Library. »•«• BlaB

OCT ^ 1962

^ecknic&l ^iote 159

A FORTRAN CODE FOR CALCULATION

OF EIGENVALUES AND EIGENFUNCTIONS

IN REAL POTENTIAL WELLS

R. S. CASWELL

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

I
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and maintenance of the national standards of measurement and the provision of means and methods for making measurements consistent

with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,- devices, and structures; advisory services to government agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; and the development of standard practices, codes, and specifications. The work includes basic and applied research, development, engineering, instrumentation, testing, evaluation, calibration services and various consultation and information services. Research projects are also performed for other government

agencies when the work relates to and supplements the basic program of the Bureau or when the Bureau's unique competence is required. The scope of activities is suggested by the listing of divisions and sectiot on the inside of the back cover.

Publications

The results of the Bureau's research are published either in the Bureau's own series of publications or

in the journals of professional and scientific societies. The Bureau itself publishes three periodicals avail- able from the Government Printing Office: The Journal of Research, published in four separate sections presents complete scientific and technical papers; the Technical News Bulletin presents summary and pre- liminary reports on work in progress; and Basic Radio Propagation Predictions provides data for determining

the best frequencies to use for radio communications throughout the world. There are also five series of nonperiodical publications: Monographs, Applied Mathematics Series, Handbooks, Miscellaneous Publicatior and Technical Notes. A complete listing of the Bureau's publications can be found in National Bureau of Standards Circular 460, Publications of the National Bureau of Standards, 1901 to June 1947 (SI. 25), and the Supplement to National Bureau of Standards Circular 460, July 1947 to June 1957 ($1.50), and Miscellaneous Publication 240,

July 1957 to June 1960 (Includes Titles of Papers Published in Outside Journals 1950 to 1959) (82.25); avail- able from the Superintendent of Documents, Government Printing Office, Washington 25, D. C.
NATIONAL BUREAU OF STANDARDS

technical ^Tiete

159

AUGUST 1962

A FORTRAN CODE FOR CALCULATION

OF EIGENVALUES AND EIGENFUNCTIONS

IN REAL POTENTIAL WELLS

R. S. Caswell

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 25 cents
Contents

Page

1. Introduction 1

2. Description of the Calculation 2

5» Search Procedure 5

k. Sample Results 11

5. References 11

Appendix 1. Evaluation of f .(outside) Ik

Appendix 2o Meaning of the Input Variables 16

Appendix 3. Format of Data Cards 18

Appendix k. Fortran Statements and Sample Output 20

ii
A Fortran Code for Calculation of Eigenvalues and

Eigenfunctions in Real Potential Wells

Randall S. Caswell

A Fortran code has been developed for the calculation of eigenvalues and eigenfunctions for neutrons

in real potential wells. A systematic procedure is

given for approximate location of the eigenvalue and an automatic search procedure to determine the exact

location. The code may be used for either bound or scattering states. In the case of scattering states,

the criterion for maximum scattering (90° phase shift)

is used to determine the energy of the state. The

eigenvalues are determined by matching the numerically

calculated logarithmic derivative (f .) inside the nucleus

to the appropriate analytical logarithmic derivative for

the region outside the nucleus. In an alternate mode of

operation, the outside value of f . may be set arbitrarily,

and a match made to this value. Sample results for a Woods-Saxon well with spin-orbit coupling for the case

of oxygen-16 are shown. The code is in Fortran and was written for an IBM 7090 computer.

1. Introduction

Nuclear shell model calculations have until now employed almost

exclusively harmonic oscillator wave functions for the evaluation of

radial integrals. This has been done both for bound states and for states

which actually belong to the continuum. The bound states evidently can be

very well represented by harmonic oscillator wave functions. Any errors

due to the difference between the "actual" optical potential and the harmonic oscillator potential will in general be much smaller than the errors

due to neglect of higher configurations. This is, however, not necessarily

true for continuum states.

The bound state wave functions have quite a different character than

the continuum wave functions. In a treatment of continuum states it is

thus desirable to have something better to work with than the harmonic

oscillator eigenfunctions. The present report is concerned with the

1
evaluation of eigenfunctions specified by a given potential well and

boundary conditions.

Calculations of eigenvalues in a diffuse potential have been reported by Ross, Mark, and Lawson (195&) where they used an analog computer and obtained eigenvalues accurate to about 0.1 Mev. The present

calculation is for a digital computer (IBM 7O9O) , is inherently capable

of much higher accuracy, and also provides numerically integrated eigenf unctions as required.

2. Description of the Calculation

This code has been developed from a nuclear optical model code

written for calculation of neutron elastic scattering which has been

described elsewhere (Caswell, I962) . As a result some of the features

are slightly more complicated than would be necessary for the present

purpose o The main calculational steps are as follows: (l) input of data;

(2) calculation of the real potential well, which is a Saxon well with

spin-orbit coupling in the present code, but can easily be changed; (5)

numerical integration of the wave functions, u(r) = rt(r) , out to a maximum radius which is taken where the nuclear potential has become negligibly

small (except during the search procedure where smaller radii are used),

and calculation of the logarithmic derivative, f .(inside) ; (k) calculation

of the required f . (outside) if it is not fixed in the input data. The

approximate location of the eigenvalues is determined by a survey versus

energy of f. (inside), f. (outside), and the difference between them. An

> automatic search procedure finds the energy for which f .(inside) =

f. (outside), and the wave function vs. radius is calculated and printed

or punched on cards for the energy of the eigenvalue.
The potential is calculated by subroutine VR3 as a function of

radius at up to 500 points, Spacing between points is determined by the

variable HI up to radius RI, by HII up to RII, and by HIII up to RMAX.

The potential used is;

V(r) - V

cp(r) - aV

c Q^ (1-3)1 M (1)

where p(r) =

-

, the "Saxon" potential; a, is the strength

1 + ***<rTV

of the spin-orbit interaction compared to the "Thomas" interaction for a

nucleon; M is the nucleon mass; V is the central real potential; h/Mc is

the Compton wave length for a nucleon; i is the orbital angular momentum

—

of the neutron; c is the Pauli spin operator of the incident neutron; R

is the nuclear radius; and a is the "dif fuseness" parameter of the potential. If j = I + Jj (chosen by letting variable MSPIN = l) , then o'i = I,

and the values of the variable VPLUS (potential for spin-orbit parallel)

for different radii are calculated. If j = i - §•, (chosen by letting

MSPIN = -1), then o-i = -(i+l) , and the VMINUS values (potential for spinorbit anti-parallel) are calculated. Other potential shapes could be used

by simple changes in this subroutine. The four arbitrary parameters, VC,

ALPHA, RADIUS, and A could be used to define the new potential with no

changes outside of subroutine VR3. Any arbitrary potential could be read

in point by point by replacing VR3 with an appropriate "READ" statement.

The calculation of f. (inside) and f. (outside), and the search procedure for matching the values is controlled by subroutine MATCH. First

the radial part of the non-relativistic Schr$dinger wave equation is
integrated out in radius for the appropriate i\

o

d U/

r) 2m f_ lU+l)h2 „, vl , v _ ,. T^~~ ^

L

E

" t 2 " V(r)

Ju

i

(r) = °" (2)

dr h 2mr

The values of the function and the first derivative at the first point

(radius = Hi) are determined from the relation u.(r) = C.r as r -* 0,

where C. is a constant. The second order Runge-Kutta method is used for

the integration (see Zurmtihl 196l) . The integration is carried out bysubroutine INTEG3 and is stopped when the radius exceeds RSTOP, and

f. (inside) = R(du./dr)/u. is calculated.,

Calculation of f (outside) . The calculation may be run in two alternative modes. When MODE = 1, if the energy is negative (bound state), the

outside wave function is required to have an f . which corresponds to an

exponential function which asymptotically approaches zero for large r.

