A COMPUTER SIMULATION OF THE EVOLUTION OF COHERENT COMPOSITION VARIATIONS IN SOLID SOLUTIONS

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DE FONTAINE, Didier, 1931A COMPUTER SIMULATION OF THE EVOLUTION OF COHERENT COMPOSITION VARIATIONS IN SOLID SOLUTIONS. Northwestern University, Ph.D., 1967 Engineering, metallurgy

University Microfilms, Inc./ Ann Arbor, Michigan

A COMPUTER SIMULATION OF THE E V O L U T I O N OF COHERENT C O M P O S I T I O N VARIATIONS I N SOLID SOLUTIONS

A DISSERTATION S U B M I T T E D TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree

DOCTOR OF PHILOSOPHY Fi e l d of Materials Science

By

Didier de Fontaine

Evanston, Illinois June 1967

Summary of the Di.ssert-ati.on,

A COMPUTER SIMULATION OF THE EVOLUTION OF COHERENT COMPOSITION V A RIATIONS IN SOLID SOLUTIONS

By

Didier de Font'aine

A m a t h e m a t i c a l m o d e l governing the b e h a v i o r o f coherent composition vari a t i o n s in cubic crystals of b i n a r y and multicomponent systems has b e e n g i v e n and computer-generated n u m e rical solutions have b e e n obt'ained in some r e l a t i v e l y simple cases.

A single non-linear part'ial differential diffusion

equation, originally derived b y Cahn, or systems o f diffusion equations, governed all processes,

from the init'ial to t'he final coarsening stages

for

any compositi.on and temperature in t'he coherent phase diagram. The case m o s t extensively studied was that o f isothermal decompo­ sition o f A^-Zn a lloys inside the spinodal.

The diffusion equat'ion was

solved for one-dimensional composition variations in periodic domains.

It

w a s concluded that the characteristics o f the composition profiles appear to var y continuously as the average alloy composition is altered fro m the center to the edge o f the m i s c ibility gap; in other words,

there is no abrupt

change in the morph o l o g y o f the resulting structures at the spinodal compos i t i o n s . • For alloys towards the center o f the misc i b i l i t y gap,

a quasi-

sinusoidal structure developed and grew in amplitude unt i l the coherent phase bound a r i e s were reached.

The r esulting structure w a s quite regular and the

average spacing b e t w e e n particles w a s practically independent of the initial conditions.

As the average composition approached the spinodal, the sinusoidal

p r o f i l e s gave w a y to composition v a riations rese m b l i n g discrete Guinier zones (central enriched regions surrounded b y depleted zones) distributed almost randomly.

A structure consisting o f regularly spaced precipitates

developed w i t h the spacing between precipitates c l osely related to the radius of the Guinier zones.

For average compositions close to the spinodal,

an initial coarsening m e c h a n i s m b e c a m e important,

and the final structures

depended critically on the initial composition fluctuations. A n experimental verification was provided b y small angle X-ray scattering studies p erformed b y Rundman.

Sequences o f experimental and

calculated intensity spectra were similar with r e s p e c t to overall shape, p o s ition of the maxima,

integrated intensity and progressive sharpening

and shift towards lo n g e r wave l e n g t h o f the m a i n satellite envelope.

Theo­

retical calculations for the continuous cooling p r o c e s s were also confirmed b y the experimental results.

Some calculations w e r e also performed on two-

dimensional and spherically symmetric composition profiles. A theoretical study of the initial stages o f spinodal decompo­ sition in n-component systems was undertaken. spinodal consists of n - 1

It w a s shown that the

surfaces enclosing r e gions in which the solid

solution becomes unstable for different sets o f directions in c o m p o s i t i o n space.

It was also shown that the coherent spinodal surfaces m u s t pass

th r o u g h points where these points, strains.

tyio

or more chemical spinodal surfaces intersect.

At

spinodal decomposition can occur initially without coherency

Ce travail est dedicace

h

Danielie.

