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to NBS TECHNICAL NOTE 936

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

A FORTRAN Program

for Desmearing Small-Angle

X-Ray Scattering Curves
NATIONAL BUREAU OF STANDARDS

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tic Sfyj§

^

A FORTRAN Program for Desmearing

Small-Angle X-Ray Scattering Curves

-t- Jom UL Khk^> ^v - ^ 3'£

Jacob Mazur

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

^°P

Co,

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

\>,S NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued April 1977
National Bureau of Standards Technical Note 936

Nat. Bur. Stand. (U.S.).Tech. Note 936, 83 pages (April 1977) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $1.45 Stock No. 003-003-01752-3
TABLE OF CONTENTS

PAGE

CHAPTER 1

1. INTRODUCTION 2

CHAPTER 2

1. PROGRAM ORGANIZATION . 11

A. Available Programs , 11

B. Instructions and Input Data 11

C. Brief Description of the Main Program and

its Subroutines 17

D. Comments on Units Employed 18

CHAPTER 3

1. THEORETICAL BACKGROUND 20

CHAPTER 4

1. DESCRIPTION OF CALCULATIONS 32

A. Preparation of Data which Characterize the Slit

Weight Function 32

B. Evaluation of the Integral . 33

C. Evaluation of the Kernel of the Integral

Equation for Various Weighting Functions .... 3A

D. Calculation of b 39

E. Solution Checking 41

CHAPTER 5

1. FORTRAN PROGRAM 42

iii
CHAPTER 6

1. TYPICAL CALCULATIONS AND DISCUSSION OF VARIOUS

ALGORITHMS 5 3

2. TEST FUNCTIONS 55

REFERENCES 58

IV
Abstract

A FORTRAN program is presented that corrects for collimation

effects in small angle x-ray scattering measurements. The method

employed in calculations is based on numerical solution of an

integral equation, which is written as a Volterra equation of the

first kind. This equation is reduced to a system of simultaneous

equations, which can be solved by back substitution. Several

different algorithms are tried, based on different interpolating

polynomials for I (x) , the true scattering intensity, which is

expressed as a function of the scattering angle x. The problem of

numerical stability, which is inherent to the computational algorithms employed in the algebraic solution equation is discussed and

exemplified with actual computations.

Keywords: Desmearing; FORTRAN program; integral equation; interpolating polynomial; numerical stability; slit correction; small

angle scattering.
CHAPTER 1

1 . INTRODUCTION

Measurement of small angle scattering of x-rays is an extremely

valuable tool for the study of macromolecular systems. In recent years,

these measurements have been extensively employed in order to determine

the three dimensional structure of triblock copolymers. These coploymers

form domains whose size is ideally suited for small angle x-ray scattering studies. As a consequence, the composition of the domains and the

changes of their morphology with variation of stress, temperature or

pressure can be easily studied. Since triblock copolymers are frequently used in the plastics, rubber and coating industry, their study

and characterizations by small angle scattering is of prime technological importance.

In x-ray scattering measurements, narrow slits are often used in

order to collimate the incident beam sufficiently to allow scattering

measurements close to the incident beam. Compared to a pinhole of

diameter approximately equal to the slit width, the slit system increases

the available scattering energy. Therefore, the experimental measurements of the angular distribution of the intensity of small angle x-ray

scattering, instead of giving the scattering at a single angle, represent

the average intensity over a range of angles. Because of this, the

measured scattering curve is often very different from the curve obtained with point collimation. Hence, the collimation effect results in

"smearing" of the scattered radiation.
When the slit is very narrow, the effect of the width of the slit

on the smearing of x-ray scattering can be neglected, and only the

effect of the slit length needs to be considered. In some cases, the

slit width produces changes in the scattering curves that should not be

neglected. Since the distortion caused by the width of the collimating

slit is small, the correction for the width effects might be well approximated using the method developed by Taylor and Schmidt [1]. In this

method the measured scattering curve is corrected for the width effect,

before it is corrected for the length effect. In the present work, the

method of reference [1] is employed. However, more than one width slit

correction is being applied to the measured scattering curve. This is

done in order to extend the method of reference [1] to cases when the

width of the collimating slits is altered at more than one scattering

angle.

The extent of the angular region over which the collimation system

averages or "smears" the scattered radiation can be expressed by a

weighting function. This weighting function represents the intensity

distribution within the primary beam in the longitudinal direction. In

many cases, with the counter tube measurements, the length of the slit

with which the intensity distribution is measured also has an influence

on the scattered intensity. If the slit length is equal to k, then the

corrected weighting function is equal to the convolution of the uncorrected weighting function with a step function which represents the

length of the slit. Thus, the corrected weighting function W is related

to the uncorrected weighting function w through
w (t) = /w(u)h(t-u)du

—00

(1)

h(u) =1 0$ u <k

h(u) =0 k< u <°°

w is a complicated function of the geometrical parameters of the

collimating system. An excellent review on the method to calculate the

weighting function from the geometry of the system is presented by

Hendricks and Schmidt [2], There is another way to determine the

weighting function with reasonable accuracy without having to compute it

from the geometry of the collimating system. In this method, the

weighting function is determined by measuring the intensity of the beam

in the plane of registration. This is done by taking various intervals

equal to the length of the scanning slit at the detector.

The integral equation which relates the experimentally determined

intensity I (x) , measured at the scattering angle x, to the true scattering intensity I (x) and to the weighting function W(y) is [3]:

oo I(x) - /~W(y) I((x2 + y

2

) 1/2 )dy. (2) /"

There is a comprehensive literature which deals with solving integral

equation (2) [4] . Most of the methods reported involve making certain

assumptions about the functional form of W(y) and I(x). An exception is

the procedure due to Vonk [5].

In reference [6] the author proposed a class of multistep methods

which like Vonk's does not require assumptions about the functional form
of I and W. These methods start with eq. (2) rewritten for positive

scattering angle x as a linear Volterra integral equation of the first

kind, [6]

h(u) = /*K(u,x)I(x)dx 0<u$x<°° (3)

where

'"I

xl(x)dx

l 2 2^/2 (x^ - u

z

) u

and

2 „ 2 a/2.

i

W((: vt o ' W((xz - u

z

) )dz ,„ v K(u,x) = 2x1 -rjz (3b)

2n1/2

"o ,°v ^ "u J\j The experimentalist provides smeared intensity data I(xq), I(xi),

. . . I(x ) where 0<a = x„ < xi . . . <x = b < °°. Generally, the x. n

1 n

J ' l are equally spaced, A square grid is set up on [a,b] x [a,b] of the

u-x plane with mesh size Ax and mesh points {(u., x.) = (iAx, jAx),

m£i,j<:N}. mAx is the lowest scattering angle at which an intensity

'Xj measurement is taken. Interpolation is used to obtain values of I(x

n

)

if measured data at angle xn are not available. Quadrature rules are

then used to discretize eq. (3). Accordingly,

N

h(iAx) / j a. . K(iAx, JAx) I (jAx)

j=i 1J

(4)

i,j = m,m + 1, . . . N-l.

To complete the set of linear equations we proposed in reference [6]

to use the equation
h'(NAx) = j [h(NAx)-h((N-l)Ax)]=K(NAx, NAx)I(NAx) . (4a)

Li A

Here I(jAx) denotes an approximation to I(jAx). The set of values

I(jAx), j=m, . . . , N will be treated as components of the solution

vector I. Ways to approximate I(jAx) and choices of quadrature rules

are discussed in Chapter 3 while the evaluation of h is discussed in

Chapter 4, Section E. For simplicity, we will frequently drop the

bar from I(jAx) when denoting approximate values.

The step-by-step discretization of eq. (3) leads to a convergent

method if in rather loose terms the difference between the approximate

value I(iAx) and the true value I(iAx) tends to zero as Ax-K) with

x=iAx fixed and xe[a,b].

Integral equations of the first kind are known to be ill-posed [7].

This means that small perturbations in h(u) or in K(u,x) may have a

large effect on the solution \. Translated into our physical problem,

one could suspect that small errors in the experimental measurements of

I(x) or in the collimation weighting function W will be greatly enhanced

in the unsmearing process. The extreme sensitivity of the solution

vector 1 to errors in the experimental data can be greatly reduced by

converting the Volterra integral equation of first kind into the equation of the second kind by differentiation. The disadvantage in making

such a conversion is that numerical differentiation of experimental data

can introduce serious errors. The method for desmearing experimental

data presented in the Technical Note should be employed in cases when

numerical differentiation of experimental data is to be avoided.
There are several sources for possible numerical errors. The first

one is inherent in the quadrature rule employed to discretize the integral equation. The second one is random errors or noise present in

experimental data. Actually, these two sources of errors cannot be

treated separately. Certain quadrature rules (e.g. trapezoid rule)

require that the data be "sufficiently smooth" to insure numerical

stability. (A function is defined to be "sufficiently smooth" if it is

continuously dif f erentiable up to sufficiently high order with respect

to its argument [8].) However, certain quadrature methods will always

be unstable. To get an insight into numerical instability, assume that

eqs. (4-4a) are solved by back substitution, starting with I(NAx). The

method is absolutely unstable if the successive values of I(x) tend to

diverge as Ax is made smaller, while keeping NAx constant. In this work

several algorithms for discretization of the integral equation will be

employed and which are not absolutely unstable. (These algorithms along

with a stability criterion will be presented in Chapter 3.) The purpose

of presenting several algorithms is to enable the user of the program to

compare results based on various algorithms and to determine whether the

maxima and minima present in the unsmeared scattering curve are due to

random errors or whether they are real.

The program presented in this Note does not provide for the smoothing of the experimental data. This should be the user's responsibility.

In particular, the use of the trapezoidal rule has been known to lead to

even-odd oscillations in the computed discrete set of values for I(x)

[8], These oscillations can, however, be reduced by smoothing the

results. The method for smoothing the computed I(x) employed in this

7
work is a modification of the method suggested by Linz [9] , and is

described in Chapter 3. We also employ an algorithm for discretization

of the integral equation which is based on forward differences in a twostep quadrature method and designated as the Nystrom algorithm. We

found that with this algorithm, no smoothing of the results was necessary, since the computed values for I(x) were relatively free of evenodd oscillations.

There is another source of error that is brought about by the

termination of the calculations at a certain point. This termination

introduces a finite upper limit to the integral equation (3) . The

perfect collimation theory calls for an infinite upper limit in this

integral equation. This error will be more noticeable in the outer part

of the scattering curve, close to its terminal. Therefore, for the

results to be physically meaningful, it is important to extend the

measurements as far as possible beyond the significant range of the

scattering angles. We did find that, if this precaution is taken, the

termination error will not affect the locations and the relative intensities of the maxima and minima in the desmeared intensity curve.

To summarize, the procedure employed in this work for the data

desmearing, based on a class of step-by-step finite difference methods,

has certain advantages and disadvantages over the procedures reported in

the pertinent literature. The advantages are two-fold: 1) No numerical differentiation of the experimental data, a process which may introduce serious error, is required; 2) No restriction on the functional

form for the weighting function is imposed. The weighting function can

be presented as a set of data which are predetermined by the experiment

8
either by intensity measurements in the plane of registration or by

calculation from the geometry of the collimating system.

The main disadvantage of the method of solving eq. (3) by finite

differences is that, even with sufficiently smooth data, many quadrature

methods, commonly employed in solving finite difference equations, will

lead to divergent results as Ax-K).

In this work we provide a few tests of the method of solving eq.

(3). These tests are of two kinds: 1) Solution checking and; 2)

testing the various algorithms against test functions for I(x), which

lead to analytically known perfect collimation intensities I(x). The

solution checking calculations are incorporated in the program. Two

test functions for I(x) are employed. One is a Gaussian function. The

other represents a scattering curve for spheres with uniform electron

densities. This function has been tabulated by Schmidt [10]. For the

first test function, the perfect collimation intensities are also of

Gaussian form. The second test function corresponds to the perfect

collimation intensity of

sin(xa) - xacos(xa)

I(x) = A

(xa)

where a is the radius of the scattering sphere and x is proportional to

the scattering angle. In both test functions, a constant weight function was assumed. Both test functions lead to satisfactory results.

The theoretical background for the methods employed in conjunction

with the numerical solution of the set of simultaneous equations (eqs.

4-4a) and the ways to cope with the problem of numerical stabilities is
reviewed in Chapter 3. In Chapter 4, the actual computations are described, followed by the FORTRAN program (Chapter 5). In Chapter 6, the

various algorithms employed in construction of numerically convergent

methods and their effect on the solution vector I are discussed and

shown in Figures 1-4. The experimental data for this investigation were

provided by J. J. Weeks of this laboratory. In Figure 5 we show how

unsmearing of data lead to a formation of peaks in the true intensity

curve. The data for this Figure were supplied by D. Mclntyre of Polymer

Institute, University of Akron. At the end of Chapter 6, calculations

based on the two test functions described above, are shown (Table 5 and

Figure 6)

.

For the person who is interested only in the preparation and use of

the program, we present in Chapter 2 the organization of the FORTRAN

program, and provide the instructions for preparation of the input data

required for the actual computations.

10
CHAPTER 2

1. PROGRAM ORGANIZATION

A. Available Programs

There are two separate programs. The first program, listed in

Chapter 5, should be used when N, the total number of experimental data

employed in calculations exceeds 150. In this case, storing matrices

would require very large computer storage. To avoid the storage problem,

the matrix elements are computed as functions for each pair of indices i

and j, with i<j<:N. This process is more time-consuming, than with the

matrix storage. Therefore, for N>150, if computer time is of prime

importance, the weighting function should be approximated by a trapezoid

or by a constant, if possible. The case where the weighting function is

represented by a set of data is, by far, the most time-consuming.

The second program is designed for N<150. Since the components of

the matrix employed In calculations (see Chapter 4 for details on the

matrices employed) are stored in the computer memory at the beginning of

the computations, they can be used in conjunction with various algorithms

employed in the computation of the solution vector 1. Except for this

detail, the two programs are identical. This second program is available

upon request. It is not listed here.

B. Instructions and Input Data

The program requires that the components of the input vector 1

a-

[ o be spaced at equal intervals of x. This increment, Ax, is customarily

expressed in milliradians. Very often, the experimentalist, at some

point, increases the spacings between two consecutive scattering angles,

11
Suppose that at a certain scattering angle equal to pAx, the spacing between scattering angles is increased from Ax to £Ax, where H is a integer.

The computer program will interpolate by quadratic forward interpolation

the input data, starting with the angle pAx to give I values at angular

increment Ax. The last interval, between x=(N-l)Ax and x=NAx, is interpolated f linearly.J The new input r vector 1 will contain Ntoo =N + (N -p)

(£-1) equally spaced angles, where N is the number of input values fed

into the machine. Only one change in data spacing is allowed.

For example, suppose that the measured data for the scattering

intensity were initially collected at a constant interval of Ax=0.2

mrad. Suppose that the experimentalist, starting with a scattering

angle equal to 20 mrads , increased the spacing between consecutive

measurements from 0.2 mrads to 1 mrad. If the initial number of experimental measurements was 200, N will be equal to 200+(200-100) • (4)=600

,

since p=100 and 1=5. The largest scattering angle is equal to 120

mrads.

The lowest scattering angle, at which the initial measurement is

taken, ordinarily exceeds the spacing Ax between two consecutive scattering angles. Let m be the ratio of the lowest scattering angle to the

spacing between two consecutive scattering angles, rounded to the nearest

integer. As a consequence of this rounding off, the measurements of

scattering angles might be off by a much as Ax/2. To avoid this error,

the lowest scattering angle should be chosen in such a way that it

equals an integer number of the spacing between scattering angles.

Thus, mAx measures the lowest scattering angle from which the measurements are taken. The array which represents the input vector ] after

12
it is read into the computer, is shifted by m locations.

Let there be gi values at initial interval Axj and g2 values at

augmented interval Ax2 , and let £=Ax2/Ax]_. The program will fill vector

and generate vector X so that one interval Ax will apply everywhere.

Only one interpolation is allowed. We have:

g! = p-m+1, g 2 =N

o -gi

.

The following diagram 1 (Diagram 1 is presented on page I) shows the

preparation of the augmented input vector T . In diagram 1, x =mAx,

x =(m+l)Ax, etc. Starting angle is equal to mAx.

'Xj Card 1 provides four input data needed to fill vector T . These

are: 1) N , total number of input data (=g]+g2) > 2) m, defined so that

mAx measures the lowest scattering angle from which measurements, taken

at equal interval of Ax, are taken; 3) p, defined so that pAx is the

starting angle for change in data spacing and; 4) the number of interpolated values required in each Ax2 interval. This number is equal

to 1-1, where £=Ax2 /Ax 1

.

These four input parameters are read as NN, ISTART, IENTR and NSUB.

They are punched on a single card, FORMAT 4110.

If ISTART=0, 1} is the intensity at the zero scattering angle

(might be obtained by extrapolation of data to zero angle)

.

The experimentalist might increase the width of the slit at certain

scattering angles, in order to provide a sufficient scattering intensity

at larger angles. In the computer program, if the card which follows

the first card is blank, no correction for the slit-width is performed.

