5911d 5tate C0mmun1cat10n5, V01. 46, N0.2, pp. 155-158, 1983. Pr1nted 1n 6reat 8r1ta1n.

0038 - 1098/83 /140155-04 $03.00/0 Per9am0n Pre55 Ltd.

8R1LL0U1N 5CA77ER1N6 1N LAYERED 57RUC7URE5

R. H0t2, J.K. Krt19er and R. 51em5

Fach6ere1ch Phy51k, Un1ver51t~1t de5 5aar1ande5, Federa1 Repu611c 0f 6ermany

(Rece1ved 15 N0vem6er 1982 6y P.H. Deder1ch5)

7he0ret1ca1 and exper1menta1 re5u1t5 0n 8r1110u1n 5catter1n9 1n 1nh0m0- 9ene0u5 mater1a15 are pre5ented: N0rma1 e1a5t1c m0de5 and 8r1110u1n 1nten- 51t1e5 have 6een ca1cu1ated f0r a per10d1c 1ayer 5y5tem. Mea5urement5 0n 5em1cry5ta111ne p01y6utene-1 are d15cu55ed 1n term5 0f the the0ret1ca1 re5u1t5.80th the0ry and exper1ment y1e1d 5evera1 8r1110u1n 11ne5 1n V V 5catter1n9 1n5tead 0f the 0ne peak character15t1c f0r hnm09ene0u5 5pec1men5.

1. 1N7R0DUC710N

7HERE ARE MANY 1N57ANCE5 0f mater1a15 w1th a near1y per10d1c 5tructure 0n a macr05c0p1c 5ca1e, 1.e. w1th per10d1c1ty 1en9th5 1ar9e a5 c0mpared t0 at0m15t1c d1men510n5. Examp1e5 are, 0n the 0ne hand, certa1n re9u1ar arran9ement5 0f ferr0e1ectr1c 0r ferr0ma9net1c d0ma1n5, mater1a15 w1th near1y e4u1d15tant 9r0wth 0r def0rmat10n tw1n 60undar1e5, a110y 5y5tem5 w1th a 4ua51per10d1c pha5e 5tructure, type 11 5uperc0nduct0r5 w1th re9u1ar arran9ement5 0f v0rtex 11ne5, 5em1 cry5ta1- 11ne p01ymer5 and, 0n the 0ther hand, mater1a15 pr0d- duced f0r techn1ca1 app11cat10n5 wh1ch c0n515t 0f a matr1x re1nf0rced 6y a near1y per10d1c arran9ement 0f a harder 5u65tance (c0mp051te mater1a15).

1n the pre5ent paper the0ret1ca1 and exper1menta1 re5u1t5 are pre5ented 0n 8r1110u1n 5catter1n9 6y 5uch 1nh0m09ene0u5 mater1a15 w1th a 5pec1a1 5tructure. After a de5cr1pt10n 0f the m0de1 and a 5h0rt c0mp11at10n 0f the 6a51c e4uat10n5 1n 5ect10n 2, the0ret1ca1 re5u1t5 are pre5ented 1n 5ect10n 3 and c0mpared, 1n 5ect10n 4, w1th 8r1110u1n re5u1t5 wh1ch we 06ta1ned f0r 5em1cry5ta111ne p01ymer5.

2. 7HE0RY: M0DEL AND ME7H0D

We 5tudy the 1nf1uence 0f macr05c0p1c 1nh0m0- 9ene1t1e5 6y c0n51der1n9 a m0de1 5y5tem c0n515t1n9 0f an 1nf1n1te per10d1c 5e4uence 0f tw0 1ayer5 a and 6 w1th d1fferent 150tr0p1c e1a5t1c c0n5tant5 C, den51t1e5 p, w1dth5 A, and e1a5t0-0pt1ca1 ten50r5 P (cf. 1n5et 0f F19. 1). 7he d1e1ectr1c ten50r5 e are a55umed t0 6e e4ua1 f0r the tw0 1ayer5.

