Note: Descriptions are shown in the official language in which they were submitted.
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This invention relates to the measurement of
gravitational fields, particularly to gravity
gradiometry, and more particularly to a method for
measuring absolutely off-diagonal components of the
gravity gradient tensor.
The gravity gradient tensor is a two-dimensional matrix
of the second partial derivatives of a gravitational
potential, Y, with respect to the Cartesian co-
ordinates, x, y, z, of some arbitrary reference frame.
It represents how the gravity vector itself in each of
these directions varies along the axes.
Accurate absolute measurements of the components of the
gravity gradient tensor I'i~ - c72i jV ( ij - x, y, z ) , taken at
some local coordinate frame OXYZ are very important to
progress in the fields of geological prospecting,
mapping of the Earth's gravitational field, and space,
sea and underwater navigation.
A method of absolute measurement of gravity gradient
tensor components was invented first by Baron Roland von
Eotvos as early as 1890, utilising a torsion balance
with proof masses hung at different heights from a
horizontal beam suspended by a fine filament. The
gravity gradients give rise to differential forces being
H applied to the masses which result in a torque being
exerted on the beam, and thus to angular deflection of
~ the masses which can be detected with an appropriate
sensor. A sensitivity of about 1 E (1 E = 1 Eotv~is = 10'9
s'2) can be reached but measurement requires several
hours at a single position due to the necessity to
recalculate the gravity gradient components from at
least 5 independent measurements of an angular
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deflection each with a different azimuth angle.
Practical devices, which have been built in accordance
with this basic principle, are large in size and have
low environmental noise immunity, thus requiring
specially prepared conditions for measurements which
excludes any possibility of using them on a moving
carrier.
A method for absolute measurement of gravity gradient
tensor components which enhances the above method was
invented by Forward in the middle of the sixties (see US
patents 3,722,284 (Forward et al) and 3,769,840
(Hansen). The method comprises mounting both a dumbbell
oscillator and a displacement sensor on a platform which
is in uniform horizontal rotation with some frequency L2
about the axis of the torsional filament. The dumbbell
then moves in forced oscillation with double the
rotational frequency, whilst many of the error sources
and noise sources are modulated at the rotation
frequency or not modulated (particularly 1/f noise).
The forced oscillation amplitude a.s at a maximum when
the rotation frequency satisfies the resonance condition
2C2=cao, where coo is the angular resonant frequency, and
the oscillator quality factor Q tends to infinity.
Unlike the non-rotating method, this method enables one
to determine rapidly the quantities r~,~, - I',~ and r,~, by
separating the quadrature components of the response
using synchronous detection with a reference signal of
3 0 f requency 2t2 .
The same principles can be directly used, as proposed by
Metzger (see US patent 3,564,921), if one replaces the
dumbbell oscillator with two or more single
accelerometers properly oriented on such a moving
platform. There are no new features of principle in
this solution to compare with the previous one except
that the outputs of the pairs of accelerometers require
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additional balancing.
Devices have been built according to this method, but
they have met more problems than advantages, principally
S because of the need to maintain precisely uniform
rotation and the small displacement measurement with
respect to the rotating frame of reference. The devices
have reached a maximum working accuracy of about a few
tens of Eotvos for a one second measuring interval, and
they are extremely sensitive to environmental
vibrational noise due to their relatively low resonant
frequencies. The technological problems arising in this
case are so difficult to overcome that the existing
developed designs of rotating gravity gradiometers are
so far only at the stage of prototypes whose measurement
accuracy is much lower than the limiting theoretical
estimates.
In a paper by A. Nicolaidis and A. Taramopoulos (I1
Nuovo Cimento, Vol. 107B, N.11, pages 1261-1266,
November 1992), the theoretical motion of a string with
fixed ends under the influence of a plane monochromatic
time-varying gravitational wave is discussed. According
to this document, a string with fixed ends may be
excited to resonance provided certain conditions,
dependent on the length and orientation of the string
and the wavelength of the gravitational wave, are met.
It is suggested that Fourier analysis of the motion of
the string could be used to extract the direction and
energy of the incident wave. The document specifically
- avoids any discussion of the technical implementation of
the theory, but it does suggest that strings a few
metres or a few kilometres long should be used for the
detection of cosmological radiation or gravitational
radiation from a black hole or supernova, as the length
of the string should be comparable to the wavelength of
the gravitational waves. For the theoretical detector
to work requires the gravitational field to oscillate in
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the form of a gravitational wave, which would not be the
case for the gravitational fields associated with massive
bodies such as the Earth.
