THE   FARADAY   BALANCE

    Dr. J. C. P. Klaasse,
    Van der Waals-Zeeman Institute,
    november 1999.

A Faraday Balance is an equipment for the detection of
magnetic moments by measuring the force on a sample
in an inhomogeneous magnetic field, generated by an
electromagnet between two dedicated, so-called,
Faraday pole caps.
The force can be measured with an analytical balance
(resolution of the order of 10-2 milligram or better)
at which the sample is suspended. If the sample is at
a position where the field gradient is known, the
magnetic moment, which is directly proportional to the
detected force, can be evaluated from the
calibration constants.

The Faraday Balance in the Materials Science group
is a high-temperature magnetometer, ranging from room
temperature up to 1250 K. The installed computer program
is capable to control a measurement at a user defined
equidistant series of temperatures without human
intervention. At each temperature the magnetic moment
is determined in an also user defined series of magnetic
fields, ranging from 0.05 T up to 1.15 T.

The sample is housed in a quartz capsule and suspends
at the end of a rod, connected to the balance beam, in
a tube which is, between the pole caps, surrounded with
a heating element in order to change the temperature.
In order to guarantee a proper force detection, the
sampleholder has to hang free from the walls of the
surrounding furnace, which latter requirement limits
its dimensions.
 
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    THE    ELECTROMAGNET

The magnet we use is a 7-inch Newport Instruments
Electromagnet type E. The field is, in principle,
inversely proportional to the gap width, with an
upper limit given by the material the yoke is made
off, in this case iron. So, fields up to some more
than 1.5 T can be generated with this type of magnet.
The water cooled coils can carry 24 A; the electrical
resistance is about 4 ohm for each coil.

The current is generated by a Bruker B-MN 120/60 A5
current supply (120 V, 60 A, water cooled).
The box is 1.40 m high, and 60*60 cm2 in circumference,
and is, therefore, housed in the cellar of the laboratory,
with a remote control unit at the magnet.

In order to be sure that the magnetic field is symmetric,
magnet coils are, usually, switched in a series circuit.
However, this showed here to be unpractical. In order to
put the wanted 24 A through both coils, some 200 V was
needed, which is too much for our current supply. So,
we decided to switch the both coils in a parallel
circuit: 50 A is no problem for the Bruker box.

Fields show, nevertheless, to be very reproducible: no
problems with this parallel circuit have been observed.

The current is computer controlled: the control signal
is generated by a 12 bit DAC resulting in steps of the
magnet current of about 0.07 A, corresponding to about
1.5 mT. Although we use only the first 700 steps of the
DAC, already resulting in 23.6 A/coil, we have decided
that these steps of 1.5 mT are sufficiently small.
 
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    FARADAY   POLE    CAPS

The inhomogeneity of the field between two pole caps is
determined by the shape of the caps. Faraday pole caps
have the property that in vertical direction z on the
symmetry axis of the magnet, where the field, let us
say, is in x-direction, the product Bx * dBx/dz is
constant over a considerable range in z.

This requirement makes measurements on paramagnets
more simple. For materials with field independent
susceptibility, the force is directly proportional
to this product, which we will call alpha, and dividing
the force by alpha, the susceptibility follows directly.
A large region with constant alpha makes the results
independent of the shape, and exact position, of the
sample.

By making the field independent of y (small gap compared
to the horizontal dimensions of the gap in y-direction) we
can restrict ourselves to solving a twodimensional problem
in the x-z plane. Simply integrating the formula
         Bx * dBx/dz = constant,
yields Bx = sqrt(a*z+b), with a and b constants.
Furthermore, if we take Bx in the gap inversely
proportional to the gap width, that means
        Bx = const / d(z),
with d(z) half the gap width at level z, it follows that
        d(z)2 = c / (a*z+b),
where a, b, and c are adjustable constants. This formula
determines the surface shape of the pole caps.

For more details, see for instance:
   M. Garber, W.G. Henry, and H,G. Hoeve;
        Can. J. Phys 38 (1960) p 1595.
    R.D. Heyding, J.B. Taylor, and M.L. Hair;
        Rev. Sci. Instrum. 32 (1961) p 161.

In practice, curved surfaces are difficult to cut, so
the surface is approximated by a number of flats.
Our pole caps have 5 flats, and follow the curve
given by Garber, et al.: d(z)2 = 50 / (8-3*z), with the
numbers in cm.
The lower 5/8 part of the cap is tapered with an
angle of 60 degrees in order to increase the
overall field strength.
In our magnet, the gap at the pole tips, arising this
way, is 36 mm. The pole tips are not at the point z=0!!
The field gradient at the pole tips is just zero, but
increases rapidly with z.
The alpha goes through a broad
maximum at 26 mm above the pole tips, with only 1 %
deviation in alpha over 5 mm.
This is the position the samples have to be placed
for optimum results.

