NEUTRON DEPOLARIZATION STUDY OF
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NEUTRON DEPOLARIZATION STUDY OF
FERROMAGNETIC DOMAIN STRUCTURES
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BIBLIOTHEEK TU Delft
P 1924 4070
NEUTRON DEPOLARIZATION STUDY OF
FERROMAGNETIC DOMAIN STRUCTURES
PROEFSCHRIFT
ter verkrijging van de graad van
doctor in de technische wetenschappen
aan de Technische Hogeschool Delft,
op gezag van de Rector Magnificus
ir. H. R. van Nauta Lemke, hoogleraar in
de Afdehng der Elektrotechniek, voor een
Commissie uit de Senaat te verdedigen op
woensdag 23 februari 1972 te 16.00 uur
door
Matheus Theodorus Rekveldt
geboren te IJsselmuiden
1972
Dit proefschrift is goedgekeurd door de
promotor prof. dr. J. J. van Loef
Aan Jet en Marlijn
Aan mijn moeder
CONTENTS.
J. INTRODUCTION.
. 9
REFERENCES.
11
II. THEORY OF NEUTRON DEPOLARIZATION IN THREE DIMENSIONS.
12
A. Polarization change in a homogeneous magnetic field, 12
B. Depolarization in a partially magnetized ferromagnetic foil. 13
C. Determination of the mean square direction cosines of the 21
inner magnetization, the mean reduced magnetization and the
mean domain size from the measured depolarization matrix.
D. Depolarization in a ferromagnet at a temperature very close 24
to the Curie temperature in the presence of a magnetic field.
E. Discussion. 26
REFERENCES.
28
III. EXPERIMENTAL SET-UP.
29
A. Survey of polarizing methods. 29
B. Polarization device. 30
C. Description of the apparatus. 36
1. Collimation and reflecting crystals. 36
2. Guide field. 36
3. Polarization turners 38
D. Sampleholder for measurements under mechanical stress. 39
E. Furnace for temperature dependent measurements. 41
F. Adjustment and calibration of the polarization turners 44
G. Automatization of the apparatus. 46
H. Experimental accuracy. 48
REFERENCES.
49
IV. RESULTS OF MEASUREMENTS IN NICKEL FOILS AT ROOM TEMPERATURE.
50
A. Measurements without magnetic field as a function of stress. 51
B. Results of measurements as a function of magnetic field and 58
stress.
V. RESULTS OF MEASUREMENTS IN NICKEL PLATES AT THE CURIE TEMPERATURE.
65
A. Measurements without magnetic field. 65
B. Measurements with an external magnetic field. 71
C. Discussion of the results. 74
1. Stability of the domain structure. 75
2. Investigation of critical exponents within one degree 76
around the Curie temperature T .
3. Depolarization within 0.2 K around the Curie temperature 86
in the presence of a magnetic field.
REFERENCES.
88
VI. CONCLUSIONS.
90
LIST OF SYMBOLS.
91
SUMMARY.
96
SAMENVATTING.
97
NAWOORD.
98
I. INTRODUCTION.
The neutron is an elementary particle with spin one half and without electric charge. The neutrality of the neutron makes it especially suitable to study the magnetic interaction between this particle and matter. The knowledge of this interaction can provide much information about the magnetic properties of matter. This interaction can be studied in different ways, first by scatter-ing experiments of polarized or unpolarized neutron beams on matter and
secondly by measuring the polarization change of a polarized neutron beam after transmission through matter. This last method is in principle an adiabatic one and it will be treated more comprehensively in this thesis.
It should be mentioned that the possibility of polarized neutron beams was suggested first in 1936 by Bloch [ 1 ] , who showed that the scattering cross section for neutrons by magnetized iron should be different for the two possible spin orientations relative to the magnetic ^ield. In 1941 Halpern and Holstein [21 derived a depolarization formula, describing the polarization change of a polarized neutron beam after transmission through an unmagnetized ferro-magnetic material in terms of the mean domain size and the spontaneous ferro-magnetic induction of the material.
since Burgy, Hughes et al [3],[4] in 1950 have done the first depolarization experiments to determine the mean domain size in iron, this type of experiments lost the interest of the neutron physisists for a couple of years. However in the midsixties a revived interest arose in this area partly because of the much improved polarized thermal neutrons becoming available and partly because neutron depolarization is a method which can give much new information about magnetic properties in ferromagnetic and even antiferromagnetic materials. At the moment four depolarization devices are in operation.
Drabkin et al [5],[6] from the A.F. loffe Physico-technical Institute, USSR Academy of Sciences in Leningrad have developed a depolarization apparatus based on the polarizing power of the reflection of neutrons by a magnetized iron
mirror. In this way he obtains a nonmonochromatic beam of long wavelength neutrons which makes his apparatus suitable in particular for studying small angle
scattering with a change of polarization.
Ranch et al [7] from the Atominstitut der Osterreichischer Hochschule Wien has built a depolarization apparatus based on the polarizing power of a
magnetized Co(92)Fe(8) single crystal using Bragg reflection. Rauch et al have carried out depolarization experiments among others in superconductors.
Berndorfer [8] from the Physik Department der Technische Hochschule, München, has built a depolarization apparatus analogous to the one of Drabkin except that in this case the reflecting mirror is a Co(5o)Fe(50) mirror which
provides a higher polarizing power. Berndorfer has used this method to study the
depolarization as a result of variations in the magnetization due to dislocations
near the magnetic saturation in nickel.
The fourth depolarization device also based on the polarization obtained
after Bragg reflection by a (200) plane of a magnetized Co(92)Fe(8) single
crystal has been developed at the Interuniversitair Reactor Instituut, Delft by
the author [9], and this apparatus will be described and discussed comprehensively
in this thesis. The neutron polarimeter is similar to the one described by Nathans
et al [10], except the two polarization turners which have been introduced just
in front of and behind the sample. The addition of the polarization turners
enables one to determine not only a depolarization coefficient but a complete
(3x3) depolarization matrix of the specimen investigated. As a consequence much
more information can be obtained with this set-up than with any of the other
ones described in the literature. It should be mentioned that Forte [11], from
Ispra, Italy, has developed independently a sort of polarization turner which he
used however for quite different purposes.
In Chapter II a theory will be developed according to which the (3x3)
depolarization matrix of the specimen can be interpreted in terms of the mean
magnetic induction B, the mean domain size 6 and the mean square direction
cosines Y
,y
X y z
and
y_
of the inner magnetization within the domains. In this
^
Chapter also the possibilities and limitations of this measuring method will be
discussed briefly.
In the following Chapter the experimental set-up will be described and the
experimental possibilities and limitations will be discussed.
In Chapter IV the results of applying this new method to nickel foils under
tension are given. They illustrate the usefullness of the method in determining
magnetostrictive properties and in understanding the magnetization process in
ferromagnetics.
In Chapter V the results are presented of investigations of the domain
structure in the critical region of nickel. From these measurements it is
concluded that the depolarization method turns out to be very suitable to
study the magnetization and the susceptibility in very small fields. Moreover
from these depolarization measurements the critical exponents Y> v and 6 can be
determined.
