Computer simulation of photons in spheric media for density gauges

COMPUTER PHYSICS COMMUNICATIONS 7 (1974) 192-199. NORTH-HOLLAND PUBLISHING COMPANY

COMPUTER SIMULATION OF PHOTONS IN SPHERIC MEDIA FOR DENSITY GAUGES E.R. CHRISTENSEN Department of Electrophysics, The Technical University, DK-2800 Lyngby, Denmark Received 14 December 1973

PROGRAM SUMMARY Title of program: MCD Catalogue number: AAUL Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue)

sidered. The photons are assumed to originate from a monoenergetic point source situated at the center. The relative photon track length, weighted with a detector efficiency spectrum, is calculated in consecutive radial intervals. In order to contribute to the track length, the elements of the photon paths must form an angle with radius vector which

Computer: IBM 370/165;Jnstallation: Northern Europe University Computing Center, DK~28OOLyngby, Denmark

isgreater than a specified cut.off angle. Method of solution

Operating system: HASP-Il

Photons are simulated by the Monte Carlo method

Programming languages used: FORTRAN IV

The scattering angle is determined by iteration from the Klein—Nishina cross section. A least-squares fit of an analytical expression to the calculated results is performed.

High speed store required: 23262 words

[11.

No. of bits in a word: 32 Restrictions on the complexity of the problem A maximum of 12 elements, 19 energy groups, 50 radial-,

Overlay structure: None No. of magnetic tapes required: None Other peripherals used: Card reader, line printer No. of cards in combined program and test deck: 631 Card punching code: EBCDIC CPC Library subprograms used: Catal. number: Title: AAUK MCS

Ref. in CPC: 7 (1974) 185

and 11 importance intervals may be considered. The medium is spheric and homogeneous. Typical running time Simulation of 24 000 histories takes about 3 miii of central processor time. unusual features of the program Responses in different radial intervals can by the principle

Keywords: Nuclear, photon scattering, Monte Carlo method, spheric medium, density gauge, principle of similitude, leastsquares fit.

of similitude be interpreted as responses in a given interval at different densities. This presupposes an infinite medium and similar radial intervals.

Nature of physical problem Multiple photon scattering in a spheric medium is con-

Reference [1] Y.A. Schreider, Ed., The Monte Carlo method (Pergamon, Oxford, 1966).

E.R. Christensen, Computer simulation of photons in spheric media

193

LONG WRITE-UP 1. Introduction The program MCD is used to simulate subsurface gamma density gauges. The program is intended to supplement the program MCS, described earlier [1], which simulates photons in two-layered media for surface gauges. Simulation of photon trajectories by the Monte Carlo method is expected to be more computer time consuming than analytical or numerical methods, However, the method is relatively simple and has some additional advantages. One example is the calculation of the radius of the sphere of importance, which is done as part of a calculation of a response curve, since life histories of The the individual source photons are the followed in detail. radius of the most distant point from the source of a photon trajectory can thus easily be identified at any moment of the photon history, and used for calculation of the contribution of different radial layers to the total count rate.

2. Simulation of photon trajectories The variables of the one-dimensional geometry are defined in fig. 1. The fate of individual source —

Loss by absorption

\~

//

photons, emitted from the center, are followed until absorption or energy degradation below the detection limit takes place. The main features of the calculations wifi be illustrated by way of an example. Let us suppose that a photon is scattered at the position r~_1with an angle Xn. .~-

//

The path length

~n

is determined from (1)

—

t’

~fl

where r is a random number between 0 and 1, generated by the IBM subroutine RANDU [2] and is the total macroscopic cross section. The new radius r,~is then given by /2 2 2~n—1~ cos Xn’ 2 rn V’~n_i + ~n + A total of 50 similar radial intervals are defined. The shells are further classified into 10 importance intervals each including 5 shells. The linear path connecting the scattering positions rn_i and r,~will in general pass through a number of different radial- and importance intervals. This path is traversed in similar subintervals i~p 1,where / is an interval index. The step length in a given interval, measured in units of a characteristic length for that interval, is a constant for all intervals. Let us assume that a path length p has been traversed from the position r~1.In the next step, the path length p’ wifi be ,

—

~ =p+Llp1. We shall consider the length q

(3)

q=p’—~.~~p1,

(4)

“

~

=

~

check whether this length exceeds the path length (5)

rn

ç

S&ittering point

— -

Vfl~

n-2

--

--

If this inequality does not hold, the intei~acentradius r, is calculated ~ (6) Further, two integers J and K are generated J = AR + BR X ALOG(RI)