For each slit-width correction, a single card is read which provides two

sets of data: the lower limit of the scattering angle (in Ax units)

,

13
from which the slit-width correction applies, and the value of M2 , the

second moment of the slit-width weighting function W , defined as: w

M2 = f t

2 W (t)dt

J

W

W is normalized, so that w

c W (t)dt=l,

w

M2 (customarily given in mrads2

) has to be given in (Ax) z units.

A separate subroutine performs a slit-width correction on the data

after the interpolation. The correction is made according to the equation

I(x)=I(x)- |M2 I"(x)

Numerical evaluation of the second derivative of I(x), based on a leastsquared smoothing technique, is given in reference [1]. Let us assume

that a series of k slit-width corrections apply, each in a different

range of scattering angles. Let the second moment of the slit-width

weighting functions in the range WjAx^x^W^Ax be M2 (l), in the range W2 Ax<

x<W3 Ax be M2 (2), etc. The last correction always applies in the range

W, Ax£x<NAx. The following diagram 2 (Diagram 2 is presented on Page I)

illustrates these corrections.

There are k cards for slit-width correction, one card for each

range of scattering angles for which M2 (i), i=l, 2, ... k, is constant.

These cards follow the first card of the program. These k cards are

followed by a blank card. A blank card must be inserted, whether the

data are corrected for the slit-width effect or not. Each one of these

14
k cards contain two numbers: W. and M2 (j), j=l, 2, . . . , k. They are

punched FORMAT 110, F10.5.

A word of warning: It is advisable not to correct data for the

slit-width effect in the steep part of the scattering curve, since the

second derivative might be very large. It is recommended to apply the

slit-width correction only at scattering angles in excess of 5 milliradians.

The program user has four directive parameters which control the

computations for the slit- length effects. These four parameters are

punched on a single card, FORMAT 4110. These are: MD, IDC, INV, MM.

MD is an integer which must have one of the values 1, 2, 3, or 4. This

integer controls the form of the weighting function to be employed in

computations. These are, in order indicated by the value given to MD:

1) Constant weighting function; 2) Gaussian weighting function; 3)

Data weighting function and; 4) Trapezoid weighting function. According to the form of the weighting function, additional information must

be supplied, which describes the shape and form of this function, as

follows

:

MD=1 No additional input data are needed

MD=2 The weight function is defined as

W. = exp - [(iA) 2 /G]

A is the interval between successive values of W and G/2 is its variance,

A is determined in such a way so that a table of 21 values of W are

computed,

W.=exp-(0.01i2 ); i=0, 1, ..., 20.

15
Thus, A, the interval between two successive values of W, is equal to 0.1

/G.

-4

W is cut-off when it is less than e . A single card giving G,

FORMAT F10.2, must be inserted.

MD=3 : The data weighting function is given by a set of values W(R) . R

should be equally spaced, but the spacing can be larger than the spacing

of x in I(x). R is given in Ax units. W can be given in arbitrary

units. A set of cards giving the pairs of R and W, one pair per card,

must be inserted, FORMAT F9.3, F11.5. As many cards as needed can be

inserted. Last card must be blank.

MD=4 : A single card giving S} and S 2 must be inserted (FORMAT 2F10.2).

Si must be smaller than S2. (The trapezoid is defined so that W=l for

r<Si and W=0 for r<S2, and it is symmetric around r=0) (illustrated on

page 33). S\ and S 2 are in Ax units.

IDC , which, if different from zero, causes the computer to print out the

values of the integrals

NAx

1 ' J (x 2 -(iAx) 2

) V2

iAx

for all values of i from m to N.

INV , which, if different from zero, causes the input vector f to be

recomputed from the output vector (inversion of calculations)

.

MM , which, if different from zero, causes the integrals h. to be calculated by the lower-order approximation (see Chapter 4 for details).

The input vector t is read after all the above instructions were

supplied. For MD=1 the input vector follows immediately the blank

16
card, which terminates the information needed for the slit-width correction.

For MD^l, the input vector follows cards which characterize the

particular weighting function employed in calculations.

The input vector '' is represented by a series of real numbers,

eight numbers per card, FORMAT 8F10.0. The components of the input

'Xj Oi 'Xj 'Xj 'Xj vector T are Ij, I2, • .., IN

• After N values of T are fed into the

o machine, the computations will start, even if there are more data

cards to be read. At the conclusion of the computations, new sets of

data cards will be read, and new computations will be initiated. The

program can handle more than one set of data, however, the instructions

and specifications must remain the same. The computations are stopped

'Xj when a blank card is read for Ij.

The instructions for the input parameters and variables are

summarized in Table 1.

C. Brief Description of the Main Program and its Subroutines

The main program reads all the necessary parameters and variables

which control the computation, followed by the input data. The input

data are first shifted and interpolated as needed. The main program

calls subroutine SWIDTH for the corrections required for the effects of

the variable widths of the collimating slits. The calculations performed

by the subroutine follow the method of Taylor and Schmidt [1], Next,

subroutines CGEN and BGEN are called to calculate the coefficients

y. , and b. in the set of linear equations

N ^ N

£ y..i. = £ b..i.

j=i 1J J j=i XJ J

17
or, in matrix form, rT =DT . The main program performs an inversion of

the matrix equation to compute T . The main program also prints out the

results and performs the back-check (optional) . Subroutine BGEN calls

subroutine WGHT23 if MD=2 or 3, or calls subroutine WHGT4 if MD=4. The

subroutines WGHT23 and WGHT4 calculate K(u,x) , the kernel of the integral

eq. (3) , employing the selected weighting function. These calculations

are performed for each pair of indices (i,j) (in notation of eq. (3),

u=iAx, x=jAx, and m<(i,j)<N, where mAx is the lowest angle at which

measurements were taken)

.

In addition, the program includes two function subroutines, DGEN

and CGEN. These subroutines transform the coefficients b. ., calculated

for the rectangle rule approximation (for which I(x) is constant in each

integral and interpolates one end point) into the coefficients which obtain when I(x) is replaced by certain interpolating polynomials. This

procedure is explained in detail in Chapters 3 and 4.

D. Comments on Units Employed

% . . T , T , and ^ are given in arbitrary units (usually as counts per

a, seconds). The coordinates of ] and of U are given in units of Ax, where

Ax=(0.

,

n -Q.) and 9. is the scattering angle at the point i. If Ax is

equal to s mrads, all other linear coordinates must be given in terms of

multiples of s. No explicit value for Ax is needed, since in machine

calculations, Ax=l. The variance in the gaussian distribution and M2 ,

the second moment of the width weighting function are in (Ax) 2 units.

For example, if M2 is measured as 0.15 mrads 2

, and Ax is equal to 0.2

mrads, then M2 , in (Ax) 2 units, is equal to 3.75.

18
Occasionally, the unsmearing of data might be restricted to larger

scattering angles, or the calculations for larger scattering angles are

to be performed separately from the calculations at very low scattering

angles. Suppose that the calculations start with the scattering angles

of pAx, at which the spacing between the consecutive scattering angles

is equal to £Ax. In this case, ISTART=p, and no interpolation of input

data is needed. All the linear coordinates employed will be in units of

£Ax instead of in units of Ax. In machine calculations, £Ax=l. Similarly, M2, the second moment of the slit-width weighting function, will

be in (£Ax) 2 units.

If the scattering intensities, starting at certain angle pAx, are

much smaller than the intensities at the lowest measured angles, then

the unsmearing of data for very small scattering angles (0<pAx) might be.

performed on a restricted range of the scattering angles, terminated at

the angle pAx. This artificial cutting off the calculations generally

will not affect the pattern of the scattering curve, that is the relative positions and intensities at various maxima and minima. In this

case, the computations should be performed separately for very low

scattering angles (9<pAx) and separately for larger scattering angles

(0>pAx) . This procedure might be preferable if the computer time is of

primary concern. By performing calculations separately for different

ranges of the scattering angles, N, the total number of the input data,

is reduced. Since the processor time involved is roughly proportional

to N

2

, considerable computer time can be saved.

19
CHAPTER 3

1. THEORETICAL BACKGROUND

The equation which relates the experimentally determined intensity

of scattered rays to the true scattering intensity ',' is given by eq.

(2). In eq. (2), W may be given either in analytic form or as a set of

discrete values. Eq. (2) is transformed as follows: both sides of eq.

(2) are multiplied by x/(x2 -u2

)

1 ' 2 and integrated over x from u to °°.

After exchanging the orders of integration one obtains eq. (3). The

total number of experimental data for I(x) is N, which are given at

equally spaced intervals Ax. We assume that, if I(x) is sufficiently

small when x>NAx, so is I(x). Therefore, both the upper limits of

integrals in eq. (3) are replaced by NAx. Both I(x) and I(x) represent

a discrete set of function values I. and I., respectively, with i = 0,

1, ..., N. Thus, i=0 corresponds to u=0, i=N corresponds to u=NAx.

Actually, the calculations seldom start with u=0 but from a small angle

equal to mAx, m>0, since l(x=0) is not known, unless it is obtained by

extrapolation of data to zero angle.

The l.h.s. of eq . (3) can be written as a :>et of N linear equations:

NAx N

h

± = / I(x)G(u,x)dx^ V* Y± jIj-^j-O for j>N, (5)

iAx j=i

where

G(u,x) = —7—; u=iAx, i=m, m+1, ...,

/..2 „2\ 1 / 2 (x z -u

z

)

20
In matrix form, h=Af. is an upper triangular matrix with entries

coefficients y... The calculations of h., though they happen to represent improper integrals, are straight-forward and will be described in

Chapter 4. Suffice here to say that in each summand of eq. (5), I(x)

is expanded in a Taylor series around x=jAx, retaining only the first

three terms of the expansion.

The right hand side of eq. (3) is expanded in terms of the discrete

values I . as

l

NAx N

J I(x)K(u,x)dx= / j b_. ,1,

iAx j = i

1J J 3

b.. are coefficients to be determined. They depend on the weighting

function W. Therefore, from (5) and (6), \, which is the N-dimensional

solution vector, is given in matrix notation by

B is the inverse of the matrix B whose elements are the sets of the

coefficents b.. of eq . (6), and h is an N-dimensional vector with components h., eq. (5). Since the matrix B is an upper triangular one, its

inversion can be performed rapidly by a back substitution, starting with

the b„ „ element of R, as follows:

N,N u

VVV'n

N (7)

I. = (h. -/. b..I.)/b.., i=N-l, N-2, . . . , m.

1 X j^i+1 ^ J 1X

The method employed in this work for solving eq. (6) is based on a

replacement of the function I(x) by an interpolating polynomial having

21
the values I,=I(x, ) on a set of points x, .

This is a commonly used procedure employed in conjunction with

multi-step methods which are based on interpolation. For a multistep method of order I , the integral in eq. (5) is replaced by a repeated

equally spaced quadrature rule defined on l+l points. Thus, eq. (3) is

written as a system of (N-i+1) simultaneous equations

N (j+l)Ax

h. = y^

J I(x)K(iAx,x)dx (8)

j=i jAx

I(x) is replaced by an interpolating polynomial having the values I,=I(x )

on a set of points x . In forward interpolation of multiplicity I, the

interpolating polynomial interpolates I(x) on points x, , x,

1

, ...,

x, . In general, the interpolating polynomial P (x) of degree I

agrees with I (x) and l+l suitably chosen points, which can be located to

the right, to the left or centered around the given point x , representing

forward, backward or central interpolations, respectively. The integral

of I(x)K(iAx,x) over the interval [jAx, (j+l)Ax] is then replaced by the

integral of P, (x)K(iAx,x) over the same range. We will consider here

only two cases, 1=1, and 1=2.

First, assume 1=1. In the simplest case, which we call the

"rectangle rule" approximation, I(x) in the r.h.s. of eq. (7) is replaced by I . . In this case,

J

(j+DAx

b. . = / K(iAx,x)dx.

IT v jAx

22
An improvement over the rectangle rule approximation for a singlestep method (1=1) would be, if I(x) in the integrals of eq . (8) are

replaced by the following averages (trapezoid rule approximation)

:

I(x) = -|(I.+I. +1 ) for jAx<=x$(j+l)Ax. (9)

In this case, the elements of the matrix g are:

m+l)Ax

b. . = J K(iAx,x)dx (j = i)

1,1 iAx

(10)

Aj+l)Ax

b. . = J K(iAx,x)dx (j>i)

1,11 (j-DAx

b =0. (j<i)

X 9 J

The question is: Is the trapezoid rule approximation (eq. (9)) really

an improvement over the rectangle rule approximation? Theoretically,

the answer should be yes. Under the rectangle rule approximation, I(x)

is assumed to be constant within each strip. Under a trapezoid rule,

I(x) is assumed to be a linear function of x in each of the integrals

of the r.h.s. of eq. (8), and then replaced by its value at the middle

points of these integrals. Nevertheless, an application of the trapezoid

rule might lead to even-odd oscillations in the computed I.. These

oscillations are associated with the presence of weak numerical instabilities. The weak instabilities, in contradistinction to strong numerical

instabilities, can be greatly reduced by either adopting certain averaging procedures or by a smoothing method, and will be discussed later.

The investigations of the numerical instabilities, associated with

the employment of various quadrature rules can be greatly facilitated by

23
examination of the elements of the matrix g . We first demonstrate the

possibility of the presence of even-odd oscillations in the computed

values of I. under the trapezoid rule approximation, and the relative

absence of these oscillations under the rectangle rule approximation.

For simplicity, assume that W, the weighting function, is a constant,

equal to W=2/tt, so that K(u,x) = 2x. Employing the rectangle rule approximation, that is, assuming that I(x)=I. in each of the integrals of the

r.h.s. of eq. (8), one obtains from eq. (8),

b. .=2j+l j^i

b. .=0 j<i

i 5 :

(11)

>. ., which are the elements of the matrix R are found to be

1,3 u

i,i 2i+l

1 i,i+l 2i+l

1. . = for -Ui+2

i,J

»-li Since J=g |j, the computed I. are seen to depend on h. and h . only.

Thus, the rectangle-rule approximation will not introduce spurious

short-range maxima and minima into the computed I(x) vs. x curves, if

the original experimental I(x) curves were reasonably smooth.

Next, consider the trapezoid rule approximation, eq . (9). Now,

24
h

±i±

- \ (2J+1) j-i

b. . = 2j j>i (12)

b. . = j<i

i,J

For large i, the elements of the matrix D are approximately,

/. for j=i+2, i+4, i+6, ...

for j>i+l, i+3, i+5, ... (13)

for j=i.

i,j 2i+l

4 i,j 2±+l

3 i,j

= 2i+l

Thus,

I. = h.g. . +^7^7 (-h. .,,+h. ,, -h. .,,+ ...) (14)

x i 1,1 2i+l i,i+l i,i+2 i,i+3

and the possibility of even-odd oscillations in the computed I. is selfevident.

Before getting into discussions of the methods that might partially

eliminate these oscillations, let us next consider the case when I(x),

eq. (8) is replaced by an interpolating polynomial having values on a

set of points x, . Let Q (x) be the polynomial which approximates and

replaces I(x) in eq. (8).

This polynomial can be expressed in terms of forward, central or

backward differences. Assume 1=2. In central difference interpolation,

the interpolating points are x., x. , and x., n . In backward interJ 3-1 J+1 polation, they are x. , x. and x.

?

. In forward interpolation, they

J J — i J — ^ are x. , x. , .. and x. , _ . J j+1 j+2

25
Consider central difference interpolation (for i~2 , this method is

equivalent to the extended Simpson rule for numerical integration)

.

Consider the case of W=constant=2/ir . The interpolating polynomial Q(x)

is given by

(I -21. +1 ) Q(x) = I. + —:L-i J ±^ (x^jAx) 2 (15)

3 2 (Ax) 2

For i>>l, the elements of the ith row of the B matrix are, approximately

b. . = i/3 1,1

b. . = 4j/3 (j=i+l, i+3, i+5, ...) (16)

i »

J

b, , = 2j/3 (j=i+2, i+4, i+6, ...-).

i> J

When this matrix is inverted, 3. ., the elements of the inverted matrix,

i 5 J are approximately given by

3. . = (-l) 1*" 1 {(-2+/3) j_i + (-2-/3) j_i}

1,J i

Thus, the absolute values of 3. . are seen to grow exponentially as one

moves away from the diagonal elements of the inverted B matrix. As a

result, the numerical calculations are affected by strong numerical

instability which renders the above method of constructing interpolating

polynomial to be totally useless.

This strong instability is characterized as a pointwise phenomenon,

as it involves error growth at fixed points as the interval between them

decreases. In other words, the multi-step solution of eq . (8), when

I(x) is replaced by the above given interpolating polynomial, diverges

as Ax-+0 [13].

