THEORY OF THE DISPERSION OF

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..
Research Laboratory of
.0j
lectro
Massachusetts Institute of Technology
THEORY OF THE DISPERSION OF
MAGNETIC PERMEABILITY IN
FERROMAGNETIC MATERIALS
AT MICROWAVE FREQUENCIES
CHARLES KITTEL
K
Of·kr3
co
TECHNICAL
REPORT
NO.
2
MAY 20, 1946
NDRC DIVISION
14
RESEARCH LABORATORY OF ELECTRONICS
MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
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'at':[email protected]?
'by
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et
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4
131
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to
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ce
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o cJ
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r
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r
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qit r.,Lrx
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1
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p
I'
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- -...t-·al ar pdi~
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X
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A,
P-dSd~t~nr,o~/.-5.
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i
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tanl
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ezmibi.!tZ,
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are erria
~. ay~, i'
J.:i&2. 15
. *trs.,;
3;'p/;$el'~ 5. .
d '
iv.f the ejC.tive
p ! X,
It b1t
-
s
.
I
t .I
that
) anh p.::Ti
Ot t
27
I
v
e
,;
5r,Trher
"/:uzf:v
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Y-irk'~,
,')47
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;,
fC' ,
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:
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o,
t,;'A.; ,,S<'
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79<
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wJ .~~~~~~~~~~~~~~~~~~~~~~~~~~~Mr
;r6 t,;ratton
C; t . rlJ ,$,
s,...v
trv:....................
't.._{
A.
...~rg*
eetromane.,,
Tner,.rcm;?.G
'r' ,
;.
?.41'82: J, Ct b8te,
p 9.5
'~*42
.
i crowve ;:;"
;
':
.,l,.1 ~ nt:in*-t
'CrAAf;,L
itA
,
'ord
20
ec
hiat&,
ac:-
.'te
c t;i,, Or
Z .:<
..
hr.
.,: ?ecto
'.r17 I
-~c~ypj
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i '
fH
=. ;.
;
-
1
9~
,?.:,,ary.
cfin'a r:
i
,r
-1;,
LJ7,
LIE
1
pitnnc
tho
*, tm,
'r .tyet, - t
':oe.lft't
9·"
.-
'
_,
:.'~"...;.9 to caxiu*a~?
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'fAt~s~'r
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:>r.,-a'
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by!e. f3'; 1te ,
.
......Cc
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t, I,2.SLV
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;;
r
.:sl.-ft.''
... 5 "'
..
,.. . ,
detr rt
I,<
,'
t
- 11.
.r..
r.d j
so tht
.ir e emfe't w,ath t
iy
/
tnh
Thi! : ,: '.
urit-iy
ve'!ue
.]Ww
reloqueAcy
cc-r e.r.
i >)
tai r./
in Table 1,
with &s:
c3!:tance from,
Eq, ((27) I'or variss value
were calcus, te;
m/
aet
vuvs
Kem Lble:s s t
'iorse, Fshbach, and. laetmuitz
Theore.: ice'
1. I.n
t!aVen and KLubens
sati.-f'yitTe,
This 2.-e,3u: -
a·,-:r,>ac.hes
.erm',
eSi..ro_,:,t't
Values of
are gI'm.
roo~a
Ice
:\ . that :~'
o'
,
catIanrd--t ri d
h
2I a,'d the tables prepared by Lowan,
I
,;a--andip,
Yvai
a re a.o
,-ven in Tabi.
i.
~- ' ;
~ac 7or.: tn.,:
he , ?el.',
,--:
.,(4>~i.
-), r where Vi
w,,. .1
deati f
accordtna
to the
.ni "~' ~'~
r--:,~.?
and: (-~le
.ol
ex)
"join
eaX
UOh1
.F
cM
exp
calc '
,',.o tha't /i
is
the reaIl
ttanh
,.ndl
is tte rea3.
.
