A Plan of Warm-Dense-Matter Experiment Using Precompressed Hydrogen Targets

J. Hasegawa, Y. Oguri, and K. Horioka (Tokyo Tech)

K. Kikuchi and T. Sasaki (Nagaoka Univ. of Tech.)

S. Kawata (Utsunomiya Univ.)

K. Takayama (KEK)

A warm dense matter experiment is planned using intense heavy-ion bunches from KEK Digital Accelerator.

• KEK Digital Accelerator (KEK-DA):

– A heavy ion super bunch is accelerated and confined by induction voltages.

– A wide range of ion species, even clusters, can be accelerated.

– The super bunch supplies a specific energy deposition of kJ/g on target.

• A WDM experiment is planned as one of the applications of KEK-DA.

– The high-energy beam from KEK-DA allows uniform bulk target heating and well-defined energy deposition.

– Measurement of hydrogen equation of state under ~200 GPa, ~6000 K requires not only heating but also compression of the hydrogen target.

KEK Digital Accelerator (former KEK-booster)

Dense hydrogen EOS is important to understand the structure of a giant-gas planet, such as Jupiter.

100-300 GPa0.5 -1 eV

To access off-Hugoniot regimes, a quasi-isentropic pre-compression scheme were proposed.

• Advantages:– The tailored driving force allows shock-free, quasi-isentropic compression.– Instantaneous bulk heating by an intense beam bunch achieves well-

defined uniform energy deposition.– The large aspect ratio of the cylindrical target guarantees one-dimensional

treatment.

Metal liner

Electro-magnetic force

Concept of the beam heating of isentropicallypre-compressed target:

Heavy ionbunch

Hydrogen EOS (SESAME5251)

Target parameter

The isentrope almost coincides with the cold curve (isotherm at T=0).

• Pressure of a solid:

• Grüneisen coefficient for solid hydrogen was evaluated from SESAME cold curve data.

• An isentropic relation between the pressure and volume:

p =pc + pT =pc + Γ(V)cvTV

: Grüneisen coefficient

p =pc + pT0

V0

V⎛⎝⎜

⎞⎠⎟

Γ(V )+1

pT0=Γ0cvT0 /V0 , T0 =10K

Γ(V )

Γ(V ) = −2

3−V

2

(d 2 pc / dV 2 )

(dpc / dV )

Equations for scale-invariant similarity solutions were solved.

dU

d lnξ+ (U −1)

dlnGdlnξ

+ (n+κ )U =0

(U −1)dU

dlnξ+ (C2 / γ)

dln(GC2 )dlnξ

+ [U(U −1 /α) + (κ + 2)C2 / γ] =0

(U −1)dln(Cμ / G)

dlnξ+ [U(μ −κ )−μ /α] =0

u(r, t) =(αr / t)U(ξ)c(r,t) =(αr / t)C(ξ)

ρ(r,t) =ρ0 (r / r0 )κ G(ξ)

ξ =r / r0t / t0

α

Reduced fluid equations:

Similarity solutions invariant:

pc =A(ρ −ρ0 )γ ≈Aργ

(Self-similarity coordinate)

Isentropic relation

pc ∝ ρ2.42

Self-similar solution of uniform compression

• Uniform density at rest:

• Isentropy:

• Cylindrical geometry:

κ =0

ε = 0, α = 1

n = 2

M=0.4 M=0.5 M=0.6

M: Mach number before stagnation

C=3.6 C=5.6 C=10

Driving current waveform required for isentropic compression was determined from the trajectory of outermost fluid particle.

• Mechanical power acting on solid hydrogen surface:

• Magnetic pressure induced by current:

• Required driving current:

• After rising gradually, the driving current increases more rapidly particularly for higher compression ratios.

• A power supply based on pulse forming network is suitable for pulse tailoring.

P(t) =2πR(t)p(t)u(t)

pM (t) =

B2

2μ0

=μ0 I (t)

2

8π 2R(t)2Q B=

μ0 I2πR

p(t) =pM (t) → I (t) ∝ R(t) p(t)

A typical example of isentropic compression and driving current waveform.

