Charge Symmetry in ¹³N and ¹³C: a Coupled-Channel Model - CaltechTHESIS

We will find that the

computed ratio agrees quite wel l with the experimental ratio; hence,
for this transition we find no violation of the charge symmetry of
nuclear forces.
on

12

In addition the bremsstrahlung radiation from protons

c will be modeled as occurring through one resonance - the J"=3/ 2-

state at 3 .509 MeV- and the incomin g plane wave modeled as the higher

16

JTI~~-state at 8.92 MeV- to the Jn=~+ first excited state at 2.366 MeV.
From the calculations of Cohen and Kurath

18

for single nucleon

transfer reactions in the lp-shell, we obtain spectroscopi c factors for
the two most important low-lying contributions to the mass 13 ground
state.

Igs> = 8(0+ ) 112C(g.s.) +nucl eon> 8(2+ ) 112C(2+ ) + nucleon>

IV.l

where 8(0 ) and 8(2 ) are the fractional parentage coefficients.

The

spectroscopic factors are given by the square of the fractional
parentage coefficients times the number of extra core nucleons as
explained in Section II B.

Cohen and Kurath determine the spectra-

scopic factors,

In coupling the
form the

13

c and

12

c(g.s.) and

12

c(2+) with single particles to

13
N ground states we proceed as follows.

The strength

of the off-diagonal mixing potential is calculated using the rotational
coupled-channel model.

The diagonal nuclear optical potentials in each

chann e l are forc e d to be equal and this strength and that of the spinorbit potential are varied to put each ground state at th e proper
eigenenergy with the correct spectroscopic factors.

These wave

functions are then used i n the computation of the El matrix elements in
conjunction with the first excited state wave functions.

Variations of

the diagonal and spi n-orbit p otential strengths for the fir st excited
state result

in the reproduction of the

12

c(p,y)

13

N capture cross

17

section (Figure 4) by estimating its value at the resonance energy and
at proton energy . 700 Me V ( center of mas s ).

The resulting norrnal i za-

tion factor gives the ground state (0 and 2 states only) spectroscopic

1T

factor s1nce the J =1/2

state is normalized to unit incoming flux.

From this computation we also ob.t ain estimates of the
S-wave phase shifts

20

12

c (p ,p ) 12c

whi ch are excellent over th e enti re ene r gy r egion

of inte rest (Figure 3)
Potential strengths identical to tho se used for the mixing and

13N, are then applied in computing the 13c

spin- orbit potentials in

ground and first excited states .

Again , the di agonal potential is

varied to put the wave functions at the right eigenenergies.

The El

transit ion width i s computed and multiplied by the ground state
spectroscopic factor estimated in the

From the

13

capture cross s ection in
width.

13N computation .

N a simple computation yields it s gamma

The peak is modeled as a Breit-Wigner form an d the gamma

width is computed assuming that the proton width i s approximately
equal to the full width.
In performing the brems strahl ung calculation a similar procedure
is followed .

1T

The J =1 /2

excited s tate potential value s computed in

the previous calculati o n are used to generate wh at will b e the final
state wave fun ctions in this problem.

Using the mixing potential

value s predicted by the rotational model we then const ruct the JTI =3/ 2-

TI

and J =1/2

- e xcited states.

7T

The J =1/2

state is u sed only to simulat e

18

the PJ. piece of the incoming plane wave.

'1T

The J = 3/2

potential values

are then varied to arrive at the best fi t to the 6 = 0° bremsstrahlung
cross section , using the El bremsstrahlun g differential cross section
formul a from Appendix B.

A factor for the final state phase space

den si ty and a simple integr al of the Breit- Wigner resonanc e shape for
the J'!T =l/2+ final state were also included.

The 6 = 90

was also determined , and is the dotted line of Figure 6 .
seen from the figure the fi t
data badly .

cross section
As can be

is qualitatively c orrect but missed some

An examination of the experimental photon l ine shapes

r eveals the reason s for the problem.

The backgroundfluctuations at l ow

energies are quite large , typi cally ten to fifteen percent of the
resonance peak values , and secondly, the line shapes deviate qui t e
severely from a Breit-Wigner form.

To correct this problem we have

computed the line shapes over the f i nal state resonance region for
selected valu es o f the incident proton energy.