I.
r IF
....
yx
tanh
n1mber w
r,he
f
caaber
h' ch s'i.:fies
t_ '
r
-w
Ma f..i
of iE
rel. pat
t arlP
--_ -t
t -V
~''~"t:.tUo 11( -anh pd/pd,
/F
''
,.i
satisfiess th
st::h
L -
the .malg:i.nr:
tanhl
-
s
-
tt
,
lil,
d
pd'itpeA
pc
; 27&2
..rt .I
:
(.tJlh
(2?-
27t.
S K_mely,
abe-:
J. .
prboli
ar : .
l
:
'-Harv~ra 'rilvex.Aity Prss
.
,ambrdgez,
'
19J4),
,
Lo,waz,
E: :K
.VXorS:,
h. . Fegbach,
au)
&
1bateWit.
.abs~ f~;r
f,
olution,. o the ,e
J;.ir, hetanfor
ar 'Ourlarie X avi.np
ljm'~etXnmeacet,
pApliec tkathenatlccs
ane Note No 1.8; Section .:,.
.'L
- sr]-1'46 - 2043
Jure, 14
unclas.:f .ed.
i
; 14
I
Ftirr Th~ci~.nrs 2, 5r; 1O.
I'erreabi i ty
Pstraet ?
Fr:quencry
0,31i
8~,
.
20O3.
0.
a67C,
'0:
77,..
.
&M.8
7:?
16
-79
~
o80
)
175CO8;0;
:I)
r
2i.'L00...
O
5
Inx Fg
4
ind
exoerimental curves for
,
are
whnich .ls the thtckn-'ss of the domaln iii.
2.48 x 10
cm; tils
experimerrta
is
.;:ata,
theo-e ical
but
f
r?.;r;t al
'he
-11tp.
e .'
domain iinmaerst¢'
· ,,.' v
t1hese
iai.
C
thlckres
variat.ic.t..v-
.. tU
:.:
the
- 3
2.6
:
,
,
0
.
, 79
i:noothed
witi. te
The arbitrary constant
as been
theory or'edicts the
and
'
,
aken to tbe
el
curvc
.n t.t,
,h
sope
to :
* he
t
:t.t
-'
i .2oT,
,s ari
h:-
t.. t,
i.s soomewh-t
,-n tne
l.lkely to i-:, as --. ur.ea
c:.-
r-tealit
.ir
:cet
itut
a
li;-:
";ow ,
'xprol-
Le etti-,
l.easoreai
s of
meac. reuit th'iTry
t
rItert.l
are st, eper tian the
th,1ckne,
ii , dcrain c.mer:ans
act
:
to
The fP:,eral natrure
in accordance Vth thne e x
',f the film ;a wthin 't.
.th
,tI
._,boa.;.ih
orer of manlit'x:e
g, curves c orr ectly,
the t1heors tica!.
ircepr.n.v
ar.at.ions wt.
,.' t ... ..
0
24,5
model,
.eab, litv chan;-e is
;Se SiAopes
r
'Th.
s
curves ceincde,
that te
seen;
the s-cna.linr between
(!: th
32
;1
o
value was chosen to iake the half-value oints or th:e
and theoretical
It
i 7
- 62:
4ottao toet>er-
ron taken from Fi?.
70
18
- i:1 1
:I. O
,
100
102
10"-
1
3n5GOt.
:.
r4;
r
t
d eri
i
,'
;¢,
t A
1
i
ox';?t
.r
i,) tL'-v
prier- t
to
0oTable
..
'¼
.at
U'C:U
ca,,
U.,.
'. tne
,.i
Imo
in
<>OCER
.> .c.Xvn
+he
ncrrfi~
r....m.,A
'&
.r:,(", mc/rec the 'Iab~"1 l;y/
i. - :rr,
k
is
c
'f
r
l ue: ze , e
:2ra
ihi:C'5
itaaginary,
].,
______i_
111II_---
gl_
Ir on
accorcdtng
yl."
S
51^' :.8£.9;
ake
n
~" e.,~ue t-les
fr
w'ni.h nma.6net'i
eaT
see pie
dr.t
r
and eddy crrents, yet.
fro
,Egrite
(tFe
4)
n;-iron at the sre
ti-
17
pertilnent wnen t
21,itt i
:C:.--O mc/'fec,
roume:nt
th
riAne
4srln-ortit
1
r,
::.
onr
The di.c
c.c.ttt.
t
t.he n.in depth
so t
interactionre .a
is.