If the energy is positive (scattering state), the criterion of a 90°

phase shift is applied to the wave function at the radius RSTOP where the

inside and outside functions must join smoothly. When MODE = 2, then f

is read in from the input data cards and no calculation of f. is made.

The outside wave function is the solution of equation (2) in the

region of space where the nuclear potential V(r) = 0, but the centrifugal

potential term is present. The solutions in general are spherical Bessel

functions. For bound states, the "outside" or asymptotic solutions are

spherical Hankel functions,

^j&M =

L

XVX) + ixn

i( x) j = F

£ + iGi

where x = ^Ajt l E l r *

For the 90° phase shift continuum case, the solutions are simply

xn^(x) = G^. However for matching we require not the function but the

k
logarithmic derivative, f . <, It is possible to calculate f. (outside) in a

number of equivalent ways, for example by first evaluating the functions

and then the derivatives from appropriate recursion relations. The calculation as done here uses the quotient functions of the Bessel functions

discussed by Onoe (1958) f° r bound states, and a relation given by Blatt

and Weisskopf (1952) for the continuum case. Details are given in Appendix 1.

3. Search Procedure

If the search for eigenvalues is carried out by matching the f *s at

some distance outside the nuclear potential, particularly for states

located deep in the potential well, there is a high probability of missing the state completely. The reason for this is that during the numerical

integration of the wave function, as soon as the first radial point outside the nuclear potential at that energy is reached, the eigenfunction

becomes in general a rapidly growing exponential, and f . becomes very

large. Only when the energy is chosen extremely close to the correct

eigenvalue does the wave function taper toward zero and have a small or

negative value of f.. The quantities f. are thus very sensitive functions

of the energy. In order to systematically find every eigenvalue, some procedure is needed to survey the eigenvalues under conditions of low sensitivity. The calculation was therefore split into two parts: first we

perform a survey, and then we use an automatic search procedure starting

with relatively small matching radius (which means low sensitivity) and

increasing it stepwise to the desired value. This requires two separate

runs on the computer, the first for the survey procedure to find the

approximate eigenvalue location, and the second to find the exact location

5
by the automatic search procedure.

Survey procedure . When the variable NOMAX = 1, only one integration

out in radius is made at each energy, and that integration is stopped at

the edge of the well (for the particular energy) for negative energy, and

at the value of RSTART for positive energy or for negative energy if RSTART

corresponds to a smaller radius than the first criterion above. The maximum radii of integration for a typical situation are shown as the heavy

black line in figure 1. The values of f. (inside), f. (outside) and the

difference between them are printed out at the specified input energies

taken typically 0.5 or 1 Mev apart. Typical results of this survey,

called the "signature of the well" are shown in figures 2 and 3 f° r the

case of si and p~/_ potential wells respectively. The small arrows

indicate the accurate eigenvalue locations found by the automatic search

procedure. The approximate eigenvalue locations corresponding to difference = 0, are used as starting energies for the automatic search procedure.

Automatic search procedure . First an integration of the wave function is carried out at the approximate eigenvalue energy. A second integration is carried out at a nearby energy. Using the differences (between

f *s) found in these two trial calculations, a linear interpolation (or

extrapolation) is made to an energy for which the difference is predicted

to be zero. This is the predicted eigenvalue energy. Integration is

carried out at the third energy, and the difference is again found. We

now know three energies and the three corresponding differences . Based

on this information, by quadratic interpolation, we predict the fourth

energy for which the difference should be closer to zero. The calculation

continues, energy prediction being made by quadratic interpolation, until

6
NOIlONDd 3AVM

(0 01 CO U <D I -H

0) B'H fi

cu -u eg cu E S "+-» CO LT\ U tC\ 0) o CO 42 • CX co 4-i vx)

CU £gCO u

.-( 4J

r-l -H

cu 4a £ fto

.-i i CO C Pc

cu poa

<uu3

CUOO

5-i a,

>,

cu>3

co

CU 43

cy co

X! m-i CO -W

oa o ft I LT\ 3 O - o

T3C

CO

O

PS o

II

•H •> ^

co

O

CO >H CH c s O ft co •H 0) rl

CJcc

CU00

•H

CU

CU

CU 43U

<4-l O

CU00

CU

cu 43

4-)

t-i o A & tc\ co

CO3 O -H

4*!O

CO

CU

cuu

43

+J

•U

CO •HCH

CU 43

4J

T3 CU

CO 43

J-) 4J

•>

m > O

CU £ <ua o co LTN43 I CO

II CU 043 > H

x> cu e G M ft CO CO CU

CO

(U 43

CU 43

4->

>%

43 • CO

(3 cu

CU 4-1 > co •H 4J bO co

COCU3 CU CO «h £ e

CO

O > <U 43

£3 43

H

CU -u

(50 CU •H CH 43 W O -P

CO

CO •H

CO3

1-) XI

CO

5-i I 3C

'H

4-1 f3OO

fto S m-i

A9SA! 'A9U3N3

•H
00

O

30N3H3ddia JO*± 8 o10 * I CM c

II>

0)

ca

CO 4J

r-l3

CO

p=5

ojV43

00 •H

»4-l s o cu

0) 42

43 4J •Hn C

CJ cu

en cu

0) £

TD 4-1 CD

r-l 43

rH

0) cu S ac o cu

ii

u

«* cu

4-1 CD m

43 •H

4-1 TJ u CU 43

H-l 4-1

QJ

h

M o 3 "4-1 13

(U T3 a c o CO Ma^—

cu ^tCU •H I

CO

4-1 3

03

O

S*_^

OJ «*

42

m

•u

a

14-1 ,.

—„ O CU T3

03 •H

4J CO i—1 C 3 •H

COQJ «^

Pi 4-1

OJ

60 •H

Pn
If)

C\J coto

en O

CJ 3

O

•H

4-1 TO in C

<u

COu g

CU in co •H

CD 43 o

4-1 O

> . <v i—i in

i ^

i—iCD > OJ O

cc ro

UJ

UJ

II

43

4-1 M

O

0)u

3

CDoo

CU>

u

3

CO

CU J3

O • CM

CO

4-1 CU rH (-1 3 3

co bO

cu -H & <4-l

30N3cJ3dJia JO '] 00
either the difference is closer to zero than the difference specified in

the input by the variable EPS, or until the maximum number of tries has

been used up. The maximum number of tries is determined by the variable

NOMAX.

The computer time for the sample automatic search procedure shown in

Appendix k t which required 21 repetitions of the integration of the wave

function and matching procedure was 0,5 min. For a systematic search procedure at 69 energies, the computer time was O.k min.

