- Email: [email protected]
No spiral vortex along c-axis in anisotropic superconductors
~ )
00351~92~.00+.00 Per~mon Pm~ L~
Solid S~PrniM ~t Grmt C°mmBritain. u ~ f ° n s ' Vol. ~ , No. 7, pp. 53%54~ 1992.
NO SPIRAL V O R T E X ALONG a A X I S IN A N I S O T R O P I C SUPERCONDUCTORS "M.L~uli~ ~ d A.Krgmer Fachbereich Phyfik, ~ d e U ~ r ~ Be~n A r ~ m ~ 14, D-1000 Berfin 33, W-Germany. (Recei~d 2 ~ u ~ y
1992 by E. M~n~i)
The pmMem of the existence of ~iral ~ s ~ati~ Mo~ t ~ ~ i s h an ~ m ~ c superconductor is reconfidered ufing the L~don ~ o a c h ~r the i~6on b~ween vortex elements. We show th~ them h ~ spirM instabifty ~r a ~n~e vo~ex ~ co~rad~t~n to m~nt m s ~ d Koyama ~ d T a c ~ . We ~ s briefly th~ a ~ h t e vo~ex lattice h Mso sta~e against s m ~ a m p l i ~ hading ~ a spirM confl~ration ufing nonbcM el~6c ~ r ~ H o w e ~ a spir£ ~gabifi~ is p ~ d ~ e ~ the p r ~ c e ~ a ~ t ~ g the v ~ t ~ fine.
The ~coordinate ~ chosen to be ~ong the ~axis, then
Re~ntl~ K~ama ~ d Tachi~ [1] ~op~ed the exhtence of s~rM vor~ces ofien~d ~ong the ~ m m ~ ~ h ~ uniaxiM sup~conductors. T~s w~ motivated by ~ e ~ s ~ t th~ the ~Wener~ ~ g r ~ g ~ v ~ c e s lying ~ the ag~ane ~ low~ than the energy d tho~ p~alld to the ~axh. In ord~ to r~xamine this pmMem we ~ t with the ~ p r ~ o n br ~ e h ~ r ~ o n betw~n vort~ elements ~ anisotm~c s u p ~ n d u ~ o ~ . In the London fimit the fr~ ~ergy ~ an ensembh ~ ~ d flux f n ~ h~ t ~ form ~
F=~. ff v~(~-G)dl,.dl~,
~= (~,-k.,O).
To account ~r spiral ~ o r f i o n the vortex ~ne ~ ~ wfi~en ~ function of a p~ameter a: .~
(3)
ARer some straightforward c ~ c ~ n the s d f energy of the s~ral vortex (1) ~ reduced to the form
F... = h(.,~)+.'~[~,(~,~)- 6(-,~,~1, (1)
(4)
where I~, I~, and I~ read
*~
h(.,~)=/ ~ / f ~ g5~'t~.~*(&~, h(.,~)=/ ~ f / ~ ~(cos~-.)) ×
: ~ : :~r oVp~~(~me : ~ t ifo~o~::~ ~:~'~et ~ e : : ~ : r ~ : : ~ : merits, ~ d a, ~ ~ {x, y, z}. In the c~e of u~axiM s~ p~conductors this p ~ e ~ i ~ is ~ven by ~ ~ a ( ~ = 1 + 1A ~ { ~ " ~ - 1 + A~A'+ A ~ } "
= ~co~ ~a~n~a)
~+~,
(~
Her< ~ = g × ~ A, = ~h, A~ = ~ - ~h, X~ is the in-#ane magnetic p e n ~ r ~ n depth, ~ d A~h the peru ~ration de~h ~r currents flowing p~pendicul~ ~ the a~#an¢ In (1) the ~mmation runs over the vortices wh~e fine demen~ d~ ~ r ~ t with e~h other by the p~e~iM ~ a ( ~ - ~ ) . S~ce we ~e h ~ r ~ d ~ the vortex sdf-en~gy we ~op the summ~bn over i,j, retM~ng oMy the douMe i~egrM ~ ~e fine donents of a ~n#e vortex. To ~mpli~ its evMuat~n and in or~r to ~ m p ~ e our resu~s with those ~ [1] we introduce ~men~omess umts g ~ ~ L dY ~ d O ~ , ~ ~ ~ / ~ , w ~ h the se~energy is #yen in u~ts of ~ / ~ A ~ .