(7)

K = (J + LDD)/LD,

(8)

Source

Fig. 1. Definition ofgeometrical variables,

indicating in which shell and importance interval the

194

E,R. Christensen, Computer simulation of photons in spheric media

photon is found. The least value of K compared to earlier values at the same trajectory is retained. Eqs. (7) and (8) conform to usual FORTRAN notation in order to emphasize the truncation by the conversion of reals into integers. In these equations, AR and BR are constants, while LDD = 9 and LD = 5 in the actual case. Cosine to the angle with radius vector

denote a number of track length units, while C is an associated number of collisions. The measured count rate F in the interval / is given by: imax

~

W~çb~1,

(16)

cos x1 is also evaluated = ~ 2 + 2 2 ~“2 ~ cosx1 ~q r1 ~ q~,. Provided that the inequality

19\

—

‘

/

cosxi>wm (10) does not hold, where Wm is a specified cut-off angle, the flux-, collision-, and importance calculation takes place. Next, eq. (3) is used to advance a new increment of the path, until the inequality (5) is satisfied. When this occurs, the following inequality is evaluated

where ‘~ is the total number of histories in all series, and W~is the detector efficiency spectrum. The weighted total number of collisions Q1 for detected photons in the interval number/is =

imax

wC

~17

i ii’

The weighted total number of track length units is in like manner imax

~ci~t>r,

(11) where is the total Compton scattering cross section, and r is a new random number. If the inequality (11) is fulfilled, scattering occurs. The scattering angle ~ and the new photon energy is determined by the subroutine COMP [1]. The new angle X0+l with radius vector is given by

F;=

cosxn+l

Hence, the average number of collisions per photon q1 is given by =~ i19 q1 ~ The count rate Zik from photons which have their most distant scattering point in the importance interval k is

=

cosOn÷lcosp~÷1 sinO~÷1 cost sinp~+1, (12) —

(18)

imax

where

Zik

cosp~+1 =

cosxn cosv~+ sin; sinv,~

(13)

and cosvn

en_i

+~

cos~~)/r~.

(14)

In eq. (12), ~ is determined by =

ir(2r

—

1),

~

=

where r is a new random number. The outlined scheme is repeated, starting with eq. (1) from which a new path length is calculated, until absorption or energy degradation below the detection limit occurs, The detected photons contribute to the arrays C11 and Ri/k,where i is an energy group index,/ an interval number, and k the importance interval number for the most distant scattering event. q and R ~,

(20)

Accumulating Z/k for all importance intervals from k to ik, we get ,

(15)

W~R11~~.

i=l

ik

=

(2

/k”

1

)

k’k

where k ~ ik

=

(J

+ LDD)/LD.

(22)

This is the count rate obtained, if all material outside the importance interval number k is removed. By normalization with we calculate the relative importance response ~fk = ~ 23 1k 1k’ /

F;

E.R. Christensen, Computer simulation of photons in spheric media

In the program MCD, the output contains the array ~/k~

Further, the vectors

F; and

l/a~=

1=

c1,

(30)

are listed. where is the calculated number of counts in the interval number /. By use of the general solution quoted in ref. [3] we find the coefficients a0, a1 and a2:

3. Least-squares fit of an analyticalexpression

,

The value of the results obtained by the Monte Carlo method is greatly enhanced, if they can be represented by a reasonably good fitting function. For this purpose, the following analytical expression isconsidered 4=a4exp(—c~),/r~1,2,.~,n.

(24)

where

(25)

x1

w

195

p,d.

In eq. (24),y is the number of counts in the interval number /, while a, b and c are parameters to be determined by the fitting procedure. In eq. (25), p1 is the density, which by the principle of similitude can be associated with interval source-detector distance.number /, and d is the average

a1

ñ~

=

a2

r~,r12),

—

(31)

~_(r2y—r1yr12),

(32)

a0 =7 —(a1~1+a2~2),

(33)

where D =1

(34)

—

r~2,

7) r 1y

=

~c1(ln ~—i~ )(ln c~—J (ln x—i )2~

i~/Ijc1

In order to perform the least-squares analysis, logarithms are taken on each side of eq. (24):

a0 + a1 in

where =

+ a2x1,

Ec,(x,—~2)Onc~—5) ~ 2,/~~(ln c—j~)2 = ~,/~c1(x1—~2)

(26)

(36)

a 0

lna,

a1

b,

c=—a2,

y1=iny).

