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Neutron Depolarization in Superconductors
N. Zhuchenko
To cite this version:
N. Zhuchenko. Neutron Depolarization in Superconductors. Journal de Physique I, EDP Sciences, 1995, 5 (4), pp.475-483. �10.1051/jp1:1995141�. �jpa-00247073�
Classification Physics Abstracts
74.30C 74.60J 74.60G
Neutron Depolarization in Superconductors
N-K- Zuucuenko
Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg district, 188350, Russia
(Received 16 November 1994, accepted 4 Jauuary1995)
Résumé. On calcule la dépendance en champ magnétique de la dépolarisation des neutrons le long du cycle d'hystérésis en termes du modèle critique de Bean, On considère aussi la dépo-
larisation dans les supraconducteurs uniaxiaux
en fonction de l'aimantation réversible, y compris
pour les supraconducteurs magnétiques. On attend une forte dépolarisation si les neutrons se propagent le long des vortex.
Abstract. The dependences of neutron depolarization on applied magnetic field are deduced along trie magnetization hysteresis loop in terms of the Beau mortel of the critical state. The
depolarization in uniaxial superconductors with the reversible magnetization, mcluding uniaxial magnetic superconductors,is also considered. A strong depolarization is expected ifthe neutrons travel along the vortex fines.
1. Introduction
Tue neutron depolarization metuod bas been intensively applied to study tue mixed state of
both uigu-T~ and low-T~ superconductors [1,2]. Trie misalignments of vortex fines from trie ap- plied field direction, trie characteristic fields of trie superconductor and trie London penetration
deptu are found by tuis technique. Tue depolarization data uas been analyzed by introducing empirical formulas [3,4]. In tuis paper we consider the depolarization by a superconductor
m terms of the critical state model. Besides the depolarization in a umaxial superconductor
with a reversible magnetization is treated. Both electronic and magnetic superconductors are considered.
2. Depolarization of Neutrons in trie Critical State Model
We restrict our consideration to the Beau model [Si. Let a superconductor be a thin plate with dimension L~, Lz > 2L as shown in Figure and trie z-axis be along trie direction of magnetic induction m a superconductor. We assume trie velocity of incident neutrons and their polarization to be along trie z-axis. So a neutron crosses trie vortex fines in this transversal
© Les Editions de Physique 1995
476 JOURNAL DE PHYSIQUE I N°4
1+8j~8+
o
Lz il
~ ~
Y
~ ~1
X 2L
Fig. l. The z-axis is parallel to both the applied magnetic field Ha and the vortex fines. The velocity
of neutrons M and polarization before the sample Po are parallel to the x-axis. The anisotropy axis c
is in the (x, z) plane.
geometry. Tue polarization components P~, Py of neutrons tuat bave passed through trie
superconductor are given in [6]
P~=Po<cosçg>, Py=Po<sinçg> (1)
~J = (~f/v)/~ B(Y)dz
= (~f/v)B(Y)L~ (2)
< sin çg >= (1/L~) /
smçg dy, < cosçg >= (1/L~) cos çg dy (3)
~ Î~
Here ~/ and v are trie gyromagnetic ratio and trie velocity of neutrons, respectively, çg is trie
neutron spin precession angle inside tue superconductor, B(y) is tue local induction m tue
superconductor. In~ equation (2) we take into account trie fact that trie local induction does not depend on z. It follows from equation (1) that trie depolarization D is [6]
D=1-(Pj+P))~/~/Po=1-(<sinçg>~+<cosçg>~)~/~ (4)
Thus trie depolarization m a superconductor is related to changes m a phase çg for dioEerent neutron patins.
We calculate tue depolarization and change in polarization P~/Po for distributions of in- duction suown in Figure 2. Distributions 1 and 2 correspond to tue first and second field increase up to 2H* after zero field cooling of trie superconductor. Here H* is trie applied field at which trie magnetic flux reaches trie center of trie plate. Then trie field is reversed down to -2H* and decreased to zero. Trie distribution upon trie third field increase in trie positive direction is shown in diagram 3, while àiagrarn 4 corresponds to trie field decrease from 2H*
to zero. Finally diagram 5 represents trie trapped flux distribution after zero field cooling of
trie superconductor. In this case Ha denotes tue applied magnetic field before it decreases to zero.