So far neutron depolarization is the only direct method known which enables
one to study domain structures in bulk matter, in particular in the very
REFERENCES.
1. F. Bloch, Phys. Rev., 50, 259 (1936); Phys. Rev., 51, 994 (1937)
2. 0. Halpern, T. Holstein, Phys. Rev., 59, 960 (1941)
3. M. Burgy, D.J. Hughes, J.R. Wallace, R.B. Heller and W.E. Woolf, Phys. Rev.,
80, 953 (1950)
4. D.J. Hughes, Pile Neutron Research, Addison - Wesley Publishing Company, Inc.
Cambridge 42, Mass.
5. G.M. Drabkin, E.I. Zabidarov, Ya. A. Kasman, A.I. Okorokov and V.A. Trunov,
Sov. Phys. - J.E.T.P., 20, 1548 (1965)
6. G.M. Drabkin, E.I. Zabidarov, Ya.A. Kasman and A.I. Okorokov, Sov. Phys.
-J.E.T.P., 29, 261 (1969)
7. H. Rauch, Z. Physik, 197, 373 (1966)
8. K. Berndorfer, Z. Physik, 243, 188 (1971)
9. M.Th. Rekveldt, J. de Physique, 32, C579 (1971)
10. R. Nathans, C.G. Shull, G. Shirane, A. Andresen, J. Phys. Chem. Solids, 10,
138 (1959)
II. THEORY OF NEUTRON DEPOLARIZATION IN THREE DIMENSIONS.
A. Polarization change in a homogeneous magnetic field.
It is useful to define first a polarized neutron beam. The neutron is an elementary particle with spin quantum number s=j. The spin of the neutron can be represented by a spinvector s, which is an operator from which only one
component along an arbitrary axis can be determined. If the average of the spin component of neutrons from a neutron beam is unequal zero then the neutron beam is said to be polarized. The degree of polarization is then given by the average value of the spin component divided by its maximum value. The polarization direction is defined as that direction in which the averaged spin component measured has an extreme. Expressed in a formula
I^ - I
+
where A is a unit vector in the direction of the polarization and I and I are the intensities of the neutrons from the beam with + and - spin component along the direction \ respectively. The vector A can also be interpreted as the mean direction of the spins and the polarization vector P as the expectation value of the spinoperator s divided by its maximum value, in formula
P = - ? - = 2( <s >T + <s >i + <s >T ) (II.2)
2 X X y y z zwhere i , i and i are unit vectors in the x, y or z direction of an arbitrary coordinate system (xyz) along each of which the three different spin components can be measured successively. After some quantum mechanical calculations it can be shown that the spin operator s obeys the classical equation of motion in a homogeneous magnetic field H
^ = Y (Ï X il) ( U . 3 )
^B
where Y is a constant given by Y = gv, -rr- This equation of motion is also valid for the expectation value of the spin operator and by using this with (II.2) it is found that
^ = Y (? X H) (II.4)
This means that the polarization vector P is precessing around the
by ü) = Y - | H | , [ 1 ] , C 2 ] . Solution of the differential equation (II.4) gives
explicitly the time- and field dependence of the polarization vector
P ( t ) = P(0)cos lot - (P(0) X n ) s i n ujt + (P(0) . n ) ii(l - cos u t ) ( I I . 5 )
where P(0) is the initial polarization and P(t) the polarization after a time
interval t, in which the polarization vector was interacting with the magnetic
field. The magnetic field is characterized by its absolute value — and its
direction by the unit vector n. Formula (II.5) represents nothing else than a
rotation of the polarization vector around the field direction over an angle
ut.
B. Depolarization in a partially magnetized ferromagnetic foil,
The behaviour of the polarized neutron beam in a homogeneous magnetic field
discussed in the last paragraph, is equivalent with that of the polarized
neutrons within one domain in a ferromagnetic material. In general a
ferro-magnet consists of many domains, in which the ferro-magnetization is homogeneous and
constant in size. The influence of the domain walls, the regions between the
domains, has been neglected, which is justified as long as the domain wall
thickness is much smaller than the domain size.
In working out the theory the following assumptions will be made which
generally are not fulfilled.
1) The correlation between the magnetization orientations of neighbouring
domains is the same as for all the other domains.
2) There is no correlation between the domain size and the magnetization
orientation of the domain.
3) A gaussian distribution function for the domain size has been assumed, from
which the distribution width is chosen.
The result of the last paragraph that the polarization vector rotates around
the field direction can be used to describe the polarization change of a
polarized neutron beam in passing through one domain of a ferromagnet. The
interaction time t depends on the domain size in the direction of propagation
of the neutron divided by the neutron velocity (t= — ) . The homogeneous field H
Bs .
^ . . . .
IS given by H = — where B is the spontaneous magnetic induction in one domain
^o ^ _
and y^ the permeability of vacuum. Expression (II.5) written in matrix form
P"(0) = fi P"(0) (II.6)
gives
P"(t) =
cos ut
sin ut
0
sin ut
cos ut
0
0
0
1
where the z"axis of the coordinate system (x"y"z") has been chosen along the magnetization direction n and the x"axis perpendicular to the z-axis of the laboratory system (xyz) (fig. II.1). The double prime at the vectors
P"(t) and P"(0) indicates that these vectors are described in the coordinate system (x"y"z"). For the coordinate systems (xyz) and (x"y"z") the following transformation matrix is valid.
y"
^"i
=
cos (}) -sin $ 0 cos B sin (J) cos 0 cos (}i -sin sin 9 sin d) sin 9 cos * cos
r "1 X y z = U 'x y z (II.7) Fig. II.1
By using this transformation matrix to describe the polarization vectors from (11,6) in the laboratory system it is found that
P"(t) = U P(t) P"(0) = U P(0)
and P(t) = U"' Ti u P(0) = D(n,t). P(0)
(II.8)
where D(n,t) is a pure rotation matrix describing the polarization change in one domain which depends only on the time t and orientation parameters 0 and (J) of n. 9 is the angle between n and the z-axis and ((> the angle between the projection of n in the (x-y)plane and the y-axis. The matrix D(n,t) can be written in the following form D(n,t) l-(l-cos ut)(1-n ) (1-cos ut)n n ' x y +n sin ut (1-cos ut)n n x z -n sin ut y (1-cos ut)n n X y (1-cos ut)n n X z +n sin ut y
l-(l-cos ut)(l-n ) (1-cos ut)n n y y z -n sin ut X (1-cos ut)n n y z +n sin ut I-(l-cos ut)(1-n ) (II.9)
where n , n and n can be interpreted as the direction cosines of the X y z ^"^
the polarization change after transmission through one single domain. The polarization change after transmission through N domains is found by multiplying the matrices in succession. In formula
P;y = DCnyy.t^) D(n.,t.) D(n,,tj) . P^ (11,10)
Here t. is the interaction time of the neutrons with the i domain. To find an
1
expression for the polarization change of the beam after transmission through a ferromagnetic foil, it is useful to subdivide the beam in a large numbers of very narrow beams called subbeams, each of which has a cross-section much smaller than the cross-section of one domain. For these subbeams equation (11.10) is valid and the resulting polarizationvector P of the whole beam is found by averageing the polarization vectors of all the subbeams, that means averageing over all possible rows of matrices. In formula
->. ^ fil* K K K
£=1 >• •" J =1 j . = l J =1 Jff ^1
'' ^ ' (11.11)
.,, D(n.^,t,p . P^
In expression (11.11) the total number of subbeams is subdivided into a large number of groups, which are numbered by i (Jl=l,...L) and in which the same time interval sequence (t t.. '-19^ ^^ present. The .'/matrices in a row are numbered by i.Within such a group all possible rows of matrices can be found by summing over K possible orientations of the magnetization direction n. in
Ji each domain i of a row, where these K orientations do not necessarily have equal probability. By writing down all possible rows of matrices in this way, the assumption is made that there is no direct correlation between the magnetization orientations of neighbouring domains and also that the distribution over the K orientations is in all the L groups the same,
Expression (11.11) can now be written in the form
''Kh
°^'«^ ••••°^'i£^ ....D(t,,) . ?^ (11.12)
where D(t.p) is defined as
° ( ' i P = i . ^ D("j.'^i£) ("-'3)
j.=l Jiand this expression for D(t.„) can be regarded as the average value of D(n. ,t..) with respect to the orientation of n. . This average value D(t.|,) can be
Ji '•''•
calculated from (II.9) where the terms with n and n average out. The average X z
value of n gives the reduced magnetization in the y-direction which can be caused by a magnetic field in the y-direction. The matrix for D(t..) becomes
D(t.,) =
l-(l-cos Ut^^)(l-Yj^)
l-(l-cos ut^jj)(l-Y )
-m sin ut
iJl
l-(l-cos ut^j2^)(l-Y^)^ (11.14) where m = < n > = — ) ( n ) .