~-~+'~ { ~ cos ~a cos ~¢+ ~ ~n ~ ~n ~¢ + ~ ~n ~(~ + ,)}. (5) Here, R(~)
~
+~~
, s(5 =~ +
,~ ~ ,~ = ~,~
~(gll, a, ~) = a[k,(cos ~ a - cos ~ ) + k~(sin~a - fin ~¢~ and ~1 = (k=, k~, 0). In the fu~her ev~uation of the integr~s in (5) cyfndfic~ coordinates k= = kll cos(O+ ~ ) , k~ = ~1fin(0 + ~ ) have b~en used. By inserting (5) into (4), and after integrations over ~ + ¢, k,, and 6 one gets the sd~energy per unR hngth
*On have of absence from Ins~tute of Physics Bdgrade Univerfit~ Bdgrade, Yugo~avig 537
538
e-AXIS
IN ANISOTROPIC
? - u = h ( a , a ) + ~-~--~[]~(a, ~, ~ = ~) + ~ ( ~ , ~ , ~)
-hC~,fl,+ = t) +,hC~,~,+)], C6) where
hCa, Z)
5(~,fl,,)
=
j~o~ ~ ~o ~ ~ e
x
-~
V~. 82, No. 7
S ~ E ~ O ~ ~
neglection of the second term in E~(]6) of [1] which clearly vanishes in first order of aft, but should contribute in higher orders. A comp~te calculation of this term leads to a result whid~ is equivalent to our exprcm sion (6). Before presenting the results of the numefcM calculation of the self-energy (6) we first derive an approximate expres~on for F,,tl. For suffidently small fl and ~11 >> 1 the arguments of the Bessd functions may be approximated by a~t or a ~ t v ~ , resulting finally in ~ V/I + ¢a~fl ~ log ~,.
~!
=
This is ~ctually the curve length of the spiral scaled by v~ in x- and y-direction. From this, and from the numer~al computations of (6) presented in Fig.1 for vafio,s values of a, ~, ~, it follows that there is no instabifity due to spiral exaltation of the straight vortex, because
/o++/:+:?-,
1,(a,Z, ~) =
+ In order to avoid the ~ng~ar b ~ a d o u r of and ~ the Cuboff ~ll is introduced in the ~ e g r ~
~, I~, over
~. We would ~ to p~nt out that our expr~s~n ~ r t~ ~ner~ ~ ! ~ t ~ s~ral vortex ( ~ ~ not the ' same as ~ven ~ K ~ a m a ~ d T ~ h i ~ [1] ~ t~ir ex~es~on (19}). T~s ~ s c r e p ~ procee~ from the
(a)50-~+
(8)
~
(9)
k.u~A4 t k.,~.~,).
The result of (9) is also confirmed by using the nonlocM ~asticity theory for small displacement ~ In that case the ~ast~ energy of the spiral vortex propagating along
the ~axB
reads
,% = -~-{+..(k, =o, fl)++,.,Ck, =o,~)},
(to)
~) 5.0
=5,0
=10.0
4.0
6=1.0
4.0
3.0 ~
3.0
~C 2.0
2.0
1.0
1.0 ~
~=0.4 0.0
"
0.0
"
'
I
0.5
r
i
~
I
1.0
'
r
I
I
1.5
I
I
0.0
I
2.0
I
0.0
[
'
I
~
0.5
Fig.1. fl dependence of the self-energy of the spiral vortex for several values of the amplitude a, and anisotropy parameter e. The Ginsburg-Landau-parameter ~, is chosen to be 20.
r
=
r
0.01 r
+.0
+
~
'
I
~.5
+
~ ~
2.0
V~, 82, No. 7
c-A~S ~ A ~ S O T R O P I C
5.0
where '
1 2 3 4
4.0
d
539
SUP~CONDU~ORS ¢=1.0 ~=0.5 ~=0. I ~=0.01
(~+~,(~+~#V,,(~+~-GoG#V,,(O)}.(11) 5.0
Here, ~ are redprocal lattice vectors which are small in the case of small magnetic induction, i.e. A~ I ~ I'~ 1. This megn.s that the summation over.~ can be replaced by an integration over ~ ~ading to the following expres~on for the das~c energy :
2.0
1.0' f=0.5
f~ = ~-~~kdk { (l + k~)(12++k~k~+ B~)
0.0
(1 + ~ + ~)(~-(1 ~)~+e~ + ~) } ,
where
~
= F,//(~),
and
B = ( ~ ,
a ~
;
;
I
I
I
0.5
(12)
a~.~.