(27)

By this transformation, the problem is reduced the determination of a linear least-squares fit of to the coefficients a 0, a1 and a2 [3] .The standard deviation a 1 fory~can be written dln4 ,

of the standard deviation 1 ‘2 —a ~‘2 /

~1

=J

(37)

(38)

1/n

i

d y. a.=—Ta., I y. I where is the standard deviation for

o/2

~c~(lnx~—i1Xx,—) 2 ~.J~c(inc_53)2 ~‘l2 ~/~c1(lnx1—i1) 2 Il/(n—1) Ec~(ln~—ii)

1

—,

4.

(28) The square

2

Il/(n—1)

s 2

(39)

=\

is thus (29)

VI

Ec,~x~—i2)

1/n

E

~,

2

________________________ Ii/(n—1) ~~(1nc1—yT~

S

1/n~~

where it has been used that the number of counts obeys Poisson statistics. In the Monte Carlo calculations described, this is only approximately true, since

The mean valuesj, i~and -

—

weighting with a detector efficiency spectrum has been introduced. The statistical weighting factor w1 to be used in the least-squares analysis is thus

y

—

~ ~lnc~,

(40)

‘

are further given by

~c

Ec1.x,

1lnx ,

~cj

i~ =

/,

~

~c

=

~c. (41)

The variance of the fit is also calculated

196

E.R. Christensen, Computer simulation of photons in spheric media

=

1/(n—3)

~c~(ln c1—(a0+a1 lnx1+a2x1))2 1/n ~

(42)

All summation symbols indicate sums from! = 1 to /n. In order to evaluate the fit of the data to the fitting function, the multiple-correlation coefficient R and the associated coefficient FR for an F test are calculated 2

Group energies and associated ht of the detector response. Energy (keY) Weight 662 629 566 509 458 413 371

.323 E 5 .348E5 .392 E 5 .474 E 5 .549E5 .651 E 5 .820 E 5

334

.101E6

301 271 244 219 197 178 160 144

.968 ES .683E5 .228E5 .000EO .000 EU .683E4 .842 E 4 .107 E 5

R =a 1s1r1~/s~ +a2s2r2y/sy, 1R2 FR

=

2

(43) (44)

(1—R2)/(n—3) The outlined least-squares analysis is performed by the subroutine CURFIT. All input data to CURFIT are explained by comment cards. The input data indude the number of counts c 1, the associated densities p1, and the source—detector distanced. The printed output from CURFIT shows the least-squares fitted values of a, b and c. Further, the number of counts calculated by the Monte Carlo method c1, as well as the fitted valuesy are listed. Finally, a condensed table of number of counts versus density is printed, in which the fitting function (24) has been used with the derived values of a, b and c.

-

of the tables from the least-squares analysis must be specified. All input data are fully explained by comment cards. It should be noted that the number of radial intervals per importance interval must be an integer.

5. The sample problem 4. Running procedure The microscopic cross-section data for the program MCD are calculated by the auxiliary program CCX [1] The cross sections are punched on cards in order to be directly usable as part of the input data for MCD. The input data for the program MCD include further a maximum radius and an interval factor determining the interval division. Importance intervals and subintervals are specified by proper integers. The medium is characterized by its density and elemental composition. In order to perform the least-squares analysis, the source—detector distance and a series of densities, derived by the principle of similitude from the interval division, are supplied as part of the input data. Further, three integers controlling the organization .

In the sample problem, the maximum radius is 21.5 cm, while the interval factor is 0.94. A number of 10 importance intervals each including 5 radial intervals are defined. There are further 2 subintervals per radial interval. As regards the detector cut-off angle, this has been chosen to be 750, This angle accounts in an approximate way for the effect of a shielding between source and detector. The medium considered is silica with a water content of 3. 26.67 per cent by weight and a keV) density Further, a 137Cs source (662 andof19 3energy g/cm groups have been introduced. The assumed lower energy threshold for the detector of 114 keY means that only the first 18 groups are relevant for the detection. By the inclusion of the energy dependent detector response, a cylindrical 6Li containing glass scintillator

E,R. Christensen, Computer simulation of photons in spheric media

(diameter: 2.5 cm, height: 1.25 cm) with pulse discriminator levels of 114 keV and 188 keY has been assumed. Group energies and associated weights of the detector response are given in table 1. The source strength is 1.2 mCi, and the response is given in counts per minute. Some results of calculations and experiments with a configuration related to that of the sample problem are given in ref. [4]. A total of 3 series of each 8,000 histories have been simulated. A part of the output from program MCD is reproduced in the Test Run Output.

