A Measurement of the 12C(n,2n)11C Cross-Section Needed for an Inertial

Confinement Fusion Diagnostic

Garrett Hartshaw, Ian Love, Mark Yuly

Department of Physics

Houghton College

Inertial Confinement Fusion (ICF)

Credit: Lawrence Livermore National Laboratory

The ρR Parameter

𝜌𝑅 ∝𝑓b

1 − 𝑓b

Determine fraction of fuel burned, 𝑓b

Credit: Lawrence Livermore National Laboratory

ICF Reactions

D + T → 𝛼 + 𝑛(14.1 MeV)

𝑛 + D or T → 𝑛′ + D′(or T′)

D′ + T or D + T′ → 𝛼 + 𝑛′′(12 − 30 MeV)

D

T

α

D

T α

n

D’

n’

n’’

Why Tertiary Neutrons

12C(n,2n)11C Reaction and Decay

𝑛′′ + C12 → 2𝑛 + C11

C11 → B11 + 𝑒+ + 𝜈𝑒

𝑒+ + 𝑒− → 2𝛾(511 keV)

n’’

12C

11C

n

n

11B

e+

νe

e-

γ

γ

Cross Sections

Likelihood of a reaction occurring

𝜎𝑛,2𝑛 ∝𝑁 𝐶11

𝑁𝑛"

Measurements for Cross Section Determination

𝜎𝑛,2𝑛 ∝𝑁 𝐶11

𝑁𝑛" 𝑁𝑛" ∝ 𝑁𝑝

Experiment at Ohio University

Improvements on prior experiments

• Ability to determine incident neutron flux

• Monoenergetic neutrons

• Proton identification

• Positron decay coincidence

SRI 2012 Experimental Design

• Design focused on use of minimal material.

• 2000μm E detector utilized.

SRI 2013 Experimental Design

Proton Telescope

Counting Station

• Pairs of NaI detectors in coincidence are used to count 11C decays in the target and shielding graphite.

NaI Efficiency Apparatus

NaI Detector Efficiency

0%

5%

10%

15%

20%

25%

30%

0 2 4 6 8 10 12 14 16 18

Ab

solu

te F

ull-

Pe

ak E

ffic

ien

cy

Distance (cm)

Measured efficiency (corrected for 1275keV)

Measured Absolute Full-Peak Efficiency

ORTEC value

Calculated Efficiency (P-to-T = 0.54,norm. to 10 cm )

Calculated Efficiency (P-to-T = 0.63,Tsoulfanidis p. 395)

S. Yalcin et al, Appl Radiation andIsotopes 65 1179 (2007) (norm. to 10 cm)

Heath, Scintillation Spectrometry, 1968(norm to 10 cm)

Monte Carlo full-peak, norm to 10 cm (P-to-T=0.71)

Monte Carlo full-peak (P-to-T=0.63)

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

5 7 9 11 13 15

Cross-Section Calculation

𝑑𝑁 𝐶11

𝑑𝑡= 𝜎𝑛2𝑛𝑁𝑛𝑇𝐶 − 𝜆𝑁 𝐶11

𝜎𝑛2𝑛 =𝑁 𝐶11 𝜆

𝑁𝑛𝑇𝐶(1 − 𝑒−𝜆 𝑡)

=𝑁 𝐶11 𝜆

𝑇𝐶(1 − 𝑒−𝜆 𝑡)

