Fortes, Wood, Oberauer: Neutrino–Earth diffraction
Using neutrino diffraction
to study the Earth’s core
A Dominic Fortes, Ian G Wood and Lothar Oberauer think through the possibilities for an entirely new
– and currently technologically impossible – means of examining the core of the Earth.
T
he various techniques for determining
the internal composition and structure
of a planet depend on a combination of
non-unique remote sensing (e.g. gravity, magnetism), cosmochemical models, and mineralogical data from high-pressure/high-temperature
experiments and calculations. Even for the Earth,
with excellent seismic data, we cannot uniquely
determine the composition of the lower mantle
and core. For the other planets in our solar system the uncertainties are even more severe. Here
we propose an ambitious project to study the
Earth’s interior using the coherent scattering of
neutrinos, and assess its technological and economic viability. In contrast to other conceptual in
situ studies of the Earth’s interior (e.g. Stevenson
2003), the physical principles of our method are
well defined, the investment is kept safely at the
Earth’s surface (or near-Earth space), and there is
a reasonable expectation of being able to measure an extremely useful property – the in situ
crystal structure of mantle and core materials.
With an appropriate radiation source, and a
suitable detector, one ought to be able to observe
a diffraction pattern from the crystalline materials that comprise the bulk of the Earth’s interior,
and the interiors of other terrestrial planets and
icy bodies. The only known particle capable of
passing through so much material without significant attenuation is the neutrino, a virtually
massless subatomic particle produced by nuclear
reactions (Winter 2000). Coherent scattering of
neutrinos from an atomic nucleus via the weak
neutral current interaction,
ν + nucleon → ν + nucleon,
although exceptionally small, is thought to be
partly responsible for the outward transfer of
momentum in supernova explosions (e.g. Freedman et al. 1977). Coherent neutrino scattering
allows for interference of the scattered waves
and the formation of a diffraction pattern
when passed through a crystal lattice. It has
been suggested in previous publications (Placci
and Zavattini 1973, Volkova and Zatsepin
1974, Nedialkov 1981, De Rújula et al. 1983,
Askar’yan 1984, Wilson 1984, Borisov et al.
1986, Tsarev and Chechin 1986, Nicolaidis et al.
1991, Kuo et al. 1995, Jain et al. 1999, Ohlsson
and Winter 2001, Lindner et al. 2003, Reynoso
and Sampayo 2004, Winter 2005) that absorpA&G • October 2006 • Vol. 47 Abstract
We explore the possibility of using
coherent neutrino scattering to measure
the diffraction pattern of crystalline matter
in the deep interior of the Earth (or indeed,
any planet). We show that natural solar
neutrinos are not a suitable radiation
source, and by calculating the structure
factors for the diffraction of neutrinos
from the Earth’s core, we constrain the
characteristics of a suitable artificial
neutrino source. The project is well beyond
our present technological capabilities, but
might be relatively trivial for a sufficiently
advanced civilization, either in Earth’s
future, or elsewhere in the universe.
tion of high-energy neutrinos could be used as a
means of reconstructing the radial density profile of the Earth (so-called “neutrino tomography”); this method, however, is still non-unique,
because any number of substances could fit such
a density profile. We will show firstly that the
natural neutrino flux on the Earth from space
is not a suitable radiation source for diffraction
experiments; we then describe the characteristics
of an ideal artificial neutrino radiation source,
and an appropriate diffraction geometry.
Solar neutrinos as a radiation source
The largest local source of naturally occurring
neutrinos is the Sun, produced as a result of
nuclear fusion reactions in its core (Bahcall and
Pinsonneault 2004); neutrinos are also produced
in a range of astrophysical sources (Weiler 2004).