The question is, how one constructs an interpolating polynomial

Q(x) that will lead to numerically stable (or convergent) method for

26
calculating the solution vector I? This question was answered in detail in reference [6]. Suffice here to say that, if the coefficients

of I in the interpolating polynomial are a, , so that I(x) in r.h.s. of

eq. (8) is replaced by

a.I.+a. ,,1..,+. • .+a. , 'I. , . i x l+l l+l x+Z ±+i

then the moduli of all roots of the characteristic polynomial

<Kn) = a.n

£ + a

i+1**~S«-..^i+£ = ° < 18 >

must not exceed unity, as a condition for convergence of the multi-step

method of multiplicity &

1

. In the above example, in which central

difference method was employed for the construction of the interpolating

polynomial, the characteristic polynomial <Kn) is given by

<Kn) = !/ 3 n

2 + V3n + 1/3

and the root condition for numerical stability is not satisfied. In

reference [6] it was shown that, if instead of central differences, one

employs forward differences, a convergent method can be obtained. (In

multi-step method, employment of forward differences is a necessary, but

not sufficient condition for numerical stability.) The interpolating

points for 1=2 are j, j+1 and j+2. The root-determining equation is

<K.n) = 7/3n 2 - 2/3n + 1/3 = o. (19)

Since the moduli of the roots of eq. (19) are less than 1, the method is

numerically stable.

^This is a very serious restriction. As a matter of fact, it is very

hard to construct an interpolating polynomial which will satisfy the

root condition, when £>2.

27
For i>>l, the elements of the ith row of the B matrix (for constant

W) are:

b. . = 7i/3

1,1

b. , = -2j/3 (jfl+1, i+3, i+5, ...)

i» J

b. . = 2j/3 (j-i+2, i+4", i+6, ...).

i j J

We designate the above-described method of constructing interpolating polynomial for I(x) as the Nystrom algorithm [12] for the

following reason:

In analogy to the Nystrom method for constructing interpolating

polynomial for solving difference equations of first order, this polynomial is based on forward interpolation and is applied to the midpoint

of each of the summands in the quadrature rule approximation of the

integral equation (because it is applied to the midpoint, the coefficient

of VI, is zero, thus insuring numerical stability).

In our computer program, this algorithm is employed in addition to

the rectangle rule and trapezoid rule approximations for replacing I(x),

eq. (6) by a set of discrete values I, .

On the basis of the above discussion on condition for numerical

stability, it becomes clear why the one-step methods are always numerically stable: for the rectangle rule approximation, the root-determining

characteristic polynomial (j>(n) is of zero order. This method is therefore the most stable numerically. On the other hand, the trapezoid rule

method implies that the characteristic polynomial is of the first order,

with root equal to -1. This represents the border-line case for numerical stability.

28
All these considerations apply strictly to the case of W=constant.

For a non-constant W we found that the stability conditions are slightly

more relaxed. Specifically, for W represented by a trapezoid, it is

easy to show that the absolute values of the coefficients b. . decrease

as j increases as compared with the case of constant W. This fact is of

importance, particularly with conjunction with the trapezoid rule method,

as it implies that a non-constant W tends to eliminate some of the evenodd oscillations in the computed I(x), as compared with the case of

W=constant

.

As expected, the Nystrom algorithm leads to results that are relatively free of even-odd oscillations. This is because 3. ., the i,i

1,1

element of B matrix, is approximately proportional to -r-| A P where

|a| is the modulus of the largest root of the characteristic polynomial

(f)(n). Since X<<1, the matrix elements 3.. are seen to decrease rapidly

as one moves, in a given row, away from the diagonal elements of the

matrix B

As demonstrated above, the solution - vector \ can display undesirable even-odd oscillations, particularly when |a|, the modulus of the

root of the characteristic polynomial, is close to one. This is particularly the case when trapezoid rule is applied to the solution of eq.

(8) . These oscillations can be greatly reduced by smoothing the results

as developed in reference [6]. In this method, one computes first the

solution vector \=\ from

-1

1° = B„h (20)

B is the matrix of b . ., calculated from eq . (7,9, and 10). The

"corrected" solution vector \ is obtained by calculating the second

29
order differences 6 2 I . '=1 .,-21 . +1 . ,, for each element I. of the

1 x-1 1 1+1 1 solution vector J

, and adding y& 2 I. to I .

' . For y , a value of

1/6 was adopted. In reference [8], y

= lM. However, we found, on the

basis of numerous sample calculations, that the value y=l/6 is preferable,

This smoothing procedure is repeated, starting with the solution vector

J , to obtain a solution vector \ . Since the computations of the

smoothed solution vectors I and I are very rapid, they are both

included in the computer program.

In the final print-out, we present solutions for the scattered

intensity, based on the rectangle rule approximation and on the trapezoid rule approximations (with and without smoothing) . This is done

for the following practical purpose: the user of the computer program

should recognize that some spurious small peaks in the calculated I(x)

curves should not be given any physical interpretation. Rather, these

peaks, or oscillations, in the computed I(x) curves result from the

presence of weak numerical instabilities. In order to be sure whether

maxima and minima in the calculated I(x) curves have physical significance, one should examine the I(x) curves obtained from the application

of the rectangle rule approximation. In the rectangle rule solution,

weak numerical instabilities, resulting in even-odd oscillations, are

usually absent, provided that the experimental I(x) vs. x curves are

also free of even-odd oscillations (which might come from noise effects

or from errors in measurements). Otherwise, even with the rectangle

rule approximation, these oscillations will be magnified in the I(x) vs.

x curves. It is, therefore, very important that the experimentally

30
determined I(x) be smoothed before it is read into the computer. This

is true in particular with respect to the trapezoid rule approximation.

The solution based on the Nystrom algorithm (multiple step method

of multiplicity 2, with forward interpolation) is more accurate than the

rectangle-rule solution, and is relatively free of even-odd oscillations, so that corrections based on smoothing the results need not be

applied to this solution. The solution based on this method should be

employed for display of desmeared intensities.

The various algorithms and their effect on the calculated I(x) are

discussed in Chapter 6, where sample calculations employing various

algorithms are presented and discussed.

31
CHAPTER 4

1. DESCRIPTION OF CALCULATIONS

A. Preparation of Data which Characterize the Slit

Weighting Function

The following slit weighting functions are employed in this program: 1) Weighting function is a constant; 2) Gaussian Function; 3)

Data Function; and 4) Trapezoid Function.

For Gaussian distribution, W(r) is computed at equal intervals

equal to A. Thus, the program computes a set of discrete values:

W. = exp(-i2 A

2 /G)

G is read from the input data card, and A is arbitrarily chosen to equal

0.1/G. The subscript i assumes values of 0, 1, ..., 20.

W(r) is represented by data function. A set of cards giving the

various pairs of W(r) and r are read into the computer. The data are

represented as ri(=0), r£, r3, ..., r ; W(ri), W(r2) , W(r 3 ) , ..., W(r ).

W(r.) are given at equally spaced intervals. The first pair of data

are stored as ?2 and W(r2) , respectively. r^ is set as equal to zero,

and W(r^) is computed according to

W(0)=W(r

1 )=W(r2 )+[W(r

2 )-W(r

3

) ]r/(r

3 -r

2

)

.

There are m pairs of data. If m<N, the weighting function W(r) for

r>r is given a constant value which is equal to W(r ). Thus, W(r, ) m

n m ' k

W(r ) for m<k<N.

m

W(r) is represented by a trapezoid, as follows

32
W(r)

W(r)=l r<S.

w(r) "s^i Sl* rSSz

W(r)=0

W(-r)=W(r)

r>S,

S. Evaluation of the Integral

oo /%, v NAx y, v xl(x) dx _ /• xl(x)

(x2 -u2

) 1/2 "#.

A [x2 -(iAx) 2

] 1/2

u ,/iAx

(21)

since l(x)=0 for x>NAx. The integral h(u) = h(iAx) is computed for each

value of i by the subroutine CGEN which is entered with a given value of

i. To evaluate eq. (21) we subdivide the integral into N-i-1 integrals

as follows:

N (j-i)Ax

i-E ../- h. = xl(x)

jAx [x2 -(iAx) 2

] 1/2

dx

(22)

I(x) is expanded in a Taylor series around x = jAx

N (j+l)Ax

^S/j=i jAx iv

I., -.-I. , [x-(j+l)Ax] 3 ^

J + j

[x-(i+l)Ax] 2

(I ..-2I.+I. .) 3+1 3 3-1 > x

(Ax) 2 [x 2 -(iAx) 2

] 1/2

(23)

dx

33
N

h. =X^ y ..I.; I.=0 for j>N. (24)

% ^

In eq. (24) and in following equations, I. = I(jAx) , etc. y. . form an

3 i >3 upper triangular matrix Q. The matrix elements y. . have different

i »3 representations depending on the following three cases: (1) j-i=0, (2)

j-i even and (3) j-i odd. In addition, in order to close the system

one has to consider, separately, the cases of j=N and j=N-l.

Sometimes the high accuracy in calculating h. is not required. One

can simplify calculations and save computer time by retaining, in the

r.h.s. of eq. (23), only the first term in the Taylor expansion of

I(x). This option is controlled by the parameter MM, read with the data

cards. Since in this case the results are simple, we spell them here.

They might be of use for a fast check of the results:

y. .=1/2 {[(j+l) 2 -i 2

] 1/2

- [(j-1) 2 - i

2

) 1/2 }Ax for N-l>j>i

1 > J

y. .=1/2 (2i+l) 1/2Ax for j=i

1 5 3

y. .=0 for i<i. i>3

i!/2 _ r^_^2_, 2l l/2-

For j=N, y. =1/2 { [ (j

2 -i 2

)

]

x/ - [ (j-1) 2 -i 2

] X/

Z

}Ax.

1 »

J

C. Evaluation of the Kernel of the Integral Equation

for Various Weighting Functions

1. Preliminary Consideration

We wish to determine the coefficients b. .in the set of simuli>3

taneous equations

JN E3 = 1

b. .1. = h. (25) i,3 3 i

34
given that

NAx

h. = J I(x)K(iAx,x)dx (26)

iAx

/-[x2 -(iAx) 2

] 1/2

K(iAx,x) = 2x / W(t)

dt

(27)

[x2 -(iAx) 2 -t 2

] 1/2

The integral of eq. (27) is replaced by the sum of integrals

j=i jAx

(j+DAx

I(x)K(iAx,x)dx (28)

We assume that in eq. (28), I (x) can be replaced by its average

value within each range and taken outside the integral sign

provided that I(x) is continuous on [jAx, (j+l)Ax].

Define

[(j+l) 2 -i 2

] 1/2 Ax t r C C W(r) dr

(j

z -i z

) 'Ax o

b. . are identical to b . . only when I(x), eq. (28), is equal to I..

Otherwise, b. . will be computed from b. ., according to the algorithm

employed in replacing I(x), eq. (28), by a set of discrete values I, .

Introduce

Rn = (j z -i 2 )" / "Ax, R = [(j+l) z -i 2_42n z ]^Ax. 1 /2 Av d ^ r^a-^2_,2i 1 / 2 ,

From eq. (29), after changing order of integration, we obtain

35
R

2 R

2 R

1 R

2 (30)

. = f drW(r) f

t

; , dt + f drW(r) f —

1/2

dt

Integrate over t, to obtain

R

2 R

l

b. . = f (R

2 2 -r2

) 1/2W(r)dr - j

(R

1

2 -r 2

) 1/2W(r)dr . (31)

o o

Write

b.. = B. ,,,-B. .. (32)

To compute B. . the FORTRAN program calls a Function subprogram

B(I,J) for each pair of values i and j. This procedure is employed in

order to prevent storing large matrices. In another program which is to

be employed when N<140, the elements b. . are stored in a matrix

i,J

array, and the FORTRAN program calls a subroutine B(I,J). The function

B(I,J) or the subroutine B(I,J) calls other subroutines for evaluation

of the integrals in eq. (31). The calculations for each pair of variables i and j are performed twice. The subroutine that computes eq.

(31) is entered with a variable R=R2 the first time, and with a variable

R=Rl the second time.

Both the Gaussian and the Data function representation for W(r) are

treated in the same way, except that, for the Gaussian W(r) , W(r) are

first computed by the Main Program. The computation of B. _ and of

B. . for the Gaussian and Data function representations of W(r)

are performed in the subroutine WGHT23. For the trapezoid W(r) , these

computations are performed in the subroutine WGHT4. For W(r) = constant,

b

±(

. -Jsj+i).

36
2. Computation of eg. (31)

a) W is given by a set of data. Subroutine WGHT23 is entered the first

1 / 7 1 / 9 time with R = [(j+l) 2 -i 2

] Ax, the second time with R = (j

2 -i 2

) Ax.

Since R is generally a non- integer, we locate an integer R, such that

Calculate W(R) by linear interpolation as f ollows

:

W(R )-W(R ) W(R) = W(R ) + ^ — (R-R ) (r >R)

or W(r )-W(r ) W(R) = W(r ) '.+ - ^-±- (R-r ) (r $R) . (34)

m r - r - m m

m m-1

W(rmm) and r is the last pair of data read into the computer. Each of

the integrals, eq. (31), is represented by a sum of smaller integrals.

B . ., . is written as i,l+l

ft rk

R

2 B

± +1

= / /

(R

? 2 -R2

) 1/2 W(r)dr + / (R 2 -r 2) 1/2W(r)dr . (35)

k=1 Vl r

£

L2

In each of the integrals in the summation, eq. (35), W(r) is approximated by a linear dependence on r within each strip (r, ,r,). The

individual integrals are expressed as follows:

x k f-n 2 2\ 1 / 2tt/' \j k k-l / r-n 2 2n1/ 2 j (R z -r^) W(r)dr = —^ / (R -r z

) dr + /

-1

(36)

2

' i^'."«

2 J

v 2 r k-l

r k-l

r k

j -w r k k-l /

: k~

rk-l J

W, -W, , / , r +r

(R 2 -r 2

) 1/2

(r- -*-£% dr,

k-l

37
In eq. (36), W =W(r. )

, etc. In eq. (36) the last term is an improvement

rC K. over the trapezoid rule approximation.

Performing the integrations and rearranging terms, one obtains

' i>J+1 > I £ M»t. 1-»

k

«

k+i

) {V sln

"1

(f)

+ Wh2 -^2 ^2

+

354

[r

2

2 -(^) 2

i 3/2

| + jVi- 7^7 R„-£A <VV-M 2

X | R^sin" 1 (M) + £A[R

2

2 -(£A) 2

] 1/2 + 3^ [R^ - (£A) 2

] 3/2

ttR 2 R +U

+ ._jWr2 + w^____ (wR2

-w£

)}.

In eq. (37), r =kA, A=r,-r, . In deriving eq. (37), use was made of

the fact that r =0, by definition. In deriving the last two terms of eq,

(37), (£-l)W A and R„W A were approximated by £W A . Similar results are

/V i— Xj Xj obtained for R.. replacing R„ . As a fast check, assume a constant W(r) =

W. In this case, b. . = -t-(R 2 -R 2

) , which also obtains directly from

1,3 4 2 1

eq. (31).

b) Trapezoid Weighting Function. Let

W(t) = 1 (t<s

1

)

W(t) = (S

2 -t)/(S2 -S1

) (S

1 <t<S2

) (38)

W(t) = (t>s

2

)

38
Substitute (39) into (30) to obtain

1) R

2 >S2

R 2_s 2)1/2 + 2s

. n

'i,j+l "l v ^2 1

-S

1 (R

2

2 -S

1 2) 1/2+R

2

l>

• -if2

" in \R )' "2 V "2 S (R 2 - S 2) 1/2

". -l/

S 2\ . -l/Ml S

2 sxn y—xn ^Js^ (39)

+| [(r2-S 2)3/2 _ (R 2_ s

2

) 3/2

] 3 LV 2 2 2 1 S 2"S 1

2) S 1

<R

2 <S2 1/2

i.1+1

= W" S

l > + V S±n \TJ +

2 V " S i(

R

2

Z - S l '©• > i2 c 2 N l/2

- R,/ sin. -1 /

S l\ S

2 2 (R2

2 -

S;L 2

) 3/2

3 (S

2 -S1

) (40)

3) R

2 <S

1 Ei,j+1

=

2 R2,

(41)

Similar expressions are obtained for B. ., with R, replacing R„

.

D. Calculations of b. . i,3

For the rectangle rule approximation, b. . = B. .. For the trapei,3 i,3

zoid rule approximation, we calculate first solution vector \ for

the rectangle rule approximation,

I (o) - B

_a

h

where R is the inverse matrix of the elements b. .. Then,

i,3

t (trapezoid) _ n

_ lT (°)

the elements of the £) matrix are

(42)

(43)

D. . = D. ... = 1/2

1 , X X , x+1

D. . = j<i+l and i>i. i,3

39
Therefore, B

= D B- F°r the Nystrom algorithm (I(x) extrapolated forward),

we employ the following approximate transformation: Form matrix f" of

elements f. .as follows:

i,3

£. ,-Ib. . 1,1 3 1,1

f, , = f b (j=i+l, i+3, ...) (44)

-1- j J -

> 1 »3

f l,J-f E

i.3

^

+2 - 1+*. •)-

Equations (45) are based on the following arguments: To insure numerical stability, the point of reference for interpolating polynomial must

be located in the middle of the integrand. Therefore, eq . (28) is

replaced by N (j+l)Ax

h_.u =Z^ = / j J I(x)l(x)K(iAx,x)dx (45)

j=i,i+2, ... (j-l)Ax

Q(x) , the interpolating polynomial which replaces I(x), is given by

Q(x) = I + (I -21

+1+I

j+2 ) (x-jAx) 2 /2(Ax) 2

. (46)

Therefore, the calculations of I. must start with i=m+l, since with this

l

algorithm, calculation of I will always be numerically unstable, as

then the root condition (eq. 18) will not be satisfied. To close the

system,

and of course,

f. , = 2/3 b. (j=N, j-i even)

f. . = 1/3 b. . (j=N, j-i odd) (47)

i,J i,3

f. .=2/3 b. (j=N-l, j-i even)

i,3 1,3

b . . = f . . , B = R

1,3 i,3

40
Therefore,

T (Nystrom) ^ 1--I1

i = h nE. Solution Checking

In order to be able to make a direct check on the accuracy of the

original calculations, we recalculated I(x) by numerical integration,

from

NAx

(N 2 -i2

) 1/2 Ax

cy o f yi(y)w(y2 - (iAx) 2

) 1/2 if w(t) (i[t 2 -(iAx) 2

] 1/2

)dt.

i ~ Ja r 2 ,. A ^71/2

dy =

•/ (48) lAx [yz - (iAx) z

] o

.2_,, a ^2m1^ .. I( [t -(iAx) z

) ] ) is calculated from the computed I by quadratic

interpolation.

