A note on the ferromagnetic Faraday effect at centimetre wavelengths

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A note on the ferromagnetic Faraday effect at

centimetre wavelengths

F.F. Roberts

To cite this version:

A NOTE ON THE FERROMAGNETIC FARADAY EFFECT AT CENTIMETRE WAVELENGTHS

By

F. F. ROBERTS.

Sommaire. 2014 On discute la rotation du

plan de _polarisation d’une onde _{électromagnétique} trans-mise à travers une matière _{ferromagnétique parallèlement} à un _{champ magnétostatiqlle appliqué} à

l’extérieur. Le mécanisme de cette rotation est différent pour les ondes optiques et les ondes

centi-métriques et l’on _indique une théorie pour l’effet centimétrique. Des mesures à 03BB ~ 3 cm _{pour quelques} ferrites de _magnésium sont en concordance _{demi-quantitative} avec la théorie. Les écarts sont peut-être

dus à la dissipation négligée.

LE JOURNAL DE PHYSIQUE ET LE RADIUM. ~ TOME

12,

MARS

1~~1,

PAGE 305.

1. Introduction. -- The

Faraday

effect to be considered here is the rotation of the

_plane

of

polarisation

of a

_{plane-polarised}

wave transmitted

through

the material on test when a

_magnetic

field is

_{applied parallel}

to the direction of propaga-tion. The associated Kerr effect is the rotation of the

_plane

of

_polarisation

of a similar wave, reflected

_nearly

_normally

from one face of the

sample,

due to an

_{applied magnetic}

field normal

to that face. Much work has been carried out on

these effects in the

_past

at

_{optical wavelengths}

_[1].

The

_Faraday

effect has been

_{investigated mainly}

for

_diamagnetic

and

_paramagnetic

_materials,

and the Kerr effect for

_{ferromagnetic}

materials. The most

_thorough

_experimental

work on the

ferro-magnetic optical

Faraday

effect seems to have

been that

_by

Cau

_[2],

and a theoretical

_analysis

for this case has been carried out

_by

Hulme

_[3].

Van Vleck and Hebb

_[4]

and J.

_Becquerel

_[5]

have discussed the correlation between the

_Faraday

rotation and the

_{magnetisation}

for

_paramagnetic

salts as functions of

_temperature

and

_applied

field

strength.

Cau’s measurements _{confirmed the very}

_large

Faraday

rotations,

about 20 0000 : mm at a field

strength

of

io ooo oe,

found

_by

earlier workers for thin iron

films,

and also showed that the trans-mitted wave was

_{elliptically polarised.}

The

rota-tions were

_{always positive,}

that

is,

in the same sense as the

_positive

current flow in the

_magnet

solenoid

_surrounding

the

_sample.

The rotations increased

_nearly

_linearly

with field

_strength

but showed

_signs

of

_incipient

saturation at fields above ro ooo Oe. The Kerr effect showed a similar behaviour as the field was

_varied,

but the rotation

was

_always

_negative,

and

_{rapidly approached}

a

limiting

value of about o.20 as the film thickness

was increased

_beyond

about

_{5o m ju.}

The reflected

wave was also

_{elliptically polarised}

to a small extent. The Kerr effect is relavant in the

_optical

case, because

_(a)

it alone has so far been measured for

a _range of ferrites and

_{ferromagnetic alloys [6],}

and

_(~)

Cau’s

_{phenomenological theory [2]}

has

shown tlat the

_Faraday

and Kerr effects should

in

_general

show similar trends with field

_strength

and

_wavelength

_except

that their

_algebraic

_signs

should be

_{opposite. Among}

the

metals,

IVIn - Bi

alloy

is the

_only

case recorded

_giving

a

_positive

Kerr rotation over

_part

of the visible

_spectrum,

but

magne-tite,

Zn-ferrite and Cu-ferrite all

_[6]

_apparently

give positive

Kerr rotations

_(and

_presumably

therefore

_negative

_Faraday

rotations)

over a band in the

_{longer optical wavelengths.}

The established

_{quantum-mechanical}

theories

_[1]

of the

_optical

_Faraday

effect

_(including

Hulme’s

theory

for

_{ferromagnetics}

_[3]

show that the

_Faraday

rotation should vary as the square of the

frequency

of the incident radiation for all

_frequencies

well below the atomic

_(visible

or

ultra-violet) absorption

frequencies

of the material. Molecular

_(infra-red)

resonances do not _appear to contribute to the rotation

_[7].

We should therefore

_expect,

even

for materials

_{giving optical}

rotations as

_large

as Cau

measured for

Fe,

extremely

small

_Faraday

rotations at the

_highest

accessible radio

_frequencies.