Superconducting gravity gradiometers are known
(see US patent 4,841,772) utilising a pair or more of
sufficiently separated superconducting accelerometers. Even
after greatly reducing the intrinsic and environmental
thermal noise factor, using stable persistent super-currents
to balance the outputs of the accelerometers, and the most
sensitive displacement sensors based on SQUIDS
(Superconducting Quantum Interference Devices), they cannot
measure the gravity gradient tensor components in their
absolute units because they are incapable of fixing a
position of the accelerometer's proof mass which is free of
all forces. Therefore, only relative displacements of the
proof masses can be measured. Rotating designs of such
superconducting gravity gradiometers are not known.
Patent Abstracts of Japan vol. 009 No. 117 (P-375)
and JP-A-60 050476 disclose a device for measuring the
acceleration due to gravity, wherein a weight is suspended
from a string. A current passing through the string causes
the string to vibrate in the magnetic field of a permanent
magnet. An amplified electrical signal corresponding to
this vibration is fed back to the string and the string
oscillates under self-excitation at a set frequency. The
acceleration due to gravity is measured from this frequency.
It is an object of the present invention to
provide an apparatus for the measurement of gravitational
fields with improved sensitivity, portability and noise
immunity over the above known systems.
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It is a further object of the present invention to
provide a novel apparatus for the absolute measurement of
off-diagonal components of the gravity gradient tensor, in
which the effect of rotation is replaced by parametric
5 interaction between the sensitive element and active force
feed-back connections, whereby enhanced sensitivity and
vibrational noise immunity are attained.
It is another object of the present invention to
provide a simple technological realisation of the above
lb apparatus utilising the advantages of the standard
superconducting techniques which have shown an ability to
reach a maximum sensitivity for mechanical displacement
measurements and to keep intrinsic noise at a minimum level.
According to one aspect the present invention
provides apparatus for the measurement of quasi-static
gravitational fields, comprising: a string composed of
conductive material, fixed at both ends, held under tension
and arranged to carry a current Io; sensing means for
detecting the transverse displacement of said string from an
unperturbated position due to a gravitational field acting
on said string; and means responsive to the detected
displacement for producing an output which is a function of
the gravitational field.
By "string" no particular limitation as to
conductive material or construction is intended. Any
elongate tension element is included which is capable of
being transversely deflected by a gravitational field and of
providing a restoring force.
i i i
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An unperturbated stretched flexible string with
fixed ends forms an absolute straight line in space going
through the points where the ends of the string are fixed.
This line can be identified as one of the axes of the local
coordinate frame, say Z, and the other two axes, X and Y,
are chosen to lie in the transverse (to the string) plane.
Any string deflection from this line is caused by absolute
values of the transverse components of the force per unit
length which is applied to each unit element of the string.
Viewed from another aspect the invention provides
apparatus for measuring off-diagonal components of the
gravity gradient tensor comprising the novel apparatus for
the measurement of quasi-static gravitational fields
described above and means to apply a force per unit length
to said string so that deflection of said string is caused
by absolute values of transverse-to-string components of
said force, in a manner such that deflection of said string
is a combination of said string's natural modes, and the
even said modes are caused only by absolute values of the
components of the gravity gradient in the string direction,
whilst the odd said modes are caused by total acceleration
in the transverse-to-string plane.
According to another aspect the invention provides
a method of measuring absolute off-diagonal components of
the gravity gradient tensor by means of the novel apparatus
wherein the deflection of the string from its unperturbated
position is caused by absolute values of transverse-to-
string components of a force per-unit-length applied to each
unit element of said string in a manner that said deflection
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is a combination of said string's natural modes, and the
even said modes are caused only by absolute values of the
components of the gravity gradient in the string's
direction, whilst the odd said modes are caused by total
acceleration in the transverse-to-string plane.
The string's deflection from its unperturbated
position can be easily detected, by any suitable
displacement sensing device.