41
CHAPTER 5 FORTRAN PROGRAM

C CALCULATION FOR SLIT CORRECTION IN SMALL ANGLE X-RAY SCATTERING

C PROGRAM AVOIDS MATRIX STORAGE

DIMENSION XI (4 00 ), YH400 ),XX(4 00 ),Z 1(4 00 ),YK( 40 ),YL( 40 0),YY( 4 00),

1 YM( 400), U( 400)

DIMENSION LW( 10 ),WM2( 10 ) COMM0N/BLOCK1 /A, N, I START, NN, MD, MM

C0MMON/BI 0CK2/RN( 400 ),W( 400 ),V( 400 ),C( 400 ),DL,G

COMMON /BL0CK3/ KB,KL

COMMON /BL0CK5/S1, S2

DATA PI/3.1415926/

DATA IPR,IRD/6,5/

C READ DIRECTIVE I NSTPUCTI ONS AND SET NUMBER OF COMPONENTS IN INPUT VECTOR

WRITE( IPR,1 ) C AT SOME POINT THE INTERVAL BETWEEN INPUT DATA IS GREATER THAN AT

C INITIAL ANGLES. THAT POINT IS GIVEN BY * IENTR'. THE NUMBER OF SUBDIC VISIONS BETWEEN CONSECUTIVE INPUT DATA FROM THIS POINT IS GIVEN BY

C 'NSUB 1 . C N IS THE NUMBER OF COMPONENTS OF THE INPUT VECTOR AFTFR INTERPOLAC TION.NN IS THE NUMBER OF INPUT DATA READ INTO THE MACHINE.

C I START IS THE RATIO OF THE LOWEST ANGLE AT WHICH MEASUREMENT IS

C PERFORMED TO THE INTERVAL BETWEEN TWO CONSECUTIVE ENTRANCES. IT IS

C GIVEN AS INTEGER

READ( IRD, 1 ) NN, ISTART, IENTR, NSUB

SSUB-NSUB

1 FORMAT (4110 )

1 F0RMAT( 1H1 // 10X,15H SPECIFICATIONS)

N-NSUB*( NN-IENTR*I START )*NN* ISTART

WRITE( IPR,30) N,NN, ISTART

30 FORMAT( 1X.2 3HN, TOTAL RANGE OF DATA- , 13, 4X, 24HNN, NUMBER OF DATA RE

IAD- , 13, 4X.24HI START, STARTING POINT," 13)

IF( IENTR. EQ.O ) GO TO 36

WRITE( IPR,35 ) IENTR, NSUB

35 F0RMAT( 1 X, 33HENTRANCE POINT FOR SUBDIVISION," , 13, 4X, 27HNUMBER OF

1SUBTABULATI0NS, » ,13)

36 K0"0

31 K0"KO*1

C DATA FOR WIDTH CORRECTION. WM2 IS THE SECOND MOMENT AND LW IS THE

C FIRST POINT AT WHICH THE WIDTH CORRECTION APPLIES. SEVERAL WIDTH

C CORRECTIONS CAN BE PERFORMED IN SUCCESSION.

READ(IRD,32) LW( KO ) , WM2( KO ) 32 FORMAT( I10.F10.5 ) IF( LW( KOKEQ.O ) GO TO 33

WRITE( IPR,37 ) LW( K0),WM2( K0)

3^ FORMAT( 5X,3 0HWIDTH CORRECTION APPLIED FROM , 15, 8X, 1 5HSEC0ND MOMENT

1 ,F10.5)

GO TO 31

33 KO-KO-1

WRITE( IPR,200)

3 8 FORMAT(/)

I0-ISTART*1

M"N-1

N1"N*1

C SELECT THE WEIGHTING FUNCTION

READ(IRD.IO) MD,IDC, INV,MM

GO TO (40,60,100,230 ),MD

C WEIGHTING FUNCTION IS CONSTANT

40 WRITE( I PR, 50 ) 50 F0RMAT( 20 X,31 HWEIGHTING FUNCTION IS CONSTANT //

)

42
cc

c60

70

80

90

C

100

110

120

GO TO 2 60

PREPARATIONS FOR GAUSSIAN WEIGHTING

130

140

C

150

160

C

170

180

190

200

21

220

C

230

24-0

REPRESENTATION

EQUAL TO 0.02

OF

FUNCTION

WEIGHTING FUNCTION ARE TERMI'

, F7. 3, 4H G- ,F7.1//)

OF DATA FOR THE DATA WEIGHTING FUNCTION

DATA FOR GAUSSIAN

NATED WHEN W( I ) IS

READ( IRD,70) G

FORMAT( 2F10.2 ) DL«SORT( 0.01 *G ) DO 80 I -1 ,21

EI-I

RN( I )-( EI -1 .0 )»DL

W( I ) -EXP( -RN( I )*#2/G ) WRITE( IPR.90 ) DL,G

FORMATC 4X,19HW IS GAUSSIAN, DL = KBM

KL»21

GO TO 2 60

PREPARATIONS

KM

K»K*1

READ( IRD,120 ) RN(K),W(K)

FORMAT (F9.3,F11.5)

IF ( RN( K).GE. 0.00000001 ) GO TO 110

IF ( W( K).GE. 0.00000001 ) GO TO 110

DL-RN(K-1 )-RN(K-2)

EN-N

IF ( RN( 2).E0.0. ) GO TO 130

FIND W(l ) BY EXTRAPOLATION

KB-1

RN( 1 ) - . W( 1 )-W( 2)*( W( 2 )-W( 3) )*RN(2 )/( RN( 3 )-RN( 2 ) ) GO TO 140

KB«2

IF ( RN( K-l ).LE.EN) GO TO 150

KL«K-1

G6 TO 1 70

EXTEND THE WEIGHTING FUNCTION TO THE END OF THE RANGE OF COMPUTATIONS

KL-EN/DLM .5 DO 160 I"K,KL

RN( I )-RN( 1-1 )*DL

W( I )«W( K-l ) PRINT OUT THE WEIGHTING FUNCTION

WRITE( IPR,180)

FORMAT (50X,23H W IS A DATA FUNCTION. //34X,13H GIVEN DATA. ,//41 X,

2 2H K,19X,2H R,19X,2H W//

)

DO 210 K-KB.KL

WRITEC IPR.190 ) K,RN(K),W(K)

FORMAT (40X,I3,14X,F9.3, 12X,F1 1. 5/ ) SK-K/5

XK-K

IF ( XK/5.0.NE. SK ) GO TO 210

WRITE( IPR.200)

FORMAT (/)

CONTINUE

WRITE( I PR, 22 ) FORMAT ( 1 HI ) GO TO 2 60

PREPARATIONS FOR THE TRAPEZOID WEIGHTING FUNCTION

READ(IRD,70) SI , S2

WRITEC IPR,240) SI , S2

FORMATC 4X.19HW IS TRAPEZOID, SI - ,F7.3,5H S2- ,F7.3//)

43
DL-1.0

DO 250 IM,N1

250 RN( I )-I -1 C READ IN INPUT VECT6R

C IF BLANK, STOP THE CALCULATIONS

C READ ONLY N COMPONENTS

260 READ( IRD.270) ( XI( I ), I - 1 , 8 ) 270 F6RMAT (8F10.0)

IF ( XI( 1 ).LE. 0.00000001 ) GO TO 800

READ( IRD,270 ) ( XI( I ), I -9 , NN ) C SHIFT THE INPUT DATA BY I START

D6 265 II *1,NN

T"NN*1 -II II -1*1 START

265 XI( ID-XK I ) C DATA INTERPOLATION BY QUADRATIC FORWARD INTERP0LATION . IF IENTR IS

C EQUAL TO ZERO, DO NOT INTERPOLATE

IF( IENTR) 276,276,271

271 S-l ,0/( 1 .0*SSUB)

N0»NN*ISTART-2

N2«NN*I START

JENTR-IENTR

K-JENTR

DO 273 I»JENTR,N0

XX( K)»XI( I ) DO 272 J»1,NSUB

T-J

K«K*1

27 2 XX( K)-XI( I )»T*S#( XI ( 1*1 )-XI( I ) )*(XI( I*2)-2.*XI( 1*1 )*XI( I) )*T#S

1 *( T*S-1 . )/2.0

273 K-K*l

XX( K)»XI(N2-1 ) DO 274 J*1,NSUB

T»J

K*K*1

274 XX( K)-XI( N2-1 )*T*S*( XI( N2 )-XI( N2-1 ) ) XX( K*l )-XI( N2)

DO 275 I-JENTR,N

275 XI( I )»XX( I ) 276 CONTINUE

C CORRECT INPUT DATA FOR WIDTH, IF NEEDED

IF(KO.GE.l) CALL WIDTHC XI , LW, WM2, KO ) DO 290 I«IO,N

CALL CGEN ( I ) C MULTIPLY MATRIX C BY THE INPUT VECTOR

C CALCULATE AND STORE AS V( I ) THE ITH ROW OF MATRIX C

C( I )-0.0

DO 290 J-I,N

290 C( I )-C( I )*V( J)*XI( J ) C MATRIX INVERSION BY BACK SUBSTITUTION, ZEROTH ORDER APPR0XIMATI9N

31 DO 320 I-IO.N

320 YY( I )-C( I ) A-l .0

DO 340 Kl -10,

N

K-I0*N-K1

DO 340 I"I0,X

IF (I.EO.K) GO TO 330

YY( I )-YY( I )-YY( K)*B( I , K )/B( K, K ) GO TO 340

33 YY( I )-YY( I )/B( 1,1 )

44
340 CONTINUE

C OPTIONAL PRINT-OUT

IF (IDC.E0.0) GO TO 390

IF(MM.NE.O) WRITE( IPR,380 ) WRITE( I PR, 350)

350 F0RMAT( ///8X, 1 1 HC-VALUES. . ///

)

DO 360 I«IO,N

II-I-l

360 WRITEC IPR.370) 1 1 , C( I ) 370 FORMATC 15X,I4,E15.5 ) 380 FORMATC 8X,35HL0W 6RDER APPROXIMATION IN C- VALUES/

)

C MATRIX INVERSION, TRAPEZOID RULE APPROXIMATION

390 DO 400 I"IO,N

400 YI( I )*YY( I ) YI< N)«YY< N)/2.0

D0 420 Kl "10,

N

K-I0*N-K1

D0 420 I-IO,K

IF (I.EO.K) G0 T0 410

YI( I )»YI( I )-YI( K)*D( I,K )/D(K,K ) G0 T0 420

410 YI( I )-YI( I )/D( 1,1 ) 42 CONTINUE

C MATRIX INVERSI0N,NYSTR0M ALGORITHM (FORWARD INTERPOLATION)

A-0.

DO 430 I»IO,N

43 YM( I )-C( I ) YM( N)»YM( N)/2.0

DO 450 Kl "10, N

K«I0*N-K1

DO 450 I»I0,K

IF ( I.EO.K) G0 TO 440

YM( I )-YM( I )-YM(K)*F( I,K)/F(K,K)

GO TO 450

440 YM( I )"YM< I )/F( I, I ) 450 CONTINUE

C SHIFT OUTPUT VECTOR YM BY ONE. SET YM( 1 ) TO ZERO, SINCE IT CANNOT

C BE COMPUTED WITH THIS ALGORITHM.

DO 451 II -10, N

I-I6*N-I1

451 YM( 1*1 )-YM( I ) YM( I0)»O.

C APPLY LOCAL PERTURBATIONS TO REDUCE EVEN-ODD OSCILLATIONS

470 DO 480 1*2,

M

480 YK( I )-( YI( 1-1 )-2.0*YI( I )*YI( 1*1 ) )/6.0

YK( 1 )-0.0

DO 490 I-l.M

490 YK( I )-YI( I )»YK( I ) DO 500 I»2,M

500 YL( I )-( YK( 1-1 )-2.0*YK( I )*YK( 1*1 ) )/6.

YL( 1 )-0.0

DO 510 1-1 ,M

51 O YL( I )«YK( I )*YL( I ) C PRINT OUT RESULTS

54 WRITE( I PR, 545

)

545 FORMAT( // 8X,12HINPUT VECTOR, 2X, 14HRECTANGLE RULE, 2X, 9HTRAPEZ0 ID,

1 2X, 1 5HTRAPEZ. SMOOTHED, 2X, 1 4HYK( I ) SMOOTHED, 4X, 2 1HNYSTR0 M INTERPOLA

2TI0N)

WRITE( I PR, 550)

55 FORMAT( /3X, 1 HI , 6X, 5HXK I ), 10X, 5HYY( I ), 9X, 5HYK I ), 10X,5HYK( I ),10X,

45
15HYL(I ),1 0X,5HYM( I )/

)

J-0

DO 560 I'ie,N

II-I-l

J*J*1

IF (J.LE.10) GO TO 560

J-l

WRITE( I PR, 38 ) 560 WRITE< IPR.580) 1 1 , XI ( I ), YY< I ), YI( I ), YK< I ), YL( I ), YM( I ) 580 FORMAT (I4,6E15.6)

C IF NO BACK CHECK, FIND OUT WHETHER THERE ARE MARE DATA TO BE READ

C IF NOT, GO TO STOP

IF (INV.EO.O) GO TO 260

C PREPARATIONS FOR THE BACK CHECK

C THE OUTPUT VECTOR TO BE CHECKED IS PUT INTO YI ARRAY.

C INVERSE CALCULATIONS BY EXTENDED SIMPSON QUADRATURE

IF (MD.EO.l) GO TO 7 30

IF (MD.EQ.4) GO TO 760

C DATA PREPARATIONS FOR THE DATA WEIGHTING FUNCTION FOP INVERSE CHECK

EN-N

KL-EN/DL*1 .5

DO 610 1-2,

N

S-I

DO 600 K-2.KL

1 1 - RN( K ) IF ( I.LE.II ) GO TO 610

600 CONTINUE

61 U( I )»( W(K)-W(K-1 ) )*( S-RNCK-1 ) )/DL*W(K-l ) U( 1 ) -W( 1 ) 620 DO 630 1*1 ,N1

63 RN( I )-I -1 C DETERMINE OUTPUT VECTOR ON WHICH BACK-CHECKS ARE PERFORMED

DO 590 I-1,N

C DETERMINE OUTPUT VECTOR ON WHICH BACK CHECK IS PERFORMED. YI( I

)

C OF THE RIGHT HAND SIDE OF NEXT CARD CAN BE REPLACED BY YY( I )

,

C YK( I ) , YL( I ) , OR YM( I ) . 590 YI< I )-YI( I ) DO 680 I-Ifl,N

S-I

ZI( I )-W(l )»YI( I )/l .5

DO 680 J-2,N1

T-J

R-SORT( (S-l .0 )**2*( T-1.0 )**2 ) Rl -Nl *1 IF (R.GE.R1) GO TO 680

DO 640 K-I0.N1

IF (R.LT.RN(K)) GO TO 650

640 CONTINUE

Yl -YK N)*( YI(N)-YI( N-l ) )*(R-RN(N ) )/( RN( N )- RN( N- 1 ) ) GO TO 660

65 Yl-YK K-l )( YI( K)-YI(K-1 ) )*(R-RN(K-1 ) )/( RN( K )-RN(K-l ) )( YI( K*l )-2.

20*YI( K)*YI( K-l ) )*0.5*( R-PN(K-1 ) )*( ( R-RN( K- 1 ) )/( RN( K)-RN(K-1 ) )-1.0)

3/( RN(K)-RN(K-1 ))

660 IF ( MOD( J, 2 ).EQ.0 ) GO TO 670

ZI( I )-ZI( I )*U( J )*Y1*4. 0/3.0

GO TO 6 80

670 ZI( I )-ZI( I )*U( J )»Y1#8. 0/3.0

68 CONTINUE

C PRINT OUT THE ORIGINAL AND THE RECONSTRUCTED OUTPUT VECTORS

WRITE( IPR,690)