To calculate tue depolarization it is convenient to introduce tue following pararneters
h = Ha/H* and à
= (~llv)H*L~ (5)
Taking into account equations (1)-(4) and tue distributions in Figure 2 tue depolarization and
change m polarization P
= P~/Po can be easily found in tue region 0 < h < 2.
2 3 ~ 5
Fig. 2. The field distribution inside the superconductor according to the Bean mortel. The shadow
area is the magnetic flux distribution at Ha = 0. The diagrams (1)-(3) are for the (1)-(3) subsequent
field increases. Distributions of (4) and (5) are for the field decrease and field increase upon the trapped
flux measurements. See text.
For trie first field increase we obtain
P = ô~~[à(1 h) + sm ôh] (6)
o $ h £ 1
D = ô~~ [ô~P~ + 4sin~(ôh/2)]~/~ (7)
P =
ô~~[sinôh sinà(h 1)] (8)
1<h<2
D = i (à/2)-1sinô/2 (g)
For trie second field increase they are
P = ô~~[smà + sin ôh 2sin (ôh/2)] o £ h £ 2 (la)
D = 1 ô~~[6 4 cos (ôh/2) 4 cos à(h/2 1) + 2 Cos à(h 1)]~/~ (ll) Upon trie third field increase after its reversai polarization P is given by equation (8) whereas
depolarization follows equation (9).
Depolarization upon trie field decrease is equal to that upon trie second field increase. Trie polarization is
P =
ô~~[2 sinà(1 + h/2) sin à sm ôh] 0 < h < 2 (12) Finally polarization and depolarization for trie trapped flux measurements are
P = ô~~ [(1 h)à + 2 sin (ôh/2)] (13)
0 £ h £ 1
D = 1 ô~~[(ô~P~ +16 sin~(ôh/4)]~/~ (14)
P = ô~~[2 sin (ôh/2) sin à(h 1)] 1 $ h £ 2, (15)
478 JOURNAL DE PHYSIQUE I N°4
.O ~
~°~ 4 (b)
3
~ ~ O.5
3
°°.5
~ O.O
-o.5
~'~
l.O 2.O ~'~ 0.5
h h
Fig. 3, Depolarization (a) and polarization (b) mrsus the reduceà fielà h at à = 5. The curves
(1)-(5) correspond to cases (1)-(5)
m Figure 2.
whereas depolarization is given by relation (11).
Figure 3 shows trie dependence of depolarization and polarization on trie reduced magnetic
field h. All trie properties of depolarization come from trie relative distributions in Figure 2. In
particular it cari be seen that trie changes in trie induction are trie same both upon trie second field increase and trie field decrease. By trie same considerations trie constancy of depolarization
upon trie third field increase is deduced.
Tue dependence of depolarization on trie phase à is derived from relations (6)-(15). Depo- larization goes to unity as tue phase increases except for botu tue depolarization upon tue first field increase and that in trie trapped flux state, where it is close to h for h < 1. In Figure 4 trie depolarization and polarization P are shown for a few phases à. With increasing à trie
depolarization increases and approaches trie asymptotic dependence mentioned above. Trie amplitude of trie oscillation of P decreases with trie field due to trie increase of depolarization,
a fact which agrees qualitatively with trie depolarization measurements in reference [3].
Note that depolarization is absent if trie neutrons travel along tue y-axis. Finally tue sub- stitution of L~ for Lz suould be made in relation (5) if trie neutron velocity is parallel to trie z-axis. This is trie case of longitudinal geometry when trie neutrons travel along trie vortex lines.