The string is formed of conductive, most
preferably superconductive material. When an electric
current flows through the string, a magnetic field
distribution is produced in the transverse plane and along
the string's direction. If the string is made of a
superconducting material, a maximum current can be carried,
and a consequent maximum sensitivity to the deflection can
be reached. A d.c. or an a.c. current may be produced in
the string by incorporating the string into a current-
carrying circuit directly or by an inductive coupling with a
pumping circuit(s), provided that the string forms part of a
closed conducting or superconducting loop. An a.c. current
may be induced in the string, for example by means of one or
more, preferably longitudinally symmetrically positioned,
coils, which may possibly be superconducting. The use of an
a.c. current is advantageous in that it allows synchronous
detection of the output signal.
When the string carries a current, the transverse
magnetic field around the string may interact with other
conductors, or superconductors, by inductive coupling. The
amplitude of the current induced in a conductor adjacent the
string will be directly related to the distance of the
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string from that conductor. Thus, in a preferred embodiment
of the invention one or more fixed pick-up coils are
arranged along the length of the string to act as
displacement sensing means, the current
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induced in each coil being directly related to the
string's displacement from its unperturbated position.
In a preferred embodiment of the invention the sensing
means comprises at least two sensors, possibly pick-up
coils, positioned symmetrically, in the longitudinal
direction, with respect to the mid point of the string.
In a particularly advantageous embodiment, displacement
sensors, for example pick-up coils, are arranged
adjacent the string in two non-parallel preferably
orthogonal, planes, so as to be capable of measuring the
string's displacement in two transverse directions
simultaneously.
It will be understood that the displacement of a string
of length 1 from its unpertubated position, for example,
in the y-direction of the above local coordinate frame
as a function of the z-position of a unit element and
time, y(z,t), can be described by the following
differential equation
2 2
il a y(z, t)~ h a 3'(z, t)- YA°1 a y(Z, t)
are at i az2
- nJy(0, t)- i11'y~(o, t) z . fL(z, t) (1)
with boundary conditions corresponding to the fixed ends
of the string, i.e. y(0, t) - y(1, t) - 0. In this
equation n denotes the string's mass per unit length, h
is the friction coefficient per unit length, the
- parameters Y, A and X1/1 are the string's Young modulus,
the area of its cross section and the string's strain
n respectively. The quantities gY(0, t) and t'yZ (0, t) are the
absolute values of the y-component of the total
acceleration and the corresponding gravity gradient
tensor component along the string, both taken at the
centre of the local coordinate frame chosen. The
function fL(z,t) represents the Langevin random force
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per unit length acting on the string due to its
interaction with the thermostat having the absolute
temperature T, with the following correlation function
C fL ( z, t ) fL ( z', t') > = 4kBThs ( z-z') 5 ( t-t') ( 2 )
where k8 = 1.4 10-23 JK-1 is the Boltzmann constant and
b(x-x') is the delta-function.
In this description, the y-direction has been chosen as
an arbitrary example to simplify the explanation of the
invention. However, the foregoing and following
analysis is equally applicable to any direction
transverse to the string or any number of directions.
Applying Fourier analysis to the complex shape of the
string caused by its interaction with the gravitational
field, the function y(z, t), can be described, in the
range z=0 to z=1, by an infinite sum of sinusoidal
functions of period 21, with appropriate coefficients
cy(n, t) . Thus a solution of Eq. (1) , which satisfies the
boundary conditions shown above, can be represented by
the following sum wherein each term in n corresponds to
one of the string's natural vibrational modes
Y(z. t) _ ~ aY(n. t) sin( nn zl (3)
n-1 11
By substituting Eq.(3) into Eq.(1) and by multiplying
its left-hand and right-hand sides by sin(rrn'z/1), and
then by integrating both sides over z from 0 to 1, one
can obtain the differential equation for cy(n,t) -
2
d C (n~ t) i 2 d ~ (n~ t)+ ~2~ (n~ t)= 2 ~(-1) n -l,g (0, t)
dt2 Y T dt Y ° $' nn
(-1) n ~nr~(0, t)
z
~1 jdzfL(z, t) sin ~ zl (4)
J0
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where the quantities
rln Y AZ
Wn- 1 p 1 (5)
represent the string's natural frequencies; z and p are
the relaxation time and the volume mass density of the
string respectively.
When n takes an even value, i.e. for those terms of the
infinite sum in Eq. 3 corresponding to vibrational modes
of the string having a node at z=1/2 (antisymmetric
modes), the term involving gY(O,t) is equal to zero.