In the

_present

Note,

measurements of considerable

Faraday

rotations at a

_wavelength

of 3 cm are

reported

for certain ferrite

materials,

and the measurements are shown to be

_{qualitatively}

consis-tent with the current theories of

_gyromagnetic

precession

resonance in such materials

_[8].

2.

_Technique

of Measurement. - The

measu-rements to be described have been confined to a

wavelength

of about

_{3 cm,}

but further measurements will be carried out in the near future at both

_longer

and shorter

_{wavelengths using}

the same

_technique

but

_necessarily

different sets of

_apperatus.

The

apparatus

employed

is

_essentially

the

_waveguide

analog

of that usual for

_optical

_Faraday

rotation measurements.

The test

_signal

is obtained from a standard low-power

velocity-modulation

tube

_{(" Klystron "}

type )

and is

_passed

into

_{rectangular waveguide}

of internal cross-section

25.![

mm X 12.7 mm, in

which is inserted a matched variable attenuator

306

of the resistive vane

_type.

The

_empty

_rectangular

waveguide

uniquely

determines the

_plane

of

pola-risation of the

_propagating

wave, and it is necessary

to _pass to a circular cross-section to enable the

Faraday

rotation to take

_place

_freely

in the

_succeding

sample-filled

section of

_guide.

A transition

_length

of

_{waveguide accordingly}

f ollows the attenuator and has the effect of

_smoothly

_transforming

the

Hol-mode

of wave in the

_rectangular

_guide

into a

H11-mode

in the 22.2 mm diameter circular

guide

cross-section at its

_output.

The

_plane

of

polari-sation

_{is, however,}

maintained

_through

the tran-sition section.

The

_sample,

in the form of

_{polycrsytalline powder,}

is

_pressed

between

_quarter-wave

slabs of low-loss dielectric

_(,,

Distrene

_")

in a short

_length

of circular

guide,

the

_sample

thickness

_{usually being}

a few

millimetres,

and this

_length

of

_guide

is

_arranged

centrally along

the axis of the

_magnetic

field coil. The free face of each of the

_quarter-wave

slabs may be coated with a uniform

_layer

of colloidal

graphite

(,,

Aquadag

")

for the _purpose of

sup-pressing multiple

reflections,

a _purpose

_largely

achieved in the

_optical

case

_{by tilting}

the

_sample

face

_slightly

off-normal to the incident radiation. Two forms of

_analyser

have been used in the

cm-wave

_{apparatus :}

_(i)

a

_simple

silicon

_crystal

detector mounted

_transversely

across the diameter of a further section of circular

_waveguide,

rotatable relative to the remainder of the

_system,

and backed with a

_{tuning piston,}

or

_(ii)

a second transition-section of

_{guide transforming}

the circular

_H11-mode

back into the

_{rectangular Hol-mode,}

followed

_by

a standard matched detector unit in

_rectangular

guide,

this transition section and detector

_being

rotatable as a whole relative to the remainder of

the

_apparatus.

The second

_type

of

_analyser

is

preferred,

as

_only

one

_polarisation

_ciomponent

is

affected

_by

residual detector mismatch in this

type.

Apart

from

this,

the use of both

_types

is

exactly

as in the

optical analog,

the

analyser

being

rotated to obtain zero or _{minimum response}

for each value of the

_magnetic

field,

the

_angle

of rotation

_being

measured relative to the

_analyser

position

obtained with zero

_{magnetisation.}

3.

_Experimental

results for certain Ferrites.

- In the

preliminary

measurements

_unexpectedly

large

rotations,

together

with

_large

_{ellipticities,}

were observed for several materials. On

investi-gation

these effects were shown to be

_largely

due

to waves

_{multiply-reflected}

between the

_wave-guide

transition sections. For most of the ferrites exa-mined the

_Aquadag

_coatings

referred to earlier have been found to reduce these reflections to

unimportant

proportions.

In some cases,

however,

especially

for materials of

_high

eff ective dielectric

permittivity,

marked

_ellipticity

remains,

almost

certainly arising

from

_multiple

reflections between the Distrene slabs and

_entirely

within the

_sample.

A set of measurements, in which the effect of

spurious

reflections are believed to be

_negligible,

are shown

_graphically

on the attached

_figure

I ,

Fig. I. - Effet Faraday h 9470 lVlcjs a 20° C.

The materials concerned are a series of

_magnesium

ferrites in which the

_magnesium

is

_{progressively}

replaced by

aluminium. The

_approximate

compo-sitions and room

_temperature

saturation

magneti-sations are

_given

in Table I. These materials

were

_{kindly supplied by}

Dr Welch of the

_Royal

College

of Science in London.

TABLE 1.

The direction of the rotation is

_positive

in all

cases, and is seen to rise for

_applied

field

_strengths

greater

than the

value,

to a constant

_angle

approximately proportionnel

to the value in each case.

4. An

_{elementary theory}

of the rotation.

