B. A. R C.-993

i

<W*M I'M

GOVERNMENT OF INDIA

ATOMIC ENERGY COMMISSION

A COMPUTER PROGRAM TO CALCULATE PHONON SHAPESUSING PLANAR DISPERSION

byA. H. Venkatesh

Nuclear Physics Division

BHABHA ATOMIC RESEARCH CENTRE

BOMBAY, INDIA

1979

B* A. R, C. - 993

GOVERNMENT OF INDIAATOMIC ENERGY COMMISSION

K

A COMPUTER PROGRAM TO CALCULATE PHONON SHAPESUSING PLANAR DISPERSION

by

A.H. VenkateshNuclear Physics Division

BHABHA ATOMIC RESEARCH CENTREBOMBAY. INDIA

1979

INIS Subject C « f gory t » 1 » A13

D«»er iptor» l

ft CODES *

DI8PKBSI0H BBUTIONS

FBWONS

SCATTEBING

CROSS SICTICHS

POTASS IDW

MBDIBOK

abstract

* program is written to calculate the shapes of phonons

scannaO by • triplw .axis neutron spectrometer. Thu program takes

into account th« population factor, 1/tO and (j£>]E ) terms of the

scattering crodS suction *nd the resolution function as formulated

by Loopur and Vat. nan sA ' to calcuiata thu intensity at aach point

of the scan. It is assumed that the dispersion relation doss not

hawa a strung curvature su that ons can make uva of planar dis-

persion. The fact that the resolution ellipsoid is highly elliptic

haa b&tsn takan cam uf to ..peed up the computation of intensity.

This computer program has been usiid to calculate the shapes of

phonona in KNO., fcr obtaining true phonon

A COMPUTER PROGRAM TO CALCULATE PHONON SHAPESUSING PLANAR DISPERSION

byA.B. Venkateah

I. IHTRUOUCTIUW

The intensity of a neutron group associated with a phonon

measured using a triple-axis nsutron-spectroaeter (TAS) is the

convolution of the resolution function of tha instrument with the

ecettering cross-section for that phonon. The Measured data haa

to be corrected .for the reaolution effecte in order to obtain

correct information rf&aut the phonon.

(2)In the «ethod of sterner and Pynn* ' uae is made of the

gradients of phonon branches* The gradient* are arrived at from

a dynaaicel matrix. This procedure cannot be •»»lly adopted for

a coaplax crystal like KNO*. slith this in view we have developed

a coaputar prograa which calculates the ahapa of • phonon* using

tha Cooper and Nathans' (hereinafter referred to as C-M) foraalisa

of the reaolution function^ . The gradients are input paraaetars

detarained froa considerations discussed in the following paragraph.

The proyraa takes into account the population factor, (O.'E ) and

1/O>.taras of uhe cross-section and calculates the intensities

tumumlnQ planar dispersion. The integration is done numerically

by Simpson's aethod.

The proyraa requirea the slope of the dispersion aurface as

mn input para«et«r. Fat acoustic phonons elaatic data haa bean

used to arrive at the slopes whereas for optic phonons the Measured

,.*'ahape of a phonon* rufwrs to 'intensity profile of the neutrongroup essociatsd with a phonun*.

s 2 i

data i t sit.-if ha a been uoud to yet the slope* line can u- e the

pruyra.Ti for comparison of data obtained by constant IJ, or

constant AL techniques. The yun^rai procedure is to calculate

tha phonon shape usiny tho program, compart* with thu .-naujured one

and obtain a yood f i t uy i/aryiny input parameters iiko slope of

the dispersion surface, pftjnon uiauevcictur q (for constant AE runs)

or frequency V ifor constant U runs). Tha one with the best f i t

yivos the true ualues of J or y for the phonon. This program has

baen uiBd to obtain corrected dispersion relations in KNU . We

have not resorted tu a iuaot ij.iuarus f i t t i n g procedure but

dut .f-in.:d the b^Jt f i t un triu basit> uf v/ioUdl jujyt-munt of jut

sh .,;• u.uul,;i;nt of i-dji-uirtLed phunon with the iDjct

the unddriyiny theory in usctian I I , the

program description i s yiwfcin in auction I I I . Auction IU deals

with input d^ta to thu pro^rjni. Typicai r e su l t s are di^cu^sed in

U and l imi ta t ions of the prujrim -are inciicat^d in ^action UI.

l i o t i n y in fLiHTil N li/ i - apptnU<jd at the end.

I I . THbuKY

a phonon ii&a»urtiinunt i j carried uut u^iny TAi the intens i ty

at any paint ( U> , «J ) of the scan is jiv/c-n by

il)

whs.re R(OJ,£) is the ru^olution function of the injtrumcntv y

definud as the probability uf ndut,ron detection as a function

Q. ( - flp+AU) when the instrument is set

a t ((*> , t j ) and i s given by,

where M, , aru th<* ON matrix elements. (Tor deta i ls see Rsf .1 } .

X,t X. , X_ are the components of A3 along the L-U axes (X. i s

pa ra l l e l to (-Q ), X_ is normal to X and l i e s in the scatter ing

plane and X_ i s normal to the plane) and X. «AtO ( f l g . i ). The

surface of constant R i s an e l l i pso id in ((O» j£) space.