Thus, for n even, cy is dependent only on I'yZ (and
thermal noise).
In practice this means that the amplitude, cY, of the
antisymmetric sinusoidal components of the deflection of
the string in the y-direction, y(z,t), is dependent only
on the magnitude of the gravity gradient tensor
component T'yZ .
The mid point of the string, z=1/2, is the position of a
node in all antisymmetric vibrational modes of the
string. If sensors are positioned symmetrically in the
longitudinal direction with respect to this point, it
will be possible to identify displacements of the string
corresponding to the string's natural antisymmetric
vibrational modes while discounting displacements
corresponding to symmetric modes, the magnitude of which
is not only affected by the gravity gradient tensor
component ryz but also the absolute acceleration due to
gravity in the y-direction, gy. .
It is particularly advantageous if displacement sensors
are positioned at z=1/4 and z=31/4, positions
corresponding to the antinodes of the first
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antisymmetric vibrational mode of the string, n=2. At
these points the displacement of the string
corresponding to the n=2 mode is at a maximum and thus
the sensing signal will also be at a maximum, giving
optimum sensitivity.
According to a further development of the invention a
conductor may be provided adjacent the conductive
string. The conductor may carry a current directly
related to the output of the sensing means, by the use
of a positive feedback loop. The current may be
activated continuously or periodically, for example in
an "off-on" manner. In this case, a small deflection of
the string due to a gravitational field will be
amplified by the magnetic interaction of the string and
conductor. In other words, the conductor will "push"
(or "pull") the string into further deflection in direct
response to a small deflection caused by a gravitational
field acting on the string. This is clearly
advantageous in that the displacement of the string is
greater by virtue of the magnetic interaction with the
conductor and is therefore more readily measurable,
improving the sensitivity of the apparatus.
In a particularly advantageous embodiment of this
development, two or more conductors, possibly
superconductors, are positioned longitudinally
symmetrically about the mid-point of the string so that
they amplify the antisymmetric modes of the string.
In overview, a preferred embodiment of the invention
provides a novel apparatus for measuring absolutely and
simultaneously a pair of off-diagonal components of the
gravity gradient tensor by means of a flexible .
superconducting current-carrying string with fixed ends,
comprising active parametric force feed-back
connections. The string is the coherent sensitive
element whose symmetric natural transverse modes are
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caused by the total acceleration in the transverse
plane, whilst the antisymmetric modes are caused only by
absolute values of the gravity gradient components along
the string's direction.
In this embodiment the string forms a low-inductance
part of a closed superconducting loop which is
inductively coupled to a high-inductance driving
solenoids) carrying an a.c. reference current from an
external pumping source with some frequency f2. The
string is also inductively coupled to two
superconducting magnetic flux transformers each
comprising a signal coil and two pick-up coils wherein
the pairs of pick-up coils lie a.n two perpendicular
planes the cross-line of which coincides with the
unperturbated string, thus forming two independent
channels of measurements. The two arms of each
superconducting flux transformer are balanced to convert
only the string's antisymmetric natural modes into
signal current in the signal coil to be measured with
SQUID's (Superconducting Quantum Interference Devices)
electronics. The output voltage of each channel, deeply
modulated with the frequency t1, is then proportional to
the amplitudes of the antisymmetric natural modes of the
string. This voltage is passed through a
differentiating and summing amplifier, and then used to
load the feed-back circuit to produce an in-line feed-
back current distribution proximate and parallel to the
string. By adjusting the feed-back current, the
effective relaxation time and the resonant frequency of
the first antisymmetric natural mode of the string
(whose amplitude depends only upon the gravity gradient
along the string's direction) can be increased and
decreased respectively, while the same parameters for
the symmetric natural modes (whose amplitudes depend
upon the total acceleration in the transverse-to-string
plane) are not changed. In use, the feed-back circuit
shifts the Brownian and vibrational noise level to far
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below the sensitivity required for industrial
applications.
A preferred embodiment of the invention will now be
described by way of example only and with reference to
the following drawings in which:
Fig. 1 is a general schematic representation of a
preferred embodiment of the invention; and
Fig. 2 is a diagrammatic vertical cross-section of
a device according to a preferred embodiment of the
invention.