't
os.,,.-{
'.r.;i
t
a.,f-.
is
'i.
:.;f the c,
' .... }Teferelnce r;
:'ith '7,z
stai
ee a
,
Lr
ife.t each
a't;s ann:; i . G.
-
:;
.
iIro
-:
A+ liet.vkrct
tA"c;
.,',;et
i
,. t
.
nnij,..Fhye9.
*
1Vj
PhS
7,
, ' (1922).
17, 98 ( 922; K.: Karts- hain
Annr d Pys, tvI
The xperieits by Kartscharin an ct tlers in, whLi '
di'.scredtd
A-;
of the .ferromagretic se2Cc, l45.6[1C
wtr' fv` lrted Olt
S
e ul
wrter b Prof
A
r, !ippe.. wt - .
t nv#.stipate
32
t, erT exeri,enta y .
i''!.~frt..rrn'; ,
. '-Xa!
cTables,
ol.~. . ~ D-,-
,'
I
oyarte,
;:
t,e:
' .-;
e.
rvoted
..
fil
or fecuncv. L;.
,,,
Uacnet^
onrlie
;
d
hp
rret
_
's.ca.
resonsa.ne henomena w-.e re,~rted are
__
t'reouency of
ener2:.icaliLy
':.ishe ifl'ren
-7,.
1'tre all
1,.ar
oc,urring when
i
Z.eit.
The interestineg
24·,
_
a
This is the fi.ti.i drile t,
r.,
ir:iider
wvln- avr!,C-,;'C
elaced
I.,ie:l.J
c:,.es.ondirn
:
Ia
houd al'z be
,equal to the Larrcr
'' ...at;.i
' r';.er',. ti~=
O
1 6 X
with
tthan the calcul.ated skin deptLh.
t
te
e!<;th it
The rKdi
[materia1s
of' ferrognetlc
the Anmermnal atFsoto:.py
aUO,
ish.n
initial pereabillity '2 of Magnetite
fi.Li
.e
tickons is .ess
o)rder of ';.A
receding
ffect of skin
m./sec, as comacred
at 1.0
D.i:
.;i..;, netiz.at-lon ir t hs
he
.
2434
,etic
resornr.ce
e.ftects
s o.f i:ron.
T he.e
are ray
e ti' rromajretic netals,
rl
tht feouecr-y of the, ap .Aied 'iid
.r
it
cc- ;iae'ued VFoehYave
The resonrance pnomenori may be ,inderstoo
atm.c suins
an
d c-or,1ete1y the
tJ.e freve
e:~.,
f ecui-c..
.-4a5u.3eeat..3
"
imortance.
t'recocun.t
hlt
.- · 5
at
tha
h:4fcxewirei.
;;t*,ezed in 2t.hcl- ,I eti,n".fP
-Xg
.r. tbat.
sl id:
I'err:,aie.et c buit have Ucw e,3i.cv'': .
ctf
be much re rater
1'.m nd at hJig
.found expertmEontall"i'
b.~
.ll
Bii:
-^ivig
ce
ect]tet
..
¢.u^:
ctfect..s are of
... a..
a
in
.
The -hrtvd t"Ion~ have all
sect, on
. 1-'
.f
.
;= f Tlec&t
sse-
r
¢uv't:e
,ly wV~?:I b)e
'
:have .:oti !e.;
Cone of the r
to see
.lt;lt
:~.)t:.:r
j,. i.rC.~zit,;:
i.
rcrsion
Sch dffect
ps
l:;,:
t ;r t
.:sio'v
'.;'.Fr
:a .:, .'e1 u?
.
i';*r
Q rh
utei
n
,',
Ir
-
-^^-----r··l----l
----I
s½sitteS.