The areas of the

resulting peaks are numeric all y computed afte r lowe r limits to the
peaks are inserted.

Figure 7 shows a set of these peaks for a range

of incident proton energi es .

The resulting final cross sections can

be seen as the solid line i n Fi gure 6 for 6 = 90° .

The lower limi ts

to the peaks were determined u sing two rul es of thumb .

First, the

line shapes should look closel y like a Breit- Wigner resonance, hence
the asymmetry of the peaks reduces the area.

Second, the background

noise results in making the placement of lower limits uncertain.

The

experimental line shapes had lower limit s less than three half widths

19

wide.

Even assuming a pure Breit-Wigner form for the peak, cutting

off the base at three half width s yields a large reduction in area.

6 shows quite vividly the large changes to the

The sol i d line of Figure

90° differential cross section.
Again the

13
13
N potenti al strengths are applied to the
c mirror

transition states and the equivalent gamma widths are computed .
iT

Tabl e 1 contains a l isting of the se widths for the J = l/2
iT

transition to the J = 1/2
iT

to J =1/2

iT

gro und state and the J =3 /2

state

state transiti on

state in both nucl ei as well as a compari son with the

experimental value s .

Table 2 contains a listing of the potential

values for all states corresponding to the widths computed in Table 1.
It should b e noted that the infrared divergence i n the
bremsstrahlung cross section seen at low energies as the dashed curve
in Fi gure 6 is a result of the breakdown of perturbation theory for
small photon energies.

The perturbation term i s a l inear appr oximation

to the matrix element of a complex valued exponential function of the
perturbation potential between the initial and final states.

Even

thoug h this linear i zed term goes to infinity for small photon energies
as do all higher terms in the perturbation expans ion, the exact term
remain s finite; hi g h e r order terms don't help the expansion converge.
Summing the peaks as done here to corre ct the bremsstrahlung cross
section also yields the added bonus of the d i sappearance of this
divergence.

As can be seen by examining Figure 7, as on e goes to

smaller photon energi es the Breit-Wigner part of the peak grows smaller

20

even though the total area under the peaks goes to infinity.

21

V.

DISCUSSION

A cursory examination of Table 2 reveals two significant features
of the potential strength values.

The first conclusion is that the spin-

orbit strength varies quite widely across states.

In performing the

calculations the mixing potentials were fixed by the rotational model
and we have truncated the full channel expansion to only two excited
states of

12

C.

Hence we shouldn ' t really expect any great consistency

in the spin- orbit strength across states.

The second major feature is

t he differences between the diagonal potentials in even and odd angular
momentum states.

The S and D shell potentials are approximately

54 MeV

strong, whereas the P- waves are bound somewhat more weakly , approximately
42 MeV .

One can only speculate as to the origin of this effect, but

again the truncation of exc i ted core states may play an important role .
In the standard derivation of the existence of the spin-orbit force
a Dirac equation is derived from principles of relati vist ic i nvariance.
The resulting spinor wave function is approximated by a scalar wave
function with the Dirac equation being transformed from a c oupled first
order differential equation to a scalar second order Schroedinger equation
with a spin-orbit potent ial and higher order terms.

That is we ' ve

reduced a c oupled wave function to a scalar wave function .

For this

r eason it is probably correct not to distort the spin- orbit force; in
fact the correct approach to deriving a coupled-channel model should
s t art with a Di rac equation, the inclusion o f core wave functions, and

22

a model of the excitation process.

Such a procedure is beyond the

scope of this paper.
As mentioned earlier, the failure of current models to adequately
model the low energy states is most probably explained by the converse
success of the niodels in consistently reproducing higher energy states.
Conservation of energy and flux fix the amplitudes of all unbound
channels whereas bound states can take on any normalization.

Thus all

that the standard models do at higher energies is determine the phase
rel-ationships between wave functions;

at lower energies when bound

states exist we must also mix in the proper amplitudes.
The resulting effects on matrix elements are unknown;

it may be

fruitful to pursue just what happens at these important thresholds.
Further insight can also be obtained by examining the role of complex
valued potentials.

It is well known that the use of an imaginary

valued potential results in the loss of flux from open channels because
the Hamiltonian becomes non - Hermetian .

Further investigation into the

use of complex valued wave funct i ons and potentials and their proper
interpretations would be of invaluable help in unraveling the mysteries
of the nucleus .

