r*5g 10/sdilr/t[u$3 696r f,rnf *ra ',-rd-,...be stabiSised by fregueney spread", The ease Mere *his...
Transcript of r*5g 10/sdilr/t[u$3 696r f,rnf *ra ',-rd-,...be stabiSised by fregueney spread", The ease Mere *his...
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1 " Introduction
Transverse *ohenent ins*abilitf-esu driven by waka fi.eldsu nay
be stabiSised by fregueney spread", The ease Mere *his spread cores
from the longitudinal momentum spreaf, 0f *he beea is straightforward,
at leas* for e *oas*j"ng beamy beeause the xongttud:inar momentum is
a eonsts3"It, wb*"eh just effeets *he eoeffiei"er*s in the equations of
notion of the *ransverse ose*llatisns, ed henee tbeir frequency,
when the freqreeney spread esms frem a nonlinearity of the focrisirgu
*he si*ustion *s less elearu because the eohered motion is thea a
smaf,l ad&i*ion to the large i"neoherent mplS,tud.es *hat mab the frequeney
spread"u end it *s j":neensis*ent *o assure thet it san be .breated as a
Sinear superposi"tionu and. *t *s not obviouq that the large*anplitude
frequeney *s the epprtpriate effeetive resoaart fregueney for ealeuj.ating
the ad.di*ioaal motion &ue to a sma3l perturbetion*
Sinee the *"esu]* fs Euftc &Sfferen* f,mm the usuaL assumption,
we try *er nah it eomvinein6-by three &istinet attaeks; in sections
21 3u A" * But the essent*alsu in e rough eoneise fom, ane in Append.ix IT.
1 We eonsi&er on-ly the
the externel foeusing systemu
: spaee*eharge,
ease wlrere *hB non*3ffir Telm *ores fro*-
not tb.e ease where it esres from the beam
2. Free osei l lat ions
3n this seetion we stuQr e, "r:ather simple si'ema*ion to show tha"b
tir.e u-su.aJ- pi-cture of uon*-l-inearity eausing frequency tc d-epend on
amplitud.eu tllen amplitud.e range aaking a spnead" of resonant frequencies,
*an be mislead-ing" Tfe shall l-ook at "bb free*ringj-qg freq-uencar nf n bea^sr
r.ihi-ch aeeupies a. r"egion of phase*spaoe 'w-ith, ilni-for"rm d_ensity"
e onsi-de:: par-i.ieles that,move aceord-ing 'ho
f f + v $ x + F ( : * ) * 6 {z,r }
" s a-p n:1" tTdul€ l-LIaJaJJ-tp
a-tqe+Tns 'u {q ienba apg.u
;ro ;: d1s noau"€+-l-11u1'is
aq u€o .{aqfi '(vl'v:f oli ?.ilq
' a-gec s J0 ac "ror{c
ptl€ v €.r? saxEar'l8s s!--l :+
( z"z\
",i"{-siMir
s"6u
Y
- t"3T,{ uT Pelar+snlTT se d:trolcaCe'r1 V aq+ i'{-}Te lroJ?dtuoc
pa+JTqs at++TT e 1sn[ sT ilTT+rIo esot{ia lrot+nqTJ+srp 'f11suap ulroJTrm E
Je?Tsuos Jar{+€J sn +eT oS 'oJaz sT uoT+ou +ua;al{oa Jo apn+TTdue aqx 6 s1
+"q+ 'aIuT+ Jo luapuadepul 'fgalalduoc s 't uoT+nqT"r+sTp tr13suap aq+ +€tt+
.rfes e qans uT spolrad pu3 se?n+TTduE ur&o rTeI{+ q+Te pm-to* oF 11e
,taql geq4 sa+€+cT? uloJoar{+ sreTTT no;1 6so1c;pxed ;:o flgsuap ulroJTun
E i{+Te d"rolcefe'r+ "
qens 'fq pasolaua €are aI{+ sTT}J suo JI
"(v)aluaolan/aTuIoJJspoTJadTlF?unoJeq'TTT&Vs+oJazIIIoJJ