Here, the p~ameter c char~t~ses the ~pe ~ the vortex l~fice. W h e n we d ~ ~th a ~ute vortex ~stem the ~wer ~ u n d ~ y ~ ~ e i~egral ~ (12) ~ zero ~cause ~ . ~ o m (12) it h ~ v ~ u s th~ ~ ( a , ~ E 0 ~r e~h value ~ ~, t ~ c o ~ r ~ ou~ prev~us ~ e m e ~ on t ~ ~sence ~ ~ e ~ inst~H~ ~ ~ e vortex~ e ~ along the ~ . Up to now we con~dered only the magnetic part of vortex self-energy. The reason for this is that in the case of large G~nsbur~Landau-parameter ~II the core contribution to the sd~energy may be .ne~ected (see for exasnple [4]).The energy of the vortex core is pr~ portional to the curve ~ngth of the sp~al, where the z- and ~-direct~ns are scaled due to the anisoWopy of the superconductor. T~s ~ads to a term w~ch has the same form as (8)~ and therefore does not qualitatively change our result. Finally, we discuss b~efly t~e p o s ~ l i t y of the s~ral vortex in~abi~ty in the presence of external forces. This idea was first daborated by Clem [3] for isoWopic superconductors, where it has been shown that a helical (spiral) in.ability occurs in the presence of a ~n~tudihal current flowing along the vortex axis. The Gibbs ~ee energy in the presence of a current f(~) at the pl~ce of the vo~ex is ~ven in reduced units by
G= ~,o~,~-
I
0.0
~ / d ~ (Y × ~ ) . ~(~). (13)
~ =0.5 ;
I
~
I
;
1.0
I
;
~
I
1.5
#
2.0
Fig.2. ~ dependence of the Gibbs freeenergy of the spiral vortex (15) with amplitude a = 0.5 ~nd longitudinal current [ -- 0.5 for several values of e. The GinsburgLandau-p~aameter sllis chosen to be 20.
Choosing f(a) = J(a)~= and I = ~fdaJ(a)/ f da log ~[I the Gibbs ~ee energy per unit ~ngth reads (in reduced units)
~,O
= ~/(~&O-~M~.
(i~
Some numerical results~r fixed a, and I as ~nc~on of e, and ~ ~ e shown in ~ . W e find a ~ m u m ~ ~ w~ch ~ s~Red to larg~ valu~ of ~ if the anisotro~ parameter e ~ ~wered. T~s ~ o~y an i~Rrat~n how the instabiH~ depends on the ~isoWo~. A study of the real ~ u ~ o n has to take into account the ~ i a l ~ b ~ n ~ ~ e cu~ent and ~ e i~eract~n wRh other vortices as w ~ discussed by Brandt ~. Our ~ m p ~ calcu~ion is wrong ~r lar~ am~Rudes a, which c~n be shown ff (13) ~ w~tten ~ the ~ p r o x ~ o n ~):
~(~ ~, d = ~I + ~
- ~,~
°-~
-~.
~g~ T~re~re we g~ no fi&te ~ s ~ e I > 0 in the a ~ a n ~
~&mum
of ~ ~r
Acknowledgements - W e thank K.D. Schotte for usefull discussions. One of us (M.L.K) ~ thankful to the Bundesmi~erium f~r Fomchung und Technologie (BMFT) for the finandal suppo~.
~0
~ A ~ S ~ ANISOTROPIC SUPERCONDUCTORS
V~. ~ , No. 7
References [1] T.Koyama and M.Tachiki, SoLSt.Comm. 79, 1¢49 (1991) [2] E.H.Brandt, Physica B165-16~ 1129 (1990); A.Sudbo and E.H.Brandt, Phys.Re~ B43, 10482 (1991)
[3] J.R.Clem, Phys.Re~Lett. 24, 1425 (1977) [4] D.Saint-Jame~ G.Sarm~ EJ.Thomas TypeII superconducti~ty (Pergamon Pre~, 1969) [5] E.H.Brandt, £ Low Temp. Phys. 44, 33 (1981)
00351~92~.00+.00 Per~mon Pm~ L~
Solid S~PrniM ~t Grmt C°mmBritain. u ~ f ° n s ' Vol. ~ , No. 7, pp. 53%54~ 1992.