^--—r- is the one-phonon coherent scattering cros^-section/unit

ce l l given by,

with the population factor as (Klo'i+0 f o r nuutrcm onergy

process and ru> for energy gain process, where

the inelastic structure factor for th« atomic model of the crystal,

\ a-yj

ami the jacobian

(5b;^The turms in equations 3,4 and b h^ue their uaual meaning)

The dependents uf the raaolutiun function oft X will be neglected

for the rust of our report as this d<-ptindoncu ia a sluwly varyiny

one. Thus changing to variables X , X, and X

The phonon scat ter ing cro.3;>-i>trction d j j ~ ^ i s f i n i t e only over

a surface i n ( CO , ^) or (X , X , X ) spaca ca l led tho dispersion

d uhose equ&tiun can be reprt tjiitGu by,

n;

Llsewhere the cross-sect ion

Thus the i n tens i t y I given by (6) i s an in tegra l over the

dispersion surface Xyi = X / (X 1 , X9) and can be wr i t t en a.-* a double

in teg ra l

I t u i l l be asdumed hcru that j ^ .C^) } (see L(i.ba) can bo

n-.placetl by a tnrin propor t ional to sums effect ive; value of

t 5 t

(U.T; ; • essp (-2 W(3,)) where -p is thH effective value of

polarisation vector of the mode and axp (-2 W ( ^ ) ) is the effective

Debye-Waller factor. In thia program \"E\ is assumed constant

over the rayion of inturast. If the variations of "J"|(j*>) a n d

exp(-2 W ( £ ) ) art negiactad which is a rea.-i.-i-vible assumption than

the scattering cross auction .JjiQ.. can iced by the

product of C 1 ^ l+'V'fci) ' i*^» and '/*** * Thus

The resolution function ii of appreciable value only over a small

ration in (tO, Q) space around the nominal setting ( CJ , J ) of

th& spectrometer. Hencu th« integration has to bs carried out only

over a small ranye of X and X..

Mowavar there i* 3ne probietn to bo solved. The locus, of

constant H is an ellipsoid in X., X2, X space caliod the resolu-

tion ellipsoid. Thtt tlllipsoid is highly aspherical uiith the major

axis lying very near K_X, plane,, Let us now coniid^r an ellipsoid

defined by fi « 1 (sau £q.2). Tha intersection of this ellipsoid

with tha dispersion surface 13 an ellipse if the latter ia assumed

planar. The projection of this ellipse on the X^ X^ plane defines

the region (again an ellipse) in which R ^ , x^, X4(x?t X2J)of E4.

has an appreciable value. Outside this ellipse it has a negligible

value. Hence if this ellipse is chosen me the area of integration

thu integral can in: numuriu i lly OV/.Jludteo" to a yiv/en accuracy with

the least numbur of divisions. An arbitrary ranye of X1 and X

aruund U ( OP- X. «= X = 0) wi l l rn.*d a larger are* to cower the

sllipse which in turn mcjns thjt a biyjer number of divisions of

thi=- are-i is rn.edtd to uv/ajluate tliti integral to the same accuracy.

(Uio diffor*nut is well brought out in F i j . 2 . Thus i t is seen

thciL trie num-jiicul »valuatiun of the integral is most efficient i f

tns allipsu defined aoova is chussn as tht> »se* of integration of

The aquatiun or this ellipse in X and X is obtained

bwlowt

I f the planar dispersion surface has CC. and CC. as the

components of ita slope j^alony X and X axes than thta ^urfjcc is

J'.:finc:d oy thu ec|u>itian (see Fi j .3 )

where X._ = ^ U V + C. A j 9 ( b )40 ^ ^ * * r»/O

from t.ct.(2) the aquation of ellipsoid of constant probability

R exp(-p/2) is git/en by

cio;

The intersection of tna resolution ellipsoid witn the. dispersion

is an dllipab. The projection of this Ellipse on the X.X

: 7 :

plane ia ivon by eliminating X. betuiben equations* '.'(a) ar'J 10 as

(11)

A -

F= PV**, -p

I f ths e l l i p i o i d chosen has an outer surfdct of very small proba-

o i l i t y ( i . e . p is large) then thd e l l ipse yiwcn by tq . {1 ) )

tnt. r jr iyt of X. , X ovisr whiuh thu i r t te j rat ion fias to Ue c

out. Tnii. e l l ipse is not, in ^snardl, cantered around X ^ X., = U

Tranjforniny to a new systcvin of coordinates X. d^d X. .seo

ss ariyirt is -it £X1bH, X2iH) and whose .i>ts are

by cin am;It: | i uiitM ryapiict to the old

2 = X^5H 4- sinyi X,' +

i 7' + cos^ Xi J

t a

Using Lq.(12) lr Lq.(11) we yet t i e equation of e l l ipse in the

new t i ansf ormtjd coordinates X. and X as

where

ftJ- Rcos^n- 8s«iV +

b ' - flirty-*-Bu>sfy.-CGosjA

D' - ( 3-R fcos^ + C stn^) X15H -t-C -BsiT -H Cc<&^)

' - P *

If the new axes X., X. were identical to the axes of the ellipse

than thu coefficients of X.X2, X. and X in E4.(13) are zero.

i.t. C1 i D ' r E ' - O

: 9 t

The first condition gives Lt , the ani^le of tilt of the ellipse

05)

The last twe conditions of ( U ) giue XISH and X2SH, the shifts of

the center of ellipse from the point (X. =U, X =U)

XI SH

X2SH

if P^V-

where P r 1ft cosy* + Cs\

M •=. -2f l5t

V/ -

W = -

The equation of the ellipse now reduces to

(I6o)

oy» M - l

t '10 I

The lengths of thu elliptic axub 2QLU and 2DLL2 are given by

and '~"

If lines are drawn tangential to this ellipse at the end of the

axes (f.tg.2) a rectangle is formed which encloses the ellipse

completely and has the least area. Thie rectangle is chosen for

numerically evaluating integral given by Cq.(8).