sapnlgdw g:o a9uex aq+ uI '(V)n1u7 Bol*ed aq+ ItT acuo +T punor
oF .(*o1aa$e;t1 s3q1 uo saToT+"red 119 '
aueld To* eq+ uT a1e19:ed € qcns Jo
ttTia srlJa+ cTtrouu€q *aqFlq aq+ Tt€us
ssa.:dxa uec
suoT+€TTTcso
*esdg11a lqSPrdn rre r{1q8no; s1
d:oqcefe-r+ eq+ Pu€ tosTe os aq
.f1"rge;: sg -rf1;*eauTT*uou ot{+ JT
scTriorrkreq *aq3;r4 + [dr + +(v)o] soc f = 16
a1o31.ted € Jo uoT+ou sq+
aro ?u€ V apnlg1dre rTaI{+ uodn puadap TTTa poprod aq4
.ra9;re1 ros .suor+errreso TT€us .1o pot'rad aq+ sT #
?u€
o=(o)ug e 6=(o)*1
asoddns u€c a& os 6acso;
Fu1"ro1sa: aq+ Jo q"-red -zeauTT-uou eq1 sluaso':da; (o)g aroq
*7,*
*3*
Ttre result of the shlft g is to make a thin lune-shaped" region
oecup iedbypa r t i e l es j us tou t s i de theA t ra j ee to r ya r r c l anempw
h:ne just i.nside it; eve:Xrwhe:"e else the density is as it was in
the unshifted case'
N o w e o n s i d . e r h o w t h e s e l u n e s n o v e w . i t h t h e p a s s a g e o f t i m e '
They nove ro'nd in tls phase pLane with the period' Zs/v(l)" Mo:re
p r e c i s e } y : t h e p o i n t s f f i w h i e h e o n s t i t l * e t h e i r b a s e l i e o n t h e
A trajeetory a-ad therefore go round with period exaetl-y zn/v(x)
f o r e v e r u w } r . i ] . e t h e p o i . a t s o * e l i e o n n e a r b y v a l u e s o f " n n p } i t u t l e
and. wilf go round' with perioas glqi.gF-gre.-Segq- to
gid"!L)"*=l:F. *PrL ql 'P4+ g ' rt is olear that the notion
of these li:aes determines the motion of'the eentroi{L sf the whole
beam, i.e* the eohererrt oseilfation" frus we have tbe soBewhat sur-
pristng result:
For e been having rmiforn phase-spaee densi\r fe^
a1.l aoplituites up to some Au *he free oseillation
period for oseillatioa of the ev€rage :ep 5-n the
linit that its amplitude is snall, is oqual to the
period. of individual palticl-es at a'uplittrde A' and
shovs no frequeireY sPreu'A**
f re queney spread or pqfll:r-g- llSl*-tl:ji*5?l19.Sifhus the
d.ensity is rt invisiblg,. r---****IbgJ notionuniform
discfose s the fre of the ertreme particles* We show i:r
Append:ix IJ how the effective
sueh E"s to give this resu-Lt"
coherent-oscillation frequencie s are
There
limit of snal'l-
are some more remarks about our valiility *in the
6 " i-n APPendix I"
is worth mentioning that this argument srith ths urliform
the lunes applies equally 'bo hi6her ortler modes like the
( tlrrobbrng-beam, bean*enrrelope , monopole) os cillatj-ons -
t +
d.ensity and
^ , , - - l . - ' - n l = nL t u a r A u p v + u l r
nu By 'rlri" we fitean 'i;here
no slow il i-ni:.uticn or
ihaL nonnail;tr resi-rl i !s
frequencf is inPrrre in
t ' ( l ) , cou , i r r 4 f r om the
is (in ihe J-imit of small coheren't ampJ-itud-e) -beating of 'che coherent arnplitude, of ,ln? SlUfrom siectnrn wielth- It remains true tha't the
the sense c'J' conta-''-ning higher imrltipl-e s of
non- l-inea.ri- tY ..