• Target conditions:

– Final target radius = 1 mm

– Compression ratio = 5.6

• Normalization factors:

– Initial target radius:

– Imploding time:

• A peak current of ~400-500 kA and a rise time of 1.5 µs is required to implode the target.

R0 =Rf ρ / ρ0

t0 =R0

cc =

Kρ

: Sound velocity

Required peak current and total energy of the driving circuit was estimated for various target sizes and compression ratios.

• Restricting conditions:

– Target radius after compression is defined by the beam radius on target achievable in the final beam focusing system. (0.1 mm ~10 mm)

– Compression factor is determined by required final pressure. (1.8 ~ 10)

– Final beam radius less than 1 mm is realistic; larger target requires MA driving current and MJ energy.

The KEK booster-PS is now being reconstructed as KEK Digital Accelerator by installing induction cavities.

ApplicationsInduction cavities

Combined functionmagnet

Ion source

200kV H.V. Terminalfor 9.4 GHz ECR ion source

Machine parametersBending radius 3.3 m

Ring circumference 37.7 m

Maximum flux density 1.1 T

Accele. voltage/turn 3.24 kV

Repition rate 10 Hz

Betatron tune x/y 2.1/2.3

near Injection

5 msec

10 msec

15 msec

20 msec

30 msec

40 msec

50 msec

10 Hz operation

Numerical simulation on induction acceleration and confinement of Ar18+.

by Tanuja Dixit

t

B(t)

Acceleration region

Injection

Extraction

Vac=ρC(dB/dt)

100 msec

0.84 T

Expected beam intensity was evaluated: 109 to 1011 particles per bunch is available depending on projectile Z.

T =Amc2 1

1−β 2−1

⎛

⎝⎜

⎞

⎠⎟

Final kinetic energy:

The space-charge-limited number of ions per bunch:

NiN p

≈AZ3 ⋅

Vi

Vp

⋅Bf( )

AIA

Bf( )RF

Bmax = 0.87 T, = 3.3 m

Np = 3ξ1012, Vi = 200 kV, Vp = 40 MV, (Bf)AIA = 0.7, (Bf)RF = 0.3

A specific energy deposition more than 100 kJ/g is available with a beam spot radius less than 0.2 mm.

• Specific energy deposition was estimated from SRIM stopping data.

• The heavier projectile can supply higher specific energy deposition.

• The minimum requirement for the specific energy deposition is about 100 kJ/g.

• A beam spot radius less than 0.2 mm is preferable.

0

200

400

600

800

1000

0 20 40 60 80 100

Atomic Number

Target : Hydrogen (solid)

12C

40Ar

238U

132Xe

84Kr

197Au

56Fe

20Ne

R = 0.1 mm

R = 0.2 mm

R = 0.4 mm

Summary

• An accelerator-driven WDM experiment using a quasi-isentropically compressed target was proposed to investigate material properties in the off-Hugoniot regime.

• Scale-invariant self similar analysis was used to evaluate required experimental conditions, such as target size, driving power, and current waveforms.

• A final target radius after compression should be less than 1 mm to design actually adoptable power supplies.

• To examine the feasibility of this scheme more in detail, MHD simulations coupled with the external driving circuit will be performed.

• Beam energy deposition by a heavy ion beam bunch from KEK-DA was also estimated. A specific energy deposition more than 100kJ/g will be available, which is enough to reach the required WDM conditions.

Methods Max. pressure Features

Diamond anvil ~ 60 GPa Static, easy control, easy measurement

Laser-driven shock wave ~ TPa Dynamic, extremely high pressure, compact equipment

Beam-driven shock wave ~100 GPa? Dynamic, well-defined shocks, bulk heating, Controllable temporal evolution.

Appendix: a concept of beam-induced high pressure field experiments.

Temperature distribution in depth

Quasi-uniform energy deposition profile

Beam bunch

Beam-driven shock Test material

High-pressure field

Beam-heated target

Appendix: an extremely high pressure field is induced in the material by intense beam irradiation.

With Pb tamper

Without Pb tamper

Controllable pressure history

Pressure in Al

1D hydro code (with radiation transport):