y K > | y J
2 1 r 2
and Y = ^ n > = — ) ( n ) . and the same for the y and z-direction. X X K .'-, X 1
j = l
The quantities Y i Y and Y are the mean square direction cosines of the inner magnetization within the domains.
Expression (11.12) can be simplified further by averageing over the different timeinterval sequences. For this purpose the collection of L rows of matrices is subdivided into a large number of Q groups which are numbered by q, within each of which the number of matrices in a row is equal to N and where the different timeinterval sequences are characterised by placing an indexnuraber q. at each timeinterval t... In this procedure the index % can be omitted. All different timeinterval sequences in the group q are now found by summing over all indices q.. In formula 1 , Q /•,•,/? Z Z
? 4 I i
<5 q = l ^^^ q ^ = l q . = l
I •••• I
I °(tA,. )•••"('•„ )••• D(t,„ )•?
q , = I ffq iq;"iq,
(11.15)where Z is an arbitrarily large number, which gives the number of possible values of t. . By writing down all possible timeinterval sequences in this way it is assumed that there is no correlation between timeintervals in a row (t. and t. ) . This, however, is not quite correct since
jqj
J,''
= ff . t = t qj_ q c(11.16)
where t is the average value of the timeinterval in the group q and t the total interaction time of the neutron beam with the foil investigated. In case the direction of propagation is perpendicular to the foil this time is given by the thickness of the foil devided by the neutron velocity (t = —) both of which are
k n o w n . It is clear from (11.16) that for small N this c o r r e l a t i o n c a n b e v e r y s t r o n g . However it is d i f f i c u l t to evaluate the effect of this c o r r e l a t i o n on t h e final r e s u l t s . E x p r e s s i o n (11.15) can also be w r i t t e n as
P =
1
(D (tJ)
N (11.17) q=l w h e r e N is still a function of q and D (t ) is defined b y qD (t ) = -f
q z
z
Ï
q,-=i°(%.5
(11.18) This average of D ( t . ) w i l l b e calculated b y assuming a gaussian d i s t r i b u t i o n^qi
which gives the probability of finding a certain timeinterval t. . Formula (11.18) can then be written as
D (t^) D(t) exp exp -4(t-t ) ' q 2 2 n t q J
-^(^-S)
2 -I 2 2 dt (11.19) dtw h e r e rit the h a l f w i d t h at h a l f m a x i m u m , has b e e n chosen so n a r r o w that the q^
contributions of the integrand for n e g a t i v e values of t can be n e g l e c t e d . T h e time-dependence of D ( t ) m a n i f e s t s itself only in the terms cos u t and sin ut and for the averaged v a l u e s of these terms it is found,
<cos u t > = cos ut . exp
<sin u t > = sin ut . exp
L-t
nut
nut
r
(11.20)
The matrix D (t ) can now be found from (11.14) by substitution of cos ut. and
q il sin ut. by the expressions (11.20)
D (t^) I-(l-<cos ut>)(l-Y ) X 0 - m < s i n u t > 0 m < s i n u t > l-(l-<cos u t > ) ( l - Y ) 0 0 l-(l-<cos u t > ) ( l - Y ) (11.21)
The N power of the matrix in (11.17) can now be calculated by diagonalizing
the matrix D (t ). Writing D (t ) for simplicity in the form
D (t^)
(11.22)
with eigenvalues A,, A and A , and eigenvectors X, Y and Z, the matrix D (t )
1
/
3 q
with the eigenvectors as basis is diagonal and given by
S D (t )S
q
(11.23)
-}• -^ 't
where S is a transformationmatrix of the laboratory sytem (i , i , i ) on to
x
y
z
- > - > • - ^
the system characterized by the eigenvectors (X,Y,Z). By means of (11.23) the
power of D (t ) can be expressed in powers of the eigenvalues by
(D*(t ) ) * = S-' S(D*(t
)f
S-'s = S-'(S D*(t )S-')^ S =
q q q
= s
-1
(11.24)
The eigenvalues A , A and A can be calculated from the eigenvalue equation
and the eigenvectors from D (t ).
"1 2 ^2 = ^2
S • 2
- E wh 2 a ^ - a j 2 e r e EA x a.s.o. The eigenvalues are found to be
(11.25)
and the eigenvectors
If?)
2
0
1+E
The transformation matrices S and S can now be found from
SX =
SY
SZ =
(11.27)
After calculation of S an S and substitution in ( H . 2 4 ) it is found
(D (t^))"
2 2E
0
-2a^ ^ - ^3
9= X * 1 *2^4 ^, - A3
^ r ^ 3 2E
. N ^ N . N ^ N
A, + A3 A, - A3
2E
2E
(11.28)
At this stage of the derivation, where the quantity A' (the number of domains in
a row) is explicitly written down in formulae, it is possible to carry out the
averageing procedure of (11.17) over the different number of matrices in a row
N.
For this purpose a gaussian distribution function is assumed for the
probability of finding a row with
N
matrices. When the mean value of
N
is given
by N, then the halfwidth at half maximum of this distribution is defined as n.N,
*
N
where n, has been chosen n/^N. For the calculation of the average of (D (t ))
, N ^ N N '^
it IS sufficient to calculate the average of A , Ao and A , because the matrix
* N
(D (t )) contains linear combinations of these powers. Although in the matrix
these powers appear as a product with terms which are also a function of
N,
it
can be shown that the neglection of these terms in the averageing over
N
has
only a very small influence on the final results. The average of A can be
written as
exp
<A.*> =
-4(,V-N)
2„2
dff
(11.29)
exp
-4(ff-N)
2„2
n|N
dN
where A^ is still a complicated function of
N.