46
690

700

710

720

72 5

72 7

726

C

730

74

C

760

77

780

790

C

800

81

F0RMAT< 1H1 // 20X.11HBACK CHECKS /20X, 5HINPUT, 5X, 1 OHREC OMPUTED

)

WRITE<IPR,710 ) FORMAT( //15X.27HI XI(I) ZI(I)/)

J-0

DO 720 I-IO.N

II-I-l

J-J*l

IF (J,LE.10) GO TO 720

J-l

WRITE( I PR, 38)

WRITE( IPR.725 ) 1 1 , XI ( I ), ZI( I ) FORMATC 12X,I5,2E15.5 ) IF(MD.EQ.4) G6 TO 726

DO 727 1-1,50

EI-I

RN( I )-( EI -1.0 )*DL

GO TO 26

DATA PREPARATI6NS F6R CONSTANT WEIGHTING FUNCTION FOR INVERSE CHECK

DO 740 1*1, Nl

U( I )-l .0

W( 1 ) -1 . go to 620

data preparations for trapezoid weighting function for inverse check

DO 790 I-1,N1

FI-I

IF (FI.LT. SI) GO TO 770

IF (FI.LT.S2) GO TO 780

U( I )-0.0

GO TO 790

U( I )M .0

GO TO 790

U( I )-( S2-FI )/( S2-S1 ) CONTINUE

W( 1 )-U( 1 ) GO TO 620

END OF CALCULATIONS

WRITE( I PR, 81 0)

F0FM\T(//20X, 10HEND OF RUN)

STOP

END

TRANSFORMATION TO TRAPEZOID RULE APPROX IMATION. CONSTRUCTION OF D MATRIX

FUNCTION D( I, J ) IF( J.GE.I*2 ) GO TO 2 D-0.5

RETURN

D-0.0

RETURN

END

FUNCTION F( I, J ) C0MM0N/BL0CK1 /A, N, I START, NN, MD, MM

IF( I.GE.N-1 ) GO TO 3 IF( J.EQ.I ) GO TO 1 IF(MOD( J-I,2).EO.O) GO TO 2 IF( J.EQ.N) GO TO 5

F--2./3.*B< I, J ) RETURN

47
1 F-7./3.*B( I.J

)

RETURN

2 IF(J.EO.N-l) G0 TO 5 IF( J.EQ.N) GO TO 4

F-8./3.*B( I,J)

RETURN

3 F-.5*B( I ,J ) KETURN

4 F-1./3.*B( I.J ) RETURN

5 F-2./3.*B( I, J ) RETURN

END

C

C

C COMPUTES B MATRIX

FUNCTION B ( I, J ) C

C »*#*«* **«*#**# #*#»*«*#««•*•«•#*** ********** ******* ##«««««#«»»«#««««»*

c C0MM0N/BL0CK1 /A, N, I START, NN, MD, MM

DATA PI /3. 1415926/

S-I

T-J

G0 TO (10,20,20,30), MD

C B MATRIX FOR CONSTANT WEIGHTING FUNCTION

10 B*PHK2.0*T-A)/2.0

RETURN

C B MATRIX FOR GAUSSIAN AND DATA WEIGHTING FUNCTION

20 R-SORT( (T-A*l .0 )**2-( S-l .0 )*»2 ) CALL WGHT23( XKI,TKS,R)

B»( XKI *TKS)/(2.0-A)

R«SQRT( (T-l . 0)#*2-( S-l .0 )*#2 ) IF ( J.EQ. I ) RETURN

CALL WGHT23( XKI,TKS,R)

B »B -( XKI *TKS )/( 2 . -A ) RETURN

C B MATRIX FOR TRAPEZOID WEIGHTING FUNCTION

30 R-SORT( (T-A*l .0 )**2-( S-l .0 )**2 ) CALL WGHT4( Bl , R ) B"( 2.0*A-1 ,0)*B1

fi-SORT( (T-l .0 )«*2-( S-l .0 )**2 ) IF ( J.EQ. I ) RETURN

CALL WGHT4(B1,R

)

B-B-( 2.0*A-1 .0 )*B1

RETURN

END

C

C #*#*«****#*###*****#****#»#*###*##*»###***# #** *»##********•»**»

SUBROUTINE WGHT23( XKI ,TKS, R

)

C

C

C CALCULATE B(I,J) FOR DATA WEIGHTING FUNCTION

C »**«**##**#*##»#*#*##»*#****##**#***» #** ###***#***#«#*###«*#**

C

DIMENSION TK(400)

COMMON/BLOCK 1 /A, N, I START, NN, MD, MM

C0MM0N/BL0CK2/RN( 4 00 ),W(400 ),V( 400 ),C( 40 ),DL, G

COMMON /BL0CK3/ KB,KL

DATA PI/3.1415926/

48
KC»KB*1

DO 40 K-KC,KL

IF (R.LT. RN(K)) GO TO 50

40 CONTINUE

WF»W( KL )( W( K.L )-W( KL-1 ) )*( R-RN( KL) )/( RN( KL )-RN( KL- 1 ) ) LA-KL

GO TO 6

50 LA-K-1

WF-W(LA )( W(K)-W(LA) )*(R-RN(LA) )/( RN(K)-RN( LA) ) 60 XK0»R#R-RN< LA )*#2

IF (XKO.NE.O.) GO TO 70

XKI-PI#R*R*0.2 5 0*( W( LA)*W(LA-1 )-( RN( LA )*RN( LA- 1 > )*( W( LA )-W( LA-1 ) )/

2( ( RN( LA )-RN( LA-1 ) ) ) ) GO TO 8

7 XK1 «SQRT( XKO ) XK2"RN( LA)/XK1

XK3 -( R*R*ATAN( XK2 )*RN( LA )*XK1 )*RN( LA )*XK1*»3/1 . 5

XK4«< W( LA-1 )-( DL/( R-RN(LA) ) )*( W( LA )-WF)-W( LA) )#XK3/DL

IF (LA.EQ.l) XK4-0.0

XK5«PI*R*R*0.250#( WF*W( LA)-( R* RN( LA ) )/( R-RN( LA ) )#( WF-W( LA ) ) ) XKI «XK4*XK5

80 TKS-( W( 2)-W< 1 ) )*R*R*R/( 1 , 5»DL ) IF (LA.EO.l) TKS-C WF -W( 1 ) )*R»R*R/( 1 . 5*DL ) KK-LA-1

IF (KK.LT.KC) RETURN

DO 110 K«KC,KK

TKD-R«R-RN< K )**2

IF (TKD.NE.O,) G0 T6 90

TKA-PI*R*R*0.5*RN( K )/DL

GO T0 1 00

90 TKA-RN( K)*R*R*ATAN( RN( K )/SQRT( TKD ) )*RN( K)*RN(K )*SQRT( TKD )< SQRT( TK

2D**3) )/l .5

100 TK( K)«( \W K*l )*W< K-l )-2.0*W(K) )#TKA/DL

TKS-TKS*TK(K ) 110 CONTINUE

RETURN

END

C

c «##**»»***## ##** *»#*#*#»*****»«*#*#«*****#*####*#****# #** *

c SUBROUTINE WGHT4(B1,R)

C

COMMON /BL0CK5/S1.S2

DATA PI/3.1415926/

C CALCULATES B( I , J ) FOR TRAPEZOID WEIGHTING FUNCTION

C

IF (R.LE.S1) GO TO 130

IF (R.LE.S2) GO TO 120

B2-S2*SQRT( R»R-S2*S2

)

B3"S1*SQRT( R»R-S1*S1 ) B6-S1*S1/B3

B7-S2*S2/B2

B4»R#R»ATAN( B7 ) B5-R*R*ATAN( B6 ) IF( SI .LT. 0.000001 ) GO TO 115

Bl-( B3*B5)*0.5*( B2-B3»B4-B5 )*0. 5*S2/( S2-S1 )(( B2/S2 )**3-( B3/S1 )##3

2)/( 3.0*(S2-S1 ) ) RETURN

115 B1-(B2*B4 )*0.5*(( B2/S2 )**3-R**3 )/( 3.0*S2 )

49
RETURN

120 IF( SI .LT. 0.000001 ) G6 T0 140

B3-S1#SQRT( R#R-S1*S1 ) B6-S1*S1/B3

B5-R*R*ATAN( B6 ) Bl*( B3*B5 )*0.5*( -B3-B5*R#R* PI/2.0 )*S2*0.5/( S2-S1 )-( ( B3/S1 )#*3 )/( 3 . 20*( S2-S1 ) ) RETURN

130 Bl "R*R*PI*0.25

RETURN

140 Bl "R*R*PI*0.25-( R#*3 )/( 3.0*S2 ) RETURN

END

C

C

SUBR6UTINE CGEN (I)

C **«*****# #** **#**#*«•*##*#*#*#**#*#*»*#***#*»#**##**«#*##*****»

C CALCULATE ELEMENTS 6F ITH ROW 0F MATRIX C

C

C0MM0N/BL«CK1/A,N,ISTART,NN,MD,MM

C0MMON/BLOCK2/RN( 400 ),W( 400 ),V( 400 ),C( 400 ),DL,G

DATA PI/3.1415926/

S-I

SN-N

C CHECK F0R LOW ORDER APPROXIMATION

IF (MM.NE.O) GQ TO 140

IF (I.EO.N-1) GO TO 120

IF (I.EQ.N) GO TO 130

DO 110 J-I,N

U-J

IF ( I.EQ. 1 ) GO TO 50

IF ( I-J ) 10,30,40

10 IF ( M0D( J-I,2).EQ. ) GO TO 20

IF ( J.EQ.N) GO TO 80

Ul-J

U2-J-2

U3-J-1

Fl -SQRT( Ul *U1 -( S-l .0 )*#2 ) F2»S0RT( U2*U2-< S-l . )**2 ) T-( Fl**3-F2**3 )/3.0*( S-l .0 )**2*( Fl -F2)-U3*( U1*F1-U2*F2 )-U3*( S-l . 0)

2**2*AL0G( ( Ul *F1 )/( U2+F2 ) )*U3*U3*( F1-F2 ) V( J)*F1 -F2-T

GO TO 110

20 Ul-J-1

U2-J-3

U3-J-2

Fl *SQRT( Ul #U1 -( S-l . )**2 ) F2-SQRT( U2*U2-( S-l .0 )**2 ) T"( Fl**3-F2*#3 )/3. 0*( S-1.0 )**2*( Fl -F2 ) -U3#( U1*F1- U2*F2 ) -U3*( S-l . 0)

2**2*AL0G( ( Ul *F1 )/( U2*F2 ) )*U3#U3*( Fl -F2 ) D-0.5#( U1*F1 -U2*F2M S-1.0 )**2*AL0G( ( U1*F1 )/( U2*F2 ) ) )-U3*( Fl -F2 ) V( J)"0.5*( D*T)

IF (J.EQ.N) GO TO 110

IF (J.EQ.N-1) GO TO 90

Ul-J*l

U2-J-1

U3-J

Fl -SQRT( Ul *U1 -( S-l .0 )**2 ) F2"S0RT( U2*U2-( S-l . )**2 ) T-( Fl»*3-F2**3 )/3.0*( S-l .0 )**2* ( Fl -F2 ) -U3*< U 1 »F1-U2*F2 ) -U3#( S-1.0)

50
2#*2«AL0G( (U1*F1 )/( U2*F2) )*U3»U3*( F1-F2)

D-0.5*( U1*F1 -U2*F2*( S-l . 0)**2*AL0G( ( Ul *F1 )/( U2 *F2 > ) )-U3 *< Fl -F2 ) V( J )-V( J )-0.5*<D-T ) GO TO 1 10

30 Ul-I*l

U2-I-1

U3"I

Fl »SQRT(4.0*S)

T«Fl##3/3.0*( S-l .0 )**2*F1-U3*U1*F1-U3*( S-l. )**2*AL0G( ( U1*F1 )/U2

)

1 U3#U3*F1

D-0.5*( U1#F1 ( S-1.0 )#*2*AL0G( ( U1*F1 )/U2) )-U3*Fl

V( J )--0.5*( D-T)

GO T0 110

40 V(J)»0.0

G0 TO 110

50 IF (I-J) 60,70,100

60 V< J ) - 1 . GO TO 110

70 V( 1 )»0.5

GO TO 110

8 F1»SQRT(< D-l .0 )*»2 -( S-1.0 )**2 ) F2-SQRTC ( U-2 .0 )*#2 -( S-1.0 )**2 ) V( J)-( F1-F2 )*0.5

G0 TO 110

90 Fl -SQRT( U*U-< S-1.0 )**2 ) F2-SQRT( ( U-l .0 )»*2-( S-l . )##2 ) V( J ) -V( J ) ( Fl -F2 )*0 . 5

GO TO 110

100 V(J)*0.0

110 CONTINUE

RETURN

C END POINTS CALCULATIONS

120 F4-SQRT(2.0*( SN-1.0)-1 .0 ) V( N-l )-0.5*F4

V( N)-0.5*F4

RETURN

C LOW-ORDER APPROXIMATION

130 V( N)-0.5*S0RT( 2.0*SN-1.0 ) RETURN

140 DO 190 J-I ,N

U-J

IF ( I-J ) 150,160,170

150 IF (J.EQ.N) GO TO 180

V( J)»( SQRT( U*U-( S-1.0 )*#2 )-SQRT( ( U- 2. )**2-( S- 1 . )**2 ) ) *0. 5 GO TO 190

160 V( J)»( SQRT( 2.0»S-1 .0 ) )*0.5

GO TO 190

170 V(J)»0.0

GO TO 190

180 V( J)-0.5*( SQRT( ( U-l .0 )#»2-( S-1.0 )**2 )-SQRT( ( U-2.0 )#*2-( S- 1 . )**2 ) ) 190 CONTINUE

RETURN

END

C

C CORRECTION FOR THE EFFECTS OF THE WIDTH USING THE METHOD OF TAYLOR

C AND SCHMIDT

SUBROUTINE WIDTH( FF, LW, WQ, KO ) DIMENSION FF(4 00 ),F( 400 ),LW( 10 ), WQ< 10 ) COMM0N/BLOCK1 /A, N, I START, NN, MD, MM

AA-756.

51
LW(K0*1 )-N*l

DG 1 K-1.K6

NW"LW( K )*2

KW-LW( K*l )-4

D6 305 J- NW.KW

305 F(J) - FF(J)-WQ* (FF( J-2 )#1 20 . -FF< J- 1 )»87.-FF< J)»96.-FF< J*l )*12.*

1 FF( J*2 )*60.*FF( J*3 )*15. ) / AA

J0-LW(K > IMAX-LW(K*1 )-l

F( JO) - FF( J0)-WQ»(3 30.*FF( JO ) -381 . #FF( JO*l )-26 4.#FF( JO *2. ) 1 56. *FF

1( J0*3 )*354.»FF( J0*4 )-195.*FF< J0*5) )/AA

F( J0*1 ) - FF( JO*l )-WQ»(225.*FF( JO )-234.*FF( J0*1 )-180.*FF( J0*2)*72.

1*FF( J0*3)*207.*FF( J0*4 )-90 ,*FF( JO* 5 ) )/AA

F(IMAX-2) - FF (IMAX-2) - WQ»( 1 5. *FF( IMAX- 5 ) 60 . *FF( I MAX- 4 ) -1 2 . #

lFF(IMAX-3)- 96. *FF( IMAX-2 )- 87 . »FF( IMAX- 1 ) 1 20. *FF( IMAX ) ) /AA

F(IMAX-l) - FF ( IMAX-1 ) - WQ#( -90. »FF( IMAX-5 ) 207. *FF( I MAX-4 ) *72 . * 1FF( IMAX-3 )-l 80.*FF( IMAX-2 )-234.*FF( IMAX- 1 )*225.*FF( IMAX )) /AA

F(IMAX) FF(IMAX) -WQ*( -1 95 . *FF( IMAX-5 ) 354. *FF( IMA X-4 ) *156.»

1FF( IMAX-3) -264. *FF( IMAX-2 ) -381 . *FF( IMAX- 1 )*330 .*FF( IMAX ) ) /AA

CONTINUE

D0 2 J*1,N

FF( J)-F( J)

RETURN

END
CHAPTER 6

1. TYPICAL CALCULATIONS AND DISCUSSION OF VARIOUS ALGORITHMS

In Figures 1-4, unsmeared data calculated from the measurements

supplied by J. J. Weeks of this laboratory are presented. The measurements were performed on melt-crystallized polyethylene.

In Figure 1, results based on very low scattering angles are shown.

Results based on Nystrom algorithm (forward interpolation) are shown by

a solid line. Results based on the rectangle rule approximation and on

the trapezoid rule approximation (uncorrected for perturbation effects)

are shown by sets of points. As is evident,' even-odd oscillations start

to be noticeable at larger angles.

In Figure 2, results for larger angles are shown (20>.OO8 rad.).

The bottom set of points represent the input data. The desmeared data

are shown for two algorithms: rectangle rule approximation and Nystrom

algorithm. While both methods of interpolations fall on practically the

same curve, the Nystrom algorithm seems to lead to more numerically

stable results (smaller oscillations from the smooth curve)

.