23

APPENDIX

A.

Numerical Solution of Coupled Second Order Differential EQuation
System
1.

Solution by iteration of coupled eQuations

2.

Boundary conditions and solution search routines

3.

Computation of external matching functions

B.

Gamma Ray Transition Differential Cross Section Formulas

C.

Nuclear and Coulomb Potential Shapes

D.

Computer Program Info rmation

24

A.

NUMERICAL SOLUTION OF SECOND ORDER COUPLED
DIFFERENTIAL EQUATION SYSTEM

l.

SOLUTION BY ITERATION OF COUPLED EQUATIONS

The system of second order coupled differential equations is given
by

(l)
where A is a matrix specified by the particular model used.

Diagonal

elements of A are the energy eigenvalues and central potentials of
each channel.

The off-diagonal elements are the mixing potentials.

In

the standard 3-point integration formula,

= 2y-+n

(2)

-+

-+

where his the step size and yn = y(r=nh).

We substitute (l) in for the

second derivatives obtaining

= (12X-l - 10) X +y

(3)

where Xn = I -+

X y

n n

-+

n-1 n-1

h2

12

An.

At the origin our starting conditions are

= -+0, and at step l we arbitrarily set x1y-+1 = -+a and define
-+

-+

Xn+ly n+l - Zn+l a ·

-+

After dropping a from all equations, we are left

with a set of matrix equations to iterate,

( 4)

with Z

= O, z1

I.

25

We use equation set

(4) when we search for the eigenvalue or

resonance and equation set (3) when our desired solution is found.

The

usual error analysis and behavior is carried over from the onedimensional case and can be found in Melkanoff, et al.

2.

22

BOUNDARY CONDITIONS AND SOLUTION SEARCH ROUTINES

For the case when all channels are bound, we must have only
exponentially decreasing wave functions at infinity, the Whittacker
function, and these are easily matched to the iterated solution in each
channel.

In all other cases we must have outgoing waves in at least

one channel, so our outgoing wave function must behave asymptotically
like

( 5)

'¥

(GR.

+iF£ ))

where s is any channel, i is the incident channel, k. the wave number in

the incident channel, F£ and G£ are the regular and irregular Coulomb
functions, respectively, but can be replaced by sphErical Bessel
functions in the case of no charge interactions; or if a channel is
bound, G£

+ iF£

is replaced by a Whittaker function.
Cs contains
amplitude and phase shift information about the outgoing state. For

the entrance channel C.

= eio sin o where cS is the commonly defined phase

shift, and o£

is the Coulomb phase shift in channel s.
In the case where at least one channel is open, we integrate our

system of equations starting from the origin so as not to introduce
numerical errors from the irregular solution.

The iterated wave

26
funct i ons are matched to standard functions at two points for large
values of r.

z a

That is,

(47r)~

;:::

k .

e io ~· F £. (a) + c .

(6)

ei 0 i(G.~~, . (a) + ~"F R, (a))

cs ei 0 S(G.~~, (cd + ~"F R, (a))

for a ;:::l,2 , the functions all evaluated at the matching points .

We can
-+

-+

write thi s in matrix notat i on and easily solve f or the vectors a and c ;:::

( . .. ,c,
... ).

zl

-+

n·~ag [ e ios (Gl£ +"Fl
~ £ ) ]

(7)

{47r) 2-+

D"~ag [ e ioS (G2£ +"F2
~ R, ) ]

k.

(4TI)>2
k.

;,

z2

;:::

io. Fl
£.

io. F2

£.

For the cas e where all channels are bound, we must integrate from
the ends toward the middle of the i nterval because of the numerical erro rs
i ntroduced by the irregular solutions to the differential equations.

From

the origin out we have the solution matrices ZK ' ' ZK+l and from infinity in
-+

we have ZK' ZK+l '

For arbitrary start i n g vectors a from the origin and

-+

b from i nfinity we have the solutions at the points K
-r , ;:::
YK

(8)

-1

'-r
ZKa

-r,
;:::
YK+l

-1

-+

~+1 ZK+l a

and
-+

YK

;:::

-1

-+

ZKb

-+

YK+l

;:::

-1

-+

~+1 ZK+lb

and K+l,

27
We have a proper solution to the differential equations when the
values at each po i nt are the same.

This impl i es that we solve the

matr ix equation

(9)

!]