From equations (11.16) and
(11.20-25) it follows that for large
N,
A (ff) can be written as
N
N
A,"(ff) = A | ' \ N ) . |Aj(N)|
-(ff-N)
(11.30)
with this expression the integration in (11.29) can be carried out. By N assuming that (11.30) is valid for all values of N, the calculation of <A > by means of (11.30) will only for small N give a small error in the correction of
N
A (N). By working out expression (11,29) it is found
|A,(«)!-(*-«)exp N N < X / (ff)> = X, (N) -4(ff-N) (njN)-^ iN
exp
-4(ff-N)(n,N)^ J
dNA|(N) exp Jg (In |A,(N)|)2 (11.31)
where the relation n has been used. In the same way it is found
N
<A2 (»)> A2(N) exp ^ (In |A2(N)|)' = A, (11.32)
<A3 (ff)> = A3(N) exp ^ (In ,A3(N)|)' (11.33)
Since A ( N ) , A (N) and A (N) differ very little from A , A and A respectively, the quantities A (N), A (N) and A (N) can be approximated by
r 2
A|(N) = Aj exp
^ ( i n iA;|)
* K 2 (11.34)X^W = A^ exp
^ ( i n lA^I)
* i s 2 (11.35)A3(N) = A3 exp
^ ( i n \x;\)'
(11.36)Having performed the averageing over the different number of matrices in a row the polarization change can now be written in a way similar to (11.28) as
follows p = N N N N A, . A 3 A, - A3 2 E N ^ N
2a,
^ - ^ 3 2E .N ^N 2 a , 4a , - a 3
A ,
^
-
^-^
3 2E 0 X,_[
+ A., 3 22E
(11.37) or in simpler form P = D . P^
0 -A, 0S
0 A 0 A (11.38)where the elements of the matrix D are defined by (11.37). The matrix D which
- > • . _ ^ . _ - * •
transforms the polarization P of the incoming beam into the polarization P of the transmitted beam through the ferromagnetic foil, will be called the depolarization matrix of the investigated specimen. The matrix D consists of four independent elements, which can be measured as will be shown in the next chapter. From these four measured quantities the mean square direction cosines of the inner magnetization Y > Y > Y 1 the mean reduced magnetization m (as defined in (11,14)) and the mean domain size & can be determined, where 6 is related to the mean number of matrices in a row N by
6 = (11.39)
and d is the distance that the neutron beam travels through the foil.
C. Determination of the mean square direction cosines of the inner magnetization,
the mean reduced magnetization and the mean domain size from the measured
depolarization matrix,
Although in the last section a depolarization matrix has been derived, from which the elements can be written as analytic functions of the domain parameters involved, it is not possible to do this in the reverse direction except in the
'^t
limiting case that ° << 1 and m = 0. For that reason the following iteration N
method will be used.
Suppose the matrix from (11.38) has been measured, then it can be seen from
(11.37) and (11.38) that
N A +A.
* 1 3
A,-A3
N
= A„
2 2
n A,+A(11.40)
•3 , ^ - S
E
3 2 " 2
where E, defined by (11.25), can be found from (11.37) and (11.38)
2A,
['fSj
(11.41)
Two cases have to be distinquished, the first that E is complex, which
corres-ponds to |2A |> I (.\ -A ) ]or to some magnetization of the sample and the second
case that E is real, which corresponds to 12A l<| (A -A )|or to very small
magnetization of the sample. In the first case, equations(11.40) can be written
in the form
.N
R exp ie
P .
exp
where R - \/(A|A3 + A, ) , e = arc cos
4 p
A,.A3
2R
,th
(XI.42)
and i is the imaginary unit. By taking the N power root in (11.42) and
combining the results with equations (11.34-36), it is found
A|(N) = exp
X^m
= exp
A3(N) = exp
2 r ^2In R _ 1^ (in
R'N 16
In R n_ [in R
N " 16 N
In R n^ fin R^^
exp
16
exp
(11.43)
Here A (N), A (N) and A (N) have been expressed in functions where N is the
only unknown parameter. In case E is real in equation (11.41) iE should be
P
replaced by E' where e' is a real number and defined as
P P
cos h •
2R
(11.44)
With the relations (11.20-25) where the parameter t has to be replaced by —;-,
q '^ N
it can be seen that the sum of the terms of expression (11.43) must be equal to
Ut o N
exp
-nut
o 4N 2Aj(N) +
X^W
+ A3(N) = 1 + 2 cos - j ^ exp - - ^ (11.45)
where ut = 5.6 B .d, B is the spontaneous induction in Gauss and d the
o
S
S
distance in meter the neutron beam travels through the material. This is an
equation with N as the only unknown parameter for a chosen n. By using a computer
a solution for N is obtained. Although in general there are more solutions
possible, the correct one should satisfy the condition that
ut
0
N
(11.46)
which is true in all the measurements described in the next chapters. It should
be remembered here that the mean domain size 6 is determined by 6 = TJ, where d
is the distance that the neutron beam travels through the specimen. After the
determination of N, A (N), A (N) and A (N)
from the relations (11.25) it follows that
determination of N, A (N), A (N) and A (N) can be calculated from (11.43) and
A|(N) + A3(N) A|(N) - A3(N)
2 2E
32
=
x^m
Aj(N) + A3(N) Aj(N) - A3(N)
2E
(11.47)
and from these r e l a t i o n s in combination with ( I I . 1 4 ) , ( I I . 2 1 ) and (11.22) i t can
be d e r i v e d that f o r the d i r e c t i o n c o s i n e s holds t h e following
'OJt o
[ N J
exp r~4N
•v„ = 1 - cos ut o exp-n ut
o[
4N
J
'ut o N exp-n ut
o 4N2
1 - cos ut 0 N expr
n ut ^
0 . 4N j2
a - cos 1 - cosut
o[ N J
fut ^
0 exp exp -, 2n ut
o4N
J
n ut 1
o4N 'J
The reduced magnetization m can be written asa. A,
ut
N exp
n ut
4N
n A,-A 1 3 (11.48) 'ut 0 !_ N J expn ut
0[ ^^
J
2
All calculations described in this section can be performed by a computer, for which an Algol program has been written.
D. Depolarization in a ferromagnet at a temperature very close to the Curie
temperature in the presence of a magnetic field.
Wlien a magnetic field H is applied on a ferromagnet in the neighbourhood of the Curie temperature T then a mean magnetic induction B is induced which is partly of ferromagnetic and partly of paramagnetic origin. The ferromagnetic part concerns the rotation of the spontaneous magnetic induction B in the domains or the movement of domain walls and therefore the induction is an average over the various domains. The paramagnetic part B finds its origin on a much smaller scale, in fact by alignment of the short-range fluctuations and therefore its value is essentially the same in all the domains. However the direction of this paramagnetic induction is not clear. From the results of the depolarization
measurements it should follow whether this direction is parallel to the applied field or parallel to the original ferromagnetic spontaneous induction.
