Polarized neutron diffraction by a vortex line lattice in uniaxial superconductors

PhysicsLettersA 161 (1992) 545—548 North-Holland

PHYSICS LETTERS A

Polarized neutron diffraction by a vortex line lattice in uniaxial superconductors N.K. Zhuchenko LeningradNuclear Physics Institute, Gatchina, LeningradDistrict 188350, USSR Received 10 June 1991; revised manuscript received 10 September 1991; accepted for publication 10 October 1991 Communicated by V.M. Agranovich

The spin-flip diffraction due to the transversal magnetic fields ofthe vortex lattice (VL) is predicted. The calculations are carried out for the YBa,Cu,O 7.~(123) and Bi,Sr2CaCu2O8 (2212) high-Ta superconductors.

The unpolarized neutron diffraction by the VL has been used widely to study the spatial magnetic field distribution in the VL, its symmetry and imperfections, the correlation to the crystal lattice in Nb, Nb— Ta [1]. However, in the case of high-Ta superconductors there is only one successful paper [2]. The theory of the unpolarized neutron diffraction from the VL has been suggested in refs. [3,4]. In this paper we consider the polarized neutron diffraction by the VL in uniaxial superconductors. Theangle differential into a small solid cLQ byscatteringcross-section the VL is given by [3,5] do clQ

fp~u,~m~ )2 =

~ 2,th2 2

ductance B. The spin functions w~and w are chosen in the form ct’0 = (?), w= (i). By substituting cuo and w into (1) and carrying out [3] the following calculations we obtain for the spin-flip cross-section: (da/dQ)~~=p[H~(Q)+H~(Q)] ~ ô(Q—~) (2)

(3)

Q~kO—k=t~k.

2/S, ~ is the flux quantum, S Here = (i~p.BV/ ~) is theparea of the~superconductor in the x, y plane, B is the induction, H~( Q) and H,,( Q) are the Fourier transforms of H~,H,,, ‘rhk is the vector of the reciprocal lattice for (h/c) reflection. Relation (3) is the Bragg equation for (hk) reflection from the two-dimensional VL.

x~JdVexp[i(k 0—k)•r]~

.

(1)

V

Here p= 1.91 is the neutron magnetic moment in nuclear magnetons ~ m~is the neutron mass, k0 and k are the neutron wave vectors before and after the scattering, wo and ware the spin parts of the neutron wave functions before and after the scattering, H is the magnetic field in the vortex, o are the Pauli matrices. The integral in (1) is calculated over the superconductor’s volume V. First we consider the spin-flip cross-section. Let the z-axis be parallel to the vortex axis, i.e. to the in-

The same calculation of the non-spin-flip crosssection results in the following expression:

—

(do/dQ)~~~=pH~(Q) ~ ö( Q fhk

Thk)

.

(4)

The total cross-section ofunpolarized neutrons is the sum of (2) and (4). Using (do/dQ)~~ and (do/dQ)~~~ one can obtain the number of scattering neutrons with spin up and down. Then the neutron beam polarization after (hk) reflection is given by [5]

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P Po

=

PHYSICS LETTERS A

(do/dQ)fl~f—(da/dQ)~f (do/dQ)~+ (do/dQ)~f

20 January 1992

1.0

—

0.5

—

__________________________

-

(5) Here P0 is the polarization before the scattering. Note that in isotropic superconductors the vortex has only a H~component. Therefore in this case the spin-flip scatteringand the change in polarization are absent. The spatial magnetic field distribution in the VL is deduced from the measurement of the integrated reflectivity [1]. By an integration of the cross-sec tion over the solid angle dQ and the rotation angle around the z-axis the integrated reflectivity is found [1]. Using this procedure we obtain the reflectivities RSf, Rfl~ffor the spin-flip cross-section and non-spinflip one and the reflectivity R, for the total cross-section of unpolarized neutrons: RSf=G[H~(thk)+H~(Thk)], Rfl~f=GH~(’rhk) , Rt=RSf+RflSf,

2A~V

G= 2iL(~t) (So~o)2

—-

—0.5

-

-________________________________ 0 I I

the integrated reflectivities for the 123 and 2212 highT~,superconductors. The explicit expressions for the Fourier transforms H( ~hk) are derived in ref. [41.It follows from ref. [4] that these values are defined by the London penetration depths A 11 and A .L• In turn A~1 and anisotropy A ~ correspond the perpendicular supercurrents to flowing the axistoand it. along

I

I 40

e

‘80

60

100

(deg.)

Fig. 1. Polarization versus the angle 0 between the anisotropy axis of the superconductor and the axis of the vortex for (01) reflectjon, B 2000 G: (1) 123 and (2) 2212 high-T, superconductors.