1n 8r1110u1n 5pectr05c0py [ 1-3 ] the 5pectra1 den51ty 0f 119ht 5cattered 1nt0 the detect0r 15 pr0p0r- t10na1 t0 the 5pectra1 den51ty 0f d1e1ectr1c c0n5tant f1uctuat10n5:

7 J(4, c0j-- c0) ~ J e-1wt(6e5a(4, 0), fe5a*(4, t ) )dt (1)

- - 0 0

w1th

fe(4, t) = f e-14rfe(r, t)dr (2) V

and

6e5~ 5 J = n~feken e (3)

Here n a and n 5 are the d1rect10n5 0f the p01ar12at10n5 0f the 1nc1dent and 0f the 5cattered 119ht. A 5Ummat10n 15 1mp11ed 6y repeated 5u65Cr1pt5 and ( ) den0te5 therma1 avera9e5, c0 a 15 the fre4uency 0f the 1nC1dent 119ht.

7he f1uctuat10n5 f1e are expre55ed, f0r 5ma11 def0rmat10n5, 1n term5 0f 10ca1 5tra1n ten50r5 e and d15- p1acement5 u 6y the P0cke15 re1at10n5

-- fe11(r, t) = PUtme1ra(r, t) (4)

w1th

e2m = •(ut,m + um,1),

and the 1atter are expanded 1n term5 0f the n0rma1 m0de5 0f the e4uat10n5 0f m0t10n

CUk1Uk, U - - p911(r , t ) = 0.

1n 9enera1, the va1ue5 0f C and p are d1fferent 1n a and 6. 51nce the m0de1 15 1nvar1ant w1th re5pect t0 ar61trary tran51at10n5 1n the x2-x3-p1ane5 (cf. F19.1,1n5et) and t0 tran51at10n5 6y D = A a + A 6 1n the x1 d1rect10n, the 5pace dependent part5 0f the n0rma1 m0de5

uJ9(4, r, t) = e1~9(4)tUJ9(4, r)

have the 9enera1 f0rm

U/9(4 , r) = e 1(02x2+4 3x3 +01Dn)~jJ9(4, ~) (5 )

w1th 1nte9ra1 n. 7he 10ca1 x1-c00rd1nate ~ = x1 - - D n

vade5 0ver 0ne per10d1c1ty 1en9th (e.9. 1 ~ 1 < D/2). 7he n0rma1 m0de5 are 0rth090na1 and were n0rma112ed 1n 5uch a way that

155

156 8R1LL0U1N 5CA77ER1N6 1N LAYERED 57RUC7URE5 V01.46, N0 2

Le [6 1 /

/ t •1

/ ' X2 X1

2 4 6 8

F19. 1. D15per510n curve5 and 8r1110u1n 1nten51ty f0r VV 5catter1n9 w1th ~4 = (0, 4., 0). AaJA ~ = 0.625; C F = 4.2C~; C2 ° = 1.9C~; ~a/1a° = 10; pa/p6 = 2. Fre- 4uency 12/21r 1n un1t5 0f C~/D, wavevect0r 4 1n un1t5 0f ~/D. 7he 5tra19ht 11ne5 1 a and 16 repre5ent the d15per- 510n 0f 10n91tud1na1 wave5 1n h0m09ene0u5 mater1a15 0f type5 a and 6. F0r 5ma11 f2 the c0mp051te 10n91tud1na1 ve10c1ty 15 3.1Ct 6 f0r the parameter5 91ven a60ve. 7he th1ckne55 0f the curve5 and the 1en9th5 0f the 1atera1 11ne5 1nd1cate the 8r1110u1n 1nten51ty (5ee text).

f .t t , • 1 p(r)UJ9(4, r)U ~ 9 (4 , r)dr = 8 f f~ (4 - - 4 )~99~. (6)

F0r an un60unded mater1a1 the c0mp0nent5 41 are c0nt1nu0u5 var1a61e5 def1ned 1n the 1nterva15 (--0% 00) f0r 1 = 2, 3 and (-- 7r/D, 7r/D) f0r 1 = 1.7he 5uper5c1pt5 ] and 9 1a6e1 the 5ymmetry type5 0f the 501ut10n5 and the d1fferent 6ranche5.

A 9enera1 d15p1acement f1e1d 15 de5cr16ed 6y the n0rma1 c00rd1nate5 Q19(4, t):

n/D

u(r , t ) : f**f d42643• f d41e1n•e(4)tUJe(4, r) -00 J, 9 -m/D

x 0J9(4, t).

