Absorption and Scattering of Acoustic Waves in NaCl

P. B. Price, U. C. Berkeley, February 3, 2005

Acoustic Scattering

In ice, it occurs at grain boundaries. In a salt dome it also occurs at cracks, layers of clay,anhydrite, and liquid inclusions.

Acoustic Absorptionoccurs by relaxation mechanisms. In NaCl the thermal phonon gas extracts energy from acoustic phonons via anharmonic interactions.

Conversion of ionization energy into acoustic energy

ocean, 15º ice, -51º NaCl, 30º

<vL> [m s-1] 1530 3920 4560

<vS> [m s-1] n/a 1995 2610

[m3 m-3 K-1] 25.5x10-5 12.5x10-5 11.6x10-5

CP [J kg-1 K-1] 3900 1720 839

<vL>2 /CP 0.153 1.12 2.87

d ≈ RMoliere 0.1 m 0.1 m 0.054 m

L≈3Xoln(Eo/Ec)1/2 10.3 m 10.3 m 3.43m

fpeak ≈ vL / 2d 7.7 kHz 20 kHz 42 kHz

disk = d/L 0.5º 0.5º 0.9º

Scattering of acoustic wave at grain boundaries(speed depends on crystallographic direction)

Rayleigh regime (/4πa > 1) Stochastic regime (0.5 < /4πa < 1)

Geometric regime (/4πa < 0.5)

(a = grain radius for a polycrystalline medium)

Acoustic properties depend on elastic constants, cij

Ice (hexagonal): c11, c12, c13, c33, c44

NaCl (cubic): c11, c12, c44

For an isotropic solid, c11 - c12 - 2 c44 = 0 no scattering at grain boundaries.

(For glass, = 0; for NaCl, = 0.111)

αscatt=43πa

3

375πk4 (c11−c12−2c44)

2

c112 1+

32

vLvS

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

5⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

=1.65×10−4 a1cm

⎛ ⎝ ⎜

⎞ ⎠ ⎟

3 f104

⎛ ⎝ ⎜

⎞ ⎠ ⎟

4

m−1

αscatt=4

525

c11−c12 −2c44( )2

c112 k2⟨a⟩

=7.21×10−4 ⟨a⟩1cm

⎛

⎝ ⎜

⎞

⎠ ⎟

f104

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

m−1

where k≡2πfvL

; ⟨a⟩= mean grain radius

Grain boundary scattering in Rayleigh regime (cubic):

Grain boundary scattering in stochastic regime (cubic):

(-51ºC)

1 South Pole ice

In top 600 m, grain diameter ≈ 0.2 cm.

• at 10 kHz, acoustic

scattering length≈ 800 km;

• at 30 kHz, acoustic scattering length

≈ 10 km

In ice with a random distribution of c-axes, scattering is a factor

2.7 higher than shown.

0.4 cm 0.2 cmdiam

For ice, acoustic absorption is due to molecular reorientations,which dominates over other modes.

In South Pole ice at -51ºC with random c-axis distribution,we predictabs ≈ 1.4x10-4 m-1

and abs ≈ 7 km

indep. of frequency,unlike most situations

longitudinal waves c-axis

abs

[

Salt evaporite beds

WIPP repository contains salt beds < 100 m thick with >1% water (mostly in liquid inclusions) and separated vertically by thin beds of clay, silt, and anhydrite (CaSO4).

Salt domes

In Louisiana, several mines have >99% NaCl, are very dry (only 2 to 40 ppm water), and have small (7.5 mm) grain size, which is good.

Section through polycrystalline halite from salt dome. Most grainshave recrystallized, and scattering can occur at their boundaries.

White lines delineate subgrain boundaries with small misorientation

Grain boundaries (up to 90º) Subgrain boundaries (~1º)

Liquid inclusions in salt domes

They scatter acoustic waves but are infrequent.

f [Hz]dp/

df [

g cm

-1 s

-1]

10-8

3x10-8

S.P.Stationnoise?

NaCl

ice

Acoustic scattering and absorption in South Pole ice and NaCl

scatt abs

104 Hz 3x104 Hz 104 Hz 3x104 Hz

Ice (D=0.2 cm) 1650 km 20 km 7 km† 7 km NaCl (D=2 cm) 6 km 0.15 km† 3x104 km 3300 kmNaCl (D=0.75 cm)* 120 km 1.4 km† 3x104 km 3300

km*Measured for salt in Avery Island dome†Dominant contribution to attenuation at peak frequency

1. But salt domes may have clay, liquid inclusions, and other minerals, which shorten scattering and absorption lengths.

2. Scattering length in salt is shorter than in South Pole ice because grain size is larger.

3. Absorption length in ideal salt is far longer than in ice, but will be reduced by heterogeneities.

4. Must measure acoustic scatt and abs over >1km in salt dome!

Optical detection

• NaCl has absorption length >100mfor wavelength >350 nm salt dome may be useful as an optical Cerenkov detector. • Isotropy of refractive index in NaCl no scattering at grain boundaries. • To calculate scattering, measure concentration of mineral inclusions and other heterogeneities.

Bergstrom-Pricemodel