𝑁𝑝𝑁𝑛

1

𝑁𝑝

11C Activation Count, 𝑁C11

𝑁0𝑒−λ𝑡 + 𝐴

𝑁11C =𝑁0𝑒

λ𝑡trans

λ ∙ efficiency

Proton Flux, 𝑁p

𝑁p =𝑁p,fg

𝑡live,fg−𝑁p,bg

𝑡live,bg

Naïve Method 𝑁𝑝 = 𝜎𝑛𝑝 0° 𝑁𝐶𝐻2𝑇𝐻 Ω𝑑𝑒𝑡

Tritium

12C

CH2

Np 𝑁𝐶𝐻2

Fd

Tt At

Ω𝐶𝐻2

Ω 𝐶12

Ω𝑑𝑒𝑡

Ed

Detector

𝑁𝐶𝐻2𝑁𝑝

=1

𝜎𝑛𝑝 0° 𝑇𝐶𝐻2Ω𝑑𝑒𝑡

𝑁 𝐶12

Analysis Goals

• Extended sources

• Cross-sections depend on energy and angle

• Collimation by graphite target

Correction 1 – Extended Targets

𝑁𝐶𝐻2 = 𝜎𝑑𝑡 0° 𝐹𝑑 𝑇𝑡 𝑟1𝑑𝑟1𝑑𝜃1cos𝜙1

𝑅12 𝑟2𝑑𝑟2𝑑𝜃2

𝑅𝑡

0

2𝜋

0

𝑅𝐶𝐻2

0

2𝜋

0

𝑁𝑝 = 𝜎𝑑𝑡 0° 𝜎𝑛𝑝 0° 𝐹𝑑 𝑇𝑡 𝑇𝐻cos𝜙1

𝑟1𝑑𝑟1𝑑𝜃1 cos𝜙1

𝑅12 𝑟2𝑑𝑟2𝑑𝜃2

cos𝜙2

𝑅22 𝑟3𝑑𝑟3𝑑𝜃3

𝑅𝑡

0

2𝜋

0

𝑅𝐶𝐻2

0

2𝜋

0

𝑅𝑑

0

2𝜋

0

𝑁𝐶𝐻2𝑁𝑝

=1

𝜎𝑛𝑝(0°)𝑇𝐶𝐻2Ω𝑑𝑒𝑡

Correction 2 – Angular Dependence

𝑁𝐶𝐻2 = 𝜎𝑑𝑡 𝜙1 𝐹𝑑 𝑇𝑡 𝑟1𝑑𝑟1𝑑𝜃1cos𝜙1

𝑅12 𝑟2𝑑𝑟2𝑑𝜃2

𝑅𝑡

0

2𝜋

0

𝑅𝐶𝐻2

0

2𝜋

0

𝑁𝑝 = 𝜎𝑑𝑡 𝜙1 𝜎𝑛𝑝 𝜓, 𝐸𝑛(𝜙1) 𝐹𝑑 𝑇𝑡 𝑇𝐻cos𝜙1

𝑟1𝑑𝑟1𝑑𝜃1 cos𝜙1

𝑅12 𝑟2𝑑𝑟2𝑑𝜃2

cos𝜙2

𝑅22 𝑟3𝑑𝑟3𝑑𝜃3

𝑅𝑡

0

2𝜋

0

𝑅𝐶𝐻2

0

2𝜋

0

𝑅𝑑

0

2𝜋

0

0

5

10

15

20

25

30

0 5 10 15

Cro

ss-S

ect

ion

(m

b)

Angle (degree)

σdt (Ed = 9.07 MeV)

0

20

40

60

80

100

120

140

0 5 10 15 20

Cro

ss-S

ect

ion

(m

b)

Angle (degree)

σnp (En = 26.4 MeV)

Correction 2 – Angular Dependence

0 5 10 15

Nu

mb

er

of

Ne

utr

on

s, N

n

DT Scattering Angle, ϕ1 (degree)

Neutrons vs Scattering Angle Nn_CH2

Nn_12C

0 5 10 15 20

Nu

mb

er

of

Pro

ton

s, N

p

NP Scattering Angle, ψ (degree)

Protons vs. Scattering Angle

Correction 3 – Graphite Effects

Tritium CH2

Detector φ1

Tt

φ2 ψ

TH

R1 R2

(r1,θ1)

(r2,θ2)

(r3,θ3)

Fd

zf

zb

Ri

12C

𝑁𝑝 =

0, 𝑟 𝑧𝑓 ≥ 𝑅𝑖 𝑜𝑟 𝑟 𝑧𝑏 ≥ 𝑅𝑖

𝜎𝑑𝑡 𝜙1 𝜎𝑛𝑝 𝜓, 𝐸𝑛(𝜙1) 𝐹𝑑𝑇𝑡 𝑇𝐻cos𝜙1

𝑟1𝑑𝑟1𝑑𝜃1 cos𝜙1

𝑅12 𝑟2𝑑𝑟2𝑑𝜃2

cos𝜙2

𝑅22 𝑟3𝑑𝑟3𝑑𝜃3

𝑅𝑡

0

2𝜋

0

𝑅𝐶𝐻2

0

2𝜋

0

𝑅𝑑

0

2𝜋

0

Correction 3 – Graphite Effects

0.920

0.940

0.960

0.980

1.000

1.020

1.040

1.060

1.080

1.100

3.07 4.07 5.07 6.07 7.07 8.07 9.07

Co

rre

ctio

n F

acto

r

Deuteron Energy (MeV)

Comparison to Naïve Approach

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.07 4.07 5.07 6.07 7.07 8.07 9.07

Nn/N

p

Deuteron Energy (MeV)

Neutron to Proton Ratio

CH2

12C

Future Plans and Current Goals

• Continue work on NaI detector efficiency.

• Conduct experiment at energies beyond 26.4 MeV

• Additional corrections?

Collaborating Institutions

Resources

• [1] D.R. Welch, H. Kislev, and G.H. Miley, Rev. Sci. Intrum. 59, 4 pp. 612 (1988).

• [2] R. Town, LLE Review, 69, pp. 46 (1996)

• [3] A.A. Harms, K.F. Schoepf, G.H. Miley, and D.R. Kingdon, Principles of fusion energy: An introduction to fusion energy for students of science and engineering (World Scientific Pub Co Inc, Hackensack, NJ, 2000), pp. 191-194

• [4] S. Skupsky and S. Kacenjar, J. Appl. Phys.52, 4, pp. 2608-2613 (1981)

• [5] J.E. Brolley Jr. et al., Phys. Rev. 88, pp. (1952)

• [6] O.D. Brill et al., Dok. Akad. Nauk SSSR, 6, pp. (1961)

• [7] P.J. Dimbylow, Phys. Med. Biol. 25, pp. (1980)

• [8] B. Anders et al., Zeit. Phys. A. 301, pp. (1981)

• [9] T.S. Soewarsono et al., JAERI Tokai Rep. 92, 27, pp. (1992)

• [10] Y. Uno et al., Nucl. Sci. Eng. 122. 27, pp. (1996)