Because it is not possible to collimate neutrinos,
one must measure the diffraction pattern of the
entire planet at once when using solar neutrinos; several problems with the radiation source,
the sample size, and the detector geometry then
arise. If we consider the Earth as a polycrystalline mixture, its diffraction pattern from solar
neutrinos will be a powder pattern, consisting
of nested diffraction cones centred along the
Sun–Earth line. In order to measure the positions of Bragg peaks from the Earth’s interior,
one needs to scan an appropriate neutrino detector through a range of scattering angles (2θ). For
the highest flux monochromatic solar neutrinos
(7Be neutrinos at 862 keV) the de Broglie wavelength is order 0.01 Å; for this wavelength, the
bulk of the scattering from the Earth’s interior
would be contained in a cone extending only ~3°
either side of the Sun–Earth line, with the strongest peaks having 2θ values of barely 1°. Unless a
detector with superlative directional sensitivity
can be constructed, the diffraction pattern will be
lost in the glare of the incident solar neutrinos.
In order for the Bragg peaks to be sharp, the
sample must subtend a small solid angle from the
point of view of the detector, which therefore has
to be placed far out in space, preferably several
million kilometres away. However, even at the
only viable location in space – the Earth–Sun
system’s L2 point – diffraction peaks from the
upper mantle would still be ~0.5° wide; in addition, this large detector distance would result in
severely reduced flux. Moreover, the very small
Bragg angles will result in the superimposition
of peaks from the whole of the Earth: crust,
mantle and core. The measured diffraction pattern would then be very difficult to interpret,
especially as the peaks would be dispersed by
the large range of pressures to which the sample
is subject. This pressure broadening would be
particularly marked for mantle materials, which
experience the largest radial pressure gradient.
These effects mean that diffraction using solar
neutrinos is not physically realizable. However,
the technique may be possible with artificially
generated neutrinos from some ideal source.
An artificial neutrino source
The most important characteristic of our ideal
neutrino source is that it be distinct from the
astronomical (including solar) neutrino flux, so
that we can measure any pattern free of background. This distinction may be in energy (i.e.
<< 0.5 MeV), in intensity (i.e. >> 1014 m–2 s–1), or
in character (antineutrinos vs neutrinos). For
greatest convenience, the detector should be at
the Earth’s surface and should remain stationary.
We, therefore, employ an energy-dispersive diffraction geometry, which requires our artificial
neutron source to be polychromatic rather than
monochromatic. It also requires excellent energy
resolution in the detectors. Finally, to limit the
problems due to sample size mentioned earlier, it
5.31
Fortes, Wood, Oberauer: Neutrino–Earth diffraction
is important that the incident neutrino beam illuminates relatively small volumes of the Earth’s
interior, of order 106 km3: this also gives us the
potential to study inhomogeneities in the Earth’s
internal structure at a useful length scale. For this
reason, and to overcome natural background,
the beam must be many orders of magnitude
more intense than the solar flux.
Our proposed neutrino diffraction experiment
consists of a nadir-pointing, high-brilliance, parallel-beam, white neutrino source at a fixed point
generating a beam that is passed through the centre of the Earth (figure 1). Antipodal detectors
act as downstream beam monitors, primarily
to normalize the measured diffraction pattern
to the incident spectrum, but also to perform
long-baseline neutrino physics experiments; a
ring of detectors placed along a great circle, at an
angle of 90° to the source, measure the scattered
flux. With a directional sensitivity of ±1° for our
detectors, regions ~200 km across in the Earth’s
core can then be studied. To observe a useful
range of d-spacings (1–3 Å) at a fixed scattering
angle of 2θ = 90°, the neutrino beam must cover
a range of very low energies from ~3–12 keV.