The matching described above is carried out at a radius determined by

the variable RSTART. RSTART should be chosen in the region where sensitivity of the f . changes with energy is low. Having determined the eigenvalue for a potential which is cut off at a value of radius determined by

RSTART, we use this energy as the first trial energy for a larger value of

r, which is determined by the variable RSTOP. The permissible size of a

step in radius may be estimated from the WKB approximation. If we let

cp = f (inside) - f. (outside), then let S(R) = -r^, a quantity which we

have previously called the sensitivity. In the outward steps in radius,

<Rn+ l)

the increase in sensitivity, — >'

^ = c , should be kept about constant

^ n' for each increase in RSTOP. Approximately we may obtain o_ as follows:

K

R . . ^_^_

£n ctr

=

f £

72m(V-E) dr

R

nR

a4 J

n+1 £y^[I! dr^/^AR

R

n

since V soon becomes small in this region outside the well. The change in

4nCJ

RSTOP is given by AR ~y2mM7£

10
where 0" is an arbitrary parameter determine i by the input. R

At the new matching radius, the first stimate of energy is the

converged eigenvalue at the previous matching radius. The second estimate

2 should be lower since VR for the well will be slightly larger, causing

the new eigenvalue to be slightly lower. At the second matching radius,

the second trial eigenvalue is generated in the same way as at the first

matching radius. At third and higher matching radii it is generated from

the relation:

AE . = i- (AE ) n+1 a„ rr

£1

where o_ is an arbitrary parameter determined from the input. Figure k

shows a typical case of converging on an eigenvalue using this procedure

of calculating at successive matching radii. The final match is carried

out at the radius RMAX. When a converged eigenvalue is found at radius

RMAX, the integration in radius is repeated and the wave function vs.

f* 2 radius is printed out. Also the val of thi u dr is calculated by the

trapezoidal rule for normalization purposes, and is printed out.

ko Sample Results

Sample results of the automatic search procedure for the case of

0xygen-l6 are shown in figure h, -' "'-?envalues are shown for the Is, 2s,

3s, the two lp states, the Id state, and the eigenfunctions are shown for

the three s states.

5. References

Amster, H. J. and Culpepper, L t fe (1957) > Westinghouse Atomic Power

Department Report WAPD-TM-87*

Blatt, J. M. and Weisskopf , V. F., (1952). Theoretical Nuclear Physics , P. 334, Wiley (New York)

.

11
A9l/\| A9H3N3

12

60
Caswell, R. S. (I962). J. Research National Bureau of Standards, Section

A, Physics and Chemistry, to be published.

Magnus, W. and Oberhettinger, F. (195*0 • Formulas and Theorems for the

Functions of Mathematical Physics , p. 19 9 Chelsea (New York)

Onoe, Morio {1958). Tables of Modified Quotients of Bessel Functions of

the First Kind for Real and Imaginary Arguments , Columbia University

Press, New York.

Ross, A. A., Mark, Hans, and Lawson, R. D. (I956). Phys . Rev. 102 , I6I5.

Zurmuhl, R. (I96I). Praktische Mathematik fur Ingenieure und Physiker,

p. 412, Springer-Verlag (Berlin, Gottingen, Heidelberg).

The author would like to thank Dr. M. Danos for the suggestion of

this problem and for helpful suggestions and discussions.

13
Appendix 1. Evaluation of f (outside)

For bound states the logarithmic derivative, f , may be obtained

from one of the quotient functions of the Bessel functions which have

been discussed by Onoe (1958)

,-du

f, (outside) = R-^^ Cdr ) R __ Ldr ( rh

i).

= X

( U

JL)R [r hjt R

x (3)

h

x (x)jx

where X = i /^|e|r.

but

dT < x V = k G*

+1*"Vx - (*+<Vx - x Wx) • w

' ' A Therefore.

'l+l - X h

X (X) ) X hjtfl

(x)

^(outside) = X T^-

(T) = Ul - ^x) ( 5 )

For a continuum state we have a similar relation in terms of the

Neumann functions rather than the Hankel functions:

x n » + i( x) /- G .\

h - »\- -$r(r * £> (6)

where the prime refers to differentiation with respect to X. For a

bound state, /,\

^(outside) = i+1 -^t(x),

in the notation of Onoe.

In the notation of Magnus and Oberhettinger (195M»

x K

u+1 (x)

C\ (l) = > \ ——. A recursion relation may be derived which

Ik
~ (1) is used by the computer to calculate x/ v

' for the appropriate value of

I: ^U+

x w.

frv-l

(7)

For continuum states, the criterion of a 90* phase shift, corresponding to a maximum value of the scattering cross section (or "resonance")

is used to determine the location of the state. The relation for f . may

be written as in equation (6) above

or h = A

z + h

tan h> (8)

V i ¥i rG4

G/+F// where exp(-2i^) = ^ . ± ^= \G^ + F

^

2 _ r=R;

and S„ = R

Vi' - fa'

1 XV + V r=R. (See Blatt and Weisskopf , 1952)

Substituting for exp(-2i§.) we have

h - h - h

<*t

G

t

/

G

i

2

" F/> + < F/

+ G/>]

(9)

Equation (9) is equivalent to equation (5)» and was the formula actually

used in the calculation. Analytic expressions for all of the quantities

in equation (9) appear in the report of Amster and Culpepper (1957)*

15
Appendix 2. Meaning of the Input Variables

The meaning of the input variables are listed here for convenience

in using the code. Format may be obtained from Appendix 5*

RADIUS radius in the Saxon potential in fermis

A diffuseness parameter of the Saxon potential in fermis

VC central real potential in Mev (is negative for attractive

potentials)

ALPHA magnitude of the spin-orbit potential expressed in number of

times larger than for the Thomas term of a nucleon

AMASS atomic mass in a.m.u. of the nucleus except the single neutron

whose states are being considered

CONV parameter which determines energy spacing between first two

trial eigenvalues, typical value 0.010

MDELTA the radial potential is printed out at every MDELTA'th point

JPRINT conditional print variable, = 1 for normal operation, = 3

maximum print out for code checking

L orbital angular momentum

EPS if the difference between f 's is less than EPS, then the

eigenvalue is considered converged.

MSPIN = 1 for spin-orbit parallel, = -1 for spin-orbit anti-parallel

HI spacing between radial integration points at radii less than

RI, in fermis

HII same, but for radii between RI and RII

HIII same, but for radii between RII and RMAX

RMAX maximum radius, maximum value of RSTOP

NOMAX maximum number of tries for convergence on an eigenvalue at a given value of RSTOP, = 1 for survey procedure) maximum value 20

JPUNCH = 1 for no punch out of wave function, = 2 for wave function

to be punched on cards

KZ number of trial energies for the given run, maximum = 100.

16
RSTART initial value of RSTOP

SIGMAR variable which indirectly determines spacing between successive

values of radius (RSTOP) at which eigenvalue search is carried

out, typically 2.0

SIGMAE variable which determines trial eigenvalue spacings after the

first two trial radii, typically 10.0

MODE = 1 for normal search for bound and scattering states

= 2 for use with arbitrary f . determined from the input

trial energy in Mev (negative for bound states, positive for

scattering states)

EN(KY)

FLOuT(KY) value of arbitrary f (outside)

17
Appendix 3* Format of Data Cards

First card ; C in column 1, any desired information describing the run (such as name, date, element, etc.) in columns 7-72.