A single channel prototype of a device according to the
invention (see Fig.2) has a flexible string 1. The
string is preferably formed of a superconducting
material such as Niobium (Nb). Niobium wire is the best
choice, having optimum elastic properties, which have
been proven to be usable at 4.2 K. The string forms the
low-inductance part Lo of a superconducting closed loop
which is inductively coupled to high-inductance driving
solenoids) Ld carrying an a.c. reference current Id(t)
from an external pumping source with some frequency t~.
The rest of the loop is provided by the casing of the
device 2,2',3,4,5.
The string has, in this embodiment, a length 1 = 24 cm,
is 1 mm in diameter and is fixed at its ends by two Nb
cups 2,2' of cylindrical shape each having a hole of 1
mm diameter at its centre. The cups 2,2' close tightly
the ends of a Nb cylinder comprising three parts 3,4,5
connected together with two Nb cylindrical rings 6,6' .
carrying a fine thread. The parts 3 and 5 also carry
threads to engage other elements of the construction.
The string's tension is produced by two Nb nuts 7,7' of
1 mm fine thread.
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The whole construction forms a closed superconducting
cylindrical cavity with the string axially positioned.
There are three spaces 10, 11, 12 inside this volume
electromagnetically insulated as much as possible from
each other by Nb partitions 9. In two of them 10, 12,
driving toroidal solenoids Ldl and Ld2, wound with 0.01
mm Nb wire and connected in series, are placed, thus
forming a large mutual inductance Ma between Ld=Lal + Lda
and the inductance of the cylindrical cavity Lo which is
of the order 10-' H for the sizes chosen. The ratio Md/Lo
is about 5x102, so if the a.c. pumping current Id(t) in
the driving solenoids has an amplitude of about 100 mA
then the induced a.c. supercurrent Io carried by the
string is about 50A peak to peak. In this case, the
corresponding circular component of the magnetic
induction B at the string's surface is nearly 200 Gauss
which is approximately four times smaller than the
Niobium first critical field BSI.
The two rectangular-type pick-up coils Lpj and Lpa of
the superconducting flux transformer, and the active
force feed-back circuit are mounted together on a
Titanium tube 8 placed inside the central space of the
construction shown in Fig.2. Titanium is chosen because
it has a thermal expansion coefficient matching that of
Niobium. The active force feed-back circuit comprises
two arms of 0.5 mm insulated copper wire stretched
parallel to the string and carrying the feed-back
E~:.rr°n t y v yi '!- Ty2
This design has particular advantages; for instance, the
closed superconducting configuration gives optimum
. shielding against external varying electromagnetic
fields. Also, the cylindrically symmetric configuration
has a small radial size which, including all integral
parts of the prototype, is no more than 3.8 cm diameter.
Thus, it is possible to utilise a standard commercial
100 litre liquid helium vessel having an input opening
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of about 4 cm diameter to cool the construction down
with a standard probe. Special helium cryostats, which
have been used for known devices, exclude the
possibility of removing the device from the cryostat's
inner volume, for example if something goes wrong, to
readjust it under field conditions. Removal of a device
from a cryostat requires a long period, of for example
several hours, to warm the cryostat contents to
atmospheric temperature so that the contents do not
explode under rapid thermal expansion. This is one of
the major disadvantages of known constructions.
However, the small input opening of the standard
commercial 100 litre liquid helium vessel prevents such
an explosion occurring, which means that the apparatus
according to the present invention can be removed from
the vessel and adjusted under field conditions.
The string is deflected by a non-uniform quasi-static
gravitational field and interacts with a variable feed-
back current distributed close to and substantially
parallel to the string. The distribution is optimum
when the feed-back current is injected at or taken from
the point in the feed-back circuit opposite the mid-
point of the string (see Fig. 1). In the local
coordinate frame chosen this point is z = 1/2. Another
requirement for optimum operation is that the two arms
of the feed-back circuit are substantially equal and
grounded at their ends. In this case, there is no
electromagnetic coupling between the feed-back current
and the closed superconducting loop in which the string
is incorporated.
The current .Io(t) flowing through the string and
interacting with the feed-back current distribution
Ty(z,t) gives rise to the following transverse component
of the force per-unit-length fy(z,t) acting on the string
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fY(z, t) = t 2 ~IpIY(z, t) sin (f2t) (6)
where ~.~,o = 4rr 10-' Hm-1 is the magnetic vacuum
permeability, d is the distance between the centre of
the unperturbated string and the centre of the wire
carrying the feed-back current and the phase of the
pumping current source 15 is chosen to be zero. The
sign + or - is determined by the output buffer of the
differentiating and summing amplifier 16 shown in Fig. 1.