III . PRUURAW DESCRIPTION

Fig.5 gives tho flaw chart of the program RcSOLN. The

program RtSQLN calculates a 3x3 matrix 018 at evary point

(Q1, 42, U3, LTHl) or t.ib scan. OIR ia the matrix used for

transforming the components of any wactor If^ftom crystal frame

to C-N frame of isferance through tha re la t ion

XCW •* OIR * /

where X C r y 3 t* 1 and X define the components of JK in" crystal

and C-N frames respactively. The Matrix OIR ia often used in this

program. The program then calls the subroutines ANGL, BRAIN and

ORIENT in that order.

Subroutine ANGL calculates the angles PS! and PHI ( <f> and

2 € respectively in raf.1 ) of the wavevector-transfer-trlangle

with tho incident wavevactor AKI, the scattered wavevector AKF and

: 11 :

wavevector transfer QU as inputs. subroutine BRAIN calculates the

mdtrix eleinonts P\{\<,i). oubroutine UflltNT calculated the half-

lunyths of the e l l i p t i c axis Dtl_1 and DLL2, their orientation

TILT ( U in the taxt) and the shifta XI ah* and X2SH of the center

op thd ell ipse from (M = 0, X2 =» 0). The program then determines

a rtcta iyle of sides 2*DLL1 and 2*D£L2 around the e l l ipse, divides

this area into (2»rit.UH)#(2*flLiH) equal rectangles and performs the

doublu integral numerically by Simpson's ru le. -4t gach mesh point

the proyram calls the subroutine Ofli'GH which calculates the integrand

(The variable XX is used for CO in this sub-routine)

If needed the program utilious th« subroutine LLIPit which

monitors the intersection of the 5U/£ resolution ellipsoid with

(X1, X2), (X1, X4) and (X2, X4) planes. The results when plotted

give thuss ellipses of intersection which taken toyethur gives an

idea about the size and orientation uf the resolution ellipsoid.

11/.

Tha program Rt^OLN reads the following para

1. ALJ, OM, OA, rtXL, 8XL, CXL all in A" (af10.5)

whore «LI is thb incident wavalbnijth,

On, OH, the monochromotor and analyser d-spacings respectively,

AXL, BXL, CXL, the unit call edyas of the sample.

J 1 2 l

2. HLFAQ, M1.FA1, *LM2, *LF^3 (all in minute's) ... (8F10.S)

are the character!otic horizontal divergence parameters of the

in-pilu, wonochromator to sample, sample to analyser and

analyser to datdttor collimatord reapbctively. <Ko , for

instance, may be defined as follows:

The transmission probability of a neutron at an angle o<o to

the central Una of thd col lima tor =« exp(-y2)» transmission

probability at zero angle, (under Gauoaiun a

CTAii, tTAA (in minutua) (0F10.5)

t'hu munochromator and analyser mosaic soruads respectively

4. CLIP5 (8HU.S)

If fLLXPS ^ 0, the prayru/n makoa use of subroutine ELIPSE to

•id us. in wisualiainy the resolution ellipsoid.

5. snt (ano.5)

This dbtucminua the size of the resolution ellipsoid over which

the integration has to be carried out. If, for instance,

UlZt » 100, thts resolution ellipsoid chosen for integration will

ba bounded by the surface of detection probability 1/1UQ.

6. UU, uU, UU (8F10.5)

0SLadefines the i0SLa axis of tha crystal, which is normal to tha

&*. '•of the

7. The next sut of four cards specific the phonon, thu

effective polarisation direction, tho scan direction and the

mesh sita for integration. Thuan cards can be repeated for

as many phonons as required (with thB same zone. axis).

(a) QIC, U2u, U3D, E, C1, C2, C3 (8F10.5)

tf1u, U20 and QiU define the wavevector transfer U such,*o

that it a components along the (Crystal axes atu

(41u*2TT /AXL)A~1 etc. t ( V in the text) is the phonon

frequency in TH2. If ono dufineii energy loas process as one

in which neutron lod|ui> enur^y to the sampie then E is

positive for energy loas and negatiua for energy gain.

U1, C2, C3 are the components of the slope of dispersion

surface along the crystal axes such that C1*AXL ate are in

( b ) SYt l lNI , iVLON2, iYLUN3 (8F10.5)

are the components alonij" the crystal axes defining the

direction of tha effective polarisation vector. For instance,

lonyitudinal phonun Jlong A axis has "j» »(1,0,0) and a

transverse phonon measured at j^» (•} ,0»4) has "P =(U,U,1 ).

(c) DtU1, UELQ2, Dt.LJ3, QLU, N (4F10.2, II0)

The first four are the steps in U1, U2, 43 and L between

neighbouring scan points; 2N+1 is the number of scan points.

In constant ^ scans OtLJ1, O£LQ2 and OtLU3 are zero and

OLL£ is specified in THz. In constant--AE scans, DELHI,

D1LQ2 and DEL43 are specified in units of (•J.fl/flW-jK*' etc.

apd Dt.Lt is zero.