At the level of maximum alpha, the highest magnetic
field is about 1.15 T (at 23.6 A per coil).
This field, however, is rather strongly dependent
on the vertcal position:
the gradient dBx/dz amounts to about 0.16 T/cm
at this point. This means that only for very small samples
(<1 mm) the magnetic field is a well defined parameter!
Fortunately, all samples for which knowledge of the field
is important because M(H) is not a straight line, have
large moments and, therefore, can be kept small...



    Faraday pole cap profiles according to Heyding, et al.
    Small hatched area indicates specimen position.


polecaps3.gif (93505 bytes)

    Photograph of the Faraday pole caps.



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    FORCE    DETECTION

For the force on a magnetic sample by the field gradient
holds:
        F = moment * dBx/dz.


As soon as the gradient dBx/dz (and/or alpha) is known, it
is easy to evaluate the magnetic moment of a sample from
the observed force. The force is usually measured with a
balance. That's why we call this kind of equipment a
Faraday Balance. The magnetic part of the force is
evaluated from the difference in apparent sample mass
with, respectively, the magnet switched on and off.

The balance, in use in our equipment, is a Mettler H16
analytical balance, with sensitivity 2*10-5 g. In normal
use, mass compensation in steps of 0.1 gram yields
the rough result, where the last 4 digits follow from the
deviation of the beam, detected by an optical system by
which a transparent scale, attached to the balance beam,
is projected on a frosted glass screen in the front plate
of the balance. The 100 mg scale on the beam is, in reality,
only about 1 mm wide, corresponding with a displacement,
at full scale, of the balance rod at the other end of the
beam over a distance of the same order of magnitude.

This small deviation, however, causes too large uncertainties
in the position of the sample, and, therefore, as we shall
see later, in the contribution of the sampleholder For
that reason, the force in our Faraday Balance set-up is not
determined from the deviation of the balance beam.
We have built a compensating coil system consisting of
two concentric coils, each 10 cm long, positioned with
the centres at 5 cm distance from each other. In that
configuration, the force between them, at constant current
through these coils, is nearly independent of small mutual
shifts along the axis. See figure.

    Force characteristics of the
    compensation coil system as a
    function of mutual displacement.

fbschema.gif (20480 bytes)

        Schematic view of the positioning of the two compensation
        coils in the balance. MD: mechanical damping; GP: glass
        plate with scale; PM: permanent mass; VM: variable mass.
        The optical system that projects the scale onto the frosted
        glass screen is not given.

                        fbbalans.gif (58910 bytes)

    Photograph of the balance and the compensation coils.

The inner coil is attached at the beam of the balance and
a constant current through it generates a constant magnetic
moment. The outer coil is fixed at the frame, and by varying
the current through it, we can generate a compensating force
which is proportional to the current through this outer coil.

It is important to keep the current through the hanging coil
constant (and to take that through the fixed coil variable)
because of two reasons.
First, the current connections of the hanging coil to
the frame (thin copper wires, diameter 0.15 mm) influence
the equilibrium position of the balance, and this influence
shows to be current dependent: the temperature of these
thin wires changes with the current.
Secondly, the magnetic moment of the hanging coil is felt
by the main magnetic field, resulting in an extra force
component. Keeping this force component constant, we will
automatically correct for it by correcting for the empty
sampleholder contribution.

The current through the hanging coil is 660 mA, that
through the fixed coil varies between zero and 660 mA.
This compensation current is generated by a 16 bit DAC
(0 .. 64000), resulting in about 10 microampere/bit.
A compensation of 0.1 gram (0.981 milliNewton) corresponds
to 36205 units of the DAC (1 unit ~= 0.0027 mg), about 8
times more sensitive as the standard optical detection of
this type of balance. The resulting current through the
fixed coil is, with the beam in in equilibrium, the quantity
determining the force.

For the detection of the occurring balance beam deviations,
necessary for automatic compensation, we use the optical
system of the balance. On the screen whith the scale
divisions on the balance beam a window is attached wich
produces a specially designed shadow on the front plate
(see figure). Two Light Dependent Resistors (LDR's) are
attached to the front plate besides the frosted glass
screen, in such a way that a change in the shadow position
originating from a moving balance beam, increases the
amount of light on one, but just decreases the amount
of light on the other LDR.
This results in a high resolution, necessary in order to
be competitive to the optical detection.