Because the neutrino beam is illuminating a column through the Earth, scattering from shallower depths arrives at the detector at shallower
angles: for example, if one wishes to study the
mantle at a depth of 1500 km, one selects neutrinos arriving in the detectors at 2θ = 53° (or 127°
in backscatter). If the neutrino beam were pulsed,
it would be possible to improve the resolution by
binning the arrival times of neutrinos scattered
from greater depths in the Earth; for example,
a neutrino scattered at 2θ = 53° will arrive in the
detector 10.67 ms before a neutrino scattered at
2θ = 90°, and 32.07 ms before a neutrino backscattered at 2θ = 127°. Coincidence detection,
combining angular and arrival-time discrimination, thus provides good spatial resolution and
a high degree of background suppression. The
resolving power of the instrument is limited
by the uncertainty in scattering angle and the
energy resolution of the detector. A directional
sensitivity of ±1° leads to a d-spacing resolution
of ∆d/d ≈ 1.75% at 2θ = 90°; better resolution is
achieved at higher scattering angles; for example,
∆d/d ≈ 0.8% at 2θ = 127°. These values are comparable to instruments such as the POLARIS diffractometer at the ISIS neutron spallation source.
Since ∆E/E = ∆d/d, the energy resolution of the
detector should be no worse than this.
We now calculate the integrated intensity
expected in the strongest Bragg peak from iron
in the Earth’s core. We assume, for convenience
(as the true structure is in some doubt), that it is
body centred cubic.For a fixed scattering angle
(i.e. using polychromatic radiation) and a powder sample, the integrated intensity is given by
(Buras and Gerward 1975),
cosq0 Dq0
Phkl = ji0(l)VN2 ​|Fhkl |​ 2 l4 __________
​ (1)
4sin2q0
5.32
Circumferential array of neutrino detectors at a fixed scattering
angle. The detectors measure the energy dispersion from
volume units at various scattering angles, therefore yielding
diffraction patterns as a function of depth in the Earth
Diffracted beam (actually a cone),
in this case at 2θ = 90° for
scattering from the centre of the
inner core
Diffracting region – a
column 200 km long and
1 cm2 in cross section
Solid iron
inner core
Incident pulsed
neutrino beam
Liquid Fe(S, Si, O)
outer core
Silicate mantle
Silicate lithosphere
Neutrino factory. Synchrotron
with vertical decay tunnel
generating an intense
polychromatic neutrino beam
1: Elements of the neutrino diffraction experiment as a tool for understanding Earth structure.
where Phkl is the total power in a given Bragg
peak (s–1), j is the reflection multiplicity, i0(l) is
the incident flux at wavelength l (cm–2 s–1), V
is the sample volume, N is the number of unit
cells per unit volume, Fhkl is the structure factor,
q0 is the scattering angle, and ∆q0 is the angular
width of the diffracted beam. The structure factor for the strongest reflection of a body centred
cubic crystal, 011, including the effect of thermal
motion, is
2
2
​|F011 | ​2 = 4b2 e –2Bsin è/ë (2)
The coherent neutrino scattering length, b, and
coherent scattering cross section, σ, are related
by σ = 0.41 × 10–42N2E2, where N is the number
of neutrons in the scattering nucleus, and E is the
neutrino energy in MeV. The Debye–Waller factor
B = 8π 2<u>2, where u is the mean square vibrational amplitude. Therefore, for iron (N = 30) illuminated with 12 keV neutrinos, σ = 5.3 × 10–44 cm2
and b = 6.5 × 10–23 cm. Under core conditions
(pressure ≈ 360 GPa, temperature ≈ 6000 K),
N2 = 5.102 × 1045 cm–3 (7 Å3 per atom) and
<u>2 ≈ 0.3 Å2 (B ≈ 24 Å2; L Vočadlo pers. comm.).
Note that this is by far the worst scenario we can
imagine, crystallographically; the considerable
thermal motion of the iron atoms diminishes very
severely the diffracted power in each Bragg reflection. Diffraction from mantle materials, at much
lower temperatures, is not so greatly affected.