----
---L---
-···l------rr-··----·------^--
--^---I-U-L-----·L1-n--·-^·---ll--
--
-·llr- ----
----------------·
ix a de'inite direction relativea io tho latice bty mcuris
of iiviu.p
coiled springs representing the spin-orbit interaction, and the magnets
will oscillate in an applied field with a component pe,'pendicular to the
rest position of the magnets.
The sine of the angle of oscillation is
proportional to the macroscopic magnetization. Resonance occurs when the
applied frequency is equal to the free period of a magret + spring unit,
The bar magnet analog supposes that the relaxation time of
the pins is eufficiently short so that gyrosoopic effects may be neglected.
It is not usually reoogized that this assumption ts being made. If this
amntion is not true, the spine will precess about the field direction
without lining up, It is indeed a prerequisite for any type of maietization
that the maeetio moments havo time in which to line up in the instantaneous
he calculation of the relevant
local field to which theq are ubected,.
relaxation time is a problem in the kinetics of thermodynmlo equilibrium.
The time-dependent processes can be described by assuming the
existence of a relaxation time, as was done b Gorter mand ronig for paramagetic relaxation, and by Landau and Lifechits for the frromagnetic case.
he quantum mechanical oalculation of the relaxation times, starting from
the detailed interactions of spine with the lattice, is extrely difficult
and uncertain, Oalculations for the paramanetic case have been made by
Waller2 and others.
No calculations have been carried out for the frromanetic case, so far s the author is Gware. It seem plausible to uppose
that the strong spin-dependent coupling in ferromanetic materials will
assure that the relaexation frequency will occur bove the microwave range,.
This question should be looked into more olosely.
It should be noted in passing that the collision frequen
of the lattice phonons at room temperature is~10 2 oollisions per second, as
estimated from Talues of the thermal conductivity of non-metallic crystals.
determine an approximate upper limit to the order of mantuldoe
This fire
of the spin relaxation frency.
The aotual spi relaxation frequenc may
be lower depasdn on the trength of the coupling between the pions and the
lattice. In metals the relaxation frequency of the lattice phonone is
estimated to be of the order of 10 soe
at roon temperature, based on
elctrial oonduotivities.
s
ef.
r exaple
Pi. Lez,
Zeit, f. Pys.
370 (1932). The
writer is indebted to Prof. L.
. iesa for sevral discussions of the
parmagntic relaxation problem.
2318
·__··
1-11·-1--·
1.
II
·- IYIIIILIIIIIU-L·IIII
_
·1_1_1 _1_1_
I
Kir
!.&.'&Te
> 1..hp
It
bt' i,,S;o.anIL. iTIn fu'co,?,:tzi.;',
s,,?trert
y
urrentS
aussccia+ted wt
ta
'LI-:
V A
tr
,
ho
sker3i d
t t e l clt
.
raagnetization in met.'u
izi
r:'ig, s.althotugh an apnrt,
has bsen suggested by
the local eddy
Mi-
'A Wet 29
.
It
.y of boura
eddy current
by the applied field is
Becker also ht':
pemlteabiL.ty,
respects st.ilar
ind
d reft. -rc
nc
eouattin in te protent
cause of disperison.
given a calculation for the dispersion of
this calcuvation leads 'to results 4r. some
to those of the
roesent
paper.
The ':beck fieL
c.. a.ed, as
n the mediuin field strength case Just mentioned,
cf Pecker'
thaory- supposes that the skin depth it rter
dimensions,
so that the cslc.ttoo
wheret he skin depth
kin depth
the
is
s ls;,
,only.C0,`
t. that
euetr-t.l.on o£ the surface dorains
iaconpllet,:
the ,;'"r
pro'agation
Er
sTil:.d be point:id ,,
current effects cnsi.dered by E-scker have
paper, according to which te
-ati'icro'vT
onc:tng. t. ess results with
on tix.e veloc
friTonhr,
nection with the use made of te
the initial
½.n? rr.
t";+ ",'tc:
the measuremrents of' Sixt-uas
)
gooc
,Iva a
at `rcqutenc es below te
f.i.ds,
4
the edcy
irscrversible d is.:-,!.caceneteL charsc;ter-
,i'
t,:c,-rg,.
pro-)ortional
rntaition; that is,
sn
t
;Bcker' s treaatment
fo.cs.