 


Fig a: Projection of the glass plate with scale and shadow of
           the window on the frosted glass screen. The positions
           of the LDR's are given by dashed lines.

Fig b: Resulting LDR bridge signal, e, as a function of the
           beam displacement, z.

 

The two LDR's are switched in an electronic bridge circuit,
and the out-of-balance voltage of the bridge is input for
a 13 bit AD convertor in the computer, where a control
unit in the computer program adjusts the coil current
until the LDR-bridge is in equilibrium. Some data on the
ADC: 1.2207 mV / step (10 V full range; 8196 steps),
low speed, about 0.11 sec/conversion, and low noise.

As can be seen in the figure, the sensitive region is only
limited to some more than 5 scale divisions (5 mg) ..
That means that larger deviations saturate the detection,
resulting in very slow compensation. Apart from a well
known PID-like algoritm, we have, in order to prevent
large deviations, installed an adaptive controller.

The balance behaves normally like a, nearly critically
damped, harmonic oscillator, with tau = 2/gamma = 1.
The period equals 5 sec resulting in omega = 1.2 /second.
The damping is a viscous damping in the surrounding gas,
so, the balance has to be operated in 1 atm gas in order
to maintain its dynamics.

The PID controller parameters are roughly determined with
the help of the rules of Ziegler and Nichols, and fine
tuned by trial and error.
The result is fast, but a little oscillatory (amplitude
between 1 and 2 current units, period 2.5 s) which
disadvantage we can circumvent by taking the average
of the last 21 calculated values of the current (repetition
rate of the AD conversion procedure amounts to about
0.12 s, mainly due to finite conversion times) for the
evaluation of the force on the coils.

Depending on the field value, one point (zero field, finite
field, and again zero field) takes between 1 and 2 minutes.

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    ADAPTIVE    CONTROLLER

In particular for large moments in high fields, the
difference in force with field on and off will exceed
the few scale divisions in the proportional region.
If saturated, only the integrator in the controller
will work significantly, resulting in a rather slow
creep to equilibrium. Speeding up this creep will
result in big overshoots, and instability.

So, we have to apply a trick to speed up the control.
In fact, the disturbance is known: a field sweep with
well known speed (about 2 A/s), resulting in a well
known response of the balance beam - provided the
moment (or susceptibility) of the sample is known.
But the latter can be learnt by the computer.


The procedure is as follows: during a field sweep,
the computer simulates the behaviour of the beam with
a preset value of a moment-parameter. The normal PID
functions are temporarily suspended. During the sweep the
out-of-balance signal is monitored and if it exceeds
a certain threshold, the value of the moment-parameter
is adjusted. In a short number of sweeps, the computer
value for the moment-parameter will converge to an
appropriate value to keep the out-of-balance signal
within the proportional region. This shows to work.

Moreover, moments of samples will, in general, not
change drastically if the temperature changes in not
too big steps. The computer can take care of the field
dependence of the moment, so, only in the neighbourhood
of Curie points we can expect problems in finding the
moment-parameter. Normally, the adaption of a new
parameter value will cause no significant delay.

The gain in time shows to be up to about 1 minute per
obtained value, a time saving of about 30%.
A complete run takes several hours, so the gain is
significant.

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    CALIBRATION

Since the force is detected in current units of the ADC
(about 0.0027 mg per step), the calibration of the field
gradient is performed in the same current units. Since
the moment is proportional to force over alpha, the
moments are, nevertheless, evaluated in normal MKSA units.

The calibration is performed by measuring the well known
susceptibility of a sphere (diam 0.4996 mm) of soft iron
(annealed in 1 atm wet hydrogen, 850 degC for 1 week).
This susceptibility equals 3.0. However, this method
works only in fields up to about 0.5 T. In higher fields,
the alpha is determined by extrapolating the results of
a paramagnetic salt at constant temperature.
Typical error of this calibration: about 3 promille.
In the lowest fields the error is somewhat larger.

We checked the calibration with a measurement on a
Pd sphere. We found the literature value for the
susceptibility within 1%, except for the lowest field;
here the error was 3 %. The reason for this larger errors
in case of Pd compared to the calibration itself are
ascribed to the necessary correction for the empty holder
contributions, which, in case of the smaller Pd moment,
are more significant than for the iron sphere.

