000 000

000 lIMO

000 0110

000 0-0-0.

U

0303 CCI 44 O 00.. 40 00) Z 4-~CO (30 =0..

~

0030

000

000

000

03

N)0-r.

0-i-.O0~ ‘-40’S

0.

‘-~0’-, 0.-43CI NO-

00103 N.003

0-.’So Ifl0300 O’SL1 ‘-bId ‘-b0-.s3

0-0.)

00’I3

.~~4L14)

‘5’-4’S

0-’S’O 00’S 1101.03 NO-N

0.,-#011-001. 0110) 3(-,’0 ..~c~-a

C-,’5 ‘50303 0..3’-4 0-,-4N.

0O~.

O-.O~’3

,-40-43

(4-4311 N3’SFI .-8~430. (‘10-10. 003,-s

0-1110 C’i,-40-I C-IC-I’S C-I-IN 001,1

143’SIf) .00~-4 N0~43 0011 00101

0—C

‘-‘-00

00110

143-I’S 011-10 0~4N r-..r..0-I F’)Nr-)

.-403-’ 00-10 001.0~ ,-4F~.1-1 0-’S03

0.0.0. o-r-.n. -(‘103 0301.0 01410

00-03 c-l~

0(1.01’)~-1’)

‘SOC-I ‘SOLO

CO’t’ ‘5-’YM “3030 0.00.. ‘SIll)

‘-~O0 11-CO ‘S’S0-i W3’S0. (10110

00110.. 0303011 —3~430) ‘00011 1)000

143(00 0-111’S

1lFl~—1 (-I’S’S

1)’S’S 0-030

000 000 000 000 000

000 000 000 000 000

000 000 000 000 000

001l

0~r-I

0011

000 000 000 000 000

000 000 000 000 000

000 000 000 000 000

000

10101(3 ‘S’S’S

101011) ‘S’S’S

000 000 000 000 000

000 000 000 000 000

000 000 000 000 000

000 0011

000 0011

000 0-0.0-

0) 0 U .0 >~

0. -4 = 0.

:~_1

>1

~

‘)~I1C)

0.. 0’

0’ C I— 03 0 4-

3: 03

‘~00

o,-~o- r..o-o

0-43’S 010.0

CO

s,-44) 00..o) 0--IC-I 0-Or.. ‘OO

-~

r~.oc~-~ COMM

‘S~0 CO.’5

U

0-’-b0-000 ‘SM’S

4000001000000-4 0000000’) 00000003 ‘—‘014C-I000LLI 40. .-40-01000 N1~’S 03 .-‘ 3: 40

03 U

10111

-4

cc

107

‘1 0..

011.43 0’S 0-00. 0)’S~-~ 03.~F’1

-.13

000000 000000 000000 000000 0000100

/ Acoustic phonon anisotropy: photon focusing

.1030

11,.4,4

01101

coo

0 0000000 0000090. 000000 0000001’) ‘50-10000

-

— 0 0.

000000 ..343..—4011 -‘‘1--0000004-000000-4 03000000) 00000003 01I’SOC-IOOLi .••‘03 01101)00 0-C,—4011 03 ‘5* -~ 03 ‘0

.3 1-L..0 4—