Behavior of Nuclear Reaction NetworksBrad Meyer

Clemson University

Answer to Binding Energy Problem

nMeV/nucleo8.6425256/981.483/)(

483.981-53.9010)(071.8*)2856(289.7*28)56,28()(

nMeV/nucleo8.7943762/251.545/)(

251.545-66.7450)(071.8*)2862(289.7*28)62,28()(

62

56

62

62

==

=−−+==

==

=−−+==

ANiB

MeVBNiB

ANiB

MeVBNiB

NSE Calculator

http://nucleo.ces.clemson.edu/pages/nse/0.1

Formal Development

∑

∑

=⇒

==

+−−=

iii

iii

dYdf

dVdT

Consider

dYPdVsdTdf

µ

µ

0

For equilibrium

∑ =

=

iiidY

so

df

0

0

µ

But we have constraints

∑

∑ ∑

=

==

ieii

i iiii

YYZ

YAX

)2

1)1

Detour (a simple minimization problem)

• Minimize the function f(x,y)=x2+y2

0,0

,

022

==

=+=

yx

so

dydxarbitrary

ydyxdxdf

A more interesting example• Minimize the function f(x,y)=x2+y2 subject to the

constraint that y=-x+b, b a constant

2/

0)22(22

022

bybyyso

yxdxyxydxxdxdf

dxdy

nowbutydyxdxdf

=⇒+−=

=⇒=−=−=

−=

=+=

A better way: Lagrange multipliers• Minimize the function f(x,y)=x2+y2 subject to the

constraint that y=-x+b, b a constant

2/2/2/

2/,2/

0)2()2(0)(

,22

byxbbbxy

yxso

dyydxxgfd

Findbxygyxf

Define

==⇒=⇒+−=⇒+−=

==

=−+−⇒=−

−+=+=

λλλλλ

λλλ

Our problem

∑

∑

=

=

ieii

iii

YYZand

YA

tosubject

fMinimize

)2

1)1

( ) 0

0)(

1

21

21

=−−

=−−

−=

−=

∑

∑

∑

ii

iii

eii

i

iii

dYZAso

hgfdthen

YYZhand

YAgDefine

λλµ

λλ

nipii

npiniiiii

npp

nn

NZ

ZAZAOthers

otons

Neutrons

µµµ

µµµµλλµ

µµλλλµ

µλλµ

+=⇒

−+=⇒=−−

−=⇒=−−

=⇒=−

)(0:

0:Pr0:

21

221

11

General Nucleosynthesis Network

np

np

NZAZZAN

ZAZAZ

µµµ

µµµ

+=−=

−+=

),(

)(),(

Chemical potentials

• Consider nuclei as Maxwell-Boltzmannparticles:

+=

2/322

),(2),(),(ln),(),(

kTAZmh

AZGAZYNkTcAZmAZ A

πρµ

The Nuclear Data Tool

http://nucleo.ces.clemson.edu/home/online_tools/nuclear_data/0.1

QSE: Quasi-statistical equilibrium

∑

∑

∑

≥

=

=

=

6,)3

)2

1)1

iZihi

ieii

iii

YY

YYZ

YA

Key results

6

:

22/

22

≥

=

=

≡

i

npkT

i

np

NSEi

ii

Zfor

RReRand

RRRQSE

YYR

µ

α

The Silicon-Burning Movies

R Process

The R-Process Movies

Alfred Russel Wallace

(n,gamma)-(gamma,n) equilibrium

• Nuclei in equilibrium under exchange of neutrons

• Beta decay:

ZZZZZ

AZ

AZZ

YYdtdY

then

and

AZYYwhere

AZYAZY

λλ

λλ

−=

=

=

−−

∑

∑

11

),(

),(),(

Tasks for today• NSE

– Compute the NSE abundances from the expressions in slides 13 and 14

– Determine at what time record in alpha-rich.fits (fits file from yesterday) that the network abundances fall out of NSE

• QSE– Confirm the expressions in slide 17 by minimizing the

free energy f subject to the constraints in slide 16– Confirm that the network abundances in alpha-rich.fits

satisfy these relations after they fall out of NSE

Tasks for Today (cont.)

• R Process– Set up the constraint equations for the

(n,gamma)-(gamma,n) phase of the r-process freezeout

– Confirm (n,gamma)-(gamma,n) equilibrium in the r-process fits file rprocess.fits.

– Confirm beta-decay steady state for a time record during the r-process phase.