NO SPIRAL V O R T E X ALONG a A X I S IN A N I S O T R O P I C SUPERCONDUCTORS "M.L~uli~ ~ d A.Krgmer Fachbereich Phyfik, ~ d e U ~ r ~ Be~n A r ~ m ~ 14, D-1000 Berfin 33, W-Germany. (Recei~d 2 ~ u ~ y
1992 by E. M~n~i)
The pmMem of the existence of ~iral ~ s ~ati~ Mo~ t ~ ~ i s h an ~ m ~ c superconductor is reconfidered ufing the L~don ~ o a c h ~r the i~6on b~ween vortex elements. We show th~ them h ~ spirM instabifty ~r a ~n~e vo~ex ~ co~rad~t~n to m~nt m s ~ d Koyama ~ d T a c ~ . We ~ s briefly th~ a ~ h t e vo~ex lattice h Mso sta~e against s m ~ a m p l i ~ hading ~ a spirM confl~ration ufing nonbcM el~6c ~ r ~ H o w e ~ a spir£ ~gabifi~ is p ~ d ~ e ~ the p r ~ c e ~ a ~ t ~ g the v ~ t ~ fine.
The ~coordinate ~ chosen to be ~ong the ~axis, then
Re~ntl~ K~ama ~ d Tachi~ [1] ~op~ed the exhtence of s~rM vor~ces ofien~d ~ong the ~ m m ~ ~ h ~ uniaxiM sup~conductors. T~s w~ motivated by ~ e ~ s ~ t th~ the ~Wener~ ~ g r ~ g ~ v ~ c e s lying ~ the ag~ane ~ low~ than the energy d tho~ p~alld to the ~axh. In ord~ to r~xamine this pmMem we ~ t with the ~ p r ~ o n br ~ e h ~ r ~ o n betw~n vort~ elements ~ anisotm~c s u p ~ n d u ~ o ~ . In the London fimit the fr~ ~ergy ~ an ensembh ~ ~ d flux f n ~ h~ t ~ form ~
F=~. ff v~(~-G)dl,.dl~,
~= (~,-k.,O).
To account ~r spiral ~ o r f i o n the vortex ~ne ~ ~ wfi~en ~ function of a p~ameter a: .~
(3)
ARer some straightforward c ~ c ~ n the s d f energy of the s~ral vortex (1) ~ reduced to the form
F... = h(.,~)+.'~[~,(~,~)- 6(-,~,~1, (1)
(4)
where I~, I~, and I~ read
*~
h(.,~)=/ ~ / f ~ g5~'t~.~*(&~, h(.,~)=/ ~ f / ~ ~(cos~-.)) ×
: ~ : :~r oVp~~(~me : ~ t ifo~o~::~ ~:~'~et ~ e : : ~ : r ~ : : ~ : merits, ~ d a, ~ ~ {x, y, z}. In the c~e of u~axiM s~ p~conductors this p ~ e ~ i ~ is ~ven by ~ ~ a ( ~ = 1 + 1A ~ { ~ " ~ - 1 + A~A'+ A ~ } "
= ~co~ ~a~n~a)
~+~,
(~
Her< ~ = g × ~ A, = ~h, A~ = ~ - ~h, X~ is the in-#ane magnetic p e n ~ r ~ n depth, ~ d A~h the peru ~ration de~h ~r currents flowing p~pendicul~ ~ the a~#an¢ In (1) the ~mmation runs over the vortices wh~e fine demen~ d~ ~ r ~ t with e~h other by the p~e~iM ~ a ( ~ - ~ ) . S~ce we ~e h ~ r ~ d ~ the vortex sdf-en~gy we ~op the summ~bn over i,j, retM~ng oMy the douMe i~egrM ~ ~e fine donents of a ~n#e vortex. To ~mpli~ its evMuat~n and in or~r to ~ m p ~ e our resu~s with those ~ [1] we introduce ~men~omess umts g ~ ~ L dY ~ d O ~ , ~ ~ ~ / ~ , w ~ h the se~energy is #yen in u~ts of ~ / ~ A ~ .