8ecau5e 0f the 0rth090na11ty re1at10n (6) the Ham11- t0n1an 0f the 5y5tem 15 a 5um 0f 1ndependent harm0n1c 05c111at0r term5 and the c0rre1at10n funct10n f0r the n0rma1 c00rd1nate5 15 we11 kn0wn. F0r 1ar9e tempera- ture5 (e.9. r00m temperature) 1t 15 a 11near funct10n 0f 7.

3. 7HE0RE71CAL RE5UL75

A5 de5cr16ed e15ewhere [4, 5] we treated the ca5e 0f 150tr0p1c e1a5t1c mater1a15 1n the tw0 1ayer51n deta11.

App1y1n9 the pr0per 60undary c0nd1t10n5 at the 1nter- na1 1ayer 5urface5 and ne91ect1n9 a11 damp1n9 effect5, exp11c1t expre5510n5 were 06ta1ned f0r the n0rma1 m0de d15p1acement5. F0r ph0n0n5 pr0pa9at1n9 a10n9 the 1ayer5, e.9.1n x2-d1rect10n [4 = (0, 4, 0)], there are f0ur 5ymmetry type5 [4-7] den0ted 6 y / = 1, t, 35 and 3a. 7hey repre5ent (1) wave5 w1th U1 ant1metr1c, U2 5ymmetr1c 1n each 1ayer, and 0"3 = 0 (1, ••10n91tu- d1na1 1n the mean••); (2) wave5 w1th U1 5ymmetr1c, U2 ant1metr1c 1n each 1ayer, and U3 = 0 (t, ••tran5ver5e 1n the mean••); and tran5ver5e wave5 w1th U1 = 0"2 = 0 and U3(~) e1ther (3) 5ymmetr1c (35) 0r (4) ant1metr1c (3a) 1n each 1ayer.

J(4, c0j -- c0) wa5 exp11c1t1y ca1cu1ated. F0r 8r1110u1n exper1ment5 w1th 4 = (0, 4, 0) and VV 5cat- ter1n9, e.9., 0n1y 1 m0de5 are 065erva61e. 0ne 06ta1n5

J(4, c0j-- ~0) ~ [8{~0 -- c09-- ~22e(4)} 9

j 1 9 2

+ 8{~0 - ~ j + 52~9(4)}] 5219

f19 are 1nte9ra15 0ver 5pace der1vat1ve5 0f the m0de d15p1acement5.7hey c0nta1n P0cke15 ten50r e1ement5.

7yp1ca1 re5u1t5 f0r a 5pec1a1 ch01ce 0f (150tr0p1c) e1a5t1c c0n5tant5, w1dth5, and den51t1e5 0f the tw0 mater1a15 are pre5ented 1n F19.1, wh1ch 5h0w5 d15per- 510n curve5 and 8r1110u1n 1nten51t1e5 f0r ph0n0n5 0f 5ymmetry type 1.1n c0ntra5t t0 the h0m09ene0u5 5y5- tem (w1th 0n1y 0ne 10n91tud1na1 ph0n0n 1n the fre- 4uency ran9e 5h0wn 1n the f19ure) there 15 n0w a mu1t1- tude 0f 6ranche5.0n the 6a515 0f the 91ant un1t ce11 (w1th macr05c0p1c d1men510n5 1n x1-d1rect10n) 0ne w0u1d c1a551fy the5e a5 0ne/-type ac0u5t1ca1 and many /-type 0pt1ca1 6ranche5 wh05e fre4uenc1e5 are a11 1n the (n0rma11y) ac0u5t1c ran9e.

F19ure 1 c0nta1n5 a150 1nf0rmat10n5 0n the 8r1110u1n 1nten51t1e5 J c0nnected w1th the d1fferent n0rma1 m0de5:1nten51t1e51ar9er than 4% 0f the 1nten51ty J0 at f2 ~ 0 are 1nd1cated 6y th1ck 11ne5.1nten51t1e5 1ar9er than 20% 0fJ0 are 91ven 6y the 1atera1 11ne5 wh05e 1en9th5 are pr0p0rt10na1 t0 J.

W1th re5pect t0 the 8r1110u1n act1v1ty there are three ran9e5 0f wavenum6er5:

F0r very 10n9 wave5 0n1y the 10we5t 6ranch can 6e 065erved. 7he pha5e ve10c1ty 15 near1y c0n5tant. 1t5 512e 11e5 6etween the 10n91tud1na1 wave ve10c1t1e5 0f the tw0 6u1k mater1a15.