The transverse motion of the unit element of the string
in the OYZ plane is then described by the following
differential equation
2 2
~ at2Y(z. t)~ h ~ Y(z. t)_ yNbg~1 ~z2Y(z, t)
a
- ngy(o, t)- nrYZ(o, t) z t 2~dIoIY(t) sin (nt)~ fL(z, t)
(~>
which will be seen to correspond to Eq.(1) with the
addition of the term of Eq.(6). Consequently, Eq.(7)
also has solutions of the form of Eq.(3). Thus,
following the same algebraic manipulation as for Eq.(1),
a differential equation for cy(n, t) can be obtained for
this embodiment. This is Eq.(8) which corresponds to
Eq.(4) but with the addition of a feedback term.
z
d ~ (n~ t)~ 2 d ~ (n~ t)+ ~z~ (n~ t)= 2 ~(-~-)n -1.~9 (p~ t)
dt z y T dt Y ° Y nn
~ (-1)i' 2lryz(0, t)
nn
t 1 u2 enloly(z, t) sin (t2t)
n 2n d
i l
~1 fdzfL(z, t) sin ~ zJ (8)
0
The quantity en relates to the characteristics of the
transducer system of the feedback loop; the longer the
length of the arms of the feedback circuit, the larger
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the quantities en are. If the arms of the feed-back
circuit are absolutely identical then the quantities en
are equal to zero for all odd n = 1,3,5.... Their
particular values for the sizes shown in FIG.2 are
determined by '
en= n/cosl 6 ~~ cos~ 56n1- 2cos~ 2 ~~ (9)
So, for the properly adjusted feed-back circuit, only
the antisymmetric natural modes of the string interact
with the feed-back current. However, only the
antisymmetric natural modes of the string are sensitive
to the absolute value of the gravity gradient tensor
component to be measured, as is seen from Eq.(4) and
(7) .
The superconducting pick-up coils Lp1 and Lp~ are placed
near the string and cause two arms of the
superconducting magnetic flux transformer to convert, if
perfectly balanced, only the antisymmetric natural modes
into the signal current Ij to be detected with the
SQUID~s electronics 13 (see FIG. 1). One uses SQUID~s
(Superconducting Quantum Interference Devices) 13 as
they are the most sensitive variable current and
magnetic flux sensors currently available. In the
prototype shown in FIG.2, the pick-up coils are made in
the form of two rectangular-type single loops of Nb wire
placed symmetrically with respect to the midpoint of the
string and connected in parallel with the signal coil Ll.
If the symmetry is perfect and the areas of the loops
are absolutely identical, then the symmetric natural
modes do not produce any signal current Ii or feed-back
current 1y. The same effect can be achieved for slightly
non-identical pick-up coils with the accuracy required,
if one uses the additional inductance(s) Lb connected in
parallel and/or series with one or both of the pick-up
coils. The inductance(s) Lb can be tuned to balance the
two arms of the superconducting flux transformer. The
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residual "zero-mode" current in the signal coil Li
corresponding to the unperturbated position of the
string can be compensated directly inside the SQUID by
an additional coupling (not shown) to the pumping
current source. If the balancing conditions are
satisfied, then the output voltage of the SQUID's
electronics 13 is determined by
VY(t)= KIoLssin (f~t)~~~nCY(n, t)~~ K~N(t) (10)
l n-1
where K is the total flux to voltage transfer function
and Ls is the SQUID's inductance. The quantities ~3n
depend on the physical design and position of the pick-
up coils and are equal to zero if n = 1,3,5.... The
function ~(t) is the equivalent-to-noise random magnetic
flux inside the SQUID loop, whose spectral density S~(c~)
determines the intrinsic instrumental limit of the
accuracy of measurements. The feed-back current IY(t) is
formed from the output voltage VY(t) by passing it
through a differentiating and summing amplifier 16 which
is loaded by resistance RY. In this case, the feed-back
current IY(t) can be represented by
I (t)= pT. a V (t)+ q V (t) (11)
' ~' R3, dt s' RY s'
where p, q and i* are constant parameters which depend
upon the design of the differentiating and summing
amplifier 16.