TABLE OP CONTENTS

I NTRODUCTION

1

1. D E R I V A T I O N OF D I F F U S I O N E QUATIONS

k

1.1. The Diffusion E q u a t i o n for Two-Component Systems

*+

1.2. The Diffusion E q u a tions for Multicomponent Systems-

9

2. S O L U T I O N F O R THE IN I T I A L STAGES OF S P I N O D A L D E C O M P OSITION

15

2.1. Spinodal Decompo s i t i o n in B i n a r y Systems

l6

2.2. Spinodal Dec o m p o s i t i o n in Multicomponent Systems

23

3. S O L U T I O N OF NON-LINEAR EQUATIONS

37

3.1. Polynomial A p p r o x i m a t i o n of Helmholtz Free E n e r g y

-U O

3.2. Discussion of General Method of Solution

Mt

3.3*

58

The Aluminu m - Z i n c S y s t e m

3.1+. Isothermal A g i n g of O ne-Dimensional Composition Variations

63

3.5. One-Dimensional Simul a t i o n of Continuous C ooling

92

•3 .6 . Solid Solutions w ith Plane-Wave Imperfections 3-7*

Two-Dimensional Composition Fluctuations

3.8. Spherically Symmetric Composition Fluctuations

99 105 . 11^4

CONCLUSION

126

REFERENCES

132

A C KNOW IE DGME NTS

135

VITA

. 137 i

Appendix A

139

: N o t ation

A p p e n d i x B : Green's Theorem

1 I+2

A p p e n d i x C : Interpretation of T; Parameters

lhh

A p p e n d i x D : Systems of Ordinary Linear Differential Equations

lh7

Appendix E

: R e l a t i o n s h i p b e t w e e n C h e m i c a l and Coherent Spinodals

A p p e n d i x F : P o l y n o m i a l Free E n e r g y Approximations

152 .

155

A p p e n d i x G : E x p ressions for M(u) and D(u).

159

Appendix H

163

: Desc r i p t i o n of P r o g r a m THERMO

A p p e n d i x I : Desc r i p t i o n of

P r o g r a m PERI0D1D

171

A p p e n d i x J : D e s c r i p t i o n of

Program FYSPIN1P

l8 l

Appendix K

: D e s c r i p t i o n of

P r o g r a m INKPLOT

191

Appendix L

:D e s c r i p t i o n

of P r o g r a m

FYFILM

197

Appendix M

:Desc r i p t i o n

of P r o g r a m

SPICOC

20l|

Appendix N

:D e s c r i p t i o n

of P r o g r a m

FYSPIMFY

211

Appendix P

:D e s c r i p t i o n

of P r o g r a m

F Y S PIN2D

217

Appendix Q

:D e s c r i p t i o n

of P r o g r a m

PLOTOODY

223

Appendix R

D e s c r i p t i o n of P r o g r a m FYSPHERE

Appendix S

: D i f f r a c t i o n Satellites f rom Spherically Symmetric C o m p o s i t i o n Variations

230

236

LIST OF FIGURES

Fig.

1.

Verti c a l se c t i o n C D through t h e .spinodal surfaces for

a ternary r e gular solution model.

3^

The full curves represent

the chemical spinodal, the b r o k e n curves represent the coherent spinodal.

Fig. 2.

Isothermal section through the chemical spinodal

3^

surfaces for a t ernary regular solution mod e l at k T / m = 1/U. Spinodal d ecomposition can occur for composition fluctua­ tions inside the dark areas, sh o w n at selected points along CD.

Fig. 3*

Helmholtz free e n e r g y data points

(open circles) and

39

fourth-degree p o l y n o m i a l a p p r o x imation (upper f u l l curve). The b r o k e n curve represents a parabolic approximation . about the p o i n t c = c Q .

The se c o n d derivative of the

fourth-degree p o l y n o m i a l free e n e r g y is shown i n the lower par t o f the figure.

Points

c'

and c " are the

spinodal compositions.

Fig.

C sCl structure represented b y a b.c.c. arr a y of tetrak a i d e kahedra instersected b y the symmetry about po i n t r I I I

*.

L

1

(110) plane.

39

Anti­

is indicated.

4 4 J

*Fig. 5. Aluminumr-rich side of the A l - Z n phase diagram.

The

57

h e a v y curve shows the coherent misc i b i l i t y gap, the

.Hi

'

'

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

* .

b r o k e n curve is the coherent spinodal.

The open circles

indicate the all o y systems investigated.

Fig. 6 .

Helmholtz free energy data (open circles) for the

...

A l - Z n system a n d the corresponding fourth-degree polynomial approximation (full curves) at selected temperatures b e l o w the critical point.

The open triangles represent

additional constructed points.