The magnetic induction present in one domain can be written in vector notation as
Bj = B + B (11.49) d s p
Since in the neighbourhood of T , B is very small the rotation angle of the polarization vector in one domain around this local field is very small also. Therefore the depolarization matrix (11.37) can be written in the simplified form P = ^1 0 - A , 0 A, 0 A 0 A .p (11.50)
where the magnetic field has been applied along the y-direction and
A = A = exp A = exp A = exp 1+Y 1+Y,. 1-Y., ( B ; B^)6d P -)6d (B' i.B^)6d P cos (c^ B d) sin (c^ B d) (11.51) and Y - 1 - 1 c = = 5.61 (Gauss meter ) o
Y is the gyromagnetic ratio of the neutron, y the permeability of the vacuum and V the velocity of the neutrons. In the limiting case of m = 0 and assuming an
— , the diagonal elements A , A and isotropic ferromagnet where Y = Y ~ ~
A represent the depolarization formula of Halpern and Holstein [ 3 ] . From formulae (11.51) it follows that
c.,d
, arc tg
(B^ + B^ - B^)6 = - ^
s p c d
where s = ln(A,A_ + A,) + In A_
1 J 4 2
and
2 In A,
T,= '
2 2
B'^+ B^
s pj
(11.52)
2 2
It IS clear that as long as B and B are not known it is not possible to
^ P . 2 2 2
determine Y , However, in most cases an estimate can be made of B / (B + B ) ,
y s p
with which Y can be approximated very well.
E. Discussion.
After the depolarization matrix has been derived and formulated in terms of
the domain parameters Y > Y . Y > m and 6, and knowing in particular how these
domain parameters have been found back from the measured depolarization matrix,
it is useful to discuss some aspects of the theory which are violated by the
appearance of a real domain structure.
Consider first the averageing over the K possible orientations in each
domain according to (11.11-13). Although in the averageing neighbouring domains
with the same orientation are not excluded, in real domain structure such a
configuration is not possible. When u is the number of configurations of two
neighbouring domains which must be excluded divided by the total number of
configurations of these two domains, then u is a figure of merit for the
assumption that there is no direct correlation between the magnetization
orientations of neighbouring domains.
For a polycrystalline material with a grain diameter in the order of the
2
mean domain size the total number of configurations is K and the number of
configurations which must be excluded is K, so that u = —. For a polycrystalline
material the assumption seems to be quite good since u << 1 because K can be
infinitely large.
For a single crystal of cubic structure with 90 walls and 180 walls
there are six possible configurations which are excluded so that u = —. The
assumption is reasonably good, although errors will occur. If in this single crystal there are only 90 walls than it follows that u = -r-, and application of the formulae will introduce considerable errors. If in this crystal there are 180 walls only than u = y- and the applicability is very bad. This can also be
o
understood since in a system with only 180 walls the rotation angle of the polarization in a certain domain is completely compensated by the rotation angle of the polarization in the next domain. In first approximation it can be said therefore that domains separated by 180 walls in a row do not affect the neutron polarization. On the other hand this property can be used in particular to determine the mean direction of 180 walls,
For a single nickel crystal with cubic structure and 70 , 110 and 180 walls it follows that there are 8 possible orientations from which one has
to be excluded, so that u = -r-. The applicability of the theory is reasonably S
good. For the case of 70 walls only, u = -j- and considerable errors by o
applying the theory can be expected. The errors in this case are such that the mean domain size calculated from the depolarizationmatrix is larger than the real mean domain size, while in the case of only 180 walls the calculated domain size is smaller than the real mean domain size.
A second effect which may influence the calculated results is the
correlation between the magnetization direction of a domain and its dimensions in this direction. If it is assumed that this correlation is such that the mean domain size in the direction of magnetization of the domain is somewhat larger than the mean domain size perpendicular to this direction and
remembering that the polarization direction of the neutron beam is not affected by domains with their magnetization direction parallel to the polarization direction, then it is clear from fig. II.2 that the polarization vector in the
transmission direction of the
.domains
— S,
Fig. II.2
neutrons will see on the average a smaller mean domain size 6
X than in the case when the polari-zationvector is perpendicular to the transmission direction, where in the average a mean domain size 6 is measured in
y
this simple model. In the previously given theory this effect has not been accounted for so that in the formulae for the depolarization in the transmission direction x, which is proportional to
c 2
combination with the depolarization in the other directions leads to an apparently smaller value for (1-Y ) and therefore an apparently larger value for Y •
X
Another interesting aspect of the theory follows from the solution of equation (11.45) for high magnetization m. For the sake of simplicity the distribution constant n will be assumed to be zero. For the limiting case that m = 1, £ would be equal ut and Y - ' and from this R = 1, the lefthand
o y
side of the equation is exactly the same as the righthand side of the equation for all values of N. This means that an unique value for the mean domain size can not be found in this case. This result is in agreement with the physical picture that at very high magnetization the domains have nearly the same magnetization orientation and therefore they are not distinguishable from one
large domain with magnetization m. For that reason the mean domain size found at high magnetization has to be considered with much reserve.
Note: In formulae (II.11),(II.13),(II.14) and (II.15),(II.17),(II.18) a weight factor should be added to account for the probability distribution of the different orientations and timeintervals respectively. In the evaluation of these formulae for this factor a gaussian distribution function has been used.
REFERENCES.
1, D, Ter Haar, Fluctuation, Relaxation and Resonance in Magnetic Systems, Oliver and Boyd, Edingburgh and London, 1961
2, 0. Steinsvoll, Kjeller Report KR-65, 1963
III. EXPERIMENTAL SET-UP.
A, Survey of polarizing methode.
A polarized neutron beam can be obtained by using the principle of inter-ference of the nuclear and magnetic scattering amplitude. According to Halpern and Johnson [1] this interference can be expressed in the coherent elastic scattering cross section (c.s.c) of an atom
a - b^ + 2bp (q.A) + p^ (^ (III.I)
where b and p are the nuclear and magnetic scattering amplitude of the atom
"*• , . . .
respectively, A is a unit vector in the direction of polarization of the incident
-> . .
neutron beam and q is the magnetic interaction vector defined by
q = £(Ë.i^) -k (III.2)
- > • . . " * " .
Here k is a unit vector m the direction of the atomic magnetic moment and E is a unit vector in the direction of the scattering vector. An unpolarised beam can be resolved in two polarized beams with A = +1 and A = -1 in some specified direction. The polarization of the beam is defined as
I -I
P = - j ^ . A (III.3) +
where I is the intensity of neutrons in the beam with mean spindirection A and
. ->
->-I the intensity of neutrons with mean spindirection -A, Then P is defined as a vector in the direction A, where A has been chosen in such a way that P has an extreme. This is necessarily the case for all directions A if P is obtained by splitting up an unpolarized beam.