.
ike a vt' i,
oualltattVie acc.tcut
fI..eld which opposes the appl.ied feld,
or
to the veloclity of tboundaoy .ve;e·,~~
currents behtse
'dty
mi.ocrasonc
ua-.t.ors of motion which is
t t:n
..
s ter
adds a sri
.t.¥rrd
dev
botunda4'!es and the
:to¢ilvatW;L ol' do: in
he
',
rotation of dot.ais eet ut:,o :raw.stis.
This back fiel
was
aroia;h
acL:rr,
Th
r:VL
iL.t;
ect,
urer
c.,y
that
J't',
than thle domain
icr.wave ranpe,
At ?ix 103 me/sec
dlmension3a
than the domain
c,
The basis
is rot applicable to t te
n
is cail-
.omain thickness
th
The present, th:ory probably could be improvec by wcrking with
a more
is
28,
i2c:
_II
I_
_1
that otf a film one domain thick,
.If the film
hacked on one side cy a maess of ferromavnetic laateriat the roti.c.r. .f tfn
2~i(aal
4c.
model tlaS
,ompii.at.eaea
In
bunda.ies
l. r
W'. Akadiew, C
.eference 3,
tb? filra wl
iu
ec ty currePrnts
i, Acad,. S
URS
'5'
Doklady2
See refere.ce 4,
H; J, van JLeeuwfer..
_11__1
_
__
Prsica
4i,
'.l4
,
ain 'he
1
4
3, (i944Y);
C
___
_I
_
_
I
acking
.
,i` ".; -4·
i..::i r· :·.R f
I
:I=·L
(
ri
.
W
l, .&
'
i
..
.·
1
, -{
-··'
·.·r
~
.;-·a
t.,
;*
>t
5 -"fAs
S
;e
._
;}n-n42j4a
pt
,
c<|4, i:lsil-'~~ ,
be du
tn
the
'.'<)
,
4
.-
,
no .=s, :,~'~;.->. :0*:'.5;
; . - ",
~
'4
i!
*i
to tho ,.:.
nc i er t,
>· .(
·
.. '
-
;ii;
'
l i~~%0.t;.R.n
~~ :.,arat,
~
4 n~ ..in,
on].y
';~e
t
i
-
~·I ·
1
o$ .h~¢of:~tT'
:
.;'
; ";.o~ -0
r0
·
s
:".·-..'':;
.~>rlt;~
.. ar
Cf~d ,SU~i:'13'
tke
.,
....
,.·
~
-t.,
'
..
-I"
-;'
-
'"
...
3,8.1:.
'-:
.,.,
,-6d~
.
a
::3,sviraw
';-,.e.%;·
r~
r
~.:a .w,-,;Una In::·
3..i?
,,,8 t,
~,c';,~..::
,rX:
,2(-.
C! rs
'
'e' + ''
t airt, alg
;
C, hS~;O^ ~
o-,,n
'r
'i
P.;
Fi>-S
;
'$t
:..
A~
.'.9e,
"¢p~-¢v.~5
t..;o
T~m~tfigcio
i·;f
:·
i,? ;.(, ~
.. r
" -t-,..
- kA "" ·
~--m'm
........
t 'C.
.-.
,
Oi R ",d- ,,f
' !?:e G
*
- .''':i
§ · )>
..
...
obta&in-s: :i-om
xtho
frat
~
;v
8
f
qthaltzct
bi
on,
w
rlt
;:",,o ;.,,::~¢
'':t,'-'
i
;.i,
.
r
i
.. :
.t...:
:,~h.:?~:
pti
de.,an b·4at,av-o
[email protected];
- .; :.
:,
...
T:Tatv+,f
(.D}*
nt¢ ..
the infuenc
omei
('.
'
li-.
~
t.,.iI
;~y ~ ,~·.·;.-r,'.:~
.....~';: ?
w'c~J~o. ~,.~r-3 '~
Cf thretco.i
.~~l
.·
i
f~'~ ;'?,~:~.
mev
....
i
r..