Fig. 7*

Coherent Helmholtz free energy curves derived f rom

the polynomial approximations of Fig. 6 b y the addition of the t erm T f Y c ( c - 0 . 7 ) .

The calculated coherent equi­

librium and spinodal compositions-are indicated b y open circles.

Fig. 8 .

Composition profile used in calculating the normal­

ization factor

aQ used

in the percent completion calcul­

ations .

Fig. 9*

Calculated values of the logarithm of the intensity

(amplitude squared) versus ag i n g time for the first three Fourier components of a n A g - A u layered structure.

Fig. 10 a. Composition profiles corresponding to the spectrum of Fig. 9 at the initiation of the calculations and at time t = 1+0 min.

Fig.

The broken horizontal lines indicate

the pure components A g and Au.

Continued on Fig. 10 b.

10 b. Continuation of Fig. 10 a.

Composition profiles

at time t = 100 and 2 0 0 min.

The b r o k e n horizontal

lines indicate the p u r e components A g a n d Au.

Fig. 11.

A m p l i t u d e spectra

(absolute magnitude) and corres­

p o n d i n g composition p r o f i l e s calculated according to Eq.^-)

for a 0.225 A l - Z n alloy aged

at

1 0 0 °C in a IfOO

A

dom a i n w i t h periodic b o u n d a r y conditions and random i n i t i a l conditions of small a m p l i t u d e .

The broken

h o r i z o n t a l lines indicate the coherent equilibrium compositions.

Fig. 12.

(Computer-generated plots)

Ampli t u d e spectra (absolute magnitude) and corres­

p o n d i n g composition p r o files calculated according to Eq.

{ jh )

for a 0.200 A l - Z n alloy a g e d at 100°C in a i+00 A

do m a i n w i t h periodic b o u n d a r y conditions and same initial conditions as in Fig.

11.

Tire b r o k e n horizontal lines

indicate the coherent equi l i b r i u m compositions.

(Computer­

g e n e r a t e d plots)

Fig. 13.

A mplitude spectra

(absolute m a gnitude) and corres­

p o n d i n g composition profiles calculated according to E q . ( 7 U) for a 0.375 A l - Z n alloy a ged at 1 0 0 °C in a 3 0 0

A

d o m a i n w i t h periodic b o u n d a r y conditions and same initial conditions as in Fig.

11.

The b r o k e n horizontal lines

indicate the coherent e q u i librium compositions.

(Computer

gener a t e d plots)

Fig. I1!.

Ampli t u d e spectra

(absolute magnitude) and corres-

p onding composition profiles calculated according to the linear diffu s i o n equation for a 0.225 A l - Z n a l l o y aged at 100°C in a h O O A domain w i t h periodic b o u n d a r y conditions and same i n i t i a l conditions as i n Pigs. 11 and 12.

The

b r o k e n h orizontal lines indicate the coherent equilibrium compositions.

Fig.

15-

(Computer-generated plots)

Amplitude spectra (absolute magnitude) a n d corres-

8l

po n d i n g composition profiles calculated according to Eq.(7*0 showing the coarsening stage for a 0.225 Al- Z n alloy aged at 100°C in a 1+00 b o u n d a r y conditions Fig. 12).

l.

domain with periodic

(the initial stages were shown in

The broken horizontal lines indicate the

coherent equi l i b r i u m compositions.

(Computer-generated

plots)

Fig. l6.

Calculated percent completion curves for 0.225 A l - Z n

alloys aged at 100°C.

83

Results for small-amplitude r a n d o m

initial conditions fall wit h i n the shaded band.

The

upper full curve is that for a large quenched-in com­ position fluctuation.

The b r o k e n curves show the corres­

ponding percent completion values calculated according to the linear equation.

The open circles indicate un­

normalized percent completion values for a spherically symmetric composition variation.

Fig. 17.

Intensity spectra (amplitude squared) calculated

a c c o r d i n g t o E q .(8 o ) for a 0.225 A l - Z n alloy aged at

85

150°C.

The i n itial condition spectrum (curve no. l)

simulates an experimental quenched-in spectrum (curve no. 1 of Fig. l8).

The critical (\e ) and o p timum (\n )

wavelengths are indicated b y a r r o w s . po Fig. l8.

Intensity specta obtained b y Rundman

f r o m small-

85

angle X-ray scattering data on a 0.225 Al- Z n all o y aged at 1 5 0 ° C .