1.0— --

(6)

In the formulae above, A~stands for the the neutron wavelength, S~,is the area of the two-dimensional unit cell of the VL. Relations (5) and (6) have been used to calculate the neutron beam polarization after scattering and

I 20

0.5

—

0.0

— -

60°

—0.5

—

—1.0

—

800 I

I

0

I

I

I I

10

2

I

20

30

(mN/mJ.)” Fig. 2. Polarization as a function of (m

1/m1 ) 1/2 at different an-

Fig. 1 represents the neutron polarization after scattering versus the angle 0 between the anisotropy axis of the superconductor and the vortex axis for the 123 and 2212 superconductors. We choose A~~=7000A,A~=l400A [6] for 123 andA~~=85000 A, A~=3400A for 2212 [7,8]. The polarization decreases sharply with increasing the angle and reaches a minimum at the angle Urn, which close to 900. The is polarization (P/Po)m at Urn is deduced if we take into consideration the small parameter z~= (AJA 2=m±/mii,where ~ and m 1i) 1 are the cor546

gles 0 for (01) reflection, B= 2000G.

responding electron masses. Let us take into account the explicit expression for the Fourier transforms H(l~hk) [4]. Then using formula (5) for (01) reflection we can obtain the values of Om and (P/Po)m: tg Urn =

2...4j

—1/2,

(~~/“o)m =

1 ) 1)2 — (,~— (~+

We can estimate

°m

and

—

—(1— 8i~).

(P/Po)m

(7)

for the 123 and

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PHYSICS LETFERS A

In fig. 2 the polarization as a function of (m11/m. )1/2 is plotted at different angles for (01) reflection. The polarization sharply 2 increases up todecreases (m 2—.10.when The (m1/m1)” 1/m.,j” calculations show that in the case of (10) reflection

_____________________________ ~

10~

R,~ Rf

j

io-’-

10

I

I

I

I

I

I I

20

I

I I

I

I

40

o

I I

60

I I

I

I

80

100

(deg.)

(b) -~ -

R~

10

R~ R,

-, -

10

10

-

I 0

I

I

I 20

I

I I

I 40

o

I I

60

I I

20 January 1992

I

I 80

100

(deg.)

Fig. 3. The integrated reflectivities versus angle 0 for a neutron wavelength of 10 A and a superconductor volume of 1 cm’, B= 2000 G: (a) 123 and (b) 2212 high-T, superconductors.

2212 superconductors using formulae (7). For example in the case ofthe 123 superconductor we have P~~00.04, 0m78.6°, (P/Po)m~ —0.7 in accordance with fig. 1. To understand the appearance of the polarization minimum the angular dependences of the polarization and the transverse field of the VL should be compared. The latter was calculated in ref. [9] for 123 at the same value of ,~.It is noted that the maximum of the transverse field occurs at U—~70°, as it does for the polarization in fig. 1. It follows from fig. 1 and relation (7) that the angular dependences of the polarization, Cm and (P/P0 )m can be used to find the ratio m1~/m.,..

the polarization decreases with (m 2 in the 11/m.)” region (mii/m.~)112~20. Fig. 3 represents the angular dependences of the integrated reflectivities for the 123 and 2212 superconductors. The calculations are carried out for a neutron wavelength of 10 A and a sample volume of 1 cm3. The value ~ approaches the total integrated reflectivity R 1 for unpolarized neutrons in the region of high angles. In this region the measurement of the spin-flip integrated reflectivity R~fis preferred to obtam the spatial distribution of the transverse magnetic field. Note that even in this angular region the transverse magnetic field is much less than the longitudinal one. According to ref. [9] in 123 the value of [ + 1/2 is approximately 30 Oe, which is only 1.5% from the longitudinal field =B=2000Oeinfigs. 1—3. In conclusion we show that the neutron spin-flip diffraction by the YL occurs in the uniaxial superconductors while the diffraction is the non-spin-flip one in the isotropic superconductors. This phenomenon can be used to determine the effectivemass ratio. In particular the total reversal of the neutron polarization is expected in the 2212 high-Ta superconductor when the vortex axis is almost perpendicular to the anisotropy axis c. Finally the polarization analysis technique is needed to find the spatial distribution ofthe small transverse magnetic field in the VL. The author is most thankful to A.V. Lazuta and V.1. Sbitnev for helpful discussion.

References [I

] J. Shelten, H. Ullmaier

and W. Schmatz, Phys. Stat. Sol. (b)48 (1971) 619. [2] E.M. Forgan, D. McK. Paul, H.A. Mook,343 P.A.(1990) Timmins, Keller, S. Sutton and J.S. Abell, Nature 735. H. [3] M.P. Ken,okliclze, Zh. Eksp. Teor. Fiz. 47 (1964) 2247 [Soy. Phys. JETP 20 (1964) 1505]. [4]V.G.Kogan,Phys.Lett.A85(1981)298.

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[5] S.W. Lovesey, Theory ofneutron scattering from condensed matter (Oxford Univ. Press, Oxford, 1986). [6] D.R. Harshmann, L.F. Schneemeyer, J.W. Waszczak and G.Aeppli,Phys.Rev.B39 (1989) 851. [7] S.G Barsov, A.L. Getalov and V.P. Koptev, Hyp. Int. 63 (1990) 87.

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[8] T.T.M. Palstra, B. Batlogg, L.F. Schneemeyer, R.B. van Dover and J.V. Waszczak, Phys. Rev. B 38 (1988) 5102. [9] V.G. Kogan, N. Nakagawa and S.L. Thiemann, Phys. Rev. B42 (1990) 2631. [10] M.M. Agamalyan, G.M. Drabkin and V.1. Sbitnev, Phys. Rep. 168 (1988) 256.