It is easily seen that the c,s,c, is different for the two cases (q.A) = +1 and (q,A) = -1
There are four different ways to obtain a polarized neutron beam. They will briefly be discussed with their advantages and disadvantages. These methods are:
1) Transmission method [2] [ 3 ] . An unpolarised beam is transmitted through an assembly of atoms with oriented magnetic moments. After simple calculations it can be shown that the c,s,c. is different for the + and -component of the incident unpolarized beam. If the total scattering of the beam is large enough, in principle the transmitted beam will be polarised. However in
addition to the low intensity of the transmitted beam, its polarization is very low. A second disadvantage is that the beam is not monochromatic. 2) Reflection by magnetized mirrors [2] [ 3 ] , This method makes use of the
wave character of the neutrons. An index of refraction can be defined and from this a glancing angle proportional to the square root of the scattering cross-section. By magnetizing the mirror in its plane two different glancing angles are found for the two polarization directions. In principle this method can give a very high polarization coupled with a high intensity of the beam, which however is not monochromatic. Moreover the glancing angle is very small in the order of minutes of arc, and this makes the method experimentally more difficult.
3) Stern-Gerlach method [3] [ 4 ] . This method makes use of the forces which are working on a magnetic moment in an inhomogeneous field. Since the magnetic moment of a neutron is very small the forces working on a neutron beam are also very small. However in a very strong inhomogeneous field (10 Oe/cm) a deflection of the neutrons with different spin components of about 3 minutes of arc can be obtained. This technique turns out to be very
difficult experimentally. Moreover the polarized neutron beam obtained is not monochromatic.
4) Bragg reflection by a magnetized single crystal with the property that the nuclear scattering amplitude b is nearly equal to the magnetic scattering amplitude p [2] [3] [5]. By magnetizing the crystal perpendicular to the scattering vector, it can be seen from (III.2) that q = -k and from (III.l) that the c.s.c. for A = +k is given by
2
a^ = (b ± p) (III.4)
In this way the reflected intensity can have a very high polarization while the reduction of the intensity is not too high. An advantage of this method is that the polarized beam obtained is monochromatic.
B, Polarization device,
The polarization device used is based on Bragg reflection by a magnetized single crystal. In this case a Co(0,92)Fe(0.08)fcc single crystal of dimensions
3 . . .
25x20x1.5 mm is used as monochromatizing and polarizing crystal with reflection plane (200) and magnetization direction along (Oil). The reflection plane is perpendicular to the surface of the crystal. A schematic view of the set-up is given in fig.III.1. After being reflected by the polarizing crystal the neutrons
Schematic view of the experimental set-up. The large arrows M and M represent the magnetization directions in the polarizing and analysing crystals, G is the guide field, D , D are the first and second polarization turner, S is the sample-holder plus sample, M the monitor counter and T the detector. The small arrows indicate how the neutron spin polarization vector could turn within the neutron polarimeter.
successively pass a guide field, polarization turner D , a field free region in which the sample holder is placed, a second polarization turner D and a monitor counter M, Then they are reflected by a second Co(0.92)Fe(0.08) crystal in (200) reflection position before being detected in the BE counter. The second Co(0,92)Fe(0.08) crystal is magnetized in a direction opposite to the first one. The two crystals have been adjusted to an optimal Bragg reflection to obtain a maximum intensity. It is useful now to follow the intensity and polarization of the neutron beam on its way through the apparatus to find the relation between the measured intensity in the counter T and the polarization change in the sample holder S.
The neutron intensity and polarization after reflection from the first crystal can be derived from equations (III.4 and III.3) respectively
I, = I r
(0^ + a )
(III.5)
I o p +
-, (a - o ) ^
P, = k P A=k 7 - — — — s - A (III.6)
1 p p p(a_^ + o_)
Here I is the intensity of the initial white beam, r is some attenuation
o P
factor in which extinction effects in the crystal are included, P is the
polarizing power of the polarizing crystal and k is a constant somewhat smaller
P
than unity which accounts for extinction and depolarization effects in the
polarizing crystal. The intensity I and polarization P,-- in front of the
analyzing crystal are given by
I, = r r I (III.7)
2 s m 1
P„. . = k a.. P, (III.8)
2iJ g ij I
where r and r are attenuation factors due to the sampleholder plus sample and
s m
the monitor counter respectively. The constant k is somewhat smaller than unity
and accounts for depolarization in the guide field and a., is a quantity which
accounts for the polarization change in the polarization turners and
sample-holder plus sample. The quantity a.. depends on state i of polarization turner
D and state j of polarization turner D and can therefore be written as a matrix.
Though the polarization vector behind the second polarization turner is not
"*" •+necessarily in the direction of P,, P_.. in front of the analyzing crystal is
^ 1 2 ij _,.
oriented along P , because of the large magnetic field in the direction of P
present at the analyzing crystal. This field averages out any component of the
polarization vector perpendicular to P ,
By writing the intensity I as
where I and I__ are defined as in expression (III.3), it follows simply that
by using (III.3)
^2
and
h---f
('-(P2ij-^))
(III.11)From these expressions the measured intensity in the counter is given by
l! . = r I., a + r I., O
ij a+ 2+ + a- 2- - (III.12)
and by working out it is found that
II. = r -^ (l+(P,...A)k P )
ij a 2 2ij a a (III.13)
Here r and r are attenuation factors which include the extinction effects a+
a-of the analyzing crystal, the depolarization in the crystal and the efficiency of the counter T, Theze factors are slightly dependent on the polarization. The constant r and k are given by
1+P
-P 1
a2 J
1+P 1-P -1 (III.14) (III.15)k is somewhat smaller than unity. P is the polarizing power of the analyzing crystal and is given by
(III.16)
By combining the expressions (III.6-8) and (III.13) the measured intensity is given by
r r r
I!. = ^ -^ " I, ( l + a . . k k k p p )
ij 2 1 ij g p a '^p a (III.17)
Here r , I. and a., are the only parameters which are not constants of the
s i ij
apparatus. By using the monitor intensity
I = (I-r )r I,
as a reference signal the two variables r and I can be eliminated, so that
l! . r r
I.. = - ^ = ,f '"-. (1 + a., k k k P P ) (111,19) ij I ('"^ ) ij g P a p a
Because the depolarization due to the different parts of the apparatus is indistinguishable it is convenient to take it all together in
a°.=- a.. k k k P P (111,20)
ij ij g p a p a
The negative sign is used because of the antiparallel magnetization of polarizing and analyzing crystal with the result that the signs of P and P are opposite to
p a each other. Expression (III.19) can now be simplified into
I.. = I (l-a°.) (III.21)
ij s ij
or written in another way
a?. = -S LL (III.22)
Here I is the intensity measured with a fully depolarized beam. The measured matrix a.. can be written in the form
ij
a ° . = (P2.,D°Pj.) (III.23)
where P,. is the polarization in front of the sampleholder with the first 11
polarization turner in position i, which means that the polarization vector from the original direction P is rotated into the direction P,-. The matrix D accounts for the polarization change in the sample holder plus sample, P^. is the polarization vector which is turned back by the second polarization turner in position j to the original direction P which is optimally analyzed by the analyzing crystal. In accordance to (111,20), the absolute values of P . and P . account also for the depolarization m the other parts of the apparatus. The matrix D is defined on an orthogonal coordinate system (i,, i«, iq)» which
1 2 3 ^