: 14 i

(d) ntsH (no)

The area of integration is divided into (2»nESH) •

(2*MLSH) aL/ual «trea* for numerical integration by

aimpaon's rule.

Mt- H * 40 huu been found to ue adequate in most c&sss. This

divider the arua of integration into (80*80) cectanyular

calls fur intttnaity calculation by iiimpson's rule. Onu

3uch calculation takes about a" of CPU tim^ in Besm-6

computer•

The program has been U3ed to obtain resolution corrected

(3 )phonun diaperaion curves in KN0_ ' . Thu room temperature phase

of KNO, i3 orthorhombic with la t t i ce parameters yiutn liy AXL =

9.1709, BXL = 6.4255 and CXL - 5.4175 A. The phonon data for

th is crystal ware collected usiny a t r i p l e axis spsctromiit sr at

Cirus reactor, Trombay. The spectrometer parameters are

(HLr/«)i = 6U , i a U,J

LTMM 3 LTrtrt s 1U

1.43 rt

oOH * .Joo94 M.

The proyrdin has list;n u»t)d to j i r i uu at thu true phonon

frequency (constant U acanj or mat/m/ector (constant ^C sc^n)

dyacribed bblow. Tha input pardintter J , f'ro^Udn^y (or thu

l 15 «

an thu case may be) of thu phonon and the diope of the dispersion

curva -Jro varied. Thu cor ro^punding calculated plionon profilus

ofj (.umpired with thu axpurimisntJl une. The one which is closest

to the uxperimentai profile givuo the true phonon fraqu^ncy or the

uuuQUdCtur of ttie phouun. Thu input parameters are chosan by trial

and error to Liu compatible with thu

(4 )

In the c i id of acoustic pfionuna elastic constants* • have baen

to doriwu the alope of tht dispersion cutva near q = 0 for

USB in thu proyi'din. for example, the transverse acoustic phonon

inbdi,ured by constant Q method with t| a (U,.1,4) has a slope of

(2.5»BXL) THz/A*"1 alony SXL axis as derived from the blastic

constants* ', The slope fud into the proyram for this phonon, as

per our geometry, is (0,-2.5,0). (A slope » (0,2.5,0) gives a

defocusad neutron yroup and io incompatible with our dota). It

in assumed here that the dispersion surface i* isotropic. Pro ra.n

outputs are obtained by varying the frequency in this case. Tht-

compariaon of acimu of the calculated intensity profiles with thu

experimental ana roe thio phonon id shown in Fig.s(a). Here

V"«22 yiwas the oa»t fit and is taken to be the true phonon

frequency. Comparison of the experimental and calculated profile*

•r« shown in Fig.6*Typical input and output ara shown at tha 9nd»

For optical phoaona the *xparim«ntal data itself is used to

deriv* tha slope of the dispersion curve for use in the program

as input parameter.

; 1 u :

V1. LXI1XiMTIuWt>

a) The program can be u&ud far cubic, tetragonal or

orthorhumbic structures.

b) ThL assumption of planar dijptsrsion madb in this program

it> nut -iluiiyi, vtilid. When aithur tho dispersion surface had a

(•truny curvature or thB resolution ellipsuid is comparable in

size to Briliouin zanu of the sanipiu crystal (dus tu poor resolution

of tha 3poctroinetar) this as>t>umptiun breaks down and one has to

include hiyher order turns in the dispersion relation (Eq.9).

c) Lffect of wurticdi divtjrysnce of collimators and samplu

mosaic on thu resolution function has baen

It in a pleaa ore for ma to thank Or. K.R. Rao far the keen

interest he has ahuwn throughout tha course of this work.

Rtft.Ht.Hu. a

^1) PI.J. Uoouer and ft. hathanb, Acta. Ccyst. 23, 35? (1967).

(2} Roger Pynn and Samuel uiernbr, Studsvik, iweden Laboratory

Report NO.<I£-FF-112 (1971).

(3) K.R. Rao, S.L. Chaplot, P.K. lyenyar, A.H. Venkatssh and

P.R. Vijayarayhavan (To be published).

(4) F. Diehard and 5.F. Pliu.ua, U.R. Acad. Sc. Paris 272,

848 (1971).

flu.11 *STR, 6STH, LSTfl »r» r«cipruc*l iatticu axu* of the

cr/»t«l| X1, X2> X^ aru Loopur-Nathans mxa*. X^ p»raliul -U ,

to X. i n th« piano nr vcat t sr iny i m

• x i « . The cpactrometer i s <ct a t fl( <ti, , Q). ^Icj ,^ , ) i * •

yunurdl po int un the r e s o l u t i o n «lllMi>uld> X , X i m thv compo-

n«int» of 4 9 " JA " Jio • lan!J X1 , X2 « > i and »v

(0,0)

L_

f lu.2t Uluattataa an •ll ipue which ia tho projuction on thw

X f , X2 plana or tha ailipaa at lntactaetlon batMaan tha caaoiution

•ll lpaold and tha diaparaton plana. .Tha ataa of intMgratlon la

tna'diiiUd tuctCngla aneloilng Urn all laaa. •taoabmn by. da*had

linaa 1» a aquara around X « »2 m 0 and thla naada* vaty larga

•raa to eneioaa tha aUlpaa.

