Few-body collisions in a weakly interacting bose gas

Few-body collisions in a weakly interacting bose gas

Citation for published version (APA):

Stoof, H. T. C. (1989). Few-body collisions in a weakly interacting bose gas. Technische Universiteit Eindhoven.

https://doi.org/10.6100/IR317773

DOI:

10.6100/IR317773

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Published: 01/01/1989

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FEW .. BODY COLLISIONS IN A

WEAKLY

INTERACTING BOSE GAS

PROEFSCHRIF'I'

WI" verkrUglng van de graad van doctor aan de Tedlnlsdle Unlver.sitelt Eindhoven, op gm:ag van de Redor Magnfflcu5. prvf. Ir. M. 'hIs, voor Ben commlssle aangewezen door het CoIIMgfl van Dekanen In het openbaar

te verdedlgfIR DP vrljdag 20 oktober 19S9 te 14.00 uur

dODr

Hendricus Theodoru5 Christiaan Stoof

gebOren te Veldhoven

Dit proefschrift is goedgekeurd door

de

promotoren

prof.

dr. B.J. Verhaar

en

beaut Hut theory. beo:tl./.se lJou """t the principle of

beauty when !Iou Qre estabL1shing fundamenta.l laws.

Str<ec

0""

h ""'rking frOlTl. a ""'-t/l.eJru:tuo.>l basts. one is

gullied Very br-gcty by the requirement of mathemc.Heo.l

b"a.u.tll. If the equ:>.t ton:'; of physics are not

IPQ.t""=tiGatt!l beautlfut thc.t d.enates an imperf"ction.

and. 1 t me.ans thc.t the theory is at f=lt and. nee<!.s

l.,pro~ement. There are o"caslons ufu:,., IPQ.tMm<>.t lcaL bCQ.Utll .houLd take prlDrHy 0""..- "9r""",,,nt with ""P"r 1men t. ( . . .

J

A

beautiful theor~ has universality and ~~ to

pred1ct. lo tnte~"t. to set up '''''''''ph'$ ¢.J1d to Irorl<

",Uh them.

Once

yOlL hc,ve the fundamen.taL

ta,.,,,.

and \lOt.<

want to appLy them. 1I0u don.' t need the prtr.ctpte of

beo.utll tu"LlJ more, because tn trea.tin.g pracHcal p~obl"",:!;

one hM to taRe. tnto accOWH """y details and tMnss

become "esslI MlI=!I.··

1. INTRODUCTION

L aasE~EINSTEIN CONDENSATION AS A SPONTANEOUS sYMMETRY BREAKING PHENOMENON

j

r .

SPIN-POLARIZED ATOMIC HYOROC£N

III. HYDROGEN ATOMS AS COMPOSITE BOSONS

IV. SCOPE OF THIS THESIS

2. SPIN-EXCHANCE AND DIPOLE REI..J..XATION RAlr.S IN ATOMIC HYOROCEN, RIGOROUS A1'P

SIMPUFIED CAI.C\JLArIONS [PhY~. Ray. 838, 46BS (1900)]

I.

INTRODUCTION

I I . TRANS ITlON AlIPLInrnES AND RATES j I I. COUl'lED--CHANl'IE1..5 APPROACH

IV. RESULTS AND DISCUSSION

V.

CONCLusIONS

:3. TlW'PINC Or NEUJRAL ATOMS WITH I!E.<iONAIU MICROWAVE llADIATlON

[Phy •. Rev. L-~tt. 62, 2351 (lSS9)] 2 5 9 16 23 24 25 27 29 33 35

4. DECAY OF SPIN-POLARIZED ATOMIC IIYDROCEN IN 'l1lE .PRESENCE OF A BOSE CO/'IOENSATE

[Pbys. Rey. A39, 3157 (l9S9)] 41

I. INTRODUCTION

I L ATOMIC HYDROGEN AS A WEAKLY INrERACTINC BOSE GAS

II I. 0 I POl.AR RELAXATION IN H H IV _ CONCLUSIONS AI'PENDIX -12 42 46 51 52

AND THREE DIMENSIONS [Phys. Rev. A38. 1248 (1988))

I. INTRODUCTION

II.

T MATRIX

Ill. LOW-ENERGY SCATTERING: ON-SHELL T MATRIX

IV. lOW-ENERCY SCATTERINC:

HALF-SHELL

T MATRIX

V.

SEPERABLE

APPROXIMATION TO

THE

T MATRIX

VI. CONCLUSIONS

APPENDIX

6. SPIt'l-l'Ol.J.RlZED ATOMIC

fI"(lJROGEN

IN VERY SfRONG MACNErIC FIELDS

[Phys. Rev. B3S. 11221 (19SS)] I.