Trie field distributions inside trie superconductor bave been lately deduced in terms of trie anisotropic Bean model [7]. If trie amsotropy c-axis is parallel to either trie z- or tue y-axis,
tuere appear tue Hi and H( applied magnetic fields. Tuey correspond to tue magnetic flux penetration up to a center of tue plate along tue
z- and y-axes, respectively. In general, only
a numerical calculation can be carried eut. However relations (6)-(15) uold true in tue applied
field region Ha < H( ifH] < Hi. On tue contrary, depolarization is close to zero for H( » Hi
and Ha < Hi.
Relations (6)-(15) can be used to find tue field H* by means of eituer a one-dimensional or
a three-dimensional polarization analysis technique. Our approach enables us to apply other models of trie critical state. To choose tue specific model of trie critical state, measurements of tue neutron depolarization dependence on tue phase à or, tuat is tue sonne, on tue neutron
wavelengtu suould be carried eut. Tue situation is very similar to tuat in magnetic materials
iO (ai ~
~~j
3
o.5
ao.5 a
~
O.o
1.O 5 2.O ~'~.O
h h
Fig. 4. Depolarization (a) and polarization (b) at Ha = 0 mrsus reduced field h for dilferent phases à, The reduced field h is that before it decreases to zero: (i) -à
= 5, (2) -à
= 10, (3) -à = 50.
wuose magnetic domain structure uas been studied by spectral neutron depolarization [8].
3. Neutron Depolarization in Abrikosov's Superconductor
Depolarization in a superconductor witu reversible magnetization is only related witu tue distribution of tue local magnetic field in tue vortex lattice [6]. Depolarization is known [9] to be very small for uigu applied magnetic fields Ha > H~i if tue polarization of incident neutrons
is parallel to trie vortex lines and neutrons cross them. Near H~i trie neutron polarization vector tutus about trie vortex line direction because it is no longer parallel to trie applied field [10].
A small depolarization was found in trie transversal geometry of trie experiment [6]. On tue contrary more depolarization is expected in tue longitudinal geometry because tuere is no
averaging of trie field along trie neutron patin.
In this section we calculate trie depolarization in trie uniaxial superconductor in trie longi-
tudinal geometry for a high applied magnetic field fia > H~i, so that trie vortex litre direction is parallel to it. Let trie anisotropy axis c be inclined with respect to z by an angle @, as shown
in Figure 1. We neglect trie transversal magnetic field in VL because it is much smaller than trie longitudinal Bz comportent iii]. Trier depolarization is given by relations (1)-(4) where trie magnetic field is
Hz(r)
= £ Hz(Q) exp (for) (16)
Q
The explicit expression for trie Fourier transform Hz(Q) bas been derived in [12] using trie London model for trie anisotropic superconductors.
Hz(Q)
= B(1+ À~mzzo~)/d
d = (1+ À~mIQÎ + À~m~XQ()(1+ À~mzzo~) À~m(~Q~Q( (17)
Here is trie average geometric London penetration depth and m~z, mzz, mzz are trie compo- nents of trie eoEective mass tensor m trie (x,y, z) coordinate system, B is trie induction in trie
480 JOURNAL DE PHYSIQUE I N°4
superconductor.
m~~ = mi cos2 + m3 sin~
@, mzz = mi sin~ +
m3 cos~
m~z = (mi m3) sin@cos@ (18)
Here m3 and mi are trie eoEective masses of a superconducting electron for trie motion along
trie c-axis and in trie plane perpendular to it, respectively, and these masses are normalized so that m3m( = 1. All tue possible Q vectors are those of trie reciprocal lattice.
These vectors rhk are given by [13]
rhk " 12x/S)jhjL/2)x + (k h/2)ayj, S
= aL/2 (19)
where a and L are trie unit cell parameters of VL along trie z- and y-axes, respectively, which
m trie field region Ha > H~i are [12]
a = (24lo/B)~/~(mzz/3m3)~~~, Lla = arctg (3m3/mzz)~/~ (20)
wuereas S is tue area of unit cell and 4lo is trie flux quantum.