*t,- ·· i
rt],r
.
W,,
'··· · ·
.·.
I
·"
~~~~~~~-- ---------r-----------
I
I
,
II
artJ lrndu
It
ancs-
o
F k
can be shoLrr t hat
;e
'eiti
considered in Section IV anr te
oFe.S3 i
e
ont.i-'aut;o
o
ts
£ti
the film to the
inductance of a cire;it ele-'ant art rel.ated .Jt.e.y
to the intrinsic
turface Lmpedance whihn is defined c-oril.ng to Ec. (22) by Z - E /Hx,
= "
evaluated at
The avrage rate of ergy' [;.(os
par tulit -rea normal .to the
y directin is given by the avr.E !ge value of the Foynting vector
when
t
s
ecalled that the film
ats to
surface,.
Now E
Z H,
so that
s
(c/4a)
-
(d) 4"(d) Re[Z],
a well-knorl result,
The cont-bututo'n
flmn to the inductance of the
magnetiing circuit is3 given by the quotient of th magnetic f.Ex through
the film by the current in the magnetizing circuftt
1
R.
w (
Now J - (-C/4x)Hx(d)
__1_____1___*__·1____
J
+ J,
and
x Uqx = + j(/,)(
~
_1_1
of te
________·^·_·_·_^1_____U__IIIVI··_
dyj'J
i
so
EA
/6y),.1~~~so t~at
= 8x RePeJz] - -
-I·ll·llilllilyl·IIX----.s---LI
s Ix.fJ.
·-- ^I
I
I
---
t*
.
J.
*·
I
'.
h."tj-;;,
~ .
~ -t-&.:tA
A
4.,a,
i
.
.. t.
.t;.~~~~.s
b
*3.p~~~~~~r
.
I
't
~tit
CI
-
r·
,',
kt,tli
1¢.t~s ;g·
1
-, * J
i;··)
1"'ii(';
4,
.*r-7;4+_
-,,r
'.
5Ce
;
;t,;
.t
1
.¢
jfI.¢
a··.
. ;.
P.
8
,
X
8
'"4 -S
,i
i[e
-s_
, -s
I
~
U
IRON
Jx.R
POTAPENO
AND
SNERRON
POTAPENKO AND SANGER-IRON
*
A
ARKADIEW-ANNEALED SWEDISH IRON
ARKADIEW -SOFT STEEL
H
O
E. MAXWELL -ELECTROLYTIC IRON
E. MAXWELL-COLD ROLLED STEEL
A 4L
O
O
A
X
-
POTAPENKO AND SANGER- IRON
LINDIMAN - IRON
HOA& AND 3ONES- IRON
HOAG AND GOTTLIEB- IRON
GL ATHART - IRON
200
.
100
F
A
;j~
I;
.I
50
AN
-
.J
-
\X
20
0.
1\
G\
E
\s NO\
10
\ A
\e 6
##L
!hl
5
I
.
.
O04
FREQUENCY
MiC/ SEC
I
_
_
NICKEL
*
*
RAD R
ARKADIEW
E. MAXWELL
A'
L.
0 LI NDMAN
A HOAG AND GOTTLIEB
0 GLATHART
100
>.
50
I.J-m 20
03.2
10
\
Gr5
wiI
M 2
L
o
0
N3
3"
R
O~1
I
l
I0 2
103
FREQUENCY
Mc/
SEC
.-
··--
.--
·------
.
-
-
-
-
-
-·-
-
·-
-
-
-
H
H-bIx
ItI4I
-
I I
I
-d
_^___11_1___11___1_
-1-
IIYPI-^---------^I
-·--I-
--1
-^11111-
·1111
._--lll-lCl-
200
m
20
EXPERIMENTAL
TREORETICAL .
\
UJI0
c
21
F RQ
I
z
CY
104
105
FREQUENCY
__I___·I_--L(-PIY-Y·IIl(I·L·IWIIIII
M C/ SEC
-li·.---·
___ II(-IYI(IIIY·II.---- -ILL-ILI--·---I_-IY-·l
·
---
----
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