- 19 -

F 1 Q . 3 I A two-dimonuional akwtch in (CO, U) space to obtain the

aquation of dispsjaion curva in the variables X1, X£ and X^. PQ is

the .point ( Cii, <4..) wher« the spactrometer is sat. Ths sllipse shown

i'» t'K* irrtansaction of bti« resolution a l i ip io id with the ( cJ» q)

ptaiM < P i«' soma point on tha diaperaion curva {liaxn-«saumiid to ba

atraight l i n t ) for which X4 and U> «ca to ba calculatad.

x4 . x4f l - c.

whara X._ •* 0

- 20 -

Fig.4: Transformation from (X,, X„} to (X' , X!). (X,1 , XI)

its origin at (X1SH, X25H) and io tilted at an anyle u with

System.respect to X ,

- 21 -

PROGRAM RESOLN

READ DATAAT EVERY POINT (0.1,02, Q3,ETHZ)

OF SCANCALCULATE DIR

CALCULATE INTEGRAL NUMERICALLYBY

SIMPSON'S RULE USING

OMEGA

PRINT

Flf.5. Flow Chmrt of the Program RESOLN

10

m

tc •

|

(SIPARAMETERS

V Ci C2 C3

( I I 0.1* 0 -2.S 0

(2) 0.22 0 -2.5 0

(3) 0.27 0 -2.5 0

CONSTANT-Q

- '

0.2 0.9 0.4 0-t O.I 0-7

U(THZ)

10

in

za>

IDtr

( I I

S.0 S.1

(b)

PARAMETERS

<>3 C, C2 C,

(1) S.2Q 0 0 -7.«

(2)S.2« 0 0 -7.6

(3) S.3B 0 0 -7.B

1.2 S.3

CONSTANT-AC

U-1.45 TH».

al £-.(1,0,031

5.4 S.S

Coiipariion of •xpatXa«nt«l<r«nd calculated profilaa of two n«utcon groups. Xn both th« ca^sa aiastic

data H»t bMn utad to dariva tfia elopa of tha diapurdion uurua for baing usad aa input paraaatera xn tha

RLSOLN. '

(a) Tranuvaraa acauatic phonon oaaaurad by constant *^ mathod at 4: 0 t . 1 f 4

(b.) l&Hjdtufflnal acouatic phonon aaasur«d by constant flEr.athod kith V i t.aSTHi

- 23 -

a

z

oJlul(E

s*Ita0ua.mC

«J3ac

aco£au•H* )(A30f,(I

0tf

u<D

c>•u*>

CM

•to

o>a

•

*CDO

•o

•r- O

09

N•3 r 3

• 3r- O

m

Output of tha Pragcaa HtSOLN for KMO3

u>«eo* on u« * & c1.43000 2.06694 2.08694 9.17090 S.H255U 5.4175

ELLIPTIC PATTERNS ML AiOTTHE. iUNE AXIS I S ( 1 0 0 )

THL RESOLUTION ELLIPiOIO Fu« Il.TEiifttTIOW HHa OUTER aURF«CE OF HHOonfaUITy 1 / ( 2 5 6 j OH t X P ( - l 1 1 . 6 3 0 ) / 2 )

HORI20MTAL OIMtflG£!»C€& OF THE CULLI^TGRa *HL KIC <iIit.uTc.Sj

60 60 60 60

i, SPRt-rtOS OF MuNbCHHOfWTOH HNO «(wLYbt.fi AM. \ln i-UUiiJi.it)

1u 1 0

Q10.00

320.10

VELOCITY IN CRYSTAL AXES4 0

Q10.000.000.000.000.000.000.000.000.00o.oc0.000.000.000.000.00

0.71355

020.100.100.100.100.100.100.100.100.100.100.100.100.100.10O.TO

J34.00o.uo

034.004.004 . 004.004.004.004.004.004.0Q4.004..U04.004.S34.004.00

E0.22

-2.5u

E 3-0.13-O.ud-0.03

0.020.070.120.170.220.270.320.370.420.470.520.57

O.Ou

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- 25 «

LIbTlWU Of THE. PROGHAH

PROGRAM RKSOl.NREAL MDIMENSION 5UMI5.00).C00PER(4i4lCOMMON/RAJU/XK3.101) ,XJK3.101) .XJ2 l3 i lOXlCOMMON/PREETI /TILT .DELI »DEt_2»XISH»X2SH»CS»5ltSIUP»PEECOMMON/COSI /POP »EN»f.VLONI t SYLON2 f SYLON3COMMON/CN/ALFAO »ALFA1.ALFA2 tALFAS »ETAMiETAA»661»G<52»GG3»JPCOMMON/PR tYA/ASTR,»BSTR»CSTR»Ql»G2»Q3»CCl»CCZ»CC3»X4ZERO»QOCOMMON/(jlRI/GG(500)iJiDELG10.DELOZOiPELG30«Cl»C2«C3COMMON/FOUR/M<4t4)«DlR<3i3)«PIFORMAT(14X«'O1 02 03 E IHTCMSIT

2 FORMAT(Bo(1H*J)3 F0RMAT(2f)X»»ELLIPTIC PATTERNS ARE NOT REQUIRED')5 FORMAT*1H1»» LAMBDA DM DA A B

1 C i)7 FORMAT!lnX»15i3F15.M8 FORMArr//»THE PLANE OF THE ELLIPSE IS"» 16)10 FORMATI8F10.5)11 FORMATJlOXtAE15.2)12 FORMAT(//10X»«H0RIZONTAL DIVERGENCES OF THE COLLIMATCJS ARE (IN MI