As we pointed above, the results for desmearing data are very

sensitive to the numerical perturbations in the input data. If the data

show excessive fluctuations (resulting from experimental error or random

noise effects), those will be magnified when desmeared. To demonstrate

this, we present, in Figure 3, results obtained from experimental data

with certain amounts of random noise (bottom curve in Figure 3) . The

upper curve and points show results of desmearing data represented by

the bottom curve. The sharp fluctuations in the desmeared data are

present, no matter what algorithm is used in the calculations. In this

53
figure, the solid line represents the results based on the Nystrom

algorithm. These results are somehow more smooth than the ones based on

other algorithms. On the other hand, the trapezoid rule approximation,

even after correction based on perturbation method (smoothing the results)

were applied, lead to the biggest scatter in the calculated data.

In Figures 1-3 a constant weighting function was assumed. In Figure

4, trapezoid weighting function was assumed. As expected, the presence

of a weighting function has an effect on stabilizing the numerical

solution. The unsmeared data are devoid of scatter, and all three

interpolating methods employed (see captions to Figure 5 for their

description) fall into identical curve.

In Figure 5 we demonstrate the effect of desmearing on peak formation

in the scattering curve. The original (input) data were provided by D.

Mclntyre of the Polymer Institute, University of Akron. The measurements

were performed on a triblock polymer of polystyrene-polybutadienepolystyrene.

The lower curve in Figure 5 represents the original data. The

upper curve shows the desmeared data, obtained with the Nystrom algorithm

(forward interpolation) . Of interest is the existence of the first peak

in the desmeared curve (indicated by an arrow) ; there is no corresponding

maximum in the input data, only an inflection point. We also notice the

following facts: 1) the maximum in the input data are greatly magnified

when these data are desmeared; and 2) the desmearing of the scattering

curve tends to shift the locations of the peaks toward slightly larger

angles.

54
In all these Figures, only a part of the scattering curves are

shown. In Table 3 and 4 typical calculations are presented, together

with the input data. Fragments of the input and output data presented

in Table 3 and 4 are depicted in Figures 4 and 5.

2. TEST FUNCTIONS

Two functions were employed using a constant weighting function:

1) I(x) is Gaussian. This function was employed in order to compare

the accuracy of the various algorithms employed and in order to check

the effect of the cut-off on the relative error. The function employed

is:

y

-i2 /1250 . . I. = e for i=0,n ,1, .... 100, im

i (49)

Tabulated values of I. /I., using various algorithms, are shown in Table

5. The exact results are computed from: — = 0.01595

I.

i

L exp[-(10,000-i2 )/1250 l)

; /io,ooo-in3/2 ^2 l

\ 1250 J

(50)

As is evident, Nystrom algorithm leads to best agreement with exact

results. At x=90 , the error because of the cut-off at x=100, is still

about 1%.

2) The function was taken from P. Schmidt [10]. It represents the

infinite-slit smeared scattering function for monodisperse spheres.

This function corresponds to the perfect collimation intensity function

with a rapid succession of maxima of diminishing intensities. Hence, it

provides a good test for the numerical method employed.

The perfect collimation intensity I(x) for spheres is given by

I. = 9 [sin(ia) - ia cos(ia)] 2 /,. 6 (51)

i via,}

55
i is a scattering angle. a=0.25; i=3,4, ..., 80. The results are shown

in Figure 6. In this figure, ia=l corresponds to a scattering angle of

4 miliradians. The solid line represents eq. (51). The points are the

calculated intensities. Method of forward interpolation (Nystrom

algorithm) was employed in the calculations.

As is evident from Figure 6, there is a close agreement between the

calculated and exact results, almost up to i=76. The cut-off of calculations was at i=80 (corresponding to scattering angle of 80 miliradians)

.

The author wishes to thank Dr. John Smith for initiating the computer

program and to Mr. James Weeks for helpful suggestions about handling

experimental information and also for providing experimental data. Mr.

Martin Cordes ' help in preparation of this Technical Note, and in particular, his help with the analysis of the mathematical background to this

work, is very appreciated.

56
REFERENCES

[I] Taylor, T. R. , and Schmidt, P. W. , Acta Physica Austriaca, 25 , 293 (1967).

[2] Hendricks, R. , and Schmidt, P. W. , Acta Physica Austriaca, 26 , 97 (1967).

[3] Guinier, A., Fournet, G. , Walker, C. B., and Yudowitch, K. , Small

Angle Scattering of x-rays , p. 114 (John Wiley and Sons, Inc.,

N.Y., 1955).

[4] Shull, C. G., and Roess, L. C. , J. Appl. Phys., 18, 295 (1947).

Krathky, 0., Porod, G. , and Skala, Z., Acta Physica Austriaca,

13, 76 (1960).

Schmidt, P. W. , Acta Crystallagraphica, 1_9, 938 (1965).

Gerold, V., Acta Crystallorgraphica, JLO, 287 (1957).

[5] Vonk, G. C, J. Appl. Crystallographica, _4, 340 (1971). In

this article a different algebraic approach for the slit-height

desmearing is presented, based on the original eq. (2), rather

than on the transformed eq. (3).

[6] Mazur, J., J. Research Natl. Bur. Stand. (U.S.), 75B, No. 3 & 4,

173 (1971).

Mazur, J., andWims, A., J. Res. Natl. Bur. Stand. (U.S.), 70A,

No. 6, 467 (1966).

[7] Delves, L. M. , and Walsh, J. , Numerical Solution of Integral

Equations , Clarendon Press, Oxford, Chapter 12 (1974).

Schmacdeke, W. W. , Approximate Solutions for Volterra Integral

Equations of the First Kind , Journal of Mathematical Analysis

and Applications, 23, pp. 604-613 (1968).

[8] Kobayasi, M. , Rep. Stat. Appl. Res. J.U.S.E., 14, 65 (1967).

[9] Linz, P. , The Numerical Solution of Volterra Integral Equations

by Finite Difference Methods , MRC Tech. Summary Rep. 825, Mathematics Research Center, Univ. of Wisconsin, Madison (1967).

[10] Schmidt, P. W. , Acta Crystallographica, I_, 722 (1955).

[II] Fox, L., An Introduction of Linear Matrix Algebra , Chapter 2,

Oxford University Press (1965).

[12] Henrici, D. , Discrete Variable Methods is Ordinary Differential

Equations (John Wiley and Sons, Inc., N.Y., 1962).

[13] Hildebrand, R. , Finite Difference Equations and Simulations , Chapter 2, (Prentice-Hall, Inc., Englewood Cliffs, N.Y., 1968).

57
TABLE 1

SYMBOL FORMAT CONDITION COMMENT

NN

ISTART

IENTR

NSUB

4110 First card.

IENTRO indicates

that there is no interpolation of

input data.

LW*

WM2>

110, F10.5

(per card)

Number of cards - number of slitwidth corrections + 1 (blank)

Last card must be

blank. If no slitwidth correction

applies a single

blank card must be

inserted.

MD

IDC

INV

MM

4110 Single card

G F10.2 MD = 2**

RN*

w*

2F10.2 MD = 3** Series of cards.

Last card must be

blank.

SI

S2

2F10.2 MD = 4**

XI* 8F10.0

card)

(per Last card must be

blank.

Dimensioned variables.

These cards are inserted only according to the directive given by the

value of the integer MD

.

58
Table 2

ARRAYS USED IN PROGRAM

SYMBOL DESCRIPTION

XI Input Vector

YY Output Vector, rectangular rule approximation

YI " " , trapezoid rule approximation

YK " " , smoothing correction applied

to YI

YL , smoothing correction applied

to YK

YM " " j Nystrom algorithm (forward

interpolation)

XX Temporary input vector, for data interpolation.

ZI Input vector, recalculated by solution

checking (used only if INV^O)

W Input weighting function

RN Input coordinates of W

U Temporary storage for weighting function

for solution checking (used only if

INV^O)

V Array of y .

. for given i

N

XJ

C /j V. . X.

3=1

59
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too rH

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}H

cO >uo

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Cfl •H •H H CO

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COo

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CO PiT3

CU eouu

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T3

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CUPu

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C/3

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ccT uH

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toq

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vO lo lo LO LO LO LO vO 00 fx ON On en en en en CTi OS en rH LO m LO LO LO LO LO m LO vO

rH

H r-\ iH T-\HH rH r-l rH

OOO o o o o O o o

r~- CO CO -d- LO 00 CO fx <-{ 00 o CO 00 LO <r vO o CO CM

«* LO LO LO vO r-* 00 rH CO LO

en ON On Cn cn cn ON o O CM H LO LO LO LO LO LO vO vO vO

rH rH <-\ f-i

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O o O o o o OO

OO

r^ vO o 00 CO r«.

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LO LO LO LO in LO LO LO LO LO H rH rH rH rH rH rH rH H rH o o O O O O o O o o

vO -sf

O r^ On vO LO

00 <- cn <f

H o rH 00 o rH CO CM cn r» LO CO rH Cn rH r~~

CTi 00 vO LO St CO CM

OO ^

LO LO LO LO LO LO LO LO LO LO

rH i-H rH rH rH rH rH rH rH rH o O O O O O O O O o

LO LO LO LO LO CM 00 00 CM r-» r^ r^ r-» r- r-- 00 <Ti vO cn LO

LO LO LO LO LO LO LO vO o <n

cn cn CTi cn en cn cn cn o CO

LO LO LO LO LO LO LO LO vO vO H rH rH T-\ rH rH rH rH rH r-i O O O O o O O O O O O O O o o o o o

CM CO <r LO vO r-^ 00 cn

60
o o o O o o o O o o o O

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61
SPECIFICATIONS • NrTOT/'L RANGE OF DATAr=10C NN» NUMBER OF PAT* RFAD=!00 ISTApTr STARTING POTNT»=

If. IS TRAPEZOID'Sl= 25.000#S2= 60. GOO

C-VALUE.S..

.984 02+02 42 .29645+^1 . 64 .92378+00

1 .83610+0? 43 .28816+01 85 .88917+00

2 .76661+02 44 .27 ,:»4 9 + n l 66 .35343+00

3 .70560+02 45 .27083+01 87 .81716+00

4 .64307+02 46 .26247+01 68 .77047+00

5 .5'7646+02 47 .254 63+01 69 .7414^+00

6 .50P16+0? 48 .24733+01 9C .70166+00

7 .43717+02 49 .24040+01 91 .66001+00

8 .37058+02 50 .23397+01 92 .61660+00

. 9 .30766+02 51 .22750+01 93 .5701 3+0

10 .25355+0? 5? .22141+OJ 94 .5197o+00

11 .2101.4 + 02 53 .21534+01 95 .46425+00

12 .17652+02 54 .20961+01 9b .40143+00

1? .15193+0? 55 ,2044'++01 97 .32734+00

1'+ .13403+02 56 .1994!; + 0.1 98 .23089+00

15 .12239+02 57 .19471+01 99 . 1156^ + 00 16 . 1] 369+02 56 .19016+01

17 .10643+02 59 .18567+01,

18 .99876+01 60 .18166+01

19 .94170+01 61 .1777'. + "! 20 , 864 60+ :H 62 .17394+01

21 .8342-3 + 01 63 .17010+01

22 .772^5+01 64 .16647+01

23 . .72106+01 65 .16265+01

24 .67074+01 66 .15fi6"7 + 0! 25 .62452+0.1 67 . 15471+01

26 .5i! 335 + 01 68 .150P6+01

27 .54593+01 b° .14685+01

2f .51182+01 70 . 14290+01

29 .48080+ri.l 71 . 13897+01

30 .45425+01 72 .13514+01

31 .43229+01 73 .13136+01

32 .41454+01 74 .12755+01

33 . 39885+0

1

75 .12373+01

34 .38337+03 7o .11995+01

35 .3b922+o'l 77 .11635+01

36 .35697+01 78 .112.84+01

37 .34^19+01 79 .in'334+ni

36 .33454+01 BO .10592+01

39 .32309+01 81 .1025^+01

40 .31417+01 82 .99ORS+00

41 .30501+01 63 ~ .95867+0

62
ARBITRARY UNITS

INPUT DAT/* EXTRAPOLATED TO ZERO ANGLE

INPUT VFCTOR 2F»OTH ORDER TRAPEZOID TRAPE70ID+PERT. "T? APF/.0ID+2PFPT. NYSTROM INTER.

y i ( i

)

YY(I) Yim YK(I) YL(T) YM ( I

)