The corresponding calculated spectra are

shown in Fig. 1 7 .

Fig. 19 a and b.

Intensity spectra (amplitude squared) cal-

88

culated a c c o rding to Eq. ( 8 o) for a 0.200 A l - Z n all o y age d first (a)-at 100°C (inside the spinodal) and sub­ sequently (b) a t 200°C (outside the spinodal).

The

critical (xc ) and optimum (>.n ) wavelengths are indicated b y arrows in Fig. 19 a.

Fig.

19 c.

Continued on Fig. 1 9 c.

Continuation of Fig. 19 a and b

: calculated

89

intensity spectra for the final aging treatment at 100°C (inside the spinodal) of a 0.200 A l - Z n alloy. The critical (lc ) and optimum (xn ) wavelengths a r e • indicated b y arrows.

Fig. 20.

Amplitude spectra (absolute magnitude) and corres-

ponding composition profiles calculated according to Eq.(7*0 for a 0.100 Al - Z n all o y aged at 100°C

(outside

the spinodal) in a 6 0 0 A domain w ith periodic bound a r y conditions.

Three one-dimensional Guinier zones constitute

91

the initial condition (at time zero).

The br o k e n h orizontal

lines indicate the e q u ilibrium compositions.

(Computer­

generated plots)

Fig. 21.

Hypothetical cooling curve f rom the solution-treatment

temperature T^ to the quenching bath temperature T^. temperature T q is located just below the spinodal.

93

The Inter­

mediate temperatures indicate assumed ranges of validity of free energy curves.

Fig. 22.

E x p o n e n t i a l cooling curves calculated according to

E q . (8 5 ) (solid curves). (l) 1 0 3 °C/sec,

The quenching rates were

(2) 5 X 1 0 3 °C/sec,

(U) 3 X 1 0 4 °C/sec,

(5) 6

93

:

(3 ) 1 0 4 °C/sec,

x 1 0 4 °C/sec,

(6 ) 1 0 5 °C/sec.

Tlie b r o k e n lines are r o u g h sketches of 2, 10 and 65 percent completion curves for a 0.225 A l - Z n alloy.

Fig. 23.

Coherent derivative modulation (a), and composition

profiles

101

(b,c,d) calculated according to Eq.(89) for

a 0.225 A l - Z n alloy age d at 100°C in a n imperfect hOO A domain w i t h insulating b o u n d a r y c o n d i t i o n s .

The initial

condition is shown in Fig. 25 a.

Fig.

2k.

C o h erent derivative modulation (a), and composition

profiles

(b,c,d) calculated according to Eq.(89) for a

0.225 A l - Z n alloy age d at 100°C in a n imperfect 1+00 A domain w i t h insulating bound a r y conditions. condition is shown in Fig. 25 a.

The initial

102

Fig. 25.

C o m p o s i t i o n profiles calculated according to E q . ( 8 9 )

103

for a 0.225 A l - Z n alloy age d at 100°C in a n imperfection/ free h O O A domain w i t h i n sulating b o u n d a r y conditions. Figure

2k Fig. 2 6 .

(a) is the common initial condition for Figs. 23,

and 2 5 .

A c r y l i c sheet and nyl o n m o d e l of a two-dimensional

108

Fourier s p e c t r u m (absolute magnitude) calculated according to E q . (90) for a 0.225 A l - Z n alloy aged at 100°C for

1 5 0 0 sec (corresponding composition profile is shown in lower frame of F i g . 27 c ).

Fig. 27 a.

Two-dimensional composition profiles synthesized

109

f r o m amplitude spectra of Eq.(90) for a 0.225 A l - Z n alloy age d at 100°C for 0 a n d 300 sec. computer-generated plots)

(Traced-over

Continued on Fig. 27 b, c

and d.

Fig. 27 b.

Continuation of Fig. 27 a.

Two-dimensional profiles

110

synthesized fro m amplitude spectra of Eq.(90) for a 0.225 A l - Z n alloy aged at 100°C for 600 and 900 sec. C ontinued o n Fig. 27 c a n d d.

Fig. 27 c.

Continuation of Fig. 27 b.

Two-dimensional profiles

synthesized fro m amplitude spectra of Eq.(90) for a 0.225 A l - Z n alloy a g e d at 100°C for 120 0 a n d 1500 sec. Continued on Fig. 27 d.