Z'

-Z

-+

-+

When the determinant of A is zero, then we have found a good
solution to the system of equations .
-+

b, simply set a

3.

-+

To find the starting vectors a and

= 1 and solve the resulting system of equations.
COMPUTATION OF EXTERNAL MATCHING FUNCTIONS

The asymptotic values of the bound channel wave functions for both
charged and uncharged states were computed using a 20- point Laguerre
quadrature formul a on the integral representation of the Whittaker
funct i on

(10)

w( n , t + ~ . 2p)

= exp( - p- n tn 2p)
r(l+£+ n)

The nodes and weights are given by Stroud and Secrest

23

The error in

the quadrature can be limited to the boundedness of the fortieth
derivative of the integrand times the error coefficient of 0 . 7254E- ll .
Representative values of these functions have been checked against
tables in Abramowitz and Stegun

17

The spherical Bessel functions are computed from formulas 10.1 . 2
in Abramowitz and Stegun

17

The irregular solution can be recursed by

28
a three-point formula up in £-values,starting being facilitated by the
analytic forms for £=0 and l.

The regular solutions are started

arbitrarily at high £-values and recurred downward.

The Wronskian is

then computed and used to correct the regular solutions.

This routine

has also been checked against tables.
The Coulomb functions are computed by a program developed at the
University of Minnesota.

Representative values have been checked against

tabl es in Abramowitz and Stegun

17

and Arnold Tubis, Tables of Nonre la-

tivistic Coulomb Wave Funct i ons , Los Alamos Scientific Laboratory (1958).
The irregular Couiomb wave and its derivative for £-values are obtained by
re curs ion.

The regular solutions are obtained by a downward recursion

and these solutions are corrected by a Wronskian calculation.

29

GM~

B.

RAY TRANSITION DIFFERENTIAL CROSS SECTION FORMULA

The derivation of the formulas used in generating the cross sections
of interest can be found in Johnson7

For the El-bremsstrahlung differential cross section, we obtain

dcr(e)
(ll)

dS"l

+ {4 Re(<~-~~ T~ll ~+>*<3/2-ll T~~~ ~+>
-21< 3/2-

II T~ II ~+ >1 2 } p 2 (cos 8)]

The El-capture involves an integration over angle and is given by

2k
= ___L { 2j< ~+II Te II ~- >1 }
cr
py
3hv.

(12)

where k

is wave number of the outgoing photon, v. is the velocity of

the incoming proton in the center-of-mass, and

(13)

where J

, J

are the total spin of the core plus nucleon system for the

initial and final states, I
W(J

is the spin of the core, and the

1 Injl; J j ) are Racah coefficients.
1 2

The reduced matrix element

30

£ -£ +1

=i 1

(14)

(-1) 1

-~ (2. +1) 2

_J;;..o2_ _

13

. ( j 1 j 2~-~ l1o) eeffky OV(n)
Jl

n£ljl

and

OV(n)

roo

J2
u £ .
n 2J2 2
r dr
eEl(r)

and

where e is the proton charge and (Zp,Ap), (ZT, ~) are the charge and
mass number of the projectile and target, respectively, the
(j j ~-~110) are Clebsch-Gordan coefficients, eE (r) is the electric
1 2
dipole operator which reduce s to r in the long wavelength approximation,

wJ 1 .

n~lJl

, u

J2
n •
are the nth channel initial and final state wave
n""2J2

functions of a nucleon with orbital angular momentum £ 2 ; and total
angular momentum j., about the core state with total spin I

to yield a

total angular momentum J ..

The bremsstrahlung differential cross section is multiplied by a
density of states to correct for all the continuum final states.

This

factor consists of two parts: the usual density of momentum in phase
space and the area under the final state resonance assuming a BreitWigner shape.
The Clebsch-Gordan and Racah coefficients were evaluated using
Rotenberg, Bivens, Metropolis, and Wooten, The 3-J and 6-J Symbols, The
Technology Press, MIT

(1959).

31

C•

NUCLEAR AND COULOMB P OTENTI AL SHAPES

The u sual diagonal central p otential is a Woods - Saxon d i ffuse edge
well

form

(15)

where a is a

= [ l + e xp( [ R- R0 ] /a) ]-l

par&~eter

which measures the d iffuseness of the edg e and

is the nuclear r adi us usually given by R

.r Al/3 .