1MUTESI»/20X.4F7.0//7/.«M0SAIC SPREADS OF MONOCHROMATOR AND ANALYSE1R ARF. (IN MINUTES)'/25Xi2F7,0|

13 FORMAT(2F10«?»I10)14 FORMATCloXt'THE RESOLUTION ELLIPSOID FOR INTEGRATION HAS OUTER SUR

1FACE OF PROBABILITY l/<•.F4.0,•> OR EXPt-t*tF7.3.">/2)•)20 FORMAT(ZOX»»THE ZONE AXIS IS (•t3I3|2X|1H>>21 FORMAT(/1H1/8O(1H*)/13X«« Ql C2 03 E'/

H0X»4F10.2)22 FORMATM VELOCITY IN CRYSTAL AXES»«3F10.2)23 FORMATUF10«2iT10>24 FORMAT(5X»4F12.2iE15*2)25 FORMATCtlO)27 F0RMAT(//23X»«COOPER-NATHANS MATRIX ELEMENTS ARE*)

PI'S.14159265 $CC«PI/10800 SPP*1BO/PIREAD 10,ALI»DM.DA.AXL.BXL»CXLPRINT 5PRINT 1O.ALT»DM.DA.AXL,BXL,CXLREAD 10iALFA0,ALFAl»ALFA2»ALFA3READ 10»ETAM»ETAAREAD 10»ELIPSREAD 1O»SIZEPEE«2»ALOG(SI2E>1FCELIPS.HO.0IPRINT 3PRINT 2READ 10*UU«VV»WWPRINT 2O.UI)«VV«WWPRINT 14.S1ZE#PEEPRINT 12.ALFA0.ALFAl«ALFA2.ALFA3.ETAMlETAAPRINT 2ALFAO«AI.FAO#CCSALFA1»ALFA1»CC$ALFA2"ALFA2*CC*ALFA3«ALFA3»CCETAM«ETAM»CC*ETAA»ETAA*CCASTR-2»PI/AXL$BSTR«2»PI/BXL$CSTR*2»PI/CXLAKI»2*PI/ALISE1«B1.79B9/Att««2THEM*ASIN(ALI/'(2»DM>)

i» CUNIiNUfcC READ Ul»O2»Q3»OtLUl»UtLU2»OtLUJ ll« IU/UMAX>ANU ttObLE <N TH^C READ THE SLOPES IN THZ/(Q/QMAX>

READ 1OIO1O»Q2O»O3O.E.C1»C2»C3READ 10»SYLONI»|YLON2.SYLON3READ 23«DEL01«DELQ2tDELQ3«DELE»NPRINT 21»Q10»Q20»O30.EPRINT 22*C1*C2«C3CCC1»C1#AXL*CCC2"C2*BXL*CCCJ"C3«CXLREAD 25»MESHPRINT 25«MESHPRINT 2

- 26 -

NSCAN=2#N+1PRINT 1DO 44 !T=1»NSC*NJP-1I-N-1Q1=Q1O+DELQ1*JPO2=Q20+DELQ2*JPQ3=Q3O+DELQ3*JPETHZ=E+DELE*JPEN=ETHZ/.241820eLO10»oi-lNT((ai*DhLQ20»O2-INT(O2l$DELQ30»Q3-lNT(«3)

33 CONTINUESQ=(O1*ASTR)*»2+(Q2*BSTR)*»2+(Q3*CSTH)»#2QOSQRT(SQ)CALCULATE DIRECTION COSINES OF C-N AXES W.R.T. TO CRYSTAL AXESZ»SQRT(UU*UU+VV*VV+V;W#WW)SQZ*QO«Z

01R (111) =-QlR/Qti*DI«(1 »2) —Q2R/00*D 11! 11» 3) =-«3K/U0DIR(2»l)=(O2R*WW-Q3R*VV)/QZSOIR(2»2)»(a3R*UU-U18«WW)/QZDIR<2»3)=(Q1R*VV-O2R*UU)/QZDIRt3»lJ-UU/Z$DlR<3«2)=W/Z»DIK(3»3)«WW/ZCCl=CCCl*DlRtl»U+CCC2*DlRtl»2)+CCC3*DIRH»3)CC^=CCC1*DIR(?.1»+CCC2*OIR(2»2>+CCC3*DIR(2»3JCC3=CCCl*D!R(3.1J+CCCH*DIR(3»2>+CCC3*DIKf3»3)EF=EI-EN$AKF=2*PI*SQRT(EF/81.7989)THEA=AStM(PI/!AKF*DA))CAU ANGL(AKI#AKF»QO»PSI»PHI)BRAIN CALCULATES M<I»J).Gl»G2.03 OF COOPER-NATHANSCALL BRAlN(TH£MtTHEAtAKI»AKF«PSI.PHI»CCl!CC2tCC:>)IF(JP.EQ.O.AND.ELIPS«NE.O> CALL ELIPSEX4iERW=2*PI*< <e-t"TH^)+Cl*<O10-Ul)+CZ*(020-U2)+C3*K030-03 >)CALL ORIENTIFJSKIP»NE.O) GO TO 44OELX1=DEL1/MESH$OELX2*DEL2/MESHINTEGRATION OF RESOLUTION FUNCTION BEGINS HERE.IXX1»2#MESH$IXX2«12*MESHIX1=IXX1+1SIX2»IXX2+1DO 66 I<=ltIXlSUMU)=0$GG(I)=0Z1=-DEL1+DELX1»(1-1)Z2=-DEL2 $J«1X1=Z1*CS-Z2»SI+X1SH