Obtained values for the magnetisation have always larger
errors because of the uncertainty of the (inhomogeneous!)
magnetic field at the sample position. Nevertheless, we
obtained Arrott plots (plots of B/M vs M2) on Fe and Ni
spheres whith well interpretable curves, and a perfect
indication for the Curie point: see temperature below.

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    TEMPERATURE

In order to prevent corrosion of metallic parts in the
sample space at high temperatures, this space is
filled with inert gas. In order to obtain a good heat
contact between the furnace and the sampleholder,
in particular at lower temperatures, this inert gas is
helium.
The disadvantage of the use of an inert athmosphere is
that the whole balance has to be placed in a vacuum vessel,
in open connection with the sample space in the furnace.

The sample space in the furnace consists of a half open
quartz tube, type test-tube, outer diameter 14 mm, inner
12 mm, at the upper open end sealed with a O-ring seal
in a water-cooled stainless steel body, that also contains
the vacuum feedthrough for the thermocouple, and the seal
for the tube connecting the furnace with the balance vessel.

The furnace element is a silicium carbide tube (Crusilite)
with helically sliced windings over a length of about 15 cm.
These elements turn out to be mechanically very vulnerable
(ceramic material!), but, once installed, they show a very
long lifetime. The inner diameter of the element is 15 mm,
the outer 21 mm. The room temperature resistance of the
element amounts to about 3.5 ohm.

element.gif (54808 bytes)

    The heating element; the ruler is for comparison

The heating element is surrounded with a water cooled jacket
with an external diameter of 35 mm. This furnace set-up
showed to meet our objectives well and, moreover, to be very
fast. With currents up to 20 A (at 70 V; 1.5 kW) a temperature
of 900 degC is attained within 10 minutes. After switching
off the power the temperature is below 50 degC within,
again, 10 minutes. The maximum temperature amounts to
above 950 degC (close to 1250 K).

In order to prevent the heat radiation in the hot region
from entering the balance, water cooled radiation shields
are applied between furnace and balance vessel, together
with a baffle attached to the stainless steel balance rod.
So, the connection is made optically closed.

The temperature is controlled with an Eurotherm type 815
controller in combination with a 240 V/ 25 A power unit.
This unit is powered from the 70 V output of a power
transformer in order to optimise the control range.
Moreover, the lower voltage works as a safeguard against
overheating. The controller is connected to the computer
by a RS232 serial connection for data exchange, in
particular the setpoint and the obtained control
temperatures.

The temperature is measured with thermocouples. A type S
Pt/Pt(10%Rh) thermocouple is installed outside the quartz
sample space close to the heater element. It is directly
connected to the input of the controller via dedicated
extension wire. This thermocouple is isolated with thin
alumina tubes within a larger (diam 3 mm) quartz tube and,
outside the furnace, with flexible glasswool tubes.

This control thermocouple is situated a few centimeters
below the sample, which latter is at the furnace centre,
because no free space was available at the centre between
quartz tube around the sample space, and the element.
For that reason, the setpoint temperature will be up to
20 degrees lower than the attained temperature at the
sample position. This deviation, however, can be accounted
for in the choice of the controller setpoints.

In order to detect the correct sample temperature we
have installed a second thermocouple inside the quartz
tube at about 0.5 cm below the sample position. This
type K (chromel alumel) thermocouple is of the so-called
thermocoax type with a stainless steel tube with 1 mm
diameter. The signal of this thermocouple is (also via
dedicated extension wires) connected to an Analog Devices
2B50B, zero point compensated, thermocouple amplifier.
The customer adjustable amplification (in our case
107.75) shows to be excellently long term stable.

However, the DC offset of this amplifier shows not to
be that excellent. From time to time we have to adjust
the readings by comparing the 2B50B output with the
Eurotherm controller display under gradient free room
temperature conditions. A simple computer generated
correction, with user installed value, corrects all
readings. Corrections, however, remain limited to
a few degrees.

The output of the thermocouple amplifier is connected
to a 13 bit AD convertor (the same as for the LDR's)
and converted to a temperature by means of tabulated
values for type K thermocouples stored in the computer
memory. The resolution is about 10 microvolt,
corresponding with about 0.3 deg.

From time to time we do checks of the temperature
readings with melting points of indium, lead, and
tin (steps in the susceptibility), and with the Curie
points of nickel and iron. In all cases, we found results
that differed not more than 0.5 % in temperature with
accepted literature values. After a recent upgrade
(installing the thermocoax instead of an until then
used type S thermocouple inside the quartz tube), we
found 356.5 degC for the Curie point of nickel.
Literature values scatter between 354 and 358 degC.