1n an 1ntermed1ate ran9e (e.9. fr0m wave1en9th5 A = 4D t0 A = 0.3D) there are 5evera1 6ranche5 w1th c0mpara61e 1nten51t1e5.80th ~2/4 and d12/d4 are d1ffer- ent fr0m the 6u1k ve10c1t1e5. Exper1ment5 1n th15 ran9e 5h0u1d 5h0w 5evera1 8r1110u1n peak51n VV 5catter1n9. F0r 5y5tem5 w1th var1a61e 1ayer 5pac1n9 the 1ntermed1ate

V01. 46, N0.2 8R1LL0U1N 5CA77ER1N6 1N LAYERED 57RUC7URE5

R 2

1

25 13H2

F19.2.8r1U0u1n 5pectra 0n me1t 0r1ented p01y6utene-1. VV, VH: vert1ca1-vert1ca1 and vert1ca1-h0r120nta1 119ht p01ar12at10n 51tuat10n. 4 11 5: ph0n0n wavevect0r5 a119ned a10n9 the me1t 0r1entat10n d1rect10n. R den0te5 the Ray1e19h 11ne (5uppre55ed 6y a fact0r 0.002). 1, 2, 3, 4 den0te 8r1110u1n peak5 w1th 1ncrea51n9 fre4uency.

re910n extend5 t0 1ar9er wave1en9th5 (1.e. t0 10wer va1ue5 0f 4).

1n the th1rd ran9e (1ar9e 4) the 065erva61e d15per510n curve5 are 9r0uped ar0und the tw0 curve5 f0r 10n91tud1na1 ac0u5t1c wave5 1n the 6u1k mater1a15.1f the w1dth 0f the 1ayer5 w1th 1ar9er 6u1k 50und ve10c1ty 15 c0mpara61e t0 0r 5ma11er than that 0f the 0ther5, 8r1110u1n 5catter1n9 0f the n0rma1 m0de5 w1th d15per510n curve5 near the 6u1k curve5 f0r the weaker mater1a1 d0m1nate5.1f the 1atter are 9r0uped c105e t09ether the re5pect1ve peak5 m19ht 0ver1ap due t0 dampm9 effect5.1n 5uch 5y5tem5, there- f0re, 8r1110u1n exper1ment5 may y1e1d 0n1y 0ne 11ne 1n VV 5catter1n9.

4. EXPER1MEN7AL RE5UL75

An 0ut5tand1n9 c0n5e4uence 0f the re5u1t5 pre- 5ented a60ve 15 the 0ccurrence 0f 5evera1 8r1110u1n peak5 1n the VV 5catter1n9 51tuat10n 1n the fre4uency ran9e n0rma11y c0rre5p0nd1n9 t0 ac0u5t1ca1 v16rat10n5. A 51m11ar 51tuat10n 5h0u1d ex15t 1n an150tr0p1c mater1a15 f0r ph0n0n pr0Pa9at10n a10n9 a pure m0de d1rect10n. 5uch peak5 were, 1ndeed, 065erved:

1n the 8r1U0u1n pr0f11e5 mea5ured 6y L1nd5ay and 5hepherd [8] 0n 0r1ented p01ymethy1methacry1ate (PMMA) f0r wave1en9th5 5ma11er than 6ut c0mpara61e t0 the d0m1nant 5pat1a1 per10d 0f the 5tructure, there are 2 we11 def1ned peak5 attr16uted t0 10n91tud1na1 ph0n0n5 pr0pa9at1n9 1n the 0r1entat10n d1rect10n. 7he1r p051t10n re1at1ve t0 each 0ther 5u99e5t5 an exp1anat10n a10n9 the 11ne5 d15cu55ed a60ve. C0mpare [9], h0wever, f0r a recent d15cu5510n 0f the5e mea5urement5.

7he appearance 0f 0ne add1t10na1 10n91tud1na1 ac0u5t1c ph0n0n 11ne 1n 8r1110u1n 5pectra wa5 a150

157

rep0rted f0r 5tr0n91y 0r1ented p01yethy1ene [11 ] (f0r ph0n0n pr0pa9at1n9 a10n9 the 5tretch d1rect10n) and f0r a p01y-4-methy1-pentene-1 me1t at the me1t tran51t10n [11 ]. F0r the5e tw0 ca5e5 the 0ccurrence 0f an add1t10na1 ph0n0n 11ne c0u1d a150 6e 1nterpreted 1n term5 0f a 5eparat10n 1nt0 am0rph0u5 and cry5ta111ne 5u6pha5e5 0f 5uff1c1ent 512e.