-A

_theory

of the rotation is

_implicit

in Polder’s

analysis

of the

_{gyromagnetic precession}

resonance

307

that the

_{applied magnetostatic}

field is

_{large enough}

to ensure saturation and

_neglects

all

_damping

effects. It is found

that,

when the static field is

along

the direction of

_propagation

of an unbounded

linearly-polarised plane

wave, the latter

_splits

into

positive-and

negative

circularly-polarised

compo-nents for which the effective

_magnetic

suscepti-bilities are

_{given by :}

-where

_Ho

and

_{J 0}

are the static field and

magnetisa-tion and ojc --- _y

He

is the

precession frequency

of

the ionic

_magnetic

_tops

in

H,,

(for

electron

_spins

~ = 2.8 Mc : s :

Oe).

The rotation of the

_plane

of

polarisation

in an axial distance z is then

2

_being

the dielectric

_{permittivity.}

For the

_sample

_shape

_employed,

the

_{demagnetising}

field will be

_large,

so that under the

_experimental

conditions we have Ú)c - = ú).

Furthermore,

for the materials considered we _may

_approximate

so that the rotation may be

_simplified

to :

where (ÜJ =

4 T:

J, y.

Thus,

within the

_{approximations}

stated,

the rota-tion should be

_positive

and

_{(i) independent}

of

ferquency;

(ii) independent

of

H,.

and

_(iii)

pro-portional

to The exact

_{analysis predicts}

interesting

behaviour for

_higher

values of

_H~.,

in

particular

a finite _range of

_Hr

over which

_only

the

_negative

circular

_component

is

_propagated,

but

_damping

_may

_prevent

this last condition from

geing fully

realized.

5. Discussion. -- The

predictions

(ii)

and

_(iii)

of the

_theory

are seen to be

_{substantially}

confirmed

by

the observations once saturation is reached in

each case, and the

sign

is also correct. Measurements at other

_frequencies

are still

_required

to test pre-diction

_(i).

Insertion of the relevant numerical constants in

_{equation (3),}

_however,

leads to

_predicted

rotations about twice as

_great

as those observed. The

_discrepancy

is

_possibly

a _{consequence of} the

neglected damping.

If a

_{damping frequency}

_{(,)1 is}

arbitrarily

introduced’in such a _way that

_{equation (1)}

is

_{replaced (consistently}

with Frenkel’s

_{equation [9]}

for the

_{linearly polarised susceptibility) by :}

then a value of WI

comparable

to w appears to be

required

to

_give

_agreement

with the observations. It remains to be seen whether

_equation

₍₄₎

will

satisfactorily

describe the

_experimental

behaviour of the rotation over a wide

_frequency

_range.

6. Conclusion. - The

Faraday

rotation

_technique

provides

a new and to some extent

_independent

method of

_{investigating}

the

_{magnetic properties}

of ferrites and similar materials up to the

_highest

accessible radio

_frequencies.

The

_degre{e

of

corre-lation between the

_preliminary

test results here

reported

and the

_theory

outlined,

.indicate the

desirability

of

_extending

both measurements and

theory

to _{fill in the gap between microwaves and} the infra

red,

taking

account in the

_theory

of

exchange

interaction at the

_higher

microwave

frequencies.

7. Remark. - Some measurements at ), = _{i o cm}

have been made since the above

_{report was}

_prepared.

The result for material NI has been added to the

original

graph.

The

_peak

of rotation occurs

approxi-mately

at the H value which

_gives

an

_absorption

peak

for a small

_sample

of the same material in

a transverse static field in a

_{rectangular cavity.}

8.

_{Acknowledgment. -}

This Note is an account of work carried out at the Post Offlce Research

Station,

Dollis

_Hill,

and the author desires to thank the

_{Engineer-in-Chief}

of the Post Office and the Chief Scientist of the

_Ministry

of

_Supply,

_for per-mission to

_publish

the material.

REFERENCES.

[1] SCHUTZ. 2014 Magneto-optik, vol. 16 of Wien-Harms’

Hand-buch der _{Experimentalphysik, Leipzig, 1936.}

[2] CAU.2014 Ann. _Physique, 1929, 11, 354-449. [3] HULME. 2014 Proc. Roy. Soc., 1932, 135 A, 237-257. [4] VAN VLECK and HEBB. 2014 Phys. Rev., 1934, 46, 17-32. [5] BECQUEREL J. - Le

Magnétisme,

vol. _{I, Paris, 1940}

(Report of 1939 Strasburg Conference), Instit. Internal.

de Coop. intellectuelle.

[6] International Critical Tables, 1929, vol. 6, 435-439. [7] LADENBURG. 2014 Z. _{Physik, 1925, 34, 898.}

[8] POLDER.2014 Phil. _{Mag., 1949,} 40, 99-115.