~-~+'~ { ~ cos ~a cos ~¢+ ~ ~n ~ ~n ~¢ + ~ ~n ~(~ + ,)}. (5) Here, R(~)
~
+~~
, s(5 =~ +
,~ ~ ,~ = ~,~
~(gll, a, ~) = a[k,(cos ~ a - cos ~ ) + k~(sin~a - fin ~¢~ and ~1 = (k=, k~, 0). In the fu~her ev~uation of the integr~s in (5) cyfndfic~ coordinates k= = kll cos(O+ ~ ) , k~ = ~1fin(0 + ~ ) have b~en used. By inserting (5) into (4), and after integrations over ~ + ¢, k,, and 6 one gets the sd~energy per unR hngth
*On have of absence from Ins~tute of Physics Bdgrade Univerfit~ Bdgrade, Yugo~avig 537
538
e-AXIS
IN ANISOTROPIC
? - u = h ( a , a ) + ~-~--~[]~(a, ~, ~ = ~) + ~ ( ~ , ~ , ~)
-hC~,fl,+ = t) +,hC~,~,+)], C6) where
hCa, Z)
5(~,fl,,)
=
j~o~ ~ ~o ~ ~ e
x
-~
V~. 82, No. 7
S ~ E ~ O ~ ~
neglection of the second term in E~(]6) of [1] which clearly vanishes in first order of aft, but should contribute in higher orders. A comp~te calculation of this term leads to a result whid~ is equivalent to our exprcm sion (6). Before presenting the results of the numefcM calculation of the self-energy (6) we first derive an approximate expres~on for F,,tl. For suffidently small fl and ~11 >> 1 the arguments of the Bessd functions may be approximated by a~t or a ~ t v ~ , resulting finally in ~ V/I + ¢a~fl ~ log ~,.
~!
=
This is ~ctually the curve length of the spiral scaled by v~ in x- and y-direction. From this, and from the numer~al computations of (6) presented in Fig.1 for vafio,s values of a, ~, ~, it follows that there is no instabifity due to spiral exaltation of the straight vortex, because
/o++/:+:?-,
1,(a,Z, ~) =
+ In order to avoid the ~ng~ar b ~ a d o u r of and ~ the Cuboff ~ll is introduced in the ~ e g r ~
~, I~, over
~. We would ~ to p~nt out that our expr~s~n ~ r t~ ~ner~ ~ ! ~ t ~ s~ral vortex ( ~ ~ not the ' same as ~ven ~ K ~ a m a ~ d T ~ h i ~ [1] ~ t~ir ex~es~on (19}). T~s ~ s c r e p ~ procee~ from the
(a)50-~+
(8)
~
(9)
k.u~A4 t k.,~.~,).
The result of (9) is also confirmed by using the nonlocM ~asticity theory for small displacement ~ In that case the ~ast~ energy of the spiral vortex propagating along
the ~axB
reads
,% = -~-{+..(k, =o, fl)++,.,Ck, =o,~)},
(to)
~) 5.0
=5,0
=10.0
4.0
6=1.0
4.0
3.0 ~
3.0
~C 2.0
2.0
1.0
1.0 ~
~=0.4 0.0
"
0.0
"
'
I
0.5
r
i
~
I
1.0
'
r
I
I
1.5
I
I
0.0
I
2.0
I
0.0
[
'
I
~
0.5
Fig.1. fl dependence of the self-energy of the spiral vortex for several values of the amplitude a, and anisotropy parameter e. The Ginsburg-Landau-parameter ~, is chosen to be 20.
r
=
r
0.01 r
+.0
+
~
'
I
~.5
+
~ ~
2.0
V~, 82, No. 7
c-A~S ~ A ~ S O T R O P I C
5.0
where '
1 2 3 4
4.0
d
539
SUP~CONDU~ORS ¢=1.0 ~=0.5 ~=0. I ~=0.01
(~+~,(~+~#V,,(~+~-GoG#V,,(O)}.(11) 5.0
Here, ~ are redprocal lattice vectors which are small in the case of small magnetic induction, i.e. A~ I ~ I'~ 1. This megn.s that the summation over.~ can be replaced by an integration over ~ ~ading to the following expres~on for the das~c energy :
2.0
1.0' f=0.5
f~ = ~-~~kdk { (l + k~)(12++k~k~+ B~)
0.0
(1 + ~ + ~)(~-(1 ~)~+e~ + ~) } ,
where
~
= F,//(~),
and
B = ( ~ ,
a ~
;
;
I
I
I
0.5
(12)
a~.~.