Equation 1 is integrated with respect to wavelength (with i0(λ) assumed constant) from
0–10 Å, and the scattering volume is assumed to
be a cylinder 200 km long with a cross sectional
area of 1 cm2 (V = 2 × 107 cm3). If we suppose,
optimistically, that we can construct 100 km of
detectors around the 40 000 km circumference of
the Earth at 2θ = 90°, we will detect ~2.5 × 10–3
of the diffracted neutrino power; for a capture
cross-section of order 10–44 cm2, and a detector thickness of 10 m, the capture probability
is ~1 × 10–19. In order to detect one neutrino
per day (1.157 × 10–5 s–1), we require P011 to be
~4.63 × 1016 s–1, necessitating an incident flux,
i0 ≈ 1.77 × 1029 cm–2 s–1. If it were possible to cover
the entire circumference with detectors (desirable also to allow measurement of any preferred
orientation of the crystals), a count-rate of 400
neutrinos per day would result; by increasing
the counting time to a decade, the incident flux
required immediately drops enormously, becoming 4.84 × 1025 cm–2 s–1. To first order, beams of
neutrinos with an average energy of ~10 keV and
fluxes of 1.77 × 1029 cm–2 s–1 (4.84 × 1025 cm–2 s–1),
illuminating a spot of area 1 cm2, represent a total
power of 2.84 × 1014 W (7.77 × 1010 W).
The limitations on the spatial resolution of the
technique are controlled to a large extent by the
neutrino flux one can deliver and the amount
of time one wishes to spend counting neutrinos
(since smaller volume elements – voxels – simply
scatter less in a given time). One can envisage a
data collection strategy in which observations are
binned in arbitrarily small voxels from the outset, the signal-to-noise ratio in each growing over
time. Initially, reasonable signal strength would
only be achieved by summing over all measured
voxels, generating a “powder pattern”of the
entire mantle. As the number of neutrino counts
accumulates, the reduction in noise will allow
data to be summed usefully over progressively
smaller voxels, and so permit quantification of
finer-scale heterogeneities.
Feasibility of proposed experiment
Purely in terms of being able to produce the necessary quantities of energy, accumulate the necesA&G • October 2006 • Vol. 47
Fortes, Wood, Oberauer: Neutrino–Earth diffraction
sary materials, build and manage the project, and
maintain it for the (possibly) decades required to
achieve a result, then the proposed experiment
is certainly formidable. The energy requirement of 2.84 × 1014 W is roughly 20 times the
power consumption of mankind (International
Energy Agency 2003); the lower requirement
of 7.77 × 1010 W is roughly 1/200 of the global
power usage. However, this does not include the
power needed to run the detectors and the ancilliary equipment related to the experiment, and
neither does it include efficiency losses (probably very significant). While the numbers are
large in comparison to our energy generation
capacity, they are very much smaller than the
radiant energy of the Sun that is intercepted by
the Earth, ~1.7 × 1017 W. Trends in energy usage
by advanced civilizations were first addressed by
Kardashev (1964): a Kardashev Type II civilization – able to draw upon the energy of a whole
star (4 × 1026 W) – would easily be able to muster
the power to carry out this experiment.
The vast quantity of material needed to construct the system, principally the detectors, is
the second major difficulty. Low-energy charged
current neutrino detectors typically employ
large-volume liquid scintillators – either water
or another organic solvent – doped (10–20 wt%)
with metals such as indium and/or ytterbium (e.g.
LENS, Lasserr 2002), gadolinium (e.g. SIREN,
Lightfoot 2004), or molybdenum (e.g. MOON,
Hazama et al. 2005). Similar large-volume scintillation detectors based on liquefied xenon (e.g.
XMASS, Namba 2005) or neon (e.g. CLEAN,
Nikkel et al. 2005) measure the occurrence of
elastic scattering events. The option of covering 100 km of the Earth’s circumference with
such detectors would require approximately
10 million tons of either In, Yb, Gd or Mo. Of
these, Gd is the most abundant in the Earth’s
crust (5–7 ppm) and In is the least abundant
(~0.1 ppm); we would need just 5 × 10–8 of the
total crustal inventory of gadolinium (assuming an average crustal thickness of 30 km), or
2 × 10–6 of the indium inventory, to manufacture
our detectors. Molybdenum is the cheapest of
the four metals; if we simply take the present
market value (~US$10 per 100 g) then the cost of
the necessary metal would be ~$1 × 1012, which
is 2.25% of the global gross domestic product
(International Monetary Fund 2006). To purchase the same quantity of ytterbium would cost
~$5 × 1013, comparable with current global GDP.