Second card:

Third card:

Variable End s in Column Example

RADIUS 8 3.150

A 16 O.65O

VC 2k -50.000

ALPHA 32 35.000

AMASS ko 15.000

CONV kQ 0.010

MDELTA 52 10

JPRINT 56 1

L 60

EPS 70 .0100000

MSPIN 72 1

Variable End s in Column Example

HI 8 0.020

RI 16 0.100

HII 2k 0.050

RII 32 1.000

HIII ko 0.010

RMAX kQ 6.4oo

NOMAX 52 10

JPUNCH 56 2

(Note: decimal point

location may be changed

from that shown here,

however print in the

output will be as shown)

10 (Note that this and other

variables without a decimal point are integers, decimal point may

not be used)

18
Fourth card : Variable Ends in Column Example

KZ 4 1

RSTART 12 4.0

SIGMAR 20 2.0

SIGMAE 28 10.0

MODE 32 1

Fifth and higher cards : If MODE = 1, list the energies with the last

character successively in column 8, 16, 24, 32, 40, 48,

56, 64, 72 5 using as many cards as required for the KZ

number of energies

ExamP le : -29.000 -29.500

If MODE = 2, list the energies, one per card as follows:

Variable Ends in Column Example

EN 8 -29,000

FLOUT 24 0.2000000E+01

The variable FLOUT is in exponential notsiior and in the example means:

0.2000000 x 10

+1

= 2,

19
Appendix k. Fortran Statements and Sample Output

EIGENVALUES, EIGENFUNCTIONS JULY 23,1962 0t>230

READ 10

10 FORMAT172H

PRINT 10 J DIMENSION VPLUS(500 ) ,VMINUS(500) , ENERGY) 10, 100), XP( 1,100),

1XMP( 1 ,100) ,XPP ( 1 ,100 ) ,AJLP { 1 -ICO) ,AJLM( 1 ,100) ,AILM( 1 ,100) , 2AILPI 1 ,100) ,FL ( 1 ,10 0) ,XX( 10 0) ,EN( ICO) ,SPHTAN( 10,100) ,XM( 1 , 1 00 )

,

3 REFLt 1,100) ,DIF( 10 ) ,HT I LDE ( 1 » 100 ) ,E ( 1 00 ) , PSKA),

AKRR(A) ,S( 1 ,100 ) , DELTA

(

1,100),TFG(1»100),FPG(1»100)»GMF(1,100)»

5P( 10,100) ,G( 10,100)

,

FLOUT! 100)

COMMON VPLUS»VMINUS,XP,XM,XPP,XMP,A-JLP,AJLM,AILP,AILM,FL,RR»X,V,

1EN, FF, ELL, FRR, RADIUS, HI , A , E , VC » ALPHA ,6 , AMASS ,EMAX ,M ,MDELTA ,R , H I I

,

2RI I ,HI I I ,RMAX,C,N,REXP,VCC,VS,AK,K.Z»KX,KP,KY,K.,XPLUS,XPLUSP,

3XMINUS,XMINUP,AI ,AI I ,AI I I ,CI ,CI I ,CI I I ,H,XX,F, I ,RI ,MPR I NT , JPR INT

,

4REFL,SPHTAN,KK,MM,C0NV,N0,EPS,ENERGY-,MSPIN,DIF,HTILDE,N0MAX,

5JPUNCH,R5T0P,DELTA ,S IGMAE , S I GMAR ,NTRY , D I FCNV , S , P , PS I , RRR , S ,GMF ,Q

,

6TFG»FPG»PMOM,MMM,RSTART , FLOUT ,MODE,PROINT ,L

CALL INPUT3

CALL VR3

CALL MATCH

CALL ENDJOB

END

Note: Identical dimension and common statements must

be included in every subroutine, but are omitted

here for brevity.

20
Appendix 4. Fortran Statements and Sample Output—continued

SUBROUTINE INPUT3

READ 1 ,RAD1US»A,VC»ALPHA,AMASS»C0NV,MDELTA, JPRINT.L , EPS. MSP IN PRINT 4501

4501 FORMATI119H RADIUS A VC ALPHA AMASS CONV MDL JP

1R L EPS MSPIN ) PR I NT 1 ,RADIUS,A,VC,ALPHA,AMASS,CONV,MDELTA,JPRINT,L ,EPS»MSPIN

1 FORMAT (6F8.3.3I4,1F10.7»1I2)

KK=L+1

READ 7,HI,RI,HII,RII,HIII , RMAX ,NOMAX , JPUNCH

PRINT 4502

4502 FORMAT (119H HI RI HI I RII HI I I RMAX NOMAX

1JPUNCH ). PRINT 7,HI,RI,HII,RII,HIII , RMAX . NOMAX .JPUNCH

7 FORMAT(6F8.3.2 14)

PRINT 4503

4503 FORMAT(50H ENGS RSTART SIGMAR SIGMAE MODE ) READ 4040, KZ, RSTART, SIGMAR, SIGMAE, MODE

PRINT 4040, KZ, RSTART., SIGMAR, SIGMAE, MODE

4040 FORMAT! 14, 3F8. 1,14)

IF (MODE-1) 4081, 4061, 4082

4081 PRINT 4504

4504 FORMAT (50H TRIAL ENERGIES ) READ 4080, (EN(KY) ,KY = 1,KZ)

PRINT 4080, (EN(KY) »KY=1 ,KZ ) 4080 FORMAT (9F8. 3)

GO TO 4085

4082 PRINT 4083

4083 FORMAT(50H ENERGY FL OUT ) READ 4084, ( EN ( KY ), FLOUT ( KY ), KY = 1 ,KZ ) PRINT 4084, ( EM(KY) ,FLOUT(KY ) ,KY = 1 ,KZ ) 4084 FORMAT (F8.3,E16.7)

4085 CONTINUE

RETURN

END

21
Appendix k. Fortran Statements and Sample Output—continued

SUBROUTINE VR3

R = HI H = HI- K = KK

M = l PRINT 2091

2091 FORMAT! 50H VPLUS VMINUS

2090 DO 2210 1=1 ,1000

1 = 1 9 C= .0110270/A

15 REXP=EXPF( (R-RADIUS) /A)

2151 VCC=VC/( 1.0+REXP)

2152 VS = AI_PHA*VC*C*REXP/ ( ( ( 1 .0+REXP )**2 ) *R-J 19 AK = ,< IF (MSPIN) 22*22.21

21 VPLUSf I )=VCC+VS*(AK-1.0)

GO TO 4020

22 VMINUS ( I )=VCC-VS*AK

VMINUS! 1 )=VCC

4020 IF (M-MDELTA) 3150,28,28

28 PRINT 30, VPLUSI I ) ,VMINUS( I ) ,R

30 FORMAT(2E15.7,F7.3)

M = 3150 M=M+1

2126 R=R+H

2130 IF(R-RMAX+.C01 ) 2140,2220,2220

2140 IF(R-RII+.001) 2150,2160»2160

2150 IF (R-RI+.001 ) 2200,2180,2180

2160 H = HI I I GO TO 2210

2180 H=HI

I

GO TO 2210

2200 H=HI

2210 CONTINUE

2220 RETURN

END

22
Appendix 4„ Fortran Statements and Sample Output —continued

SUBROUTINE INTEG3

5 R = HI MMM = K = KK

AK = K IF (JPRINT-2) 35.35.4400

4400 PRINT 4401

4401 FORMAT(50H SUBROUTINE INTE63 CALLED

35 XP(K,KY)=(HI**K)*1.0E-4

37 XPP(K,KY)=AK*(HI**(K-1) )*1.0E-4

39 XM(K,KY) =XP(K,KY)

41 XMP(K,KY)=XPP(K,KY)

51 H=HI

52 F=AMASS/(AMASS+1. 00898)

54 FF = F IF (MPRINT-1) 4090.4088.4090

4088 PRINT 4089

4089 FORMAT ( 50H WAVE FUNCTION VS RADIUS

PROINT=0

IF(JPUNCH-l) 4090.4090,4087

4087 PUNCH 4362

4362 FORMAT (50H ENERGY

PUNCH 4361. EN(KY)