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    SAMPLEHOLDER

Due to the high reactivity of most samples at high
temperatures, they have to be sealed in evacuated
quartz capsules. In order to be mounted free from
the walls of the furnace (absolutely necessary for
good results) the outer diameter of this quartz capsule
has not to exceed 6 to 7 mm. This means internally
about 4 to 5 mm, which is the maximum diameter for
samples to be accomodated.

The design of the capsules, which have to be manufactured
for each sample again, is known at our glass workshop.
Most convenient is a capsule of about 100 mm length,
at one side elongated with a quartz wire with hook for
attachment at the end of the sample rod. Total length
is limited to between 145 and 150 mm, in order to remain
within the limits of an adjusting screw in the sample rod
for positioning the sample at maximum alpha position
between the Faraday pole caps.

Although this positioning is not critical with respect
to the sample position (1 mm accuracy is sufficient),
it is very important with respect to the position of the
sampleholder: in high magnetic fields, a shift of 1 mm
introduces an error up to about 100 current units (about
2.5*10-7 N). The spatial extent of the holder makes its
effect strongly dependent on its position, and only if
the sampleholder contribution is nearly negligeable, we
can ignore these effects. This can be the case for
ferromagnets, and for situations where only temperature
dependences are studied. For these situations a default
sampleholder contribution is implemented in the computer
program.

If the sample holder contribution is not negligeable, we
have to measure it separately at exact the same position.
In order to make life easier, we have a ruler attached at
the wall, with a small hook, at which we can hang our
sampleholders in order to detect the relevant vertical
dimensions. From these, we can calculate the vertical
adjustment (the number of turns of the screw)
in the balance.

Some typical values for the empty holder contributions:
-temperature dependence is small: less than 1 %.
-effect in 0.05 T: -0.6*10-6 Am2.
-effect in 1 T: -5*10-6 Am2.

The field dependence of the contributions does not agree
with the observed alpha values, but this will not be
surprising in view of the spatial extent of the holders -
and all other known and unknown effects as, for instance,
the contribution of the compensating coil in the balance..

Disadvantage of the use of this capsules is that orienting
the samples is quite impossible. The samples are free to
rotate in the holders which, in itself, are not stabilised
within the furnace tube. Even irreversible rotations during
a run are possible.
Maybe, a solution for the problems to this point can be
developed in the future.

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    SAMPLES

The maximum moment that can be measured in the Faraday
Balance in 1T amounts to about 50*10-6 Am2. This means
about 30000 current units. Although the range is 64000
units, in reality we have to work with this lower range
because the zero point is not known in advance. This can
be at any position on the 0 .. 64000 scale. Although we
can always shift over 0.1 gram, you cannot choose for any
wanted initial current value. However, in case of
emergency some tricks are available.

In 0.05 T (the lowest calibrated field) the maximum moment
without scale overflow amounts to about 20 times as high:
10-3 Am2. Whether these moments are field induced, or
permanent, doesn't matter. For ferromagnets, these numbers
point to rather small quantities of material. A saturated
piece of iron with a moment of 50*10-6 Am2 has a volume
of only 0.025 mm3.

Although the remarks, made above, limit the use a little,
some very interesting applications are realised. Most
simple is the determining of the Curie point of new
intermetallic compounds. Moreover, spin reorientations
are detectable, although great care has to be taken in
the interpretation of results on ferromagnets below the
Curie point.

See for instance:

S.J. Hu, X.Z. Wei, D.C. Zeng, Z.I. Liu, E. Brueck,
    J.C.P. Klaasse, F.R. de Boer, K.H.J. Buschow;
    Physica B 270 (1999) p. 157.
S.J. Hu, X.Z. Wei, O. Tegus, D.C. Zeng, E. Brueck,
    J.C.P. Klaasse, F.R. de Boer, K.H.J. Buschow;
    J. Alloys Compd 284 (1999) p. 60.
L. Zhang, D.C. Zeng, Y.N. Liang, J.C.P. Klaasse,
    E. Brueck, Z.I. Liu, F.R. de Boer, K.H.J. Buschow;
    J. Alloys Compd 292 (1999) p. 38.

Very interesting, however, may be the application of the
Faraday Balance as a furnace for high temperature relaxation
(or otherwise in time changing) effects with real time
monitoring of the state of affairs as a function of time.
Dedicated computer programs are available for automatic
measurements of time-dependent magnetisations at constant
temperature. For other applications, customer designed
programs can be implemented without hardly any restriction.

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