os
ecToq3
ox=]t 4;o+>+JoJ
:uot+nTos affes €q+ 11 an13
sra os uoSlenba au€s aI{+ sT (t-f ) o+ irBI{+ raTTreo + roJ I p*Y
'(V)a *o; a pa1d3:csqnsun ue 9u111'ri& ar€ eu 'd1;..re*e[ ro'{ '(e"Z) u'r
pauoT+uau .fpea"r1e sulJa+ cTuolIIJ€I{ ;raq91q fi{+ s+uosa*da; "q'q eJ'?r#e'
"T+{+ td+sdJsooY =ox
EoT+nTos aq+ a+Trla au (z"f ) Pat'rn+*adun aqq 'ro6
"rroT+ounJ €+Te? osrT0 eI{+ sT ( )S arat{&
(o+-+)gS=(x)*+x$rt+5
trq (l"f ) gtrlceltla-r deslndrnT uE Jo IIuoJ eI{+ uT uoT+€qrnp:ad
€. Jo +ce#Ja er{+ +SJTJ JepTsuoc o+ sT asn aa poq+aE er{f S
.g o+ tstroT+Jodo.rd oq TTTIa $s :C11eu1.r:
" tx Jo +r€d luepuadalug-g f,ne sploae ox aqeprdo':dde aq+ Jo
"+T uT JepJo +sJTJ uaql .raqpTq slera+ doap pue TTws g aumss€ €i&
o:r-x= r:f
rrTe+qo o+
* (o:c)'t + oxgd +o*
Jo uoT+n1os e+€Tddo'rdde aq1 +oar+qns pu€
+ogd:cs*=(x),*+x$n+5
aaTos TTPt{s a& oS "uoT+€qJn1'rad pu"ra}xa Tt?$s € o+
aplgred epnlgldue-egae1 € #o asuodsar aq1 aorni o+ peou ail
esuodse.: t[ouenbe.r,; eq6 "{
*5*
t6 the right
most general
hand. sid.e of
solution near
( l"t.)t o x o :
is again
The two quantities
and ean be found. bY
This requires
Differentiating
(in general K
For times later than
zero and we take the
fo r t> te s
*= (A+ 6A )cos [ v ( *+aa ) t + f +s r / ] +h "h ' ( l " e t )
5A and. 6tP are
satisffing (1"+)
constants which are treated as sma1I'
in the neighbourhood of t = to "
t - to+) t = t o * ) = O:c\x(
v l
(: 'z)+ + = t 6 - ) = 8 "
(1"5) and- introdueing
Aqu =P d A
d.epend.s on A)
the abbreviation
...-.:Ft t t
f o r * p t o
ana {1.?) gives us twof
eouations for 64 and 6itr lv-ith solutions
we
x
have
- X 0
+
ae u (t+r)
6ATrz tK
6pAv
sin( ut
eos(r"t
cos( ut
+ w)- , 1 \
+ tlt)
( : 'a)
l @ N d '
t^ . the time of aPPlication
i t" l " i tg i -ndePendent of the*
Th* quanti'bies EA
of Lhe imPulse, but
tine t, o
andu r l s J
6p depend onare ccnstan'Ls
!r't)+Pgdxa g
"q*q "l'
/\- ztt * z* { 4. * u\ = $x
+ffgdxas _*;*"E--
\)i '/
sprarr +rurr ;raddn eq+ +€ uor+€nr€^" t:="^:"""T,::1T#-Ti:ril--JJ:fiilT.:t;li, "il":; ";1;*oo" ,r*,s s+carra s+r ?u" 6 uo11eq'rnq"'xacl
,il-rat{ns J"orrTT 'fq uol1e1na F+"uITxoJdG? 1s'rr'} v+ "r'*-- --*--^-p 6"r..Tnrsod..radns J"orrTT 'fq uol1e1na
aJ{+ aunss€ a$ +€ri+ 1ce; a{* dq pag'rqsnf 6uo111so
o,r9aruT Ptls o1ofdxa {q
;::T I l_H u::*
;"""*- *,.."*o*, o*n a1e"r3a1u1 pue olordxa dq
r rr.r tla esec eq+ o&Tos -r€c s sT s 5tt''l' rad cTnou"req*aldurs r{+Tia asec eq+ o&Tos
itldrlgru +snq aD (u"6) uoT+€qrn+r-o "!*": -l-'-:ar.iln raorrartrau {
"I t;:; l"*-"**tad uor+cunJ*€+Top aq+ roJ sT srt{+ ;aquauau
"s uT -rapro +SJTJ aq+ s€
.I€Js€,t1uou1e19r3g€(6"f)uTs.Io+gEtT6ouePaI{+?a?u€dxaaaeqaAsJaI$|'
"r{'q +
a(ox -1)eoc( o+ a ** *
(ot"e )n("+ *+)rrr*(z/T - t) t
= sx r o+ € +
0= ryr o+>+
(6"( )
ty aog 9u1*15
g =Sg
(diffilffir*
-9*
g. =Yg
*7*
T h e l o w e r } i i t r i t i . s m o r e a w } m a r & * a s t h e r e a r e t e m s t h a t o s c i l l a t e
infiir-itely as t6 -+ - eoe but they 41tr eonverge to zero*if we assume
t h a t a l h a s a t } e a s t a l i t t l e u e g a t i v e i n a g i n a r y p a r t , l @
Tf one puts K * 0 (*oa h"h' = 0) : 'n (J- ' t l ) ' one reeovers
the f**lliar 1inear resonengs* The extra factor (t - VZ) in the first
line is possibly only an unrinportant eomeetion if K is snall, but the
seeond J-ine has a d.ifferent order sf (o * ') * elependenee' and is
something new"
&-* The unriforno-densitJr ease
frIo
phase*spaee t
would be to
by "d
; but we can easiS-y be a little more preeise' C
unperturbed. oseillation of (3.5) t
x s= A cos f r r t + rP ] +h *h *
@
& = ^A v sin[Yt + qf] + h"h"
The ellipse that fits the fundamental evid.ently has area n A"v
ou * plane, anf,
wan* to averege (3"tt) over a r:nifornly oeeupied region of
in order to eompare the result with that of section 2' Sinplest
multiply by srclAz *, integrate from zero to , anil iliviite
der the
Amonsi
( t * . t ;
in the
\
(:*"21/'d ( n A E u ) = 2 w ( 1 +
\.