4361 F0RMAT1E13.5)

4090 DO 4210 1 = 1 ,1000

1 = 1 CALL KRR1

4126 R=R+H

4127 IF(NOMAX-l) 4130,4128,4130

4128 IF (R-0.2*RADIUS) 4130,4129,4129

4129 IFIEN(KY)-V) 4220,4130,4130

4130 IF(R-RSTOP+.001) 4140,4220,4220

4140 IFIR-RI I+.001 ) 4150,4160,4160

4150 IF(R-RI+.001 ) 4200,4180,4180

4160 H=HIII

4170 GO TO 4210

4180 H=HI I 4190 GO TO 4210

4200 H=HI

4210 CONTINUE

4220 IF (MSPIN) 4310,4230,4300

43 00 PROINT=PROINT+(H/2.0)*XPLUS**2

GO TO 4320

4310 PROINT=PROINT+ ( H/2 . ) *XM I NUS**2

4320 IF( MPRINT-1 142 30,4330, 4230

4330 PRINT 4340

4340 F0RMAT(5CH NORMALIZATION

4350 PRINT 436CPR0INT

IF(JPUNCH-l) 4230,4230,4351

4351 PUNCH 4340

PUNCH 436CPROINT

4360 F0RMAT(1E15.7)

4230 RETURN

END

25
Appendix 4. Fortran Statements and Sample Output—continued

5000 SUBROUTINE KRR1

ELL=K-1

IF (MSPIN) 5060,4125,5020

5020 XPLUS=XP(K,KY)

PROINT=PROINT+H*XPLUS**2

5040 XPLUSP=XPP(K,KY)

56 RR=R

57 V=VPLUS( I

)

59 X=XPLUS

61 CALL FR

62 AI=FRR*H**2/2.0

63 RR=R+H/2.0

65 X = XPLUS+ XPLUSP*H/2.0+AI /4.0

67 CALL FR

69 AI I =FRR*(H*#2 1/2.0

70 RR=R+H

71 X=XPLUS+XPLUSP*H+AI

73 CALL FR

75 AI I I=FRR*H**2/2.0

141 XPLUS=XPLUS+H*XPLUSP+(AI+2.0*AI I )/3.0

143 XPLUSP =XPLUSP +(AI +4 . 0*A I I +A I I I ) / ( 3 . 0*H

)

180 XP(K,KY)=XPLUS

184 XPP( K,KY)=XPLUSP

GO TO 4110

5060 XMINUS=XM(K,KY ) PROINT=PROINT+H*XMINUS**2

5080 XMINUP=XMP(K,KY)

77 RR=R

98 X=XMINUS

100 V=VMINUS( I

101 CALL FR

103 CI=FRR*H**2/2.0

105 RR=R+H/2.0

107 X=XMINUS+XMINUP*H/2.0+CI/4.0

109 CALL FR

111 CII=FRR*(H**2 ) /2.0

112 RR=R+H

113 X=XMINUS+XMINUP*H+CII

115 CALL FR

117 CI I I =FRR*H*#2/2.0

149 XMlNUS = XMINUS + H*XMINUP+(CI+2.0-x-CI I )/3.0

151 XMINUP= XMINUP + ( C I +4.0*CI I +C I I I ) / ( 3 . 0*H

)

188 XM(K,KY)=XMINUS

192 XMP(K»KY)=XMINUP

4110 MMM=MMM+1

IF(JPRINT-l) 4114,4114,4120

4120 I F( 1-3 ) 4112,4112,4114

4112 PRINT 4113,XP(K,KY) ,XM(K,K,Y) ,XPP(K,KY) ,XMP(K,KY)

PRINT 4113,XPLUS,XMINUS,XPLUSP,XMINUP

PRINT 4113,AI,AII,AIII,CI,CII,CIII,H,R

PRINT 4113,FRR,V,FF,X,RR

4114 IF(MPRINT-l) 412 5,4115,4125

4115 IF (MSPIN) 4118,4118,4117

4117 PS I (MMM)=XP(K,KY)

RRR(MMM)=RR

24
Appendix h. Fortran Statements and Sample Output —continued

GO TO 4119

4118 PSI (MMM)=XM(K,KY)

RRR(MMM)=RR

4119 IFIMMM-4) 4125,4122*412 5 4113 FORMAT (8E15.7)

4122 PRINT 41 16* PS I (1)*RRR(1).PSI(2)»RRR(2)*PSI(3)»RRR(3)»PSI(4) ,RRR{4)

4116 FORMAT ( E13.5 . F5 .2 . E13 . 5 ,F5 . 2 ,E1 3 . 5 , F5 . 2 , E13 . 5 . F5 . 2

)

IF( JPUNCH-1 )4135»4135,4132

4132 PUNCH 4116»PSI(1)»RRR(1)»PSI(2)»RRR(2)»PSI(3)*RRR(3)»PSI(4)»RRR(4)

4135 MMM=0

4125 CONTINUE

RETURN

END

SUBROUTINE FR

44 FRR=-.04826*FF*(EN(KY)-2 0.721*ELL*( ELL+1.0 ) / ( FF*RR**2 )-V ) *X

• RETURN

END

25
Appendix k. Fortran Statements and Sample Output—continued

SUBROUTINE MATCH

DO 6200 KY=1»KZ

MPRINT=2

KY = KY

RSTOP=RSTART

CALL INTEG3

PRINT 6180

6180 F0RMATI119H ENERGY DIFFERENCE FL INSIDE FL OUT

1SIDE WAVEFNPLUS(R) WAVEFNM I NUS ( R ) R ) NTRY=1

NO = l 203 ENERGY(NO»KY)=EN(KY)

IF (MSPIN) 208.208.204

204 AJLP(K,KY)=XP(K,KY)*XPP(K,KY>

AILP(K.KY)= XP(K.KY)**2

REFL (K,KY)=R*AJLP(K,KY)/AILP(K,KY)

GO TO 212

2 08 AJLM(K,KY)=XM(K,KY)*XMP(K,KY)

210 AILM(K.KY)= XM(K»KY)**2

REFL (K,KY)=R*AJLM(K,KY)/AILM(K,KY)

212 IF(MODE-l) 218.218.214

214 DIF(NO)=REFL(KK.KY)-FLOUT(KY)

218 IF(EN(KY)) 219.219.220

219 PMOM=SQRTF(-.04826*F*EN(KY) ) IF (MODE-1) 2191.2191.6001

2191 XX(KY)=R*PMOM

HTILDE(1,KY)=1.0+XX(KY)

SPHTAN(1»KY)=-XX(KY)

DO 6000 K=2»KK

AK = K HTILDE(K,KY)=(2.0*AK-1.0)+(XX(KY)**2)/HTILDE(K-liKY)

6000 SPHTAN(K,KY)=AK-HTILDE(K,KY)

DIF(NO)=REFL(KK,KY)-SPHTAN(KK,KY)