CALL OMEGA(XI»X2>Z2=DEL2 SJ-IX2Xl»Zl»CS-Z2*Sr+XlSHX2=Z1*SI+Z2»CS+X2SHCALL OME6A(X1»X2)P-4DO 77 J«2»IXX2Z2«-DEL2+DELX2*IJ-l)X1«Z1»CS-Z2#SI+X1SHX2«Z1»SI+Z2*CS+X2SHCALL OMEGA(XI»X2)SUM<H=SUM(I)+P#GG(J>P«6-P

77 CONTINUESUM( n = (SUM(IHGG(l)+GG( tX2> )»L>ELX2/3

66 CONTINUESUMTOT-0p»4DO 88 JI=2tIXXlSUMTOT«SUMTOT+P*SUM<JI)P-6-P

88 CONTINUESUMTOT=ISUMTOT+SUM<1»+SUM<1X1))»DELXl/3PRINT 24»01»02fQ3iETHZ»SUMT0TIFfP TO 44

- 2T -0 0 26 1-1*4DO 26 J - 1 . 4

26 C0OPER(I.J)-t i (I»J)IF(DELE.EQ.O) GO TO 55WIDTH»l/(Pi»GG3>GO TO 44

59 KIDTH«l/<PI»GV>3*A8SICl+C2*C3>J44 CONTINUE

PRINT 10«wr0THPRINT 2PRINT 27PRINT lliUCOOPEK(ItJ)*J*lt4>»I-l»4lIF(ELIPS.CO.0|GO TO 99PRINT 21.010*020.030»EDO 4 LMN*1»3PRINT 8»LMNDO 4 K-1.101IF1XJHLMN.O.E0.0I GO TO 4PRINT 7.K,XI(LMN.K|»XJiaMNiKJ»XJ2aMN,K)

4 CONTINUE» • PRINT 2

GO TO 18ENDSUBROUTINE BRAIN(THETAM.THETAA.K!tKF.PSItPHI»C1»C2»C3JCOMMON/CN/ALFA0,ALFAliALFA2.ALFA3.ETAM,ETAA.GGl»GC2.663»JPCOMMON/FOUR/M(4.'»I.DIR(3»3I .PIDIMENSION D(5 ) .F (5 ) .H (5 lREAL KI.KI-.LAMDA«MASS.MMASS-1674.8SPLANK-6.625H8AR«PLANK/t2«PI>A«SINIPHI+PSn 5 B"COS«PHI*PSnSHALLA>SIM(PSI) S SMAIXB*COS<PSI JLAMDA«ICI/KF $ ALFA»SIN«PHI) $ BETA-COSCPHIIC»-(LAMDA-BETA»/AUFA S E»-(BETA«LAMDA-1)/ALFAA1-TAN(THETAM1/(I:TAM»K1I S A2-1/<ETAK«KI) S A3-1/(AUFA1*KI)A4»1/«ALFA2*KF) S A6"TANITHETAA)/«ETAA»KF) * A 6 * - 1 / ( E T A A » K F )A7»2»TAN(THETAM)/(ALFA0»KI) S A«-l/<ALFAO»KIIA9-2*TANCTHETAA»/IALFA3»KK) S A1O»-1/ULFA3»KF)BO»A1*A2+A7#A8 5 Bl-A2»«2+A3»»2+A8«»2B2*A4«*2+A6«*2+A10»»2 $ 83»A5«»2 *A9»»2B4-A5#A6 +A9»A10 $ B5»A1*»2 • A7»»2

GO-Bl-«BO-t-Bl*C »*2/APRIMK61-B2-JB2»E+B4<'LAMDA»»«2/APRIHEG2-B3-(B3 «LAM0n*»4#E»»»2/APRIME

G4—2»(Brt+Bl*C'* IB2*E+D4»LAMDA)/APR IMEG8—2»(BO+B1»C'•IB3»LAMDA+B4»E»/APRIMEDID-B/ALFA » Dt2)»-A/ALFASD(4I-MASS/(ALFA*HaAR»KF*l,OE+OA)FID-SMALLB/ALPA S F (21 "-SHALL A/ALFAF14) «BETA»MASS.' (ALFA*HBAR«KF«11OE+041D(3)«0 S F(3l«0 K HC1I-0 S H(2)*0 S H(3 l *0HU)"-MASS/(HBAR*RF»l«0E*04JDO 40 K-l»4DO 40 L » l » *

, 21H<L)+F|O*M<K)

40 CONTINUEIF(JP.NE*O) GO TO 50G61>l/<M(ia>+Cl*Cl*MU»4>-2*Cl*M(l*4>|GG2-l/IM«2»2)*C2*C2*H{4»4)-2*C2»Hl2»4)-GGl*CM(l.2t-Cl»MC2.4)-C2»(

lM(lt4)-Cl«M(4»4>l>*»2>GG3«M«4.4)-GGl*«Hll.4)-Cl»Mt4.4»>*»2-G62»<-C2»M(*.4)*M<2.4»-GGl*(

lMCl»4l -Cl«M(4(4 l )«(M( l»2| -Cl*Ml2»4) -C2»IMat4)"Cl«M(4»4»MI**2GG3-SORT«GG3>

10 CONTINUERETURN

ENDSUBROUTINE ANGL<AKItAKFiQQfPSt»PHllPI=2#ASIN(I.)IF(AKI+AKF-GQ>3»3tl

2 IF(AKF+OO-AKI)3»3»44 CONTINUE

SAKI=AKI**2$SAKF*AKF»*2S SQ>QQ*«2S-(SAKI+SQ-SAKF)/t2*OO#AKI»P«(SAKI-SG+SAKF)/t2*AKl*AKF>PSI«SQRT11-S*S)/SSPHI"SORT t1-P«P >/PPSI=ATANIPSD$PHI=ATAN«PHHIF(PSI)31»32»3Z