So the response faetor averaged orrer
{ i . t r ) and ( t r . , z ) , to f i rs t o rd 'e r in
l \ \
2 I
the
T { i <
u AdA
u::-iform-lY fj-l1-ed area, from
* Th"u.t i'- s by Zsr Adi.
potTTJ ,{1uuo;:pm sq+ +€r{+ }Insar aq1 uiese a"=q nr-*"qg, ,anpr6#.r- o;raz"s€q-aila-*"q+
fG;t) +{*;; "r{+-"ni",i"*+b4"'saot
i"rqi ,,, uaql p.ro^,Bo1 pted .,treur8eus ar++TT e ,o anlg .:o uo11e;rFaluT Jo q*)&
"u* urroJap
uec pue *(f"t) ur sro+€ugtrouap o.rez lag aar d Jo a8ue;r oq+ uT salTpue T€a.r sT fD Jf * Er3
; Jo oa ;- o4 penba +ou sr m peplao,-rd
NI
,@*.dI
or r u Er| .* :*\ | :-;
u LzVn ) vI'Y
oq. lenba sp (t"t) oS
uuI4,VtL
b
wp
' ttv)a' roJ s?u€+s *n araqrr
. oo\,t,
/JqE
Yz (*"ro;* *- +-*)=
ffi - Jn$-ip. Er - ,sg-ikg =
,,P *z*
.Y''a
re?TsuoC 'pa1e.r9aquT oq uao sTqfr,
&' 8F4 *o*(
-8a
*?*
=gi:1 b9lgves as thou-6-h all the particles -hpd -q,.-IpF-.,a-narl-t..fg-€-qy_9-lgy
c*o.ye s po nd i n g,t_o - *the- p*e_rjp_-d at- the- . grea! g. g !- 9_ryp-l i t ud e "
l*::.1-.*"*"y.:-!"S*--hg:-PJ-'1-11kl--*'l-H1'e: However, real beams donot have a un-iforn phase-spaee density with a sharp cut*off, so rre
must extend. the theory to more general iiensity distributions before
seeing its practical irnplications*
5" Non*uniforn d.ensity
let the phase-space d.ensity be g:iven by
of partieles present is then, from (&"2)
p(A) - The number
p(a) d(rAeu)
-' $) u"o(e)
N= I' 0
@
fI f a
/ \ II
/ q , r \
/ c r )
and. the response
mo&tfication of
factor averaged.