FLOUT (KY)=SPHTAN(KK,KY)

GO TO 6001

220 PMOM = SQRTF( .04826*F*EN(KY)

IF (MODE-1) 2201.2201.6001

XXIKY )=R*PMOM

S( l.KY)=XX(KY)

DELTA( 1»KY)=0.0

DO 226 K=1,KK

AK = K S(K+l.KY)=(XX(KY)**2)*S(K,KY)/{ ( AK-DELTA ( K , KY )

) **2+S ( K, KY ) **2

DELTA (K+1»KY). = (AK-DELTA<K,KY) ) * ( S ( K+l ,KY ) /S ( K , KY ) )-AK

P(1.KY)=1.0

P(2.KY)=1.0/XX(KY)

Q( l.KY)=0.0

Q(2»KY)=-1.0

DO 254 K=3»KK

AK = K P(K.KY)=(2.0*AK-3.0)*P(K-1,KY)/XX(KY)-P( K-2.KY)

Q(K,KY )=(2.0*AK-3.0)*Q(K-1 .KY ) /XX ( KY ) -Q( K-2 »KY ) DO 268 K=1.KK

GMF (K,KY)=(P(K,KY) **2~Q ( K , KY ) **2 ) *COSF ( 2 . 0*XX ( KY

)~2 .0*P ( K ,KY )

1Q(K,KY)*SINF(2.0*XX(KY)

2201

221

222

223

224

226

240

244

246

248

250

251

252

254

255

256

26
Appendix h. Fortran Statements and Sample Output--uontinued

2 58 TFG(K,KY)=2.0*P(K,KY)*Q(K,KY)*COSF(2.0*XX(KY) ) + ( P ( K,KY ) **2

1-Q(K,KY)**2)*SINF(2.0*XX(KY) ) 268 FPG(K,KY)=P(K,KY)**2+Q(K»KY)**2

FLOUT(KY)=DELTA(KK,KY)-S(KK,KY)*TFG(KK,KY)/(GMF(KK,KY)+FPG(KK,KY) ) DIF(NO)=REFL(KK,KY)-FLOUT(KY)

IF(JPRINT-2) 6001,6001,271

271 PRINT 272,XX(KY) ,S(KK,KY) ,DELTA(KK,KY) ,P(KK,KY) ,Q(KK,KY)

,

1GMF(KK,KY) ,TFG(KK,KY) ,FPG(KK,KY)

272 FORMAK8E15.7)

6001 IF(NTRY-2) 6002,6002,6075

6002 IF(NO-NOMAX) 6004,6100,6100

6004 IF(NO-2) 6005,6080,6090

6005 IF(DIF(NO)+1.0) 6010,6010,6020

6010 IF(DIFINO)-l.O) ' 6030,6040,6040

6020 EN(KY)=( 1.0-CONV)*EN(KY)

GO TO 6100

6030 EN(KY ) = ( 1.0+CONV )*EN ( KY

)

GO TO 6100

6040 IF(DIF(NO)+0.1 ) 6050,6050,6060

6050 IF(DIF(NO)-0.1) 6070,6100,6100

6060 EN(KY)=( 1 .0-0.2 *CONV ) *EN ( KY

)

GO TO 6100

60 70 EN(KY)=(1.0+0.2*CONV)*EN(KY)

GO TO 6100

6075 IFJNO-2) 6076,6080,6090

6076 EN(KY) = (E(NTRY-l)-E(NTRY-2) ) /S IGMAE+E ( NTRY-1

)

GO TO 6100

608 EN (KY )=( ENERGY ( NO-1 »KY)*DIF (NO) -ENERGY ( NO »KY)*D I F(NO-l) )

/

1(DIF(N0)-DIF(N0-1)

)

GO TO 6100

6090 EN(KY)=( ( ( ENERGY ( N0-2 »KY ) *D I F ( NO-1 ) -ENERGY ( NO- 1 ,KY ) *D I F ( N0-2 ) )*

lDIF(N0)/(DIF(N0-l)-DIF(N0-2 ) ) )-( ( ENERGY ( NO-1 ,KY ) *DI F ( NO

)

2-ENERGY(N0,KY)*DIF(N0-l ) )*DIF(N0-2 ) /

(

DIF ( NO )-DI F ( NO-l ) ) ) )/

3(DIF(N0)-DIF(N0-2)

)

6100 PRINT 6160, ENERGY! NO, KY) ,DIF(N0) ,REFL(KK,KY) , FLOUT (KY) ,XP(KK,KY) , 1XM(KK,KY) ,R

6160 FORMAT(F15.7,5E15.7»F8.3)

IF (NO-NOMAX) 6114,6200,6200

6114 IFIDIF (N0)+ EPS) 6120,6120,6115

6115 IFIDIF (NO)- EPS) 6130,6120,6120

6120 IF (JPRINT-1) 6123,6123,6121

6121 PRINT 6160, EN(KY)

6123 CALL INTEG3

NO=NO+l

GO TO 203

6130 E(NTRY)=ENERGY(NO,KY)

IFfRSTOP -RMAX) 6131,6132,6132

6131 IF(JPRINT-2)6150, 6132, 6132

6132 MPRINT=1

6140 CALL INTEG3

6150 MPRINT=2

IF (RSTOP-RMAX) 6165,6200,6200

6165 RSTOP=RSTOP+LOGF(SIGMAR) /PMOM

IF (RSTOP-RMAX) 6168,6168,6167

6167 RSTOP=RMAX

6168 NTRY=NTRY+1

CALL INTEG3

NO=l

GO TO 203

6200 CONTINUE

RETURN

END

27
ooooooooooo CtTOOOOOOOOOOO

• ••••••••ft

T3

<D3

c

•wu

co

u

IIu

3

ex u

3o

ft

CO CO•oC

CO

C

cau

4-1 o

XIc

<ua.a<

~ in in in in in in in in m in in

QC o o oi o o o o o o o o

L0 LU LU LU 'jj LU JJ

1 jj

1 LU1 1 1 JJ LU LU 3 o o o o o o o O o o o

z o o o o o o o o o o o ft o o o o o o o o o o o

X o o o o o o o o o o o

z o o o o o o o o o o o

UL o o o o o o o o o o o

LU CM CM CM CM CM CM CM IN) CM CM CM > • • • • • • • • • • • < o o o o o o o o o o o

z «* «* »*• >r >* <*• >*• >! •* •* \r\ »-4 -^ o o o o o o o o o o o

a ai 1 1 1 1 1 1 1 1 1 1 1 CM 00 —i •^ LU LU LU JJ LU LU LU LU LU JJ LU

en Z oo fO O* CD m r~ cm CM n r- o« -< a o 3 m o -? r- en in nO in cm oo m

o o _J CO C"> f~ o O CO 00 in CM <-* 00 en o Q. o m ct« in «H 00 in m hit*

«\i i/i o Z r«. in oo o — >* o o m + ci

in a o LU >0 cm >* vO •o -* CM in «*>* oo O LU o

• o

X

LU>

<

3

-* CM CM

• • o o o

CM

• O

CM -i

* • o o

CM

• O

—

• o

-1 -4 ,©

• • • o o o

_j O <Jz LU o o o O o o o o o o o

3 Q

a. *—

LU LU LU LU LU LU LU LU LU LU LU

QC •^ -^ r\j 00 •O CM C7> CM en o —

—4 r- o o

Ql h- m oo cm CM »* in m CO en l> en "J X< OOK1N in o

>o o in o o o

_j a COOh —1 <-* o en CM rfH>4

a —1 o iH _j m G O en m en en en o

z z LU ^- >* <«

• • • * «* in• • on• in• m m mo• • t > O X O ooooooooooo

z —4 < o 1 1 1 1 I l I 1 l l l _J o o z >* -J u • ac t LU o vO o o o O o o o o o o o

