Probing the isospin dependence of nucleon effective mass with heavy-ion reactions

• Momentum dependence of mean field/– Origins and expectations for the momentum

dependence– Experimental observables

• Experimental results

Z. Chajecki, D. Coupland, W. Lynch, M. Tsang, M. Youngs Work performed at NSCL and Department of Physics and Astronomy

Michigan State University

• Symmetry energy calculated here with effective interactions constrained by Sn masses

• This does not adequately constrain the symmetry energy at higher or lower densities

Central question: How does EoS depend on and ?

E/A (,) = E/A (,0) + 2S()

= (n- p)/ (n+ p) = (N-Z)/A

a/s

2 A/EP

-20

0

20

40

60

80

100

120

0 1 2 3 4

symmetric matter

NL3Bog1:e/aBog2:e/aK=300, m*/m=70Akmal_corr.

E/A

( M

eV)

/0

BA,Z = av[1-b1((N-Z)/A)²]A - as[1-b2((N-Z)/A)²]A2/3 - ac Z²/A1/3 + δA,ZA-1/2 + CdZ²/A,

Brow

n, Phys. Rev. Lett. 85, 5296 (2001)

Symmetry energy at

0

=1=0

O

Key uncertainty: What is the potential energy of nuclear matter?

• EoS (T=0): E/A () = <KE>/A + <V>/A:

• In the mean field approx., <V> is obtained from mean field potentials for the nucleons. e.g. in a semiclassical approximation

– local two body interactions → ~ linear dependence on ; local three –body int. → ~ 2 term.

– p (momentum) dependence can come from:• range of NN force• exchange (Fock) term• intrinsic mom. dep. of NN interaction. • ...

– p (momentum) dependence implies an additional density dependence

ij ijki j i j k

V v v ...

n SM symU r, p U r ,p U r ,p,

Momentum dependence of mean fields

• Momentum dependence of the mean field (real part of optical potential) is well established for symmetric matter. – At low energies, it can be described by

effective mass, m*:

– Momentum dependence increases with ρ, is maximal at p=pF and vanishes as p→.

• Is the symmetry potential mom. dependent?

2

2 2* 2

p 0

1 1 Up p2m 2m p

USM

(MeV

)

Un-U

p (M

eV)

H. W

olter Nusym

13 (2013)

Consequences of momentum dependence of isovector mean fields

• For an expanding and statistically emitting source, it is easy to show that that the n/p ratio depends on n and p, at low T (in the effective mass approx. and neglecting VCoul).

– The effective mass effects dominate at early higher energies, corresponding to early emission times when density is higher.

• Trend is well supported by transport theory and by simple dynamical arguments.

n p

22F,pF,nn n

n pn,eff p,effp p t t t

ppdN texp / T exp c2 / T

2m 2mdN t

Kinetic Esym +Mom. Dep.

Symmetrypotential

Rizzo et al., PR

C 72, 064609 (2005)

6

2sym sym

*

if U U , p

d Uaccel.

dt m

B. Liu et al. PR

C 65(2002)045201

From dynamical point of view

Central 124Sn+124Sn CollisionE/A = 120 MeV/A

Rn/

p= Y

(n)/Y

(P)

• m*n<m*

p - neutrons more easily accelerated to high energies

p

n

• m*p<m*

n – protons more easily accelerated to high energies

n

p

Y. Zhang., private comm

. (2013)

Experimental LayoutPhD theses: Daniel Coupland & Michael Youngs

• Courtesy Mike Famiano

• Wall A

• Wall B

• LASSA – charged particles• Miniball – impact parameter

• Neutron walls – neutrons• Forward Array – time start• Proton Veto scintillators

Experimental observables

Somewhat problematic: - neutron measurements have known efficiency ~10%

- Effects we are going to measure are often of the same order

n/pR Y n Y p

Rn/

p(124 S

n+12

4 Sn)

More robust:- reduces systematic uncertainties- reduces differences in energy calibration- Coulomb “cancels out”

124 124n/p

n/p 112 112n/p

R Sn SnDR

R Sn Sn

DR

n/p

p

n

Y(p,124)R (124 /112)

Y(p,112)similarly : R (124 /112)

Rp(1

24/1

12)

Y. Zhang, Z. Chajecki. Private comm. (2013)

ImQMD05_sky: incorporates Skyrme interactions

Predicted incident energy dependence

• Possible explanation: Decrease of symmetry energy effects with incident energy may be the effect of increasing temperature.

Skyrme S0(MeV) L (MeV) mn*/mn mp*/mp

SLy4 32 46 0.68 0.71SkM* 30 46 0.82 0.76

Y. Zhang, private comm

. (2013)

• Coalescence and thermal models → t/3He is derivable from n/p

• Is t/3He a surrogate for n/p?