Fig. 27 d.

C o n t i nuation of Fig. 27 c.

Two-dimensional profiles

111

synthesized f r o m amplitude spectra of Eq.(90) for a 0.225 A l - Z n alloy a g e d at 100°C for 180 0 sec.

Fig. 28.

Amplitude spectra

(absolute magnitude) and corres-

122

p o n d i n g spherically symmetric composition profiles cal­ culated according to Eq.(l06) for a 0.225 A l - Z n all o y age d at 100°C (inside the spinodal) in a 300 A radius spherical domain.

I n itial condition is a small composition

fluctuation a t the origin.

The b r o k e n horizontal lines

indicate the e q u i l i b r i u m compositions.

(Computer­

generated plots)

Fig. 29.

Amplitude spectra (absolute magni t u d e ) and corres-

12h

po n d i n g spherically symmetric c o m position profiles cal­ culated according to Eq.(l06) for a 0.100 A l - Z n all o y age d at 100°C

(outside the spinodal) in a 300 A radius

spherical domain.

Initial condition (at time zero) is

a spherically symmetric Guinier zone.

The broken h orizontal

lines' indicate the equi l i b r i u m compositions.

(Computer­

generated plots)

Fig. Al.

Interpretation of the T]j (j = l,...,n) parameters

for multicomponent systems.

A h y p o t h e t i c a l lattice

parameter versus composition surface

is shown in p e r ­

spective for the t e r n a r y subsystem (j,k,n).

The plane

tangent to this surface at a Q (average lattice parameter) is also indicated.

lh5

Fig. A2.

Ideal e n tropy (full curve) and its fourth-degree

lb 5

Legendre p o l y n o m i a l approximation (open circles).

Fig. A3.

Diffu s i o n equation coefficient

d (u ) (solid

curve)

160

and its parabolic T a y l o r ’s .expansion about u = 0 (c 0 = 0.225 at. fract. Zn) in the case of a n A l - Z n solid solution at 100°C.

True spinodal compositions

are indicated b y open circles, incorrect ones

(derived

from the parabolic expansion) b y open triangles.

Fig.

Ah.

Photographic reproduction of typical Fourier

spectrum a n d composition profile plots generated b y the subroutine GRAPH used in conjunction with programs F Y S P I N 1 P or FYSPIMPY.

The symbols A X M I N A ,

DAXA, WAX, AXMINU, D AXU 'were added manually.

182

LIST OF PRINCIPAL SYMBOLS

First used on page

A

Amplitude vector

19

A

Time derivative of amplitude vector

19 19

A°

Initial amplitude vector

A

Amplitude vector in diagonalyzed sy s t e m

27

Fourier coefficients, amplitudes

17

A°

Initial amplitudes

19

Ak

Fourier cosine coefficients

65

A

Sums of products of amplitudes

1+8

A

k

k

a aj

_

5

Lattice parameter Coefficients o f fourth-degree p o l y n o m i a l in u A = BA

1+2

B

M atrix in system

19

B

Diagonalized m a t r i x B

27

Bk

Fourier sine coefficients

65

Bj

Sums of products of amplitudes

ll 8

b KL Elements of m a t r i x B

19

Ck

61+

C o m p l e x Fourier coefficients Sums of products of complex Fourier coefficients

^ i x ’C i s ’ C44

65 6

Elastic constants

c

Composition (atomic fraction) in b i n a r y system

1+

cQ

Average composition in bi n a r y system

It

ct c°

Composition (atomic fraction) in multicomponent system Average composition in multicomponent system

9 9

c 7 ,c"

Coherent spinodal compositions

^3

0 ^, 0 ^

Coherent equi l i b r i u m compositions

^3

xii

Page . D

Diffusion equation coefficient

B o j D i ,D 2 D^, D*

1+5

Coefficients of expan s i o n of D

1+5

Tracer diffusivities

‘

7

E

Diagonalizing operator of matrix B

27

Ej

Eigenvectors of matrix B

30

F

Free energy of non-uniform system

F

Polynomial approximation of free energy

F

Matrix of second derivatives of free energy

31

F

Matrix of minors of matrix F

32

Fk

Amplitudes for spherically symmetric fluctuations

5 f

F k>k' Fourier coefficients of coherent second derivative f

Helmholtz h u l k free en e r g y

f 0 ,fg,f ",f'",fo

1+2

109 17 5

Bulk-free energy and derivatives at u = 0

16,56

fjj

Second derivatives of free energy (multicomponent systems)