It must be noted that a mismatch between the two arms of
. the feed-back circuit always exists. The design shown
a.n FIG.2 uses two identical feed-back resistances, RYI
and RY2, one for each arm. In this case the mismatching
can be easily compensated by tuning one of the
resistances, say Ry2, to obtain the optimum case.
Equations (7) and (11) represent a closed infinite set
of differential parametric-type equations. Careful
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analysis has shown that we can ignore the terms
involving the quantities cy(n,t) with n> 2 in the right-
hand side of Eq.(11). The reason is that just one mode
can be made "soft" i.e. the most sensitive to the
gravity gradient, namely cY(2,t). If the string's
natural frequencies are high enough and separated by
single octave gaps, only second-order corrections are
required which can be easily taken into account along
with analysis of other instrumental errors. Then, as
follows from Eq. (7), the self-consistent equation for
the gravity gradient sensitive mode, n=2, including
unavoidable fundamental noise sources can in turn be
written in the form
z
d z cY(2' t) * ( T - 2 a * 2 acos (2L2t) ~ dt cY(2, t) *
dt
.(wz- lu>2. lw2cos(2i2t)- la~sin(2Llt)~c (2,t)= If (0,t)*
2 2 2 Y II YZ
1 (
* ~1 fdzfL(z, t) sin' 2~ z~.(back-action noise ) sin (tlt) (12)
°°
where
2
w2= ~s2~2qK1 n 2n d Rs a w2lqlT. (13)
Y
and it is assumed that the true sign of the feed-back
current has been chosen.
If some easily carried out conditions are satisfied,
which are
r dt « w2 2 w2 sn2 ~ wi i1z ~ I ( 14 )
then one can show that the self-consistent output
voltage is
1 r (0, t)
P ( t) ~(3zKI°L - ~ . brownian noise sin (t1t) * KEN( t) (15)
n w2- _1 ~2
2 2
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where under "brownian noise" the combination of thermal
and back-action noises is implied.
It is of interest to estimate the limiting accuracy of
measurements of this embodiment of the invention, which
can be represented by the value of a minimum detectable
gravity gradient
n l6kBT W4 E~cn> 9 EOtVOS
to r - . io (16>
min' 1 m T eff 2 ~ 2 1' a'I0 Hz
where zeff = T/(1-at/4) is the effective relaxation time,
m is the total mass of the string, and E~,(n) is the
energy resolution of the SQUID. Using the following
practical parameters: 1=0.24m, m~1.6 10'3 kg, Tegg~ 104 s,
(32~~ 4x103 m-1, Ls~~5x10w1 H, Io~ 50 A, (w22-X2/2) 1~2/2rm 2 Hz,
w2/2rm 40 Hz, n/2rt 2 2x102 Hz, E~(n) ~ 2x10-31 J/Hz (d.c.
biased SQUID), one can obtain from Eq.(16)
r~n~~ 0 . 4 Eo'tvd's ( 1.~ )
Hz
It can be shown, that a range of the parameters z, cat, ca,
a and n exists where the string's response described_by
Eq. 12 is stable. For example, for quasistatic gravity
gradients and sufficiently high pumping frequency n one
can ignore the oscillating terms containing Cos(2nt) and
Sin(2nt) in the right side of Eq. 12.
There are a number of detecting strategies which can be
employed by the present invention at this stage, which
are dependent on the initial mechanical parameters of
the string and the application for which the apparatus
is intended. It is preferable to use a string with a
high mechanical stiffness and a short relaxation time in
order to increase immunity to vibrational noise, which
is the main noise source in industrial applications,
particularly in mobile gravity gradiometry. On the
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other hand, the stiffer the string, the stronger the
feedback force that has to be applied to the string to
s
soften the signal mode, and the larger the back-action
noise associated with the feedback current.
Additionally, the shorter the string's relaxation time,
the stronger the influence of thermal fluctuations of
the string on the measuring accuracy since the mass per
unit length of the string will normally be quite small.