197

References [1] E.R. Christensen, Computer Phys. ~ommun. 7 (1974) 185. [2] Random number generation testing, IBM Publication No. GC2O-8011-0 (IBM, Newand York, 1959). [3] P.R. Bevington, Data reduction and error analysis for the physical sciences (McGraw-Hill, New York, 1969). [4] E.R. Christensen, Nucl. Eng. Design 24(1973)431.

198

E.R. Christensen, Computer simulation of photons in spheric media

TEST RUN OUTPUT SERIES

NO.

3

HiSTORIES ABSORPTIONS L OW—ENERGETIC ESCAPES

8000 617 7383 0 7248

DETECTIUNS IMPORTANCE

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.59 0.87 0.93 0.96 0.95 0.98 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

INTERVAL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

MAP

0.0 0.0 0.0 0.0 0.0 0.66 0.82 0.91 0.94 0.97 0.98 0.99 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.57 0.80 0.86 0.90 0.93 0.95 0.96 0.97 0.97 0.98 0.98 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

PHOTONS 2189 2847 3536 3838 4654 5310 6070 6665 7546 8190 8597 9778 9670 10059 10679 11316 11368 11841 12667 12360 12656 12620 12763 12678 12573 12076 12369 12466 12301 11515 11707 11214 10682 10734 10381 10v11 9814 9348 8996 8391 7902 7866 1493 7245 6912 6691 6370 5772 5246 5189

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.54 0.71 0.82 0.85 0.89 0.90 0.92 0.94 0.94 0.96 0.96 0.97 0.97 0.99 0.98 0.99 0.99 0.99 0.99 1.00 1.00 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 00 0.0 0.0 0.0 0.0 0.0 0.52 0.68 0.76 0.80 0.83 0.85 0.87 0.90 0.92 0.93 0.94 0.94 0.95 0.96 0.97 0.97 0.96 0.97 0.98 0.98 0.99 0.98 0.99 0.98 0.99 0.99 0.99 0.99 0.99 0.99

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.Q 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.47 0.64 0.69 0.79 0.80 0.83 0.84 0.86 0.89 0.90 0.92 0.91 0.92 0.93 0.94 0.94 0.95 0.96 0.95 0.96 0.96 0.97 0.96 0.96 0.96

COUNTED PHOTONS 0.133838 04 0.194486 04 0.213918 04 0.246356 04 0.30974E 04 0.376688 04 0.394496 04 0.46895E 04 0.501896 04 0.572486 04 0.589946 04 0.679188 04 0.659616 04 0.66977E 04 0.723766 04 0.735616 04 0.717196 04 0.790548 04 0.781806 04 0.789648 04 0.782096 04 0.765248 04 0.798886 04 0.788396 04 0.82074E 04 0.731296 04 0.73644E 04 0.754706 04 0.781526 04 0.687796 04 0.683966 04 0.658816 04 0.608036 04 0.633226 04 0.595126 04 0.621586 04 0.572026 04 0.557896 04 0.529166 04 O.49248E 04 0.462776 04 0.471026 04 0.423926 04 0.417716 04 0.403416 04 0.355536 04 0.366248 04 0.286798 04 0.274736 04 0.252866 04

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.47 0.0 0.62 0.0 0.66 0.0 0.72 0.0 0.75 0.0 0.81 0.53 0.80 0.61 0.84 0.67 0.84 0.69 0.87 0.74 0.89 0.76 0.88 0.79 0.90 0.79 0.90 0.80 0.91 0.81 0.88 0.81 0.92 0.85 0.92 0.84 0.90 0.83 0.92 0.84

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.49 0.0 0.64 0.0 0.64 0.0 0.68 0.0 0.71 0.0 0.68 0.45 0.76 0.63 0.73 0.61 0.75 0.62 0.75 0.61

COLLISIONS 10448 13516 16811 17938 21598 23454 26476 28519 31310 33369 34472 37907 34953 37390 39441 41191 40435 40442 42528 39788 40720 39288 39592 38441 37532 35616 35711 35474 33330 31119 31421 29687 27451 26797 25269 24041 23481 21813 20912 18732 17509 17067 16088 15417 14491 13929 12850 11408 10494 10277