F0ur 10n91tud1na1 ac0u5t1c ph0n0n 11ne5 1n a 8r1110u1n 5pectrum were recent1y f0und 1n a 5pectrum 0f me1t-0r1ented p01y6utene-1 (P8-1)* 1nve5t19ated at r00m temperature (F19.2). 7he 8r1110u1n 5pectra were determ1ned w1th a h19h prec1510n 5-pa55 8r1110u1n 5pectr0meter de5cr16ed e15ewhere [13, 14]. 7he accuracy f0r the determ1nat10n 0f ph0n0n fre4uenc1e5 15 a60ut 0.01%. A free 5pectra1 ran9e 0f 50 6H2 wa5 ade4uate f0r the mea5urement5. A 1a5er vacuum wave1en9th 0f 514.5 nm wa5 u5ed. F0r the 90 A 5catter1n9 9e0metry [13, 15] u5ed, a ph0n0n wave1en9th 0f 363.8 nm re5u1t5, wh1ch, f0r VV-5catter1n9, d0e5 n0t depend 0n the effect1ve refract1ve 1ndex 0f the 5amp1e. 7he 5cat- ter1n9 cr055-5ect10n 0f P8-1 15 rather 1ar9e f0r the VV-5catter1n9 51tuat10n. H0wever, 6ecau5e 0f the 1nter- ference 0f d1fferent 5pectra1 11ne5 a re915trat10n t1me 0f at 1ea5t 2 hr wa5 u5ed f0r any 5pectrum t0 5h0w c1ear1y a11 5pectra1 feature5. F19ure 2 5h0w5 the VV- and the VH-5pectrum f0r wave pr0Pa9at1n9 a10n9 the d1rect10n 0f me1t 0r1entat10n. 1t 5h0u1d 6e n0ted that n0ne 0f the 065erved ph0n0n5 1n the VV 5pectrum c0rre5p0nd5 t0 a para51t1c ph0n0n fr0m a 6ack5catter1n9 51tuat10n (ref1ex-ph0n0n) [12, 13]. 7h15 wa5 pr0ved 6y u51n9 91a55 511de5 and an 1mmer510n 114u1d 0n the f11m 5urface5. Wh11e the VH-5pectrum d0e5 n0t 5h0w any ph0n0n 11ne at a11 (weak e1a5t0-0pt1c c0up11n9) f0ur ph0n0n 11ne5 are pre5ent 1n the VV-5pectrum. A5 w111 6e de5cr16ed e15ewhere [12, 16] th15 P8-1 f11m c0n515t5 0f need1e- 11ke cry5ta15 em6edded 1n an am0rph0u5 matr1x. 7he axe5 0f the need1e5 are para11e1 t0 the d1rect10n 0f me1t 0r1entat10n. Macr05c0p1ca11y the P8-1 f11m ha5 theref0re a un1ax1a1 5ymmetry w1th a ••nemat1c-11ke•• 5uper5tructure [16]. 7he f11m had a ery5ta111n1ty 0f 54% -+ 10%. 7he cry5ta1 need1e5 have p5eud0- hexa90na1 5ymmetry w1th the hexa90na1 ax15 a10n9 the need1e ax15.7he need1e d1ameter 15 a60ut 100,8, [16 ].

A 51mp1e add1t10n 0f the 8r1U0u1n 1nten51t1e5 0f the tw0 pha5e5 (wh1ch w0u1d anyh0w 6e perm155161e 0n1y 1f the un1f0rm re910n5 were 5uff1c1ent1y 1ar9e) y1e1d5 0n1y tw0 10n91tud1na1 ph0n0n 11ne5 f0r wave pr0pa- 9at10n a10n9 the un14ue ax15 0f the f11m even 1f the tw0 pha5e5 are e1a5t1ca11y an150tr0p1c. Furtherm0re, e1ectr0n m1cr05c0p1c 1nve5t19at10n5 5h0w that there are 0n1y tw0

* 7h15 f11m wa5 k1nd1y 5upp11ed 6y Dr. Petermann, Fachr1chtun9 1 1.2, Un1ver51ty 0f 5aar6r11cken.