31 PSI-PS1+PI32 CONTINUE

33 PHr=PHI+PI3* GO TO 103 ST/JP10 RETURN

ENDSUBROUTINE OMEGA(XI»X2JREAL MCOMMON/COSI/POPiEN »SYLON11SYL0N2•SYL0N3COMMON/PR IYA/ASTRtBSTU»CSTR,Ol»O2»Q3»CCl»CC2»CC3»X42ERO»O0COMMON/GIRI/OG(5OO).J»0ELO10»DELQ20»DELO30.CltC2»C3C0MM0N/F0UR/M(4»4)*DIR(3t3)tPIYl»DIR<l»l)»t-QO+Xl>*01Rt2il1«X2Y2=OlR<l»2)*(-Q0+XlJ*DrR(2i2»#X2Y3-DlR(l«3)*f-00+Xl)*DlRt2»3l«X2QDOTSY«<Y1*SYUON1+Y2»SYLON2+Y3»SYLON3»»»2Y1=Y1/ASTR$Y2«Y2/BSTRSY3«Y3/CSTRYl*Yl-INT(Yl««Y2«Y2-INTtY2)«Y3"Y3-lNT(Y3> .

12 X4»X4ZERO-(Cl*iYl-DELQ10)+C2*(Y2-OEtQ20)+C3*IY3-OELQ30))«2*PIXX=)(A+EN*«2*X82*2*PIIF<ABS(XX».GT«.1> GO TO 2IF(J.EOtl>GO TO 4GG<J1«GG(J-1>GO TO 4

2 CONTINUEIF(XX|13»14»14

13 POP=0GO TO 15

14 POP-113 CONTINUE

XX*A6S(XX)

U«l,054*XX/(1.38*301XX=1«/XXPOPFAC»l./tEXP(U)-l)+POPCNFAC-EXP <-•5»EXFACIGG(J > «CNFAC*POPFAC»XX»ODOT SYRETURNENDSUBROUTINE ELIPSEREAL MC0MM0N/F0UR/MUi4l»DIR<3»3| »PICOMMON/RAJU/XI(3»l01)»XJll3»101)»XJ2t3»101lLMN«lSDEtTA».OlDO 4 I » l » 200 4 J»2»4IF(J.LE»I» CO TO 4IF(I .EQ,3«0R.J .EQ.S l GO TO 4DO 1 K>1»IO1XIILMNtK'"DELTA*<K-51)AEO«M(J.J)

- 29 -

BEO=(M(I,J)+M(J»I>)*XI(LMN»K)CEO=M(I.I)*XI(LMN.K)«*2-1.386QUIZ=8EQ**2-4»AEQ*CEQIF(QUIZ}1»2»2XJl<LMN*K)»(-BEQ+S0RTCQU!Z}#/<2»AEQ)XJ2(LHN*K)*(-BEQ-SQRT(QUIZM/(2«AEOI1FO«NE*4) GO TO 1XJl(LMN»K)«XJl<tMN»K>/<2*PI)XJ2(LMNiK)«XJ2(LMNtK >/(2#PIJCONTINUELMN-LMN+1CONTINUERETURNENDSUBROUTINF ORIENTREAL MCOMMON/FOUR/M(«»4>»DIR(3»3J »PICOMMON/PREETI/TILT,DELl.DEL2»XlSH»X2SH.CS»SItSKIP.PEECOMMON/PRIYA/ASTR,BSTR,CSTR.01»Q2»Q3.CC1«CC2.CC3«X'»ZERO»QOA«M(ltmMU»4)*CCl**2-2»MU»«>*CClB«M (2.2 >+M( it* 4 )«CC2**2-2#M< 2 »4»«CC2

E-2»X*ZERO*(M(2»4)-CC2»M<4t4>)F«M(4»4)«X4ZERO**2-PEETN=C/IA-B)TILT»ATANCTN)/2CS»COSITILT)SSI«SIN(TILT|P«2»A*CS+C«SIQ«2»B»SI+C»CSR=D*CS+E*SIU»C*CS-2*A»SIV*2«B*CS-C»SIW=E#CS-D*SIQU0«P*V-O«UX1SH*(Q»W-R*V)/QUOX 2 SH*(R*U-P#W)/QUOXNUM=A*X1SH**2+B*X2SH*»2+C*X1SH»X2SH*D#XXSH+E»X2SH+FVARl*XMUM/(A*CS#*2*B*Sr«»2*C*CS»SI»VAR2*XNUM/(A*SI**2+B*CS*»2-C*CS*SI»IF{VAR1.GT.O«OR.VAR2.6T«0> GO TO 3DEH«SORT(-VAR1>0EL2=«SQRT(-VAR2)

GO TO 13 SKIP*11 RETURN

END