AVG. CDLI... PER PHOTON 0.440328 01 0.425448 01 0.422178 01 0.424648 01 0.427986 01 0.391588 01 0.395336 01 0.376458 01 0.37594E01 0.352638 01 0.359258 01 0.346618 01 0.344638 01 0.332758 01 0.328226 01 0.334398 01 0.329616 01 0.303388 01 0.304746 01 0.281448 01 0.292826 01 0.285876 01 0.280206 01 0.278848 01 0.272048 01 0.276938 01 0.268276 01 0.266858 01 0.247348 01 0.250878 01 0.254958 01 0.247086 01 0.242366 01 0.229606 01 0.222488 01 0.213598 01 0.221466 01 0.216878 01 0.215578 01 0.208378 01 0.203698 01 0.201298 01 0.203348 01 0.200698 01 0.191566 01 0.202048 01 0.188988 01 0.197648 01 0.196938 01 0.203208 01

E.R. Christensen, Computer simulation of photons in spheric media N

~0

NT =

30

05

2.0805

01

1.155E

00.

CM

YAM*X**BM*EXP(—CM*X) WHEI4E AM,3M.CM RMUL I I 2 3 4 5 6 7 8 109 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 41 48 49 50 I I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 28 27 28 29 30

=

=

Q.98899E N

3.0005 00 2.8205 00 2.0515 00 2.4925 00 2.3426 00 2.202E 00 2. OTOC 00 1.945E 00 1.8296 1.119E 00 00 1.6166 00 1.5196 00 1.4286 00 1.342E 00 1.2625 00 1.1866 00 1.1155 00 1.0486 00 9.8506—01 9.260E—01 8.1006—01 8.1806—01 7.6906—01 7.2305—01 6.SOOC—O1 6.390E—01 6.000E—0l 5.6405—01 5.3106—01 4.990E—01 4.690C—01 4.4105—01 4.1405—01 3.890E—01 3.6606—01 3.4405—01 3.230E—01 3.040E—01 2.8605—01 2.0905—01 2.5206—01 2.3706—01 2.2305—01 2.1005—01 1.9705—01 1.8505—01 1.7406—01 1.6405—01 1.5406—01 1.4506—01 R G/CM**3 1.0006—01 2.00OE—0~ 3.0005—01 4.000E—01 5.0006—01 6.0006—01 7.0005—01 8.0006—01 9.0005—01 1.0006 00 1.1006 00 1.2005 00 1.300E 00 1.4006 00 1.5006 00 1.6006 00 1.7006 00 1.800E 00 1.900E 00 2.0006 00 2.1005 00 2.2005 00 2.3005 00 2.4006 00 2.SOOE 00 2.600E 00 2.700E 00 2.BOOE 00 2900E 00 3.0006 00

9.768E 00

F

02. =

MC

0.10492E

04

L5—FIT

1.3380 1.945D 2.1390 2.4630 3.097D 3.767D 3.9450 4.6890 5.0190 5.7250 5.8990 6.7920 6.5980 6.6980 7.2380 7.3560 7.1720 7.9050 7.818D 7.8960 7.8210 7.6520 7.9890 7.8840 8.2070 7.3130 7.3640 7.5470 7.8150 6.8780 6.8400 6.5880 6.0500 6.3320 5.9510 6.2160 5.7200 5.5790 5.2920 4.9250 4.6280 4.7100 4.2390 4.1770 4.0340 3.5550 3.6620 2.8680 2.7410 2.5290

03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

V 1.975E 3.8185 5.2926 6.4026 7.1596 7.7006 7.9846 8.0836 8.036E 1.8756 7.6295 7.3196 6.966E 6.585E 6.187E 5.784E 5.383E 4.9906 4.6086 4.2436 3.8956 3.5666 3.2576 2.9696 2.700E 2.4525 2.2226 2.011E 1.8176 1.8396

03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

1.639E 1.9716 2.3326 2.121E 3.1345 3.5606 3.997E 4.4426 4.8786 5.3075 5.7206 6.IIIE 6.4745 6.8086 7.1045 7.3665 7.5865 7.7665 7.9056 8.0045 8.063E 8.0845 8.0705 8.0225 7.9435 7.8365 7.7005 7.5456 1.3755 7.1836 6.977E 6.7615 6.531E 6.2976 6.0636 5.8235 5.578E 5.3436 5.108E 4.8756 4.632E 4.4015 4.1906 3.9826 3.7685 3.564E 3.373E 3.1955 3.0136 2.8475

03 ~3 0.. 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

6.8246—02 52

=

0.25241E02

199