374000+02 .Ql+l^crn-Ol .166500+02 .168500+02 .168500+02 .000000

1 138P00+C2 .14-7069+01 .1986 02+01 .42.916^+01 .584756+01 ,326784+01

C * 1C3P00+02 .779972+00 .9S5_7 56+00 .106887+01 .153638+01 .102463+01

•7 868000+01 .569320+00 .604! 59+00 .661160+00 .69978,2+00 .656559+00

4 785C0O+01 .4-71755+00 .534451+00 .525176+00 .529571+00 .515821+00

5 667000+01 .395878+00 .409059+00 .425564+00 .433837+00 . 4*3296+00

6 « 581 0+01 .348198+0.0 .382697+00 .375591+00 .373827+00 .372754+00

7 4 890 0+01 .283215+00 .313699+tJO .3.15037+00 .315518+00 .315484+00

fc < 41.10000+01 .236164+00 .252731+00 .2573 7 0+00 .25^510+00 .260375+00

9 320000+01 .181866.+00 .21 9598+00 .? 12543+00 .21.0040+00 .207721+00

10 .240000+01 .132104+00 .144134+00 .162701+00 .156433+00 .155251+00

13 1*1000+01 .93578"- 01 . 1 20075+00 .115253+00 .114719+00 .111273+00

12 1320 011 + 01 ,631338-Q] .670 825-01 ,74598,4-01 .781095-01 .770785-01

13 990000+00 .(+27127-01 .591851-01 .580106-01 .543833-01 .516637-01

11 74-0000+00 ,260245-Cl .262403-01 .316592-01 .341725-01 .331945-01

15 595000+00 .163302-01 •258007-01 ,233878-01 .232936-01 .216987-01

If 520000+00 .144720-01 . 1U8516-01 .146513-01 .160945-01 .162414-01

17 4-65000 + 00 .12357°-P1 .18092.4-01. .14-9741-01 .140311-01 .132851-01

ie 420000+00 .102354-01 .66231.7-02 .973880-02 .10*674-01 .111945-01

is. 390000+00 .974 181-02 .138477-01 .112750-01 .104518-01 .985671-02

2(J 359000+00 .823824-02 .563590-02 ,7«71 c'P-0? .871470-02 .904751-02

21 341000+00 .949694-02 .1O34C6-01 .952526-02 .9071.50-02 ,89988°-02

?2 314000+00 .771770-02 .815329-02 .0455^8-02 .845811-02 .054245-02

£3 273000+00 .71 Q 86°-o? .728211-02 .739950-02 .749181-0? .748865-02

24 255000+00 .637677-02 .711527-02 .689691-02 .680443-02 .677101-02

2b 2.20C00+00 .550336-0? . 563 p 26-0

2

.583946-02 .591585-02 .5 Q l?50-02

26 201000+00 .464911-02 .536845-02 .524031-02 .521537-02 .516678-02

27 < 177000+00 .429566-02 ,a?2Q 76-02 .449151-02 .455437-02 .458257-02

28 < 160000+00 .360249-02 .426156-0? .41 19^0-0? .406970-02 ,405°75-02

29 140COO+00 .319542-02 .334341-05" .344711-0? .347680-02 .3^8*36-02

30 1240 00+00 .261458-02 ,304744-02 .pc.5248-02 .292646-02 .287215-02

31 11 0000+00 ,210894-02 .215172-02 .230175-02 .235937-02 .232659-02

32 leipoo+oo .164520-32 ..''0361 7-0? .199677-02 .200765-02 .107443-02

33 9500 00-01 . 17716^-02 .165423-02 .175709-02 .179647-02 .181919-02

34 690000-01 .159073-02 . 18.894-3-02 .175366-0? .170054-02 .167821-02

35 520000-01 .138472-02 .131002-02 .143149-02 .147628-02 .148201-02

36 775000-01 .129005-02 . l45°4]-0? .137606-02 .135751-02 .1^3433-02

37 735000-01 .119334-02 .112068-02 . 120136-02 .123090-02 .124*21-02

3c < 700000-01 . 114633-02 .126600-02 .1201*9-02 .118016-02 .117199-02

39 < 6-65000-01 .104-308-02 .102666-02 .107202-02 .108583-02 .109211-02

4 630000-01 .972527-0 3 .1 05949-02 .1 02503-02 .101549-02 .100420-02

"1 6050 00-01 .904241-03 .865560-03 ."?0776-03 .934114-03 .931*36-03

42 580C 00-01 .862476-03 .922921-03 .896546-03 .892015-03 .88272?-03

43 1 565( 0-01 .670865-03 .802034-03 ."45126-03 .860664-03 .871119-03

44 545000-01 .8502.42-03 .939697-03 .886935-03 .866692-03 .86370^-03

4b 520000-01 .810829-03 .760788-03 .8ri7286-03 .820963-03 .831494-03

4b 495C0C-01 .757400-03 .860871-.03 .809700-03 .791928-03 .783*44-03

47 470000-01 .70510+-03 .653930-03 .705478-03 .723491-03 .730357-03

48 4-50000-01 .666619-03 .756278-03 .709334-03 .694042-03 .6*3863-03

49 .430000-01 .620734-03 .576961-03 .6?1438-03 .637783-03 .642580-03

63 JOT-
50 .^15000-01 .609593-03 .664^07-03 .631612-03 .621426-03 .615614-03

51 .400000-01 .577740-03 .5*4679-03 .580671-03 .589174-03 .594325-03

52 .384000-01 .56370 4-0^ .600800-0"* .^B07H3-03 .574135-03 .572.98 q -03

53 .370000-01 .533132-03 .526609-0"! . ^41148-03 .544431-03 .547322-03

54 .352000-01 ,49095"*-03 .5.39654-0-* ,<^Pl2'J6-03 .515779-03 ,510«65-03

55 .340000-01 .472415-03 .4'-l-2?52-03 ."68540-03 .478327-03 .4*1280-03

56 .329000-01 .44866:>-03 .^02579-03 ,4745 c: 4>-03 .465284-03 .460255-03

57 .-318000-01 .431413-03 .394756-03 .4249"6-03 .435754-03 .441402-03

56 .310000-01 .4210b9-03 .468067-0 3 .440183-03 .429970-03 .426153-03

59 .299000-01 .387283-03 .374071-03 .394141-03 .400785-03 .402^00-03

60 .291000-01 .37611,0-03 .400496-03 .3«7963-03 .384961-03 .381785-03

61 .286000-01 .363479-03 .351723-03 ."*63770-03 .367908-03 .369300-03

62 .280000-01 .354498-03 .375234-03 .3644 04-03 .361029-03 .358317-03

63 .276000-01 .346108-03 .333761-03 ."44789-03 .348922-03 .349680-03

64 .272000-01 .345351-03 .358455-03 .349971-03 .347699-03 .346750-03

65 .269000-01 .346963-03 .332246-03 .341520-03 .344248-03 .346888-03

6b .26'i000-01 .339672-03 .361680-03 .349438-03 .344647-03 ,34231 q -03

67 .258000-01 .323494-03 .317664-03 ."23610-03 .332783-03 .334301-03

6a .254000-01 .330633-03 .339323-03 .332817-03 .330728-03 .330437-03

69 .249000-01 .320897-03 .321944-03 .3244°2-03 .324945-03 .325892-03

70 .243000-01 .315007-03 •319P51-03 ."18885-03 .318525-03 .318766-03

71 .238000-01 .305^84-03 .311964-0^ .311113-03 .310696-03 .310042-03

72 .232000-01 .297^71-1^ .249005-03 .300820-03 .301803-03 .301022-03

73 •228000-01 .294354-03 .296^37-03 . 2964^1-03 .296437-03 .296411-03

74 .223000-01 .290231-03 .291771-03 .202110-0"* .292062-03 .293161-03

7b .210,000-01 .28349^-0^ •2886 Q l-03 .287473-03 .286657-03 .286917-03

76 .212000-01 .271°85-0 3 .278305-03 ,'77q?q-03 .277754-03 .278203-03

77 .207000-01 .264347-03 .265664-03 .'6,73^2-03 .268239-03 .268631-03

76 .20^000-01 .259156-03 .263030-03 .?6217-'-03 .261903-03 .264378-03

79 .1990 0-01 .251700-03 .256282-03 .2553"'9-03 .255137-03 .254322-03

80 ,195000-01 .2415bl-03 .248] 18-03 .247126-03 .246824-03 .245810-03

81 . 192000-01 .234621-03 .235003-03 .237062-03 .238189-03 .237057-03

82 .'190000-01 ,23242'-i-0 3 .234239-03 .233763-03 .2337^,0-03 .233632-03

83 .188000-01 .228652-03 .2"06]9-O" .''30566-03 .230254-03 .230467-03

84 .185000-01 .22114U-P3 .226685-03 ,2?54ou-o3 .225178-03 .224443-03

6E ,183000-01 .218829-03 .215604-03 .218526-03 .219658-03 .220067-03

86 .181000-01 .214170-03 .222054-0" .216351-03 .217154-03 .216^*67-03

87 .179000-01 .212 531-03 .206286-03 .2109«5-03 .212501-03 .23 3285-03

bt .177000-01 .20670^-03 .2.18777-03 .212672-03 .210677-03 .209057-03

89 .175000-01 .20577'4-C3 .194641-03 .502375-03 .205210-03 .20628^-03

90 .374000-01 .204590-03 ' .2l6°07-03 .2090 Q 0-03 .206415-03 .204789-03

91 .172000-01 .202395-03 .102273-03 .109753-03 .202387-03 .203O&0-03

92 .1710 0-01 .203755-03 .2.12517-0" .206222.-03 .204387-03 .204361-03

93 .170000-01 ,2062fc: f:>-03 .194903-03 .201679-03 .204083-03 .205349-03

94 .169000-01 .210806-03 .2175°6-03 .211560-03 .209984-03 .200818-03

95 .168000-01 .221114-0 3 .204026-03 .21 19^2-03 .215804-03 .192538-03

96 .167000-01 .246700-03 .238202-0"* .2353^8-03 .237600-03 .186856-03

97 .166000-01 .314886-03 .255199-03 .272262-03 .279718-03 .343^93-03

96 .165000-01 .372314-03 .374^73-03 .353924-03 .281326-03 .723^43-03

99 .164000-01 .370056-03 .370056-03 .000000 .000000 .111279-02

64
BACK CHFCK^

INPUT Rf COMPUTED

XT(I) ZI(T)

.374-00+0-? .24612+0? 50 .41500-01 40985-01

1 .13800+02 .1309^+02 51 .40000-01 391.21-01

2 .10300+02 .10145+02 52 .3840 0-0 1 37905-01

3 .86800+01 • 85G4<~ + 01 53 .37000-01 36136-01

4 .76500+°1 .7F566+01 54 .352.00-01 34649-01

5 .66700+01 .66062+01 55 .34000-01 33254-01

6 .58100+01 .58016+01 56 .32900-01 323 3^-01

7 .48900+01 .40465+01 57 .31800-01 31097-01

8 .40000+01 .39053+01 56 .31000-01 30489-01

o .32000+01 .31603+01 59 .29900-01 29151-01

10 .24000+01 .23713+01 60 .2910 0-01 28603-01

11 .18100+01 .18002+01 61 .28600-01 27Q22-01

12 .13200+01 .13040+01 62 .28000-01 27502-01

13 .99000+00 .9R528+00 63 .27600-01 26951-01

14 .74000+00 .7188 a +CC 64 .27200-M 26724-01

IF. .59500+00 .59830+00 65 .26900-0 1 26321-01

16 .520 + C' .50683+00 66 .26400-0 1 25902-01

37 .46500+00 .46964+00 67 .25800-01 25224-01

la .42000+00 „4<!<jc;,u + 00 68 .25400-jl ,24954-01

19 .39000+0° .3953°+00 69 .24900-0.1 .24336-01

20 .3590 0+0° .34873+00 70 .24300-^1 .23853-01

21 .34100+00 ,3a;\4P + 00 71 .23600-01 .23?6t;-01 22 . U4CO+00 .30369+00 7? .23200-01 •2274U-01

23 .2730 0+01 .27738+00 73 .22HO°-01 .2231^-01

24 .25500+00 ,24^3-'+00 74 .22300-01 -2164C-01

25 .22000+00 .22063+00 75 .21600-01 .21330-01

26 .20100+0° ..I0793+OO 76 .21200-01 ,20730-01

27 .17700+00 .17623+00 77 .20700-01 .20233-01

28 .16000+00 .15302+0° 76 .20300-01 .l a 036-01

29 .14000+Q0 .13.'.l2 r'+0 70 .19900-01 .19407-01

30 .12400+00 .12246+00 80 .19500-01 .lioQO-01

31 .11000+00 .10826+00 81 .19200-01 .18691-01

32 .10100+00 .99984-01 82 .19000-01 .18521-01

33 .95000-01 .^3872-01 83 .18300-°! .18377-01

34 .890 0-01 .88021-01 P4 .13500-01 .1815 c>-01 35 .32000-01 .30444-01 85 .18300-01 .18125-01

3b •77500-°1 .76642-01 86 .I8IOC-11 .17839-01

37 .73500-01 .72082-01 87 .17900-01 ,17872-01

3° .70000-0! .69315-01 88 .17700-01 .17,11-01

39 .66500-01 .65177-01 . 89 .1750 0-01 .17711-01

4 .63000-01 .62344-01

41 .60500-01 .5932.0-0.1 90

91 .17400-01

.17200-r

1

17776-01

,17007-01

42 .58000-01 .57343-01 9.2 .17100-01 17 ; i37-01

43 .5650 0-01 .5550^-01 93 .17000-01 1° 133-01

4 4 .'54500-0! .538 33-01 94 .16900-11 ,18216-01

45 •52000-ji .51002-01 05 .16800-01 .18418-01

46 .49503-01 .48799-01 96 .16700-01 , ip,o5^_03

47 .47000-01 .46074-01 97 .16600-01 .19310-01

48 .45000-01 .44404-01 98 .16500-^1 .19162-01

49 .43000-01 .42071-01

QFIH

.16400-01

F.MD OF RUN

,13487-01

MIFQ

65
\D<rH<rrvOsrcor~.irioooHoo<rH

CM H (Ti H O \D fO LD coHN\OMHHo\onrsovc^\oinin

<t-d- mvO-d-CMCMCMi—li—IOn

O CO NO N CO (M co <i- vo ro in lo <f ON H CO VD CN H H

<r <r h

inOCN100C7N00CMOv£>

r-- rOCMrHOOOr^vOvO vOvOindNCSHHON

ONCOHOMTIiTiONN m o m i—i o on en o oo cm co o m m

HrlCvlvOcnHHCOCNOOCNMCO

m <r cm vOvomcocMCNi

rv in o

CMCONCOOOCMsTOO

cm oo <r cm *£> in in CMCMON^mHHinHin-d-COOONvO

U~| <T CM NvOmolCMCMrlr-1

CMOvO^OrsON-JOi

rH<rooa\cococna\mCT\\ooo<rcM<fcM v£)oo^Doco<rm • COCO<|-vO<rrHTH-d-CjNvO^D00ONCr>v£)CN m-d-co ooinmcocsjcNiHi-Hi—i

iNorsNooommoNrcon^cMco r^ONOr-^om-X) in<roominHrH^o\a»cTiooovo\ocM

m-d-co a^inincocMNNHH

N-JONCOHOOnONNNcn^OCOv! CO 0> r^aNr^minm^oo • • • • • • • • rNLnwinmHHHcoHNcooNCMinco

in <T CO lOvOsTCMNNHH

w

PQ<

oo

CM vooococoaNOLncMn c^ m co oo i— *o cm ovoommr—ii—i -< co o m on on <r vo<r-<r in vo -j- cm cm cm

HOMnincnoNco<f

m <r

66
SPLCIFICATIONS

IWTOTAL RANGE OF DATAr=135 NMf NUMBER OF DATA RE n ri=128 IST*RT»STAR T ING POTNT»= i W TS CONSTANT

ARBITRARY UNITS

INPUT VECTOR ZERQTH ORDER TRAPEZOID TRAPEZOTO+Pe'RT. TP ^P^ZOTn+^PEPT. NYSTROM INTER,

XKI) yyit; YKI) YK(I) Yi.m YM(T)

7 .607600+04 .141759+03 1800^2+0-3 . 1 37279+03 .116902+03 .000000

6 .577700+0* .121642+03 103465+03 . l??.2 clc, +07 . 1?5?03+03 .131.1 23 + 03

9 .557700+04 .112883+03 139319+03 . l?47"?+03 .119938+03 .116242+03

10 .536100+04 .967098+0? 3594 75+0? .°A51^6+0? .107257+03 ,loao67+03

11 .522200+04 .979292+02 1 07472+03 . 100704+0^ .989904+02 ,970185+02

12 .51.0900 + 04 .915276+02 . 883-O63+0? •°?6144+02 .936497+02 ,9aa?0^+02

13 .493000+04 .860158+02 946683+02 ,o ( i737(j+02 .3°6208+02 .8«6 c;87 + 02

11 ,452600+04 .331039+02 773628+0? ,8?1603+02 .833034+02 ,P44'»16+02

15 .469800+04 .799909+02 363451+0? . ^^9800+0? .824256+02 .814134+02

16 .459400+04 .782903+02 711363+0? .764727+0? .78^733+02 .7917?5+02

17 ,4noooo+04 ,767<i37+02 854438+0? .003 5o?+o? .784069+02 .776704+02

lb .433400+04 .752085+02 680«36+0? .733310+02 .750852+02 .759470+02

19 . 4?.8800+04 .734904+0? 323734+02 .770241+02 .751095+0? .7371 16+02

20 .418800+04 .712085+0? 646074+02 .607680+02 ,71«654+0? .717^61+02

21 .414800+04 .762621+02 778096+0? .750934+02 .745773+02 .7^2880+02

22 .411(100 + 04 ,609377+02 . 7)4 7146+0? .773215+02 .783779+02 .770O9o +02

C w .40530 0+04 .894163 + 0?. 872607+0? .PS8832+02 .857960+0? .8H6668+02

24 .307900+04 .100717+03 915719+0? .°390 1.8+0? .°43?21+0? .9RRQQq+ pj> 25 .300700+04 » 111732+03 109T62+0 3 .107437+03 .106197+03 .107830+03

2o .346800+04 .115263+03 113601+03 . 1.1 3532+03 .ll?69?+03 .116810+03

27 .294700+04 .111571+03 1.16^26+03 .1 I.4507+03 .ll?17?+03 .1168?o+03

2c .222100+04 .056671+02 1 062.1 5+03 .101151+03 .977818+02 ,985?68+0?

29 .13670 0+04 .445148+02 651186+02 .651002+0? .6491 15+02 .622325+02

30 .791000+03 .153256+0? 239110+02 .270171+02 .309654+02 .275775+02

31 •5ft3r00+03 .500530+01 674 r 2 8+ 01 .°02374+01 .113056+02 .921 071+01

3£ .558000+03 ,318811+01 327031+01 .3821 '4+ 01 .4673R o +01 .380135+01

33 .577000+03 ,418990+01 310590+0] .349464+03 .3°.1836+01 .363?08+01

34 .609000+03 .586739+01 527389+01 .511039+01 .500126+01 .506047+01

35 .628000+03 .750582+01 646089+01 .661137+01 .66^931+01 •677?17+01

36 .639000+03 .896337+01 855P75+0.1 ,O339°«+01 .821320+01 .8?7R3U +01

37 .630000+03 .958440+01 937600+ni . c'307n? + n 1 .o?6041+01 ,9?«786+ol

3o .614000+03 .104151 +0? 979279+01 .003075+QI .006^81+01 .100^53+02

39 .589000+03 .108794+02 110374+n? .1 n7b'<-0 + 02 .106464+02 .107^48+02

40 .551000+03 .111942+02 1^64 16+0? . 1 08917+02 .100171+02 .111611+0?