To overcome both of these problems a best mode of
carrying out gravity gradient measurements according to
another embodiment of the present invention uses
variable feedbacks in an "off-on" manner. In this case,
the feedback force is initially not applied to the
string for an 'off-period' during which the string
reaches thermodynamic equilibrium. The feedback force
is then quickly activated for an 'on-period' during
which the effective natural frequency
2 _ 1 2
weff ~ ~2
and the effective relaxation time
QCT '1
Teff ~ T(1 4 ) (19)
become substantially smaller and longer respectively
compared to the corresponding initial parameters of the
string. The feedback is adjusted in such a way that the
effective relaxation time becomes much longer than the
on-period. Measurements are carried out during the on-
period only, in which the string never reaches
thermodynamic equilibrium. For example, the fluctuation
dissipation theorem is no longer applicable to the
string during the period of measurements and its
response to all external noise sources is changed (see
V.B. Braginsky and A.B. Manukin, Measurement of Weak
Forces in Physics Experiments, Ed. by D.H. Douglass,
I ~ I i i n , I
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University Press of Chicago, 1977).
One can show that in this case the least gravity
gradient detectable by this embodiment of the invention can
be represented by
rmin - ~ Y~S) lm zzT + 2 2 L I (~) log ~ S ~ 1 ~9 EotvoS ( 2 0 )
m ~ 2 s 0 eff
where
Y(S) - Z~~ logC~~+ ~~~ [~ +log~~~~ (21)
2
Tm is the measurement time (on-period), m is the
total mass of the string, E~(S2) is the energy resolution of
the SQUID at the frequency S2 and b is a statistical error of
the first kind. The value of b is the likelihood that the
equivalent gravity gradient noise will exceed the value
represented by the left side of Eq. 20 for the period of
measurement.
Using the following practical parameters: 1=0.24
m, m=1. 6x10-3 kg, i =0.5 s, T m=1 s, i eff= 104 s, 13z- 4x103 m-1,
LS= 5x10-11 H, Io= 50 A, (aeff/2H= 3 Hz, wz/2II= .80 Hz, S~/2II >-
104 Hz, E~ (S2) - 5xlO-3z J/Hz ( 500 n d. c . biased SQUID) , one
can obtain from Eq. (20)
2O hm~n - O.OZ EOtVOs
In both the above embodiments, the desired signal
is obtained from the output voltage by synchronous detection
in synchronous detector 14 with a reference signal taken
from the pumping source 15, and the invention allows
I i i I
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calibration of the desired signal in gravity gradient
absolute units without rotation as has been proposed for
known rotating gravity gradiometers. As for rotating
designs, the
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invention allows the movement of the noise spectrum to a
frequency range at which 1/f contribution is
sufficiently small. Natural vibrations of the string,
which occur during the time of measurement (on-period),
do not cause a problem since they can be filtered out
from the desired signal provided that the on-period is
chosen to be much longer than the period (2II/GJeff) of
such vibrations.
Vibrational noise immunity is improved by the factor
~~eff/~1) 2 which can be made as small as 10-a.
One must consider inductive cross-coupling between the
feedback currents and each pair of the pick-up coils and
cross-coupling between the pick-up coils themselves,
both of which act like negative feedback loops. On the
one hand this leads to unnecessary renormalisation of
the amplitudes of the output signals until the gain of
the SQUID~s electronics exceeds some critical value. On
the other hand in the case of double-channel
measurements, the output signal of each channel contains
a linear combination of each gravity gradient component
to be measured. It can be shown that each of such
components can, nevertheless, be measured separately and
simultaneously, if a proper data acquisition system is
used. The effect can be easily eliminated by organising
additional positive feedback to counteract this negative
feedback, for example by connecting, via a weak
inductive coupling, each feedback current with each
SQUID.
In practice, the apparatus according to the invention
can be used to determine in absolute units the off-
diagonal components of the gravity gradient. By
conducting a gravity survey over an area, small
differences in absolute gravity gradient can be
detected. Such small changes may indicate variations in
local geological features, for example, the presence of
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minerals, gas or oil.
Repeated readings over time at a single locality could
indicate changing geological status of an area, such as
s
rising magma. Clearly the invention enhances
prospecting and other data gathering pursuits where
accurate gravitational field measurement is required.
Use of absolute values enhances the information that can
be determined from the data measured. A gradiometer
according to the invention can be used while moving,
which allows the gradiometer to be used on vehicles
whether land, sea or air vehicles. For example, the
device can be suspended from a helicopter and used while
the helicopter traverses a selected area.