coincides with our laboratory system (x,y,z). The polarization vectors P . and P . do not necessarily coincide with this coordinate system since the adjustment of the polarization turners introduces certain errors. By describing P . and P.. on these axes
5,, = ^ = i k \
(i"-2^)
k
?2j = S e ! , t ^ (111,25)
it is found with help of (III.23)
«?j = J ']n
^Ik
S i ("^-"^
where D ° ^ = (^^0°^!,) ^"d Ê^^. = e.^_
Written in terms of matrix multiplication
a ° . = ( E ' D ° £ ) . . = ( E ' D ° E ) . . (III.28) ij Ji iJ
D°. = (E'''
a°
Ê ' ' ) . . (III.29)
Remembering that the matrix D accounts for the polarization change in the
sampleholder plus sample it is convenient to split this matrix into
D° = V(H) D V(H) (III.30)
where V(H) is a pure rotation matrix. The latter describes the rotation of the
polarization vector around the magnetizing field in the sampleholder the
rotation angle being proportional to the magnetizing field H and D is the
depolarization matrix of the sample which gives the polarization change in the
sample investigated. By combining (III.29) and (III.30) D is found to be
D = V(-H) E'~'
a°
ê~' V(-H) (III.31)
where a can be measured and the other matrix quantities can be determined by
measuring a at different magnetizing fields without sample in the sampleholder
(see section F),
C,
Description of the apparatus,
1) Collimation and reflecting crystals. The white beam of neutrons coming from the core of the reactor has an angular divergence of one degree,a
cross-2 8 section of about (2,5x2) cm and a total intensity of about 10 n/s incident
on the polarizing crystal. In the white beam a monochromator and polarizing CoFe single crystal is placed. This crystal is mounted in a permanent magnet which delivers a magnetizing field of 3000 Oe.The whole system is turnable around two axes, one of them coincides with the magnetization direction of the crystal perpendicular to the scattering area, the other is perpendicular to the first one allowing for making corrections in case of a wrong
orientation of the monochromator crystal. The monochromatic polarised neutron 2
beam obtained has an intensity of about 3000 n/cm s.The cross section of the monochromatic beam at the sample holder and polarization turners has been
2
reduced to 1 cm . This has been done to be ensured of the homogeneity of the magnetic field across the beam in the polarization turners» the guide field and in the sampleholder. The quality of monochromatization is given by
^ = cotn e„ de (III.32)
where X is the wavelength (1,19 A) of the neutron beam and dX the spread in it. The Bragg-angle 6„ is 19,5 degrees and dO is the divergence of the beam,
D
Substituting these values in (III.32) the result is
f = 5 , 0 10-2
The beam is reflected by a second CoFe crystal with the same properties as the polarizing crystal. Both polarizing and analyzing crystals have a mosaic spread of 10 minutes of arc derived from the full-width at half maximum of the rocking curve measured from the analyzing crystal in parallel position with the polarizing crystal. The angle of the analyzing crystal with respect to the incoming neutrons can be changed automatically in minimum steps of 1.5 minutes of arc.
2) The guide field. The guide field consists of two parts, both of which should prevent depolarization of the beam. For this purpose it is necessary that the magnetic field across the beam is as homogeneous as possible. In the first part the field is of the order of 120 Oe which is large enough to keep the polarization of the neutrons in the field direction even when small inhomogenities are present. The second part of the guide field being the transition region between the field of the first part and the zero field of
the polarization turner and sampleholder is more complicated. The field gradient has to be chosen very carefully, because a decrease of the field in the z-direction gives an inhomogeneity in the x-component of the field according to the Maxwell equation rot (H) = 0 and
dH x dz dH z dx
and from which the inhomogeneities can be determined. It can be calculated [6] that a decrease of the field intensity from 100 Oe to zero over a distance of 25 cm gives a depolarization smaller than one percent with 1 A neutron wavelength if the height of the beam in the field direction is smaller than 1 cm.
Another point of interest is the stability of the polarization direction after passage through the second part of the guide field. Because of the decrease of the field to zero an angular deviation between polarization and field direction will occur, and rotation of the polarization vector around the field will change the polarization direction. In first approximation the polarization vector turns about 10 times around the magnetic field in the second part of the guide field. The magnetic field is created by permanent
-3 . magnets (ferroxdure) with a temperature coefficient of about 2 10 . This means that with one degree temperature change the polarization vector varies about 7 degrees around the magnetic field. To minimize this effect the guide field is temperature-stabilized to within 0.1 centigrade.
O
( r-_c
^ r
®
~i r^
n,
DC
®
y,'///.'A—ty'////\—TZZZ2Z2i~
sideview® ® ©
Z l
V , . . v . . ^ V
croM-section.F i g . 111,2
Schematic side and front view of the guide field,
1,2 first and second part of guide field 4 permanet magnets 3 first polarization turner D. 5 soft iron strips 6 correction magnets,
2 The inner cross-section of the guide field is about 4x4 cm ,
The construction of the guide field will be discussed briefly [ 7 ] , The magnetic field provided by the permanent magnets is homogenized by fastening
2 the magnets on to a soft iron strip with a cross section of 50x5,5 mm , To minimize errors due to edge effects, correction magnets are used on the sides of the guide field (fig. III.2), The second part of the guide field only consists of soft-iron strips without permanent magnets, which are magnetically closed at the ends by a soft iron box of the first polarization turner (fig, 111,3).
Fig. III.3
Polarization turner D . 1 soft iron rectangular tube 2 field coils
3 connections for the field coils
3) Polarization turners. The polarization turners consist of three pairs of coils perpendicular to each other. In each pair of coils a magnetic field can be created up to about 50 Oe. The influence of the magnetic guide field on the first polarization turner is minimized by putting the coils in a soft-iron
box of dimensions (60x40:.45 mm ) which is left open in the transmission direction of the neutrons (fig. III,3),This soft-iron box also should localize the magnetic field of the coils and shortcircuite the magnetic flux from the soft iron strips of the second part of the guide field. In this way a field can be created in any required direction which gives the
possibility to get in the sample holder a polarization vector of any direction wanted. The second polarization turner is constructed in a similar way, though with different dimensions required by the specific dimensions of the sample holder used,
The adjustment and calibration of the polarization turners is only possible using the magnetizing coils of the sampleholder. This will be discussed later in section F.
D. Sampleholder for measurements on foils under mechanical stress.
The purpose of this sampleholder was to be able to investigate the depolarization of a neutron beam passing in different directions through ferromagnetic foils under the simultaneous application of a magnetic field and mechanical tension on the foil,
Fig, 111,4
Central part of the sampleholder in fig. III. 5. 1 foil to be investigated 2 magnetizing coil
Cross-section. A-A
Fig. I I I . 5
Sample holder in side view and cross-section
1. transmission direction of the neutron beam
2. soft iron magnetic shielding
4. sample holder( central part)
5. magnetizing coil
For simplifying the interpretation of the measurements the stress on the sample was applied in the direction of the magnetic field. The applied magnetic field across the beam has to be homogeneous. Besides the distance in the transmission direction of the neutrons over which the field is present, has to be constant across the beam. This problem was solved by winding a flat coil in which the foil is fastened.The ends of the foil are pinched in a soft-iron yoke which serves as a short-circuit of the total magnetic flux from coil and foil (fig.III.4).