The off-diagonal and sp i n - orb i t potential are g i ven in ter ms of the
deri vative o f t he central pot ential

(16)

df
-=dr

The spi n - o r bit force picks u p an addit i onal 1/r dependence as presented
in Table l.

The Coulomb potential i s the us u a l uniform sphere

val ue inside the nuclear r adiu s and the standar d 1/r dependence outs i de ,

(17)

V (r)

={

ZZ ' e 2
(3 2R
R2

r 5 R

ZZ' e 2

r > R

32

D.

COMPUTER PROGRAM INFORMATION

The programs were run on the CDC 7600 high-speed digital computer
at Berkeley through the remote terminal in the Downs-Lauritsen physics
building.

The programs can be contai ned in two boxes of cards, but

testing and production were done using the program storage system at
Berkeley.

The high speeds and inexpensive costs in running the programs

make any timing estimates superfluous.
The linear equation solver package LINIT is a Berkeley computing
center program.
All numerical and physical constants values are from F . AjzenbergSelone and C. L . Busch, Nuclear "Wallet Cards", University of Pennsylvania, Philadelphia (1971).

33

RE;FERENCES
1.

E. K. Warburton and J. Weneser, The Role of Isospin in Electromagnetic Transitions, edited by D. H. Wilkinson (North Holland ·
Publishing Company , Amsterdam, 1969), p. 174

2.

F. Ajzenberg-Selove, Nucl. Phys. Al52 (1970) 1

3.

F. Ajzenberg-Se love, Nucl. Phys. Al90 (1972) 1

4.

P. M. Endt and C. Van der Leun, Nucl. Phys. A214 (1973) 1

5.

K. W. Allen, P. G. Lawson, and D. H. Wilkinson, Phys. Le tt. 26B
(19 6B) 138

6.

S. W. Robinson, C. P. Swann, and V. K. Rasmussen, Phys. Lett . 26B
(1968) 298

7.

J. D. Seagrave , Phys. Rev. 84 (1951) 1219 and 85 (1 952) 197

8.

D. F. Hebbard and J. L . Vogl, Nucl. Phys. 21 (1960) 65; W. A.
Fowler and J. L. Vogl, Lectures in Theoretical Physics, VI
(University of Colorado Press, Boulder , 1964), p. 379

9.

F. Riess , P. Paul, J. B. Thomas, and S. S. Hanna , Phys. Rev .
176 (1968) 1140

10 .

C. Rolfs and R. E. Azuma, Nucl. Phys. A227 (1974) 291

11 .

D. F. Measday, A. B. Clegg, and P . S. Fisher, Nucl. Phys.
( 1 965) 269

12.

J. B. Marion, P. H. Nettles,
Phys. Rev . 1 57 (19 67 ) 847

13.

I. M. Ge l' fand, G. E . Shilov , Ge n eralized Functions, Vol. I
(Academic Press, New York , 1964) .

14.

T. Tamura , Rev . Mod. Phys. 37 (1965) 679

15 .

D. L. J ohnson, thesis , University of Washington (1974)
unpublished

16.

H. J . Rose and D. M. Brink , Rev. Mod. Phys. 39 (1967) 306

17.

Abramowitz and Stegin, Handbook of Mathematical Functions (1 964)

18 .

S. Cohen and D. Kurath , Nucl. Phys. 73 (1965) l; Nucl . Phys . A10l
(1967) 1

61

c. L. Cocke, and G. J . Stephenson,

34

19.

o. Mikoshiba, T. Terasawa, and M. Tanifuji, Nucl. Phys. Al68
(1971) 417

20,

H. L. Jackson and A. I. Galonsky, Phys. Rev. 89 (1953) 370

21.

D. Kurath, submitted t o Phys. Rev. Comments (1975)

22.

M. D. Melkanoff , T. Saweda, J. Raynal, Methods in Computational
Physics, Vol. 6(Academic Press, New York,l967)--

23.