INTRODUCTION

I I.

METHOD

I I L RESULTS

IV. CONCLUSIONS

55 56 56 58 59 61 53 63 67 6S 68 69 10

7. REsoNANCES II'! RECOMBINATION OF ATOMIC HYDROGEN DUE TO LONCE-RANCE

Ha.

IIOLECI.JLA.!/.

STATES [Phys. Rov. B (aeeept~d for p~blle~tlQn)]

I.

iNTRODUCTION

II. RESONATING-GROUP TaEOR"t

III.

RESULTS

AND

DISCUSSION

IV. CONCLUSIONS SlJlOJMAl1. y SAIIrnV A IT INC 73 H 75 79 86 89

ClI.APTfR 1, INTROWCTION

I. BOSE-EINSTEIN COt1IlENSATlOO AS A SPONTANEOUS S"tMMETl!Y BREAKI HG PHENO/IIENON

The initial successes of quantum mechanics we~e associated with systems havln~

~ fL~~t~ ~~~h~( Qf degreQ~ Qf ft~~dOm_ Th~ su~~es~ful explanation of the properties

of such systems led to a genecal belief in the correctn~ •• of thQ q~ant~~ me~h~~t~al

th~o,y. V~,y d1ffQ,~nt ~h~nom~n~ have .Ince then been explained without

modifications of the basic formalism. However. there are physical phenom~n~. khQ~

~oll~~t~vely ~~ ~~(ro~¢op~~ q~~~tu~ effectg. whcse explanation requires the extension

or

quantum mechanics to syst9m~ wIth ~~ 1pfln~l~ p4rnbe~ Qf ~~g~ees of freedoM. The first develcpments leading to this extension of quantum mechanicst

~hich i~ ~omQtim~~

called

~~.

dato from the fiftles.l R.eent monogcaphs by

Strocchi 2 and Sewel13 summarize the pre.ent .tat~. of QM~.

One of the phe~omena typIcal for infinite quantum systems is spontaneous

symmetry breaking. Suppose that th~ system ~nd~r eQn$leerat~Q~ pO'3~".es a symmetry.

b~~au~e the Harollt~nla~ ig invariant under the transformations of a group~. In the c~se of qu~ntum m~ch«nl~s with ~ flnlt~ ~uMbe~ of degrees of freedom this implies

th~t the (in mogt cases unique) ground state is also invariant. However, for QM~ ~Q

h~vQ ~n g~n~r~l ~ $et ~f ~~e~ulvalent ground states and all we deduce from the symmetry of the system is the Invariance of the complota ~ot ~nd net no~~~~~rlly of ~n ind1v1Q~al ~l~m~ot, A$ ~ eQ~$~q~e~ce the symmetry may be broken spontan8ou81y~

Although the dynamics of the system 1~ ~~~tr1e. 1t~ ero~ne stat~ m~y not h~ve th1.

property. However I the possible ground states are ~onneGted to each other by the

element. of the symmetry erou~ ~.

An important example in e18mentary-partI~lo phy.l~~ 1. th~ U(1)xSU(2)

eleetroweak ~auge theory. 1" which .pont.neous symmetry bre.king in combination wIth

gauge invariance gives rise to the mass of intermedIate vector boson~ and of spin;

parti~I@

•. 1 Anoth0r

cxampl~.

which wo di.cus. below. Is Bose-EinsteIn condensation.S

associated with a macroscopic occupation of the Qne-partl~lo ~ro~nd ~t~t~ In ~ ~a~

of we.kly intacaeting bOBon3. To place the l.tter eK.mple in the wider GQntext of

QM~ it is useful to skatch btlefly ~o~ aspect. of this general theory. For a more

systematic introduction we refer to the Refs. 2 ~n~ 3.

The dlffecences between QMoo and the quantum theory involvinJ only a finito

number of de~ree~ of fr8~~¢m e~n be traced baCk to the following theorem due to Von

Neu"a"".S In the case of " finIte system the algebra ,1 of th0

o;oo'~in"t".

'la Mol

1

[<j«,q~]-~, [p""p~L..o

(qa·PIlL"'~Ma;8 .

3

(1)

ha. only lrre~ucl~le representation' wh!~h are (un~tariJy) equ!vaJent to the usual

Schroding;"r r"pr"santaUon P« ... iI18Iilq" on th" HUbert ;;pa.:e of squara-int"l:rabla

functions. S;tn~e a un,itary traJlsfarmatio;n. does

not

change the expectation valueg of

physieal quantitlc= the choice of a representation 1$ just a matter of convenlenee.