3
The inner dimensions of the coil are 1x10x30 mm , while the outer dimensions are 8x18x30 mm . The dimensions of the foils investigated are 0,125x9,5x40 mm . The maximum field intensity obtained in the coil is about 150 Oe, The magnetic yoke does not only minimize demagnetizing forces due to the foil, but also eliminates the back flow of flux through the air. Without the yoke this should give a rotation of the polarization vector in the sampleholder that is opposite to the rotation within the coil, and which amounts to about 20% of the total rotation,
To minimize stray magnetic fields on the sample the whole assembly is surrounded by three concentric boxes from which the inner one is a high mumetal box of 0,5 mm wall thickness. In the boxes a gap is left for the neutron beam
(fig,III.5),
Different directions of neutron transmission relative to the plane of the foil can be obtained by making coil and magnetic yoke turnable around an axis perpendicular to the direction of the neutron beam and the magnetic field direction of the coil,
The foil can be put under tension by pulling on one side. The magnetic yoke is constructed such that two parts are movable in each other with one freedom of motion in the required direction of tension. The friction of the moving parts of the magnetic yoke is less than 0,05 kgf,
The electric current through the magnetizing coil has been made proportional to the output voltage of a ten-turn potentiometer which is connected between - 1 5 V and +15 V, which voltage can be changed automatically.
E, Furnace for temperature dependent measurements.
This furnace is especially made for very accurate temperature measurements in a small temperature region, in particular within a few degrees around the Curie point T of nickel. This furnace has been built as a double furnace, the
c ' outer furnace is employed in order to have a coarse temperature control whereas
the inner furnace in which the sample is placed has been used as a fine one (fig. III.6).
Fig. III.6
Furnace for temperature dependent measurements. 1 transmission direction of the neutron
beam
2 soft-iron magnetic shielding 3 outer furnace
4 inner furnace
5 sample to be investigated
6 magnetizing coils
7 heating wire
8 location of temperature sensors 9 tubes for water cooling
10 vacuum transits for connections.
The advantage of this construction is that one can get a small heat current between the inner and outer furnace only and a relatively small heat resistance. This results in a homogeneous temperature distribution of about 0.01 centigrade across the sample coupled with a relatively rapid temperature adjustment of about 5 minutes for a temperature change of about 0.1 centigrade. The temperature
of the outer furnace has been controlled proportionally by using the difference signal between a reference voltage and the voltage of a chrome1-alumel thermo-couple, which is placed in the outer furnace, and amplifying the signal in a power amplifier before it is supplied to the heating element of the outer
furnace. The temperature can be held constant within a few hundredths of a degree over several hours .
The temperature of the inner furnace has been controlled proportionally by comparing the voltage over a N.T.C. resistance mounted in the inner furnace with the voltage over a reference resistance, both of which are part of a double Kelvin bridge. This voltage difference is amplified by a John Fluke 0 detector
(type 845) and a power amplifier and then supplied to the heating element of the inner furnace (fig. III.7). From fig. III.7 it can be seen that the temperature
f
Sx'I
BJ
T
4—<r^
>
Furnoc»Ï
Heater 0-detectorF i g . I I I . 7
Diagram of the temperature control of the inner furnace.
is changed by varying resistance R-, The range over which the temperature can be varied by R is determined by resistance R . For the resistance R a ten-turn-potentiometer has been chosen which can be changed automatically in the desired region. In this way the temperature can be held constant within a few milli-degree during the measuring time of about half an hour. The temperature
measurements go in a similar way as the temperature control. The voltage over a platinum resistance in the inner furnace is compared with the voltage over a reference resistance in a double Kelvin bridge. The difference signal is ampli-fied by a John Fluke 0-detector and after adding a constant voltage to it to ensure a positive output, supplied to an analog-digital converter. The frequency of the converter is counted by a scaler and printed automatically after the measuring time is over. From the pulses counted during the measuring time the mean temperature of the sample can be calculated. However directly after the warming-up of the furnace from room temperature to 360 C a drift has been observed in the platinum resistance which corresponds to a drift in the measured temperature of about 0,1 C in 24 hours. This drift was very constant during a measuring cycle and it diminished after operating the furnace at about the same temperature during several weeks.
Although the furnace has not been built for specific field-dependent measurements, it was possible to place some magnetizing coils in it, which could give a magnetic field up to about 10 Oe at the sample. The homogeneity of the field is not so good as in the other sampleholder but it is sufficient to allow a qualitative explanation of the experiments performed, A more detailed description of the furnace itself is given by H,K, Bakker [8],
F. Adjustment and calibration of the polarization turners,
The precise adjustment of the polarization turners is of paramount importance for the experiments described here. Since both polarization turners affect the neutron polarization vector it is difficult to know what its orientation is between them. However by means of a magnetic field with a known orientation at the sample position, it is possible to solve that problem. By taking a field strength corresponding to a rotation of 90 degrees of the polarization around that field in an arbitrary direction i and by measuring the polarization with an alternating plus and minus field direction, it is clear from fig, 111,8 that all components of P.. t perpendicular to i, are averaged out. The procedure followed can also be understood from expression (111,26) where the matrix D is the average of a pure rotation matrix over +90 degrees added to a pure rotation matrix over -90 degrees around the axis i, , That means that all elements of the
k
matrix D., are zero except D, , = 1. The measured quantity a., is referring to
ij kk ij "
(III.27)
a. . = E ' E, . (III.33) ij jk ki
and in particular
a = r' r = r' r
kk kk kk kk kk (III.34)
Fig. III.
By adjusting the polarization turners D and D„ in such a way that ct is a maximum, which according to (III.22) means that the measured intensity is a minimum, it is found that a,, is nearly one. From this it follows that E, , is also nearly one, so that the polarization directions P and P should approximately be along i . The adjustments of the polarization turners corres-ponding to the direction i shall be called the positions k of the polarization turners. The different positions of a polarization turner can be chosen auto-matically if desired. Fig. III.9 shows how the four most important elements of a.. vary with the field in the y-direction in the empty sampleholder described in section D. The curves have been measured at two different angles of trans-mission of the neutron beam with the normal on the flat coil. From the nearly perfect coincidence of the a and a curves and the 90 phase shift with
XX zz
respect to the a it is concluded that the adjustment is quite good,
The calibration of the polarization turners is performed by determining the matrices E ' and E, This can be done by using the adjustment procedure with the resulting formulae (III.33) and (III.34). From (III.34) it is seen that c and E ' cannot be determined independently. Only a lower limit of their value can be given which is of the order of 0.95. Without making a significant error it is assumed for convenience that
E = e' =Va°
With this assumption for the diagonal elements the other elements of the £ and E ' matrices can be determined from (III.33) by measuring the a., with different field direction i . — • . . ,, , . .. — -100 0 100 —.- H(o«) -'00 0 1 0 0 _ — H ( O l ) Fig. III.9 o Calibration of the polarization turners. The four most important elements a^. are given as a function of the applied magnetic field in the y-direction at two different transmission angles ij) of the neutron beam direction with the empty sampleholder.
G, Automatization of the apparatus,
Since the depolarization measurements are performed during a few days cycle with the applied magnetic field and the temperature as variables, it is useful to automatize the measurements,
Both the room temperature measurements with varying field and the temperature dependent measurements have been carried out with quite a similar automation as