36 LUQ LU LU LU LU LU LU LU LU LU LU LU Z M Ol,Ofl(MN lOOO' r- <» m ovio M o 00 •* c> o O O •* .—4 •O -* >t ««• X 00 o t—

in Z ^ -< «0 —4 o> r«- oo s- m co oo < < o I-* o »-m ffM«- CM >0 CM o sO <o -* r- 00 Z • X t O O O ( 00 >t Cj> CO -^ CM O o -^ CM On < m o tn m in in in in in in m in in in m -j o> oo o> >o •O 00 o >} <t CIH z •M as en CO en CO m 00 CO CO m co m co fO LU oo in >*• >t <r r- >4 >f m m r- LU LU O a o o f—

-4 CNI CM m m >!• >t in in . o o o c o o o o o o o

>- o —4 o o •^ i l 1 1 1 l 1 i I l l X < o *-* o s: o i o or o o o -: —4 i—1 O i—

<* ** —

o i-t CM <-> Q. • « CM <M CM t\J (NJ CM cm CMCMCMfMCMCMLUOOOOOOOOOOO

—i in —4 LU o 1 i 1 t 1 1 1 1 I i 1 • 1 o 1 1 I 1 1 CM < en < • LU LU LU LU LU LU LU JJ LU LU LU LU LU Z LU LU LU LU LU LU LU LU LU LU LU O s: o oooooooooooooiuo-<r>-p-om a* o •c in cm

0> o •=1 vo o o a •-> i-« (O CM 00 o CM CM -4^-1 ^ O o •—

o i—

3 o o O O o o O o o O o O o LU O oo r- O r- cm •-4 -* O oo en • > o —

in 00 z o o o o o o o o o o o o o LU co m cm r~ in en O o oo en r»

«fr o X o •-« o o o o o o o o o o o o o a. o >*• in >t r«- -* o * oo in co CM • • 2:0000000000000 m «<• CM 00 m «» tn m <-* O* 00 -^ o o a: o > o o o o o o o o o o o o O Q ** ~> en r-t CO CM en 0> o» «» -^

_l m < • 3 l s: (M o o o o o o o

-5 o l 1 1 1 1 ( 1 1 1 1 1 1 1 l 1 1 1 1 1 » 1 1 < o «—

o *-^ m a: o 00 00 cm CM (\J (Nl IN CM rsi CM —1 —J -* o o Ohff o oo >r m <7> 09 00

_l o r-l LU o o O 0> CM O «<- f- -4 -O CO CM «* _l • • t~^

1 o o> >o o cn >* r- esi in oo oo ai o o 1— o u LU JJ LU JJ LU JJ 'JJ JJ -U LU 'JJ JJ JJ o o t> o >a- •« o ro CM -^ en 2 a • 0£ —

en ~-l in O o (N in vO o CM -H •o > o er> m o CM CM r~ co CM -O >0 to < -J- LU oo r- r-4 in CO o CT. cr •*• >o in o> oo oo o O 00 o •o •o o n 00 ^ * «»• < >— z 3 m CD o cr r~ 00 en * CM r«- o> in o ct: O CM «» •T * * r- in m in in O (/) o o oo LU O _J rn CD >r m o> >0 00 o o r- 00 O CM LU # • • • • • • • • t 3 m i—

CM cc o a. <r> in o o vO -o —f«- <-> o m ca "^ z 0> C3> C7> O o 0« O CT> CJ* O O

1/1 l-H -h X O _J o > t> cc r- >*• CO o ,-* CM co >* <3 r^ o LU CM CM CM CM CM CM CM CM CM CM CM a • • 00 < • >J- •J- >r -J- en en cm -* •c m •^ r- fc 1 1 1 1 1 1 1 1 1 1 1 at < en o o <-» i—* o> • • • • • t • • • • • • • of z C£ cm ooooooooooooo LU l- 1 t 1 1 1 1 1 1 1 ( 1 1 1 1

28
oooooooooo

• ••••#•••• mminmmm>o>o»o>o

in in o o

l I LU UL> o o

o o

o o

o o

o o

o o

CM CM

m m in in o o o o

i i i i LU LU LU LU o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

CM CM CM CM

in m

o

1 I LU LU o o

o o

o o

o o

o o

o o

CM CM

in in o o

l l JJ LU o o

o o

o o

o o

o o

o o

CM CM OOOOOOOOOO

<D3

3

3au

3O

I

co•a3

<0 u

CO3

COuu

uo

in in in in in in in in m in oooooooooo

LU LU LU jj LU LU LU LU LU LU

CM m r» r- r» p- O f- CM CM

CM CO sO m —

i

>r i-i —J CO O —> tNfff1 CD CM 0> o O

oo co o ^-t 0> —> p- in CD p-

•a CM ro CO —4 PO a» CM <M CM CO in in sO CD CO CD 0> 0> CT> r- p- p~ PO PO eo —

(

—J <-* —> oooooooooo

o o o o o o o o o

LULULULULUJJLULULULU

CM«*00p«.CMCO<O^CMCM

rt«OIM<t<tOOOO »H^-^mmmo>o>»CT>o>

_i <-•i-toOOOOO^O^

OOOP-f-r-cocOcoco

vO'O-o-o-o^or-p-p-r—

• ••••••••• oooooooooo 1 I I I I I I I I I oooooooooo

LUJJLULULULULULULULU o^wmiflOONrMH eOOOWMJ>Om(MOO>

Ocor-r--*—'iniM—io OinHIMOMOOOH Mfl-t^^OClHOO

CMOOCMr-r--o^-«4-«iOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ^pOLnp-c>^pOLnp~o^»^poLnr~o>^coinr^c>^roinp~c>-*poinp<-o> '-«po o o o o o (MNN(MMflffimWl»l^*<t<t-tlfllfllfllf\l/>^<0

m>4'<4- >4--4'»i-POPOpOrOf0.4-<*-^.4'-4, <r-4' oooooooooooooooooo I I I I I I I I I I I I I I I I I I IU iJJ JJ UJ llf ill UJ UJ tU 'M " f IM 'M 'I f Ul 'U 'If UJ inHsO^^ocoor-cocciiM^rMON+co

•oomoCT>incMCDCMin-<<^yor^in«toocD oO'O^oin^OineoN'tnhO'inooofO iMMomfHOooooooo-HNwrni/i

* >r •4- -4- «* >f •f in in in m in in m oooooooooo o o o o

LU LU LU UJ 'JJ LU JJ LU LU LU LU jj LU LU o o co in >o •—4 in co CO >* in >r CO o o o> ooooo-t CD r» 0> + <0(Mr->0

p» o po m •c eo c PO o CM <-> PO CM vO r- >-j in o >0 CO o in co sf PO «* p- *** CO CO CM CM ^ r-t 1—