( t - : ) :
over the part ieles is the appropriate
1N
t t vl (ry**t ""I l v - - u -U
j-nteresting toT'c i- s integrate this b"y parts" n ^ 1 av d f J
^ [
p (A) z.n
] =v u
u
"arraru.g puE rqgru"olrroJJ s+uauuros rnJasn srrros p€q osr' J 'rrrJoJl.rn Jaq+€J sr f+rsuapao€ds-es€qd aq+ JT uan.€ 6suo.rle11Tcso
TeuorsrTaurp_aarq+ Jo .-ojr+
Jo as€c eq+ uT FtrSdurap nepu€T Fu;*19 ur aAT+caJJe aq f,r:ur f11-reau11*uou ruorJ salcuanba;J Jo pEarde ari+ +eq+ +no pelulod aH "+xe+ aq+ JouorsJa.rl dreuufig1a.:d e FulpeeJ JoJ uo+Jol[6d quEi{+ o+ a^eq J
s4uaue$preTlorqcy "9
"+T pulJ o+ TT€J fllensn qceo"rdde +atrTp arom e asn oqla sraq+o +nq{poqlau uoT+€nba--d.os?Tl\ aq1 Fug.sn ,fq +t punoJi or{ra 6
tr.TE +a ++oTsETq13u saa.rge (t"E) 1e.r8a1ug uols.radsgp sit+ uT .\; rng
*e splap{ (r"5) pue .f,.ress"oo "*'ffi"iil:I;ffi=";i;-::; -€'LroJap rr€rus rsnsn eq+ o = f1 +8 ora'-uou sr ,d Jr "(1.5) o+rrt(5-V)S- =, a Fu11+nd lfq (5't) reaooar uec auo d4caqc e sy .Fugqlou
e+nqTJ+uoc ifllsuep EroJFrnl .1o suolg* 61 uopcas ur se : 1e.:ge1u1qgq+ opTsuT uoT+cun.f gu-r+q€Ta&, eq+ sT . d +ou ,W/op
+eq+ acT+og
st0n 0,-
/s I JVLa
"]
w 'o..-1d (v). d oF gi-
zY 4 ' '" _J;\y
q+Te &I8Toc*a pepgno.rds+TnTT
e.r€ oa (?"t) 9ugsg "qgnoua IT€usq+eq +€ seqsT[s^ a.ra+ +srTJ sr{f
trfP a sn
*O!*
o+ T€nba sT(z'6 ) oeql
4 4
Appendix Ts@
Snall perturbations" . - . - - , . *
* f f i
N o a p o } o r y i s n e e d ' e d f o r e a l c u l a t i r r g t h e c a s e o f s n a l l p e r -
turbations (srnal3" g on pagp 5 p srnall B' SA' 6{r' in section J)'
for we are nainly interested i-n eriteria for stabiJity of statS-onary
i n i t i a l & i s t r i b u t i o n s , a n d f o r t h i s t h e b e h a v i o u r i n t h e l i n i t o f
srnall pertuxtations is prinorcl-tal* One leaves until later the
b e h a v i o u r o f u n s t a b l e b s c i l l a t i o n s a s t h e y g r o w i n t o t h e r e g i o n o f
non-snall amFlitude, and the possibility of finite oscillations
grow'ing in a systeo where snalL oaes do notn 'Some care is however
n e c e s s a r y w i t h t h e t i n e v a r i a b l e . F o r i n s t a n c e , w h e n c o m p a r i n g ] c
with xo on page 5 we effectively made use of
cos [v(a + an)t + r/i
*o" [u(A)t * sr] *6ut sin[e(a)t + 4']f t r \
'T+L V
i s
d' cos[u(a)t + 4']is trrre that ffieorrect in the lirit of snall
is - #- sin[v(l)t + {l ,
fA , But clearlY it is in
t l
I t l -
"o ( : , t )
fact
lor" t l <. 1
-Lhat is- i ;he condit ion for (r . t ) to be a good approximation"
So we InaY saY:
f,or any ltl one can ehoose 6p small enough to
nake the aPProxination good"
- But, for any 16' l Sreater than zero the
approximation will fail at su'fficiently large
behaviour is typical of' a'pproxiroate solutions to
equaticns "
(r"z)
This so:r-L af
r1i-f f eren'rial
-3
aq+ +€ +urod ?axTJ aTqe+sun u€ pu" aTq"+s € errETd-as"qd 8uT+€q'o-r a{4+
uT sT aral& uoT+"iuTxo'rdde lensn aq+ o' " (,
'&roaqi suTl{c€u J€auTT*uou
uoJJ uisorr{ sT acueuosal € qons o+ uo Jo asoTo rlaql s8ul:q apn111due
uo acuapuadep-n asor{ir selcl4*red Jo JnoTr€qaq aq+ .'raaerog "sa1a11*ed
ffi .:o.g; poo8 aq o+ uoT+€ugxo'rdde Jno JoJ q9noua TT€lIIs g a{€w uec auo
+fl{+ snoT^qo +oTr sT +T os 6 r" o+ s?ua+ 4 s? ,t+pr'rul o+ s?ue+ +€q+