Standard routines for tue Fourier transform are designed for tue two-dimensional Bz(r). To calculate tue depolarization tue dioEerent neutron patus crossing only one unit cell are taken and averaging in relation (4) is carried out by direct summation over ail of tuem. We also
calculate tue depolarization in tue transversal geometry to compare it witu depolarization in
longitudinal geometry. Tue former is given by [6]
D = i Jo()H~(roi) (21)
where Jo is the spherical Bessel function of zero order.
The dependence of depolarization on is easily found from relations (17)-(21) in the transversal geometry. This anisotropy of depolarization is D(90°)/D(o°)
= mi /m3 if D « 1.
With the notation e
= À~/Àab = (m3/mi)~~~ where À~ and Àa~ are the London penetration depths along the anisotropy axis c and perpendicular to it respectively, we have mi
=
e~l~/~)
and m3 " e~/3 and
= (À(~À~)~/~ [12]. Then it follows from relations (17)-(20), that only
À~ and Àab are needed to carry out the calculations. Trie values of À~ and Àa~ for the differ- ent superconductors are presented in Table I. All the following calculations are related to a
superconductor with a thickness of L~
= 1 cm as well as to the neutron wavelength of1 À (v = 3.959 x las cm/s).
Figure 5 represents depolarization as a function of the angle between the vortex lines and the anisotropy axis c for different HTSC for both the longitudinal and transversal geometry.
Depolarization for the longitudinal geometry is much larger than the depolarization for transversal geometry. Its angular dependence can be used to measure the amsotropy parameter
Table I.
Superconductor Àa~( À~( À( Ref.
Lasr 1400 4400 2500 [14], [15]
123 1400 7000 2400 [16]
2212 3400 85000 9900 [17], [18]
1-O lO
(ai (bi
O.8 la -2
~
3 ~
'~ 10
°
cJ
O.4
10
O.2
10
~'~
40 60
~ (~~~ j 8 (deg.)
Fig. 5. Depolarization as a function of the angle 0 for the longitudinal (a) and transversal (b) geometry, B = 2000 G: (1) Lasr, (2) -123, (3) -2212.
e. Another davantage of longitudinal in comparison with transversal geometry is that the much
more imperfect VL is appropriate to observe neutron depolarization. This failure of transversal geometry has been pointed out in reference [6].
Another interesting subject for the depolarization study is magnetic superconductors. The interaction between a regular magnetic lattice and superconducting electrons results m many
more changes in induction as compared to usual electromc superconductors [19]. Then an
increase of neutron depolarization should be expected.
To calculate the depolarization we neglect the transversal induction component in a uniaxial magnetic superconductor because it is small in comparison with the longitudinal one [20].
Relations for the Fourier transform of the longitudinal Bz component have been derived in reference [19]
Bz(Q)
= B l~ + ecos~ (j £)j
C C
where
ÎÎÎÎ~~IIÎ " ~( ~~~~sill~~ QS Q2)
l 47rxi ' ~ l 47rxi)À2 (22)
Here xii and xi are the magnetic susceptibility along the c-axis and in a plane perpendicular
to it, respectively.
The anisotropy of depolarization for the transversal geometry follows from equations (21) and
(22) in a direct and simple manner if D < 1
D(90°)/D(0° = il + £(1 1/C)i~ (23)
To calculate the depolarization we use trie unit cell parameters of magnetic superconductors
obtained in reference [21].
482 JOURNAL DE PHYSIQUE I N°4
.O
o 0.5
O.O
40 60
~ (deo.)
Fig. 6. Depolarization ~ersus the angle 0 for the magnetic superconductor with e
= 0.9, 47rxi " 0, S/A~ = 1, = 1000 À: (1) longitudinal, (2) transversal geometry.
Figure 6 represents the depolarization both for longitudinal and transversal neutron travel-
ling. Depolarization is rather large even for transverse geometry. Trie study of depolarization
can be used to measure the amsotropy parameter e.
Acknowledgments
This work has been supported by the Russian National Program of High Temperature Super- conductivity (Grant 93050) and the International Science Foundation.
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