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Finding the Incompressibility Coefficient of Nuclear Matter
Jennifer KachelMarietta College, Marietta OH
Texas A&M University, Cyclotron REU Program
Advisors: Dr. Shalom Shlomo, and Mason Anders
MotivationThe incompressibility coefficient of nuclear matter is largely important to the area of physics because it allows scientists to have a better understanding of properties of neutron stars, supernova explosions, and heavy ion collisions. By measuring the incompressibility coefficients of nuclei, nuclear matter’s incompressibility coefficient can be extracted using an A-⅓ expansion analogous to that of the semi-empirical mass formula.
Classical idea of Compressibility
In introductory physics, when studying springs: F = kx and U = ½kx2.
If you exert the same force on two different springs what happens to the potential energy of the spring with the smaller spring constant?
Take:F1 F2 k1x1 k2x2
k1 2k2 x1 12
x2
Classical Idea of Compressibility
Substituting in these values:
Conclusion: Springs with smaller spring constants can store more potential energy
Supernova explosion
U2 12
k2x22 1
212
k1
(2x1)
2 k1x121 2U1
Nuclear Matter’s Equation of State
Equation of State defined is as the binding energy per nucleon(E/A) as a function of the matter density(ρ) [1].
Saturation point (E/ρ0 , ρ0)
ρ0= 0.16 fm-3 is the density at saturation point
E/A(ρ0)= -16MeV, extrapolated from the mass formula
E/A
[M
eV]
ρ[fm-3]
Equation of State
Definition: The compressibility coefficient K is directly related to the curvature of the EOS
Using the Taylor series expansion, the first derivative of the EOS is zero and we get:
EA
[] EA
[0 ] 118
K 0
0
2
...
K k f2
d 2 EA
dkf2
k f
92d 2 E
A
d 2
0
The Giant Monopole Resonance The density oscillates as a
function of time
Isoscalar monopole: both protons and neutrons oscillate in phase
Classical picture of the breathing mode.
(r, t) 0 (r)cos(t)
EGMR
Measuring KA
Before the compressibility coefficient of nuclear matter(Knm) can be found, the compressibility coefficients of a set of nuclei(KA) must be measured
The approximation of KA is given by the expression:
Where m is the mass of the nucleon, EGMRis the Energy of the monopole and rm is the mass radius
222 mGMRA rEmK
Determining the Mass Radius The proton radius
is accurately determined by electron scattering[2].
We approximate the difference between the proton(rp) and the neutron(rn) radii by
Solving for the neutron radius:
The mass radius(rm) is then obtained from:
rp rp2
rn rp r
r rn rp 0.01Z 1.2 N ZA
A rm2 Z rp
2 N rn2
The Mass RadiusNucleus rp rp error Δr Δr error rn rn error rm rm error40Ca 3.371 0.001 ‐0.045 0.010 3.326 0.011 3.349 0.00648Ca 3.368 0.001 0.155 0.043 3.523 0.044 3.460 0.02648Ti 3.489 0.002 0.053 0.027 3.542 0.028 3.518 0.01656Fe 3.639 0.001 0.035 0.024 3.674 0.026 3.658 0.01458Ni 3.678 0.001 ‐0.012 0.017 3.666 0.018 3.672 0.01060Ni 3.716 0.001 0.027 0.023 3.743 0.025 3.730 0.01490Zr 4.184 0.001 0.070 0.032 4.254 0.033 4.223 0.019110Cd 4.495 0.002 0.083 0.035 4.578 0.038 4.542 0.022116Cd 4.550 0.002 0.138 0.044 4.687 0.047 4.631 0.028112Sn 4.515 0.002 0.058 0.031 4.573 0.033 4.547 0.019116Sn 4.548 0.001 0.095 0.038 4.643 0.039 4.602 0.023124Sn 4.598 0.001 0.162 0.049 4.760 0.050 4.695 0.030144Sm 4.871 0.006 0.088 0.038 4.959 0.044 4.921 0.027208Pb 5.435 0.001 0.163 0.052 5.598 0.053 5.534 0.032
Calculating KA
A search of the literature was required to find the energies and the error bars of the giant monopole resonance for each nucleus (data from Professor Youngblood group at the Cyclotron Institute)[4-10].