Experimental resultsR

i(124

/112

)

DR

n/p

E/A=50 MeV E/A=50 MeV

p

p,124 p,112 p

n n

Y(p,124)R (124 /112)

Y(p,112)

exp / T exp / T

similarly : R (124 /112) exp / T

124 124n/p

n/p 112 112n/p

n,124 p,124 n,112 p,112

n p

R Sn SnDR

R Sn Sn

~ exp / T

exp / T

D. Coupland, M. Youngs , Ph.D. (2013)

Ri(1

24/1

12)

E/A=50MeV E/A=50MeV

Ri(1

24/1

12)

DR

n/p

Comparison of (n,p) to (t,3He) observables

3

3

3

124 124t / He

t / He 112 112t / He

n,124 p,124 n,112 p,112

124 124n/p

112 112n/p

R Sn SnDR

R Sn Sn

~ exp / T

R Sn Sn

R Sn Sn

3

3

3

3He

p,124 n,124 p,112 n,124

p p n p

p

nHe

Y( He,124)R (124 /112)

Y( He,112)

exp 2 2 / T

exp / T exp / T

R (124 /112)

similarly : R (124 /112) R (124 /112)

0

D. Coupland, M. Youngs , Ph.D. (2013)

E/A=120MeVE/A=120MeV

Ri(1

24/1

12)

DR

n/p

E/A=50MeV E/A=50MeV

Ri(1

24/1

12)

DR

n/p

Comparison of (n,p) to (t,3He) observablesD. Coupland, M. Youngs , Ph.D. (2013)

Comparisons with transport theory: n,p

ImQMD:- Cluster production does not have the

correct binding energies for light fragments.

- Test semi-classical dynamics by constructing “coalescence invariant” nucleon spectra, which represent flows prior to clusterization.

Coalescence invariance: - Coalescence protons or neutrons

spectra include both free neutrons and protons and those within clusters. This is done for both experiment data and theoretical calculations. It is essentially an observable constructed from measured spectra.

Free particles

E/A=50MeV

ImQMD05_sky: incorporate Skyrme interactionsY. Zhang (2013) Private CommunicationTsang (2013) Private CommunicationD. Coupland, M. Youngs (2013)

124 124n/p

n/p 112 112n/p

R Sn SnDR

R Sn Sn

Coalescence invarient n/p

E/A=50MeV

E/A=50MeV E/A=50MeV

Ri(1

24/1

12)

DR

n/p

Comparison of independent particle ratios

Rn(1

24/1

12)

Rp(1

24/1

12)

ImQMD:- Soft sym energy approaches free

data at high energies, but differs at low energies where clusters contribute.

- Including free and bound nucleons in the observable reduces the discrepancies

D. Coupland, M. Youngs , Ph.D. (2013)D. Coupland, M. Youngs , Y. Zhang. (2013)

15

Comparisons of n/p double ratios

• ImQMD:- Cluster production for alphas is not

realistic- Possible solution: Ignore the cluster

production mechanism and look all the light particles (neutrons and protons) at a given velocity

• Coalescence invariance: - Coalescence protons (neutrons):

Include protons (neutrons) from within clusters with the free proton (neutron) spectra

- Possibly a better match between simulation and experimental data

• Free particles

• E/A=50MeV

• Coalescence particles

• E/A=50MeV

• Y(n)/Y(p); 124Sn+124Sn• Y(n)/Y(p); 112Sn+112Sn

• DR(n/p)=

E/A=50MeV

Energy dependence

E/A=120MeV

ImQMD05_sky: incorporate Skyrme interactionsY. Zhang (2013) Private CommunicationTsang (2013) Private Communication

D. Coupland, M. Youngs , Y.Zhang. (2013)

Z.Chajecki - NuSYM 2013 16

Comparison with transport theory: clusters

np

nN

tapn

pN

n

• Includes dynamical production of clusters up to A=3 (but not beyond)

• m*=0.7m0, m*p= m*

n

• Calculations underpredict the double-ratio

Alpha production not included in the model => alpha ends up being t or 3He

t

Solution: combine experimental alpha spectra with tritons and helium-3 and compare to the model predictions

M. Youngs, Z. Chajecki (2013)

17

Summary Momentum dependence can be expected. It will influences dense

mater within neutron stars. Calculations show that n/p and t/3He ratios are sensitive to

momentum dependence and symmetry energy. There is a clear connection between n, p, t and 3He spectra that can is

qualitatively similar to behavior expected from chemical potentials and from transport theory.

n/p observables are cleanest at high kinetic energies where cluster production can be neglected. At lower energies, the trends are consistent with coalescence invariant analyses. Improved cluster production would allow more careful comparisons at lower energies or using t/3He ratios.

Cluster comparisons for 40,48Ca reactions

18

48,40Ca+48,40Ca @ 80MeV/A 112,124Sn+112,124Sn @ 50MeV/AM. Youngs, Z. Chajecki (2013)