41 .500000+03 .105743+02 1 17468+C? .11 1718+02 .108808+0? . 10°6R4+02

42 .43300 0+03 .930339+01 940185+01 .075986 + 0.1 .op?5«4. + oi ,9O?-«00 + 01 43 .368000+03 .772316+01 9?0492+01 .074302+01 .856978+01 ,8 r'?07l + 01

4 4 .305000+03 .608601+ )1 624.140+01 .£.£,83 c ?+0! .683034+01 .630561+01

45 .252000+03 .449584+01 5Q3062+0 5 .560415+01 .536894+01 ,5?474l+ni

46 .207000+03 .298367+01 3061 06+01 .361353 + 0.'! .36 Q 00?+01 .366807+01

67
47 . l"7Mino + Q3 .185556+01 .290628+01 .2581*4+01 .251962+01 .235690+01

48 .'15&C00+03 .11701 4+01 .R04843+C0 .127685+01 .147460+01 .145925+01

49 .150000+03 .769560+00 • 153543+n .115838+01 .106458+01 .924647+00

5C .l4<~O00 + 03 .658164+00 .368735-0? .477137+00 .665146+00 .6°962l+00

51 .152000+03 .801045+00 .131264+01 .923950+00 .806459+00 .732213+00

52 .165000+03 .906958+00 .269449+00 .665817+00 .738759+00 .842497+00

53 .158000+03 .100458+01 .152447+01 .1.14534+01 .102107+01 .937598+00

54 .160000+03 .114847+01 .484689+00 .879247+00 .102744+01 .109*59+01

55 .165000+03 .154620+01 .181225+01 .150231+01 .138592+01 .136143+01

56 .163000+03 .145475+01 .128015+01 .142703+01 .146365+01 .149111+01

57 .160000+03 .163028+01 .162935+01 .1571^6+01 .155924+01 .154507+01

58 .159000+03 .166619+01 .163120+01 .164255+01 .164419+01 .164009+01

59 .154000+03 .180312+11 .170117+01 .172349+01 .17^044+01 .173020+01

60 .150000+03 .183011+01 •1°0606+01

'

.1*4610+01 .182246+01 .1*2221+01

61 .142000+03 .189638+01 .175516+01 .182688+01 .184750+01 .187047+01

62 .134000+03 .1863C6+01 .203561+01 .193135+01 .188777+01 .190305+01

63 .122000+03 .176046+01 .169050+01 .177434+01 .178822+01 .1*3029+01

64 .109000+03 .148404+01 .184842+01 .170064+01 .164033+01 .162052+01

65 .949000+02 .119156+01 .111966+01 .126509+01 .131772+01 .133342+01

66 .849000+02 .981093+00 .126346+01 .114537+01 .110994+01 .108476+01

67 .758000+02 .759434+00 .698729+00 .813085+00 .851320+00 .862485+00

66 .682000+02 .551062+00 .820139+00 .71.0212+00 .679618+00 .651927+00

69 .635000+02 .438280 »-00 .231986+00 .423776+00 .473783+00 .4*658*+00

70 .598000+02 .279286+90 .594575+on .437380+00 .388945+00 .345623+00

71 .581000+02 .237834+00 -.360034-01 .160372+00 .234530+00 .253"*76+00

72 .537000+02 .235595+00 .5il672+00 .328367+00 .271787+00 .237143+00

73 .595000+02 .275524+00 -.404817-01 .ittR-'V+OO .221258+00 .250751+00

74 .599000+02 ,247940+00 .591630+00 .371665+00 .302165+00 .261446+00

75 .612000+02 .356068+00 -.956501-01 .169453+00 .2 c 7e03+00 .300740+00

76 .622000+02 .328169+00 .*P 7785+0 .497341+00 .393691+00 .340124+00

77 .620000+02 .433273+00 -.151446+00 .203332+00 .324941+00 .3*7075+00

76 •637COC+02 .465672+00 .101799+01 .63897Q+00 .505755+00 ,445721+00

79 .609000+02 .446827+00 -.866470-01 .2752*4+00 .390367+00 .4616R2+00

80 .617000+02 .541131+00 .980301+00 .656086+00 .545470+00 .400597+00

81 .590000+02 .476466+00 . 101of,2+'" ."73103+00 .459122+00 .505120+00

82 .569000+02 .490132+00 .851Q10+00 .605876+00 .525614+00 .479348+00

83 .550000+02 .451570+00 . 129254+00 .3569*5+00 .436188+00 ,4"'4625 + 00 84 .535000+02 .524485+00 .773885+00 .583313+00 .51*267+00 .493457+00

85 .512000+02 .458518+00 •275084+^0 .419363+00 .^62768+00 .489053+00

86 .475000+02 . 4147053+rin .641952+00 .515841+00 .475609+00 .464204+00

87 .4^5000+02 .413575+00 .252153+00 .370927+00 .410215+00 .433371+00

86 .423000+02 .396640+00 .574997+00 .461738+00 .421547+00 .404407+00

89 .394000+02 .319288+00 .218283+00 .311404+00 .342740+00 .357776+00

90 .366000+02 .307685+00 .420293+00 .3490*9+00 .326109+00 .315393+00

91 .346000+02 .243915+00 .195077+00 .248892+00 .265388+00 .273774+00

92 .320000+02 .206338+90 .^°2763+00 .247669+00 .233550+00 .221680+00

93 .308000+02 .158938+00 .119924+0 .161733+00 .178021+00 .181672+00

94 .2^8000+02 .163684+00 . l«7O52+ r .1735^+00 .166597+00 .l61°59+00

95 .295000+02 .138160+00 .129416+0" .143754+00 .147630+00 .14*271+00

9b .286000+02 .126659+00 ,146°04+00 .1.37241 + 00 .135c;6 c;+00 .13113^+00

68
97 .2^:6000 + 02 ,12Pc »47+00 .106414+00 .120674+00 .126791+00 .l?0533+00

9H 2£ 60 00+02 . 1.42012+00 .151480+00 .140813+00 .136082+00 .135279+00

99 2rtS000 + 02 ,123136+00 .13254 4+ CO .132564+00 .132417+00 .130"'77+00

1 2.35000 + 02 . 13344 + n .11 3727+O0 .133434+00 .127932+00 .127739+00

101 289000+02 ,137271+30 .153154+00 .141239+00 .137673+00 .135051+00

102 >2 ,J 0000 + 02 .153683+00 .121389+00 .137443+00 .143121+00 .145601+00

103 2'-)300n + 02 ,lb32(J2 + n .185977+00 .167647+00 .161362+00 .157793+00

104 243000+02 .176538+00 . I.aa583+'t0 .160136+00 .166733+00 ,16Q258+00

105 .293000+02 .188575+Pfj .212488+00 .192534+00 .186941+00 .134700+00

|06 2^4000+02 .220392+00 .164661+00 .19137^+00 .193591+00 .203830+00

107 287000+02 .202513+00 .277122+00 .23351 1. + 00 .21°097+00 .210248+00

10a 2^1000+02 .237065+00 .127914+00 .189186+00 .210914+00 .22.1963+00

109 278000+02 .242651+00 .346215+00 .275310+0" .250024+00 .241802+00

110 261000+02 .2474 81+0 . 130086+PP .209739+00 .232567+00 ,248°23+00

111 249000+02 ,240043+00 .355375+00 .281135+00 .253574+00 .244534+00

112 2 24000+02 ,197238+oq .1.24221 + 00 . 1 87160+00 .207392+00 .217560+00

133 2 070 0+02 .176144+00 .2702^5+00 .214546+00 .196812+00 .186424+00

114 1 ^2000 + 02 .148384+00 .320339-01 •1355°l+00 .153449+00 .161655+00

13b 17900H+02 ,12°0B'l + O0 .214735+00 .164063+00 .147731+00 .133613+00

116 170000+02 .111267+00 .434341-01 . O n5953-0l .111635+00 .11°?00+00

117 162000+G2 .926072-01 .179101+10 .127658+00 .111084+00 .100585+00

118 1^6000+02 .Bb0^84-01 .61175&-02 .612730-01 .808624-01 .878161-01

119 1P8000+02 .844897-01 .164083+00 .111224+00 .940922-01 ,83000°-01

120 lS«000+02 .857636-f1

] .489619-0* .5A3849-01 .772763-01 .842332-01

121 .162000+02 . 104273+00 .166641+0 .1 18894+00 .103765+00 . 961^77-01

122 .164000+02 .119697+00 .4 19046-01 .086250-01 .103699+00 .112752+00

123 164 000+02 .129221+00 . iQ7aP9+oo .148302+00 .131465+00 .122800+00

124 .159000+02 .12.5052+00 .609521*01 .105075+00 .121152+00 .128343+00

12b lbOCOO+02 ,159223+00 .169151+00 .157809+00 .146820+00 ,14370 Q +00

126 154000+02 .145321+00 .1292«5+00 .144613+00 .148344+00 .151653+00

127 lui+QOO + 02 .154748+np .161346+00 .153605+00 .181714+00 .150595+00

128 .139000+02 .148464+00 . 148150+00 ,15045/i + QO ,151080+00 .153043+00

129 1260CO+02 . Ib5336 t-CO • 148770+00 .150860+00 .151.849+00 .152258+00

130 .122000+02 .154366+00 .1618,03+0 .157198+00 .155001+00 .14283°+00

131 109000+02 .149861+00 .1.46839+00 .150355+00 ,152900+00 .131625+00

132 103000+02 .173593+pn .152083+00 .158779+00 .161875+00 .191206+00

133 976000+01 .18 Q 43 r,+nfj .194302+00 .185776+00 .150314+00 .370020+00

134 951000+01 .184567+00 .1^4567+^0 .neoooo .ooooop .562712+00

69
BACK CHECKS

TMPUT RECOMPUTED

XT(I) ZKT)

7

- .60760+04 .60833+04 57 .16000+03 .16285+03

ft .57770+04 .57555+04 58 . 1 5900 + "^ .15931+03

9 .55770+04 .55991+04 59 .lS'+oo+o's .15^24+03

10 .53610+04 .53151+04 60 .15000+13 ,15n??+03

11 .52220+04 .52649+04 61 .14200+03 .1425^+03

12 .51090+04 .50973+04 62 .13400+03 . 1340°+O3

13 ,<t9300 + 04 .49539+04 63 .12200+m .12225+03

14 .48260+04 ,4ft071+04 64 .10900+03 .10986+03

15 .469S0+04 ./j700^ + 04 65 .9490 0+0? ."5175+02

16 .45940 + 0'* . 4573^+04 66 .84900+02 - .r^os^ + O?

17 .44900+04 .44081 +04 67 .75300+0? .7541^+02

la .43840+04 .4373? + 0<J 68 .68200+0° .690-(5 + C2 19 .42980+ J4 .42^83+04 69 .63500+0° .63201+02

?o .41680+04 .41798+04 70 .59800+02 .60210+02

?i .41480+04 .41841+04 71 .58100+0° .58481+02

22 .41140+04 •410&4+04 75 .58700+0° .59777+02

23 .40530 + 1.14 .40657+04 73 .59500+0? ,59?8^+0?

24 .397Q0+04 .39P07+04 74 .59900+02 .61252+02

2.5 . 38070+ O'l . 3ft26P+04 75 .61200+02 .61092+02

26 .34680+04 .34734+04 76 .62 20 0+02 .63436+02

27 .29470+04 .29710+0^ 77 .62000+0° .61860+02

28 .22210+04 .22003+04 78 .63700+02 .6549<^+0?

29 .13670+04 .13°97+04 79 .60900+02 .6016^+02

30 .79100+03 .7668°+03 ftO .61700+0? .64256+02

31 .56300 + 7. .57"32+03 81 .59000+0 2 .5*137+02

32 .55800+03 .55513+0 3 82 .56900+02 .58663+0?

33 .57700+03 .67777H03 83 .56000+0? . 5^951+0?

34 .60900+03 .60^7^+03 84 .53500+0? .55344+02

35 .6280 0+0 3 .63053+03 8^ .51200+0? .50700+02

36 .63900+03 .64291+03 86 .4750 0+02 .49136+02

37 .63000+03 .63142+03 87 .4530 0+0? .451.00 + 02 30 .61400+03 .61612+03 86 .42300+0? .43^29+02

39 .58900+03 .5894°+ 03 89 .39400+0° .39237+02

4 .55100+03 • .55315+03 90 .3660 0+0° .38287+02

'a .50000+03 .50053+03 91 .3460 0+0? .34447+02

4? .43300+03 .4323^+03 °2 .32000+0? .3324^+02

43 .36800+03 .36940+03 P3 .30800+02 .30847+0?

44 .30500+03 .30?9'J + 03 Q4 .29800+0? ,3111°+02

45 .25200+03 .25226+03 ^5 .29500+02 .29707+02

46 .20700+03 .20566+03 96 .20600+0? .2965^+02

47 .17600+03 .17619+03 97 .28600+02 .29247+0?

48 .15600+03 .15670+0 3 98 .28800+02 .?9ft21+02

49 .15000+03 .15142+03 99 .28b0 0+0° ,?<ao27+02

50 .14800+03 .14752+0 3 100 .28500+0? .2^468+02

51 .15200+13 .15435+03 101 .28900+0° .2952^+0?

52 .15500+03 .1540^+03 102 .29000+0° .H9796+02

53 .15800 + 0"^ .15932+03 103 .29300+02 .2^902+0?

54 .16000+03 .15877+03 104 .29300+0? .3002^+0?

55 .16500+03 .16685+03 105 .29300+0° .30221+02

56 .1630^+03 .1622^+03 106 .29400+0° .29903+02

70
107 .28700+0? .29450+02

108 .28100+02 .29189+02

109 .27800+02 .28518+02

110 .26100+0? .26693+0?

111 .24900+0? .26281+0?

112 .22400+0? .22509+02

n-3 .20700+0? .22001+02

at .19200+0? .19557+02

115 .17900+0? .19204+02

116 .17000+n? .17167+0?

117 .16200+0? .1737J+0?

118 .15800+0? .16407+02

119 .15800+0? .16970+0?

120 .15000+0? .16623+02

1?1 .16200+0? .17601+02

122 .16400+0? .17245+0?

123 .16400+0? .17726+02

124 .15900+02 .16658+02

125 .16000+02 .17817+02

126 . 15400+0? .16269+02

127 .14400+0? .16060+02

128""' .13900+0? .15013+02

129 . 12800+0? .1446F+0?

130 .12200+02 .13669+02

131 .10900+0? . 12612+02

132 . 10300+0? .11813+02

'133 .97600+01 .10611+02

134 .95100+01

END OF RUN

.76184+01

OF IN NIFI3

71
100

in

>-cr<

m

<

10

1 -

-

i 1 1 1 1

: D

- rectangle rule approximation; * 0- trapezoid rule approximation, corrected by smoothing a the results;

- f x

-

V

s

trapezoid rule approximation.

«

-

\*

-

— ex

_

43 ©

-

ED X © XV © Vx _

«Hm CD ©\«© CD 5* x

Xv_ x

-

-- CD '^Svim ©

-

X

-

1

CD

1 1 1 ©I 10 20 30 40

x(x^milliradians)

50 60

Figure 1. Scattering from melt-crystallized polyethylene, at very low

angles (20^0.008 radians). Weighting function is constant.

Solid line-Nystrom (forward) interpolation.

72
v/

6

4

if)

>~<

m

<

2

. 1

8

6

4

2

10 -2

»S§

Bottom curve (represented by Q points) give the input

data. O - Nystrom (forward) interpolation.

X - rectangle rule approximation.

8©

x

© ffi X**

© »©

x

&©

*x

© ©

X «»»©$®®x©

© ©

"Q„ © »

DE

CQmitoramn

J(D

tD %

tnaa

©x

mm111

40 50 60 70 80

x(x^ milliradians)

90 100

Figure 2. Data from the same experiment as in Figure 1, but for

scattering angles in the range of (0.008<26<:.02 radians)

73
if)

<

crm

<

10

cWnffln m

m

1

1 T

- O - rectangle rule approximation.

X - trapezoid rule approximation, corrected by smoothing the results.

V L \ x / x

cdo^h

Illtsmaj)

nmmQ^ujH

J nano

ana

am

50 60 70 80 90

x(x5

, milliradians)

100 110

Figure 3, Unsmearing of experimental data which were not smoothed out

for random noise effects. Bottom curve (represented by

points) show the unsmoothed input data. Solid line-Nystrom

(forward) interpolation.

74
CO

Z)

>-<

h:

m

<

10

10

O - rectangle rule approximation.

X - trapezoid rule approximation, corrected by smoothing

the results.

20 40 60 80 100

x(x^ milliradians)

120

Figure 4. Scattering from melt-crystallized polyethylene. N=1UU,

trapezoid weighting function. Bottom curve (represented

by points) show the input data. Solid line-Nystrom

(forward) interpolation.

75
100 -

oo

>-<

h:

m

<

10 -

1 -

0-1

100

x(ARBITRARY UNITS)

Figure 5. Scattering from triblock copolymer. Constant weighting

function. Bottom curve shows the input data. Upper

curve shows desmeared data, Nystrom (forward) interpolation.

76
4 8 12 16 20

Angle(in radians X250)

24

Figure 6. Scattering intensities by spheres with uniform electron

densities. Solid line-exact results. Points represent

computed results, based on Nystrom (forward) interpolation,

VU.S. GOVERNMENT PRINTING OFFICE! 1977-240-848/52

77
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-936

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A FORTRAN PROGRAM FOR DESMEARING SMALL-ANGLE X-RAY

SCATTERING CURVES.

5. Publication Date April 1977

6. Performing Organization Code

7. AUTHOR(S)

Jacob Mazur

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. •5110110

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as 9.

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) A FORTRAN program is presented that corrects for collimation effects in small angle

x-ray scattering measurements. The method employed in calculations is based on numerical solution of an integral equation, which is written as a Volterra equation

of the first kind. This equation is reduced to a system of simultaneous equations,

which are solved by a matrix inversion method. Several different algorithms are tried, based on different interpolating polynomials for I (x) , the true scattering

intensity which is expressed as a function of the scattering angle x. The problem

of numerical stability, which is inherent to the computations algorithms employed

in the algebraic solution of integral equation is discussed and exemplified with

actual computations.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

FORTRAN program; integral equation, interpolating polynomial; numerical stability;

slit correction; small angle scattering; unsmearing.

18. AVAILABILITY [j^ Unlimited

~J For Official Distribution. Do Not Release to NTIS

| y| Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. C13. 46:936

Order From National Technical Information Service (NTIS)

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19. SECURITY CLASS

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83

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PATENT REVIEW IS NOT REQUIRED.

USCOMM-DC 29042-P74
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Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or final reports on work performed by NBS for

outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey

bibliographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service. A

literature survey issued biweekly. Annual subscription: Domestic, $20.00; Foreign, $25.00.

Liquified Natural Gas. A literature survey issued quarterly. Annual subscription: $20.00,

Superconducting Devices and Materials. A literature survey issued quarterly. Annual subscription: $20.00.

Send subscription orders and remittances for the pre- ceding bibliographic services to National Bureau of Standards, Cryogenic Data Center (275.02) Boulder,

Colorado 80302.
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National Buraau of Standards

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