apnlgldrne alclped e saa18 (ul'f ) +eu+ s1 'f,11nc1JJTp rat{+ouv -;
"TTEIns aq. o+ peJepTsuoc aq u€c asTolr Jo
uoT+"TnuIT+s TEuJe+xa TaaaT +€q$ o+ dn elelncT€c u€o auo 3a1e': Sulduep
uTE+Jac E q+T& lfllgqep slcrPa"rd '&:oaq1 aq+ JT ''f1:e11u15
'+nE+suoca0T+ aq+ sT t a;aqla
! >> J"rtg
'r)
-aET+ lral € +s€e1 +€ roJ ift1e11ueuodxa gupm;r9
L >> +P"49 rf
sT sTq+ dsluelsuoc
ueaq s€r{ dg JT pu€
'l+l eerelpeuroJisP "f1sso:r9
l?,l ttne "ro; '+ng -
"uoT+stIJoJe?pug Jo.rJa
puno.r oB
.fut .rog * u.ec ar l+l
: {a9ed
s{J"uer 1e11e.red fllctxg
w/ (v)a +
QT
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resonant amFlitude, with topologY like Fig"2'
Fig. z
l$o part ic les go
a patte:'n ean be
without eausing
to unlim.itecl aqflitude, e.ren at large tu and such'
immersetl in a region of urriform phase*space density
anything special j-n the behaviour of the average x"
Finally a remark on the systematic disregard of the
harxron-ic ter-ms and the fact that we have wo:"lmd onf.y to the
in the coeffici-ent K* In a nachine wrth a hiSh Q-value '
lnerl:-um a"nd- high fiel'd- range of a machine with Q of orily 5
higher
f i rst order
or i n t he
o r s o , i t
"lf Jo sJo&od laq8pl pue scguou.req
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* 15 *
Lp:gndjx II
Average frequency
The behariour of the eentroid. of a reg'ion occupied. by particles
can be irrferrecl, when the ctensity is uniform, fron the bound.aries aLone "This is what we ditl. with fig""l, in section 2, But another anrL more
general method is to work out r*rat happens to the snaL] tlisplacernent
vector g for a general particle, and then average this (weighted
with the dtensity if it i.s. not uniforn) over the whol-e oocupierl area,
We do this for a notlel in which all partieles move in cireles
in an x* p plsse-plane, 1ib
* = y (a )p
$ = -u("t) x
with u rtepen&i"ng on the anplitude A" Tkis is
mod.elu but Hanrjltonia^ns ea3 be invented. to d.o
a little coherent oscillation in ad.dition will
( x+5x ) " = v (n+dA) , (p+
(p n sp)u = * u(e + m) " (x +
(rr , t )
a somevhat artificial
it* Now a particle w'ith
have
op)
s*) (='"z)
Subtraeting autt roz{d.ag to first ord.er in $*quantities we get
u( l )sp + pau
=-u(a )sx -x6v
u(l+a*)* v(a)cly .^Er o^
{S snA
5'x
s"p(m. t)
where
tTrom. the flefiniti-on of K^
6u
. . .
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JoAo FuT9?JaAV "aTeJTc
e o+ BulPuodsa"r.roc d
11c safcTlred .;:o 1es
v ar{+ puno"I
puB x q+Ta
€ oi€1 ie6u s!!asald ug
+nq. 6v ptr€ {dg'
9r:o1e ,tg'xg
;:6 , ''_
aa*@
=uYo
Exga
r:r
.II)
sqroceq (f"ff) os Pu€
:...,.
6d6x uoT+ceJTP or{+
ro+coa aq1 Bnppear Sq. (t"€?g) dr4ma8 aq+ trord
g'6,t
-91 *
*17-
px -'
2x " +
pa '+
0
L'/2
L'/2
and (fg 6) e so averaged, gives
cEx
a
ip
J i
So u( r +
for tb
ft (ul") = av
* ZAtr
i.ntegral and
u(t + r/z)op
+(t + r/z)s*
d v as the effecti've frequencY
of, the alPl-itutle-A Particles"
density of partieles from anpl"itutle
eoryfletely clear what is the interpretation
, but Te can anJrwaY calculate it :
(rr"s)
1T K/21 =-hv
A" F I h
#- | , (r+r/z)znaem uo
"" cli l' l l - @{ ' s aa
(r " vz)
nd.
Ar --'l m
lus" lL J O
(n.T)
t. n/t)*l = fu*Av "n
= u(nr)
*
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