Here, the p~ameter c char~t~ses the ~pe ~ the vortex l~fice. W h e n we d ~ ~th a ~ute vortex ~stem the ~wer ~ u n d ~ y ~ ~ e i~egral ~ (12) ~ zero ~cause ~ . ~ o m (12) it h ~ v ~ u s th~ ~ ( a , ~ E 0 ~r e~h value ~ ~, t ~ c o ~ r ~ ou~ prev~us ~ e m e ~ on t ~ ~sence ~ ~ e ~ inst~H~ ~ ~ e vortex~ e ~ along the ~ . Up to now we con~dered only the magnetic part of vortex self-energy. The reason for this is that in the case of large G~nsbur~Landau-parameter ~II the core contribution to the sd~energy may be .ne~ected (see for exasnple [4]).The energy of the vortex core is pr~ portional to the curve ~ngth of the sp~al, where the z- and ~-direct~ns are scaled due to the anisoWopy of the superconductor. T~s ~ads to a term w~ch has the same form as (8)~ and therefore does not qualitatively change our result. Finally, we discuss b~efly t~e p o s ~ l i t y of the s~ral vortex in~abi~ty in the presence of external forces. This idea was first daborated by Clem [3] for isoWopic superconductors, where it has been shown that a helical (spiral) in.ability occurs in the presence of a ~n~tudihal current flowing along the vortex axis. The Gibbs ~ee energy in the presence of a current f(~) at the pl~ce of the vo~ex is ~ven in reduced units by
G= ~,o~,~-
I
0.0
~ / d ~ (Y × ~ ) . ~(~). (13)
~ =0.5 ;
I
~
I
;
1.0
I
;
~
I
1.5
#
2.0
Fig.2. ~ dependence of the Gibbs freeenergy of the spiral vortex (15) with amplitude a = 0.5 ~nd longitudinal current [ -- 0.5 for several values of e. The GinsburgLandau-p~aameter sllis chosen to be 20.
Choosing f(a) = J(a)~= and I = ~fdaJ(a)/ f da log ~[I the Gibbs ~ee energy per unit ~ngth reads (in reduced units)
~,O
= ~/(~&O-~M~.
(i~
Some numerical results~r fixed a, and I as ~nc~on of e, and ~ ~ e shown in ~ . W e find a ~ m u m ~ ~ w~ch ~ s~Red to larg~ valu~ of ~ if the anisotro~ parameter e ~ ~wered. T~s ~ o~y an i~Rrat~n how the instabiH~ depends on the ~isoWo~. A study of the real ~ u ~ o n has to take into account the ~ i a l ~ b ~ n ~ ~ e cu~ent and ~ e i~eract~n wRh other vortices as w ~ discussed by Brandt ~. Our ~ m p ~ calcu~ion is wrong ~r lar~ am~Rudes a, which c~n be shown ff (13) ~ w~tten ~ the ~ p r o x ~ o n ~):
~(~ ~, d = ~I + ~
- ~,~
°-~
-~.
~g~ T~re~re we g~ no fi&te ~ s ~ e I > 0 in the a ~ a n ~
~&mum
of ~ ~r
Acknowledgements - W e thank K.D. Schotte for usefull discussions. One of us (M.L.K) ~ thankful to the Bundesmi~erium f~r Fomchung und Technologie (BMFT) for the finandal suppo~.
~0
~ A ~ S ~ ANISOTROPIC SUPERCONDUCTORS
V~. ~ , No. 7
References [1] T.Koyama and M.Tachiki, SoLSt.Comm. 79, 1¢49 (1991) [2] E.H.Brandt, Physica B165-16~ 1129 (1990); A.Sudbo and E.H.Brandt, Phys.Re~ B43, 10482 (1991)
[3] J.R.Clem, Phys.Re~Lett. 24, 1425 (1977) [4] D.Saint-Jame~ G.Sarm~ EJ.Thomas TypeII superconducti~ty (Pergamon Pre~, 1969) [5] E.H.Brandt, £ Low Temp. Phys. 44, 33 (1981)