Note that we have not considered the costs of the
liquid component for the scintillators or the costs
of the tanks, electronics, or even the real estate.
The difficulties highlighted are such as might be
addressed only by a significantly more advanced
(and rich) civilization than our own, which we
might also expect to be an accomplished spacefaring society. In this case, there is nothing to
prevent us from considering the benefits of
moving the entire operation into space. In fact,
A&G • October 2006 • Vol. 47 certain problems are diminished in low Earth
orbit (LEO). Placing the neutrino source in LEO
effectively provides radiation shielding (the
atmosphere) to protect the terrestrial populace,
and an environment where energy generation is
much more straightforward, using either solar
power or artificial nuclear fusion. Placing the
detectors in LEO keeps the project from ruining
a very large area of the Earth’s surface (in terms
of the detector footprint and the mining needed
for the materials), and also allows us to consider
seriously having a complete circumferential ring
– or more – of detectors, possibly mounted on
a “space elevator” style filament (e.g. Edwards
2000). Of course, this increases the mass requirement, but it is much more practical to source
material in space (the Moon or asteroid belt)
than raise it from the Earth’s gravity well.
It is already possible to generate extremely
intense beams of artificial neutrinos using synchrotrons; these so-called super-beams and betabeams are passed along chords through the Earth
to investigate the fundamental properties of neutrinos themselves (Dydak 2003). Generating a
neutrino beam with a peak flux at sub-20 keV
energies needs the decay source to be moving
away from the irradiated area at very high speeds
to provide the requisite Doppler shift. This is
beyond current technology, as is the high energy
resolution required for the detectors. Given that
the energetic and logistical problems can only be
addressed by a more advanced civilization, then
these technological issues become secondary; it is
probably only a matter of time (rather than novel
physics) before it is possible to produce very high
intensity keV-range white neutrino beams for use
in geophysical prospecting.
Interest within the particle physics community
is, however, driving the development of small
low-energy neutrino detectors with the specific
goal of being able to observe – for the first time
– the coherent scattering of neutrinos (Collar
2001, Barbeau et al. 2003). The experimental
observation of coherent neutrino-nucleus scattering will place constraints on the neutrino
magnetic moment and provide a sensitive test of
non-standard particle physics models (Barranco
et al. 2005). Large-scale planetary neutrino diffraction experiments, of the kind described here,
may one day be motivated by the drive to better
understand the structure of the universe, with
more clarity over the structure of the Earth’s interior being a secondary (but welcome) bonus.
Summary
Neutrinos, as the only subatomic particles capable of passing right through the Earth, are the
sole means of extracting crystallographic information on the substances in Earth’s interior, short
of taking the planet to pieces. Our experiment is
technologically ambitious but, apart from greatly
improving measurements of neutrino properties, could also allow us to solve long-standing
problems relating to the structure and possible
anisotropy of the inner core, study the D” layer
at the core–mantle boundary, and investigate
small-scale heterogeneities in the mantle. Furthermore, refinement of the Debye–Waller factor
from the diffraction data will allow (by reference to anharmonic models) the Earth’s radial
temperature profile to be inferred independently
of the geophysical models used today. Neutrino
diffraction has the potential to revolutionize our
understanding of planetary physics in much the
same way that helioseismology has transformed
our view of the Sun’s interior. ●
A Dominic Fortes ([email protected]);
Ian G Wood, Dept of Earth Sciences, University
College London, Gower Street, London, WC1E
6BT, UK; and Lothar Oberauer, Physik Department
E15, Technische Universitat München, JamesFranck Strasse 1, D-85747, Garching, Germany.
Acknowledgments. We wish to thank Lidunka
Vočadlo and Kevin Knight for useful discussions
and comments on the paper. Hans Henrik
Andersen provided some very useful criticisms of
the manuscript. ADF is funded by PPARC, grant
number PPA/P/S/2003/00247.
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