1

00 c m ST CO CM CM oooooooooooooooooooooooooooooooo

oomminmininininmmminininmminmmmminmmminmmmmm

OCM"4"vOCOOCM«4->OCOOCM-4, >OaOOCM>J->OCDOCM<4-»OCOOCM<4- >OCOOCM

o o o o o ^

m .*• «* »* «t -4- o o o o o

1 I I I I I LU LU LU LU LU LU in o ro oo r» oo co r» m oo oo <o

j> <t c> o o co N(NJ^>OS0O

NNNNNCIflUlffim + ^^^^inUMAinifl-O-O

•4- COo

1 I LU UJ

CO CJ> r~ p- m **

CO o

0> —i

CO PO CO ^ o o o o

I I I I LU LU LU LU

P*» PO PO CO O O cm r» o oo m o> O o o 0> _i —t -J o

o o

I I UJ UJ

CO o

"4- -4- o

1 I 'JJ LU

oo r~

co o

O -4-

<4- m

P- sO

4- «* o o

I I LU LU ro r- IM PO

«-< *

m *

o o

l I LU UJ

in >o

fj> CM

PO PO

o o

1 I I UJ LU UJ

CM <4" 0> CM O CM

r- r- in

>o <-• Pcm cm -»

o o

I I LU UJ

<o CO o o O CM

m in m

o o o

l l i UJ UJ LU o e> •

CM >0 CM l*IO + O CM P~ o r- in

m m

o o

i l m mo

1 i UJ LU CM >*•

oo o>

co CM

CM CM oooooooooooooooooooooooooooooooo

_ ooooooooooooooooo

OCM<4-<OCOOCM-4-OCOOCM>4->OCOOCM<4'>OaoOCM'4-<OeOOCM'>*-<OOOOCM

OOOOOOOOOOOOOOO

•H3

o o

I I o -* o o

I I LU LU O CO

co co -* o

p~ rP- CM

>o ^> CM CM

t • o

1 I

o o o

I I I o o o

I I LU LU UJ

CO -+ *?

0> fj> pO

r- >» o

a> r» r» o CO CM

-* \t\ ** o o o

I I I

o o

I I CM O

o o

I I LU JJ o> <o O CM

-* CO r~ •» co o> o PO -^ CM

• • o o

o o

I I —i CM o o

I I LU LU

CO -t

CM 00

>0 CO

PO O

-« O

Po o o o o cMCMCMCMCMropoporopo^^^^^mmmmmvO^

o

1 I

i-i in o • < o

or

o o

l l LU LU PO PO

oo in co o

o> o> -4 PO

o o o

1 I I I LU LU LU LU

po <© o in

PO —I O •£> CO >0 00 CM o cm in vO m p- co cj>

c*> poo

1 I LU LU

0> 00 * -*• PO P- o o

ro co po o o

1 I I LU LU UJ O CO 0> PO «* >* CO o -* o o o

o o o

I I I LU LU LU o -* t-

•* CO o

P~ >t CO

^t -O P»

o> CO Po o

1 I I LU LU UJ CO 0> p00 O* CO

r~ co m

r- eo o

«o m «»

o o

I l LU LU

CO CO m o

.» co

•4- CO

O O

I I UJ LU

0> «o -> P-

<-* 00 CO CM

CM CM

>* ^t >*• in o o o

1 I I I UJ LU LU LU

O- -* «*• 00 O1 5* ^ fO

-t eoom

oo -j* *-> in f* «H ^H o

m in o o

I I LU UJ

><- -o

-O CO

cm r-

•o o

in ino

1 I UJ LU

CO <f

0> 0> CO >* 00 00

•4- CO

m ino

1 I UJ UJ

-* >* o —

•C CO o >»• CO CM ooooooooooooooooooooooooooooooo

>OHfiinp-o>H ininmmmininminmininmminmininmmmminmm ninsOHfciifir-rA^foifisoHCiinNOf-ifftinNO^^

<o m co

p» <o CO

in co pin cj* -^

«* >o P» m m in m in in

• • • O o> tj\ CM CM CM

I I I

CO vO CO CO P» CM

r«- r» p- —

CM PO

P- CO CO in in in

in m in

• t • o* o> r/1 CM CM CM

I I I

-* m

m -i p- o

PO «f co co in m

in in

• t 0> CT> CM CM

I I

-i >D o o»o

in << ao co

in m

in in

• •

CM CM

1 t

ZOO

oi— m >»• h- O O O I I Z LU UJ 3 in cm

u. o> m

o> o> LU O >t > PO -J < • t zoo

o o o o cMCMCMCMCMcocoPOPOPO.*.4-.4-'>*'>4, mmmmm>o

«*-«4-<4-«4-<4-pocococo<o>r-4, <4-^«r«4->4- «»

, «i->4- <4- >4-«*--4-inmmmmm oooooooooooooooooooooooooooooo I I I ( I I I I I I I I I I I I I I t I I I I I I I I I I I LU LU LU LU LU LU UJ UJ UJ JJ 'U IJJ UJ UJ 'JJ HI 'U Ul 'U 'IJ 'U 'U fP il> \U ' UJ UJ UJ UJ incOP~OC>0>00e0«0C0CM>*- -4"0»0>00<4- '-<<»- CMP~P»<4''4'P"»^P-<J-0><

^incDPOina>r,-poc7i ooa>inin-oocoo>or,,-0'-<cMCMoo>4'co<ooin romcocoo^—i-ocovO<MmvOO—ip-opoom--imp-o-H>OPOCMr->3-p- >tCM00CMcOOOOOO>0C0rj>OO'-<cO«0O«4'C/> mCMOO'4'rHOCMm

min*a>o>^i-»"<HHij»cop-p-'Oifi<j, mN(\i-^HH^oD«in4-cflN

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

o o

- I t— UJ < m

INI O I-. p~

-J CO < -I

s: >*

0H CM O • z o

29

ft U. S. GOVERNMENT PRINTING OFFICE : 1962 O - 656388

U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized researcn, development, and engineering in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

front cover. WASHINGTON, D. C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements.

Dielectrics. High Voltage.

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering Metrology.

Mass and Scale. Volumetry and Densimetry.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics,

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment.

Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard Refer- ence Materials. Applied Analytical Research. Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics. Rheology. Combustion

Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental Research.

Metallurgy. Engineering Metallurgy. Microscopy and Diffraction. Metal Reactions. Metal Physics. Electrolysis and Metal Deposition.

Inorganic Solids. Engineering Ceramics. Glass. Solid State Chemistry. Crystal Growth. Physical Properties.

Crystallography.

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Metallic Building Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research. Data Processing Systems. Components and Techniques. Computer Technology. Measurements Automation.

Engineering Applications. Systems Analysis.

Atomic Physics. Spectroscopy. Infrared Spectroscopy. Solid State Physics. Electron Physics. Atomic Physics.

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation. Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy. Molecular Kinetics. Mass Spectrometry.

Office of Weights and Measures. BOULDER, COLO.

Cryogenic Engineering Laboratory. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryogenic Technical Services. CENTRAL RADIO PROPAGATION LABORATORY

Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research. Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Vertical Soundings Research.

Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements.

Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Modulalation Research. Antenna Research. Navigation Systems.

Upper Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Radio Physics. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency and Time-Interval

Standards. Millimeter- Wave Research.

Circuit Standards. High Frequency Electrical Standards. Microwave Circuit Standards. Electronic Calibration

Center.
NBS