2l+

G

Coefficient matrix for multicomponent systems

25

G°

Coherent second derivative matrix for multicomponent systems

28

G jj

Elements of matrix G

25

Gn

Function x _1 sin x integrated fro m

J

Flux for b i n a r y systems

Jk

Flux for multicomponent systems

k

Boltzmann's constant

k

Wave index

M

Mobility in b i n a r y systems

M

Mobility matrix in multicomponent

• 0 to m n

117 ^ 12 7 61+ 5

systems

25

Mo,Mi,Ma , Coefficients of expansion of M

^5

M kl Elements of mobility matrix

12

Ny

7

Number of atoms per unit volume

9

n

Number of components in multicomponent

systems

n

M a x i m u m number of Fourier coefficients

retained

p

Percent completion

67

Q

Activation energy

62

Q

Quadratic f orm

£S

Q

Source t erm vector

19

Qk

Fourier coefficients of source term

q

Source t e r m i n binary systems

qk

Source ter m in multicomponent

6k

' 17 5

systems

lh

62

R

Gas constant

R

Radius of spherical domain

115

r

Radial polar coordinate

llL

r

Position vector

T

Absolute temperature

t

Time

U

Free energy of volume element of non-uniform system

U0

Free energy of volume element of uniform system

u

Composition variation c - c 0

1+

"

7 ■■ h

6 10 1+

u

Composition variation vector in multicomponent systems

28

u

Normal composition variation vector

29

u0

Initial composition fluctuation

9

Uj

Composition variation c t - c°

9

V

Domain volume

17

v

Function r u

1-15

W

Coherent strain energy

5jl0

Xj

Coefficients of fourth-degree polynomial F

1+2

x

C a r t e s i a n distance coordinate

63

Yk

S t r a i n energy parameter

o'*

E i genvalue of matrix B, amplification factor

ofk

Eigenvalue of ma t r i x B, amplification

o£

A m p l i f i c a t i o n vector

p

W a v e number

5

factor

' "

21,50 26

1+3 17

Critical 'wave number

20

(3B

O p t i m u m wave number

90

At

Iteration time step

21

7|

Linear expansion per unit composition

1|

Vec t o r of comp'onents 1^

change

5 31

Tlj

Ti parameters for multicomponent systems

h

Gradient energy coefficient in binary systems

H jj

Gradient energy coefficient in multicomponent systems

13

X

Wave l e n g t h

61+

Xa

Op t i m u m ’wavelength

cp

Potential for bi n a r y

cpt

Poten t i a l for multicomponent systems

0

Integrated

a0

Normalization factor for percent

13 5

53,60 systems

5

(c - c 0 ) 2

11 67

completion

t

Re d u c e d temperature k T / m

u>

Interaction parameter for regular solution model

67 33

DO 2 0 L = 1 »N A1 = ALPHA( 2 »L ) $ A2 = ALPHA( 3 » L) KL = K- L $ LK = K+L I F( KL) 2 3 , 2 0 , 2 1 21 C ( K ) = C ( K ) + DBK# t ( A 1 * C 0 ( K L ) + A 2 * C T ( K L ) ) * C 0 ( L ) 1A2* CT( LK) >* CONJG( CO( L) )) GO TO 2 0 23 KL = - KL

( A1*C0(LK)

+

C(K ) = C (K) + DBK*((Al*CONJG(CO{KL)) + A2*CONJG(CT(KL)))*C0(L) 1(A1*C0(LK) + A2*CT(LK))*CONJG(CO)) 2 0 CONTINUE IF CCABS( C C1 ) 1 . GE. 1 0 . 0 ) GO TO A ... IF-( MOD ( AM»APRI NT ) ) 1 6 , 1 7 . 1 6 17 TIME = AM*DELT C C

FOURIER SYNTHESIS DO 50 K=1»N A ( K) = REAL( C( K) )

B(K ) = AIMAG(C (K)) I NT( K) = A ( K) * A ( K) + B ( K ) * 6 ( K ) 50 AMP( K) = SORT( I NT( K) ) SUM = 0 . 0 DO 81 K=1»MAX 81 SUM = SUM + I NT( K) PER = 2 0 0 . *SQRT