A. H. Stroud and Don Secrest, Gaussian Quadrature Formulas
(Prentice Hall, Englewood Cliffs, N. J.:i966). ---

35

TABLE 1
WIDTHS

1/2+ + 1/2

3/2 -

-+

1/2+

THEORY

EXPERIMENTAL

(13c)

.617 eV

.44 ± .05 eV

( 13N)

.701 eV

.64 ± .07 eV

Ratio of Strengths

2.52

3.2 ± 0.7

<13c)

2.5 meV

6.6 meV

r y (13N)

4.2 meV

4.4 meV

.240

.095

Ratio of Strengths

It should b e noted that the 3/2- to 1/2+ transit i on widths
are in mill ielectron volts ( l 0- 3 ev) .

In thi s transition strong

inte rference between channels yields a lar ge error in the
theoretica l width values.

36

TABLE 2

STATE

MIXING

POI'ENTIAI.S
SPJN-QRBIT

CENTRAL (MeV)

SPECI'ROSCOPIC
FACI'ORS

13
C(g.s.)

2.22

.93

40.207

(.8252, 1.6091)

13N(g.s.)

2.22

.93

40.257

( • 88121 l. 5531)

13c(l/2+)

2.73

.31

53.304

(.905, .095)

13
N(l/2+)

2.73

.31

54.263

( .922, .078)

13c(3/2-)

1.6

0.

42.557

(.142, .858)

13N(3/2-)

1.6

0.

43.355

( .136, . 864)

The full mixing potential is a prcxluct of the factor given above
times (hc/2mc 2 ) , the Ccmpton wave length of the reduced mass, t.irres the
derivative of the central Wcxrls-Saxon fonn.

Similarly the spin-orbit

potential is a product of the above factor times (h 2c 2 /2rrc 2 ) times
(-;·L)/r t.irres the derivative of the Wcxrls-Saxon fonn.
radius is 22.35 :Em.
JTKXlel with s=-l/2 .

The matching

The mixing strength corresponds to a rotational

37

Figure 1.

Ratio of corresponding El strengths in light mirror nuclei.

The data have been obtainErl from the data canpilations of
refs. 2-4.

'Ihe indicatErl ratio for A = 15 is an upper limit.

::> 4

(l)

0::

ll..

12

A= IS
(EI -lo- 6 w.u.)

RATIO

A=l3
(EI-0.1 W.u.)

RATIO OF EI-STRENGTHS

0.1

A= 17-43
(EI-I0-3 -lo-6 w.u.)

I~EXPECTEu

EI-STRENGTHS IN MIRROR NUCLEI

OJ

39

Figure 2.

Isobar energy level diagram for A = 13.

Note that no correction is made for the Coulomb energy difference
between

13

c and 13N.

4o

I SOBAR 01 AGRAM

2+

A = 13

4.439

6.864 5/2+v /

12C+n

3.547 5/2+
/1 3.509 3/~
//' 2.366 I /2+

o+
(4.947)

//

3.854 5/2+ // / / /
3.684 3/2- /
"'
3.086 1/2+

(1.944)

12C+p
(2221)

1/2-

·~

41

Figure 3.

Comparison of the experlinental S!:! phase shifts with the

rrodel calculations for the
taken from ref. 20.

12

C(p,p)

12
C reaction.

The data are

42

eo
ILL

(f)

(f)

<(

Cl..

300

500

700

PROTON ENERGY (keV)

900

43

Figure 4 .

Capture cross section near the resonance peak.

Dots are data of Rolfs

10

The solid curve is the theoretical fit.

44

100

V)

,_

..c

:t.

450 500
Ec.m. (keV)

550

600

650

45

Figure 5.

S-factor given by S = oE exp (21Tn) •

Solid curve is theoretical fit. The circles are the data of
Rolfs

10

The plus signs are the data of Vogl

curve is a single channel fit with no mixing.

The dashed

46

12C (p, y) 13N

• Rolfs data
+ Vogl data
100

(f)

c:
.J:)

10

Q)

--..::s:;

(/)

100

200

300

400

500
EcM (keV)

600

700

47

Figure 6.

Differential cross sections for bremsstrahlung at 0° and 90°.

The data are from Rolfs

10

The dashed curve is the preliminary fit

using a Brei t-Wigner form to represent line shape.

'Ihe solid

curves are the result of integration over the calculated line
shape.

48

......--.

Q)

-~

.,.

. . ----·--

L[)

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The garrma ray line shapes for bremsstrahlung at 90°.

The

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