Using this data and having calculated the mass radius, we can now find the experimental values of KA
Experimental KA
Nucleus EGMR +ev ‐ev rm rm error KA + error ‐error40Ca 20.42 0.89 0.36 3.349 0.006 112.67 10.22 4.3848Ca 22.64 0.27 0.33 3.460 0.026 147.83 5.72 6.5148Ti 20.25 0.99 0.28 3.518 0.016 122.29 13.07 4.4956Fe 19.57 0.73 0.16 3.658 0.014 123.47 10.18 2.9958Ni 20.81 0.9 0.28 3.672 0.010 140.69 12.94 4.5660Ni 19.54 0.78 0.23 3.730 0.014 128.03 11.17 3.9690Zr 18.69 0.65 0.3 4.223 0.019 150.13 11.76 6.14110Cd 15.58 0.4 0.09 4.542 0.022 120.66 7.38 2.58116Cd 15.02 0.37 0.12 4.631 0.028 116.58 7.14 3.26112Sn 16.05 0.26 0.14 4.547 0.019 128.34 5.23 3.32116Sn 16.13 0.2 0.2 4.602 0.023 132.78 4.60 4.60124Sn 14.96 0.1 0.11 4.695 0.030 118.88 3.10 3.26144Sm 15.12 0.3 0.3 4.921 0.027 133.41 6.78 6.78208Pb 14 0.2 0.2 5.534 0.032 144.66 5.82 5.82
Experimental KA
100
110
120
130
140
150
160
170
0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31
KA
[MeV
]
A-1/3
KA vs. A-1/3
The Semi-Empirical Mass Formula Binding energy per nucleon as an expansion
with terms based on the factors due to attractive and repulsive forces in the nucleus:
Using a fitting technique and comparing to experimental data on binding energies, scientists have found the values of the constants above.
There is a large amount of precise data to fit so the values for each constant are accurate.
BACVol
CSurf
A13
CCoulZ(Z 1)
A43
CSym (A 2Z)2
A2 Cp
A74
...
KA Expansion
The expansion for the compressibility constant is analogous with the empirical mass formula:
Also has parameters based on their attractive and repulsive forces in the nucleus
A statistical fit can be used to find the parameters of the expansion just as they did with the mass formula
KA KVol KSurf
A13
KCurve
A23
KSym KSS
A13
N ZA
2
KcoulZ 2
A43
...
Finding the Parameters
Each of the parameters were found by a method of least squares fit: minimizing reduced chi squared values.
Utilizing Microsoft Excel’s solver tool, the experimental data was fitted.
Also using solver, the errors on each parameter were calculated.
2 1NData NParam
KAExp KA
Theory
2
Calculated KA Expansion Constants
σ of KA is measuredNumber of Parameters Kvol ± Ksurf ± Kcurv ± Ksym ± Kss ± Kcoul ± Χ2
4 ‐82 10 555 44 0 0 489 380 0 0 19 2 4.255 ‐269 8 950 37 0 0 ‐2191 320 15140 1600 42 2 3.346 ‐2036 7 11313 34 ‐17293 150 1062 290 11197 1440 114 2 3.16
σ of KA is 104 128 11 2 48 0 0 0.01 606 0 0 0.01 3 1.655 119 12 23 50 0 0 2 520 807 2551 0.5 2 1.946 ‐1542 9 8852 39 ‐13796 159 ‐0.0005 421 12455 2075 88 2 1.45
Fit to microscopic theory gives: Kvol= 240 ± 20 MeV
Conclusions
More data points and better accuracy are needed to produce a better fit
EGMR for smaller nuclei needs to be measured more accurately
Solver Theory is incomplete meaning something
needs to be changed before incompressibility coefficients can be extracted
References[1]D.C. Fuls. Microscopic Description of the Breathing Mode and Nuclear Incompressibility (2005) [2]I. Angeli. Atomic Data and Nuclear Data Tables 87 (2004) 185-206. [3] Y. Tokimoto, Y-W Lui, H. L. Clark, B. John, X Chen, and D. H. Youngblood. Phys. Rev. C 74, 044308 (2006) [4]Y-W Luiet al. Phys Rev. C 83, 044327 (2011)[5]Y-W Lui et al. Phys. Rev. C 69, 034611 (2004)[6] Y-W Lui et al. Phys Rev. C 70, 014307 (2004) [7]D. H. Youngblood et al. Phys. Rev. C 69, 054312 (2004)[8]D. H. Youngblood et al. Phys. Rev. C 63, 067301 (2001) [9] Y-W Lui et al. Phys. Rev. C 73, 014314 (2006)[10] D.H. Youngblood et al. Phys. Rev. C 69, 034315 (2004)
Acknowledgements
Thanks to Dr. Sherry Yennello, Larry May and Leslie Spiekes for organizing the REU program and keeping it running smoothly this summer. Special thanks to Dr. Shalom Shlomo and Mason Anders for all of their help with my research. This program was funded by the National Science Foundation and the Department of Energy.