158 8R1LL0U1N 5CA77ER1N6 1N LAYERED 57RUC7URE5 V01. 46, N0.2

d1fferent pha5e5 1n the P8-1 f11m [12, 16]. We 6e11eve, theref0re, that, a5 de5cr16ed a60ve, the 5pec1a1 9e0m- etry 0f the tw0 pha5e5 5tructure 15 re5p0n5161e f0r the unu5ua1 5pectra1 feature5 0f F19.2.

1n c0nc1u510n, a5 a c0n5e4uence 0f the the0ret1ca1 re5u1t5 pre5ented 1n the f1r5t part 0f th15 paper, 0ne expect5 t0 f1nd 5evera1 8r1110u1n 11ne5 1n mater1a15 w1th a macr05c0p1c per10d1c1ty. 1n 4ua11tat1ve a9reement w1th th15 expectat10n there are a num6er 0f 8r1110u1n exper- 1ment5 1n wh1ch, 1ndeed, 5evera1 ph0n0n 11ne5 were 065erved, wh1ch c0u1d 6e exp1a1ned ne1ther 0n the 6a515 0f an an150tr0p1c h0m09ene0u5 m0de1 n0r 0f a 51mp1e tw0 pha5e m0de1. F0r a 4uant1tat1ve c0mpar150n 0f the0ry and exper1ment ca1cu1at10n5 f0r a m0de1 w1th a tw0-d1men510na1 per10d1c1ty are needed, wh1ch w0u1d 6etter repre5ent the 9e0metry 0f actua1 5em1cry5ta111ne p01ymer5.5uch ca1cu1at10n5 are under way.

Ackn0w1ed9ement - 7h15 w0rk wa5 perf0rmed w1th1n the frame 0f the 50nderf0r5chun956ere1ch ••Ferr0- e1ektr1ka••, wh1ch 15 5upp0rted 6y the Deut5che F0r5chun959eme1n5chaft.

1.

REFERENCE5

6.8.8enedek • K. Fr1t5ch,Phy5. Rev. 149,647 (1966).

2. 8. Chu, La5er L19ht 5catter1n9, Academ1c Pre55, New Y0rk (1974).

3. 8.J. 8erne • R. Pec0ra, Dynam1c L19ht 5catter- 1n9, J. W11ey, New Y0rk (1976).

4. R. H0t2 • R. 51em5 Recent Deve10pment51n C0nden5ed Matter Phy51c5, (Ed1ted 6y J.7. Dewee5e, L.F. Lemm0n5, V.E. van D0ren • J. Van R0yen) V01.4, Henum Pre55 (1981).

5. R. H0t2, J. Kr119er • R. 51em5 (t0 6e pu6115hed). 6. 7.J. De1ph, 6. Herrmann • P.K. Kau1, J. App1.

Mech. 46, 113, (1979); 47,531, (1980). 7. 5.M. Ryt0w, 50v. Phy5. Ac0u5t1c5 2, 68 (1956). 8. L1nd5ay • 5hepherd, 0r1entat10n Effect5 1n 5011d

P01ymer5, (Ed1ted 6y 6.80der), p. 85. J0hn W11ey and 50n5 (1972).

9. J.K. Kr119er • M. P1etra11a, P01ymer 23, 3 (1982). 10. J.K. Krt19er, A. Marx • L. Peet2, Ferr0e1ectr1c5

26,753 (1980). 11. J.K. Kr119er, L. Peet2, M. P1etra11a • H.-6. Unruh,

P01ymer 8u11et1n 4, 591 (1981). 12. J.K. Kr119er, R. H0t2 • R. 51em5 (t0 6e pu6115hed). 13. J.K. Kr119er, L. Peet2 • M. P1etra11a, P01ymer 19,

1397 (1978). 14. J.K. Kru9er, R. K1mm1ch, J. 5anderc0ck • H.-6.

Unruh, P01ymer 8u11et1n 5, 615 (1981). 15. J.K. Krf19er, L. Peet2, M. P1etra11a • H.-6. Unruh,

C011• P01ym. 5c1. 259,215 (1981). 16. R.M. 60h11 • J. Petermann,J. Mat. 5c1. (1982)

(1n pre55).