HoweveI'~ in QM~ t.he algebra r.4 has inequivalent irredu{;lble representations. The

various representations can l:onerally be dlstinl:Uished by meanS of a maeroscopie

quantity, commuting wIth the complete algebra d and consequently T~duclng to a

eonst3nt in an Irreducible represent~tion. Duo to this the ~lfferent representatlon~

correspond to macroscopically distinct classes of states of the system. which may

have very different behavior because also the oynamles (the Hamiltonian) depend$ on

the particular representation chosen. As a result QM_w displays interesting

prqperties such as macroscopic d.egeneracy. st:lontaneOllB synmetry breaking and pbase tran~itioM, ... hieh Ua elosely related to MGh oth,,~ and ;lra ;lb.aht in ordinary

quantum mechanics.

To ~ MOr$ $~elfle W$ eot'l$id"r a non-rl1llaUlllstlc many-body problem, ,,;sl~

$ceond ~uantl~ation and formulating the algebra d in term. of the field operators

>/-(x) a.nd / (x) obeying the usual (anti)"ommutation relations in the ease of

(fermion.) bo~o"._ For clarity r~a$On$ po.olblo Indlee$ eo~re.~ohdlhl: to .pln

dCl:.ee. of freedom a.re suppressed. Note that the algebra d can also be generated by

the creation anO annihilation o~r.tor$ of deflt11te mo~¢htU~ .t .. t~., at(k) and a(k).

r~.\:>Cet1

... ely. A$ usud th"y are relat"d to the field operatora / (x) and t(x) by

means of a Fourier transform.

For a lar~ class of systems. in particular systems showing superfluld or

superconductive behav1Qr~ a~y 1,~cduelble representation of ~ can be formulated as

an operator algebra in a Hilbert spaee ~, wher$ the latter eorresponds to a ground

state !~o> and dl IU d"", .. nlary ""citaUons. (TeGhniGally thIs is Immm a. a

Foek-.~pr~~~ntation.) For a ~haracteri%3tion of the repre.entation or equiv.let'ltly

the Hilbert space

W

with ground state I~O)' we ~o".lder oo-e.l1ed H •• g-eh'rg~. of

the fonn

whl~h v~n1~h¢s ~nle5s Yl~"'Y~ and ~l'" '%M ~ra in ~ fl~lte env1~onment of x. Such

operator. can be shown7 to commute with the aigebra d. In an irredu~lbl~

representation they ~r~ ~q~al to a ~o~pl~x ~on~tant which can therefore b~ ~~Qd tQ

l.b~l that r~p,e~ent~tlQn. In tho 5CS-thoQryS of $~petco~u~tivity a Haag-charge of Q

02 type plays a central role. It is equal to the well-known .... n<>riY $:1IP" ~n the

quasi-particle di$p¢,~lon rol~tlon and ean be ~$~d to solve this model exactly in

the thermodynamical limit.

We now turn to th~ form of th~ Ha~g-eharg~ in the case of Bose-Einstein

~ondenaatlon. Assuming two-body interactions only and denoting the Fo~rler transform

of the interpartlC1Q potQntlal bY Vq th~ H.~1Iton13n of N bosons in a volume V with

poriodic boundary conditions is given in usual notation by

H (3)

o

7-2

whe,Q e~~-~ 12m i. the kin .. tic energy of a parti~IQ with mQrn~~tum hk. Following

Strocchi.2 we ta~~ a~ a Haag-Charg~

(1)

and require that the states of the gas form an irreducible representatlgn of ~. so

that Q ceduces to a eon.tant. Clearly, In the ease of a non-vanishing value of Q, w~

have 3 ~~~~O~~Qpl~ numb~r of partlGlea with .oro mQm~ntum a~ the corresponding

t~prQ.ontation of ~ desccibas the Bose-condensed phase of the gas. Hen¢~ w~ write Q as

(5)

t ~ -HI

nO being the density NOIV of condensed pscticles and in Eq. (1) repla~e ilO by NO~

and a_O by the e¢mplc~ ~Qnjy~atG number. Thi~ .ubstltutlon breakg the U(I) gauge

~Y~otry (~enerated by the pactlcle numbec opecatoc) of the Hamlltonl~n. Howevec. as

an observable con~~rvl~g th~ nUmb~~ of p~~t1~le~. the e~ersy Is gauge Invariant ~n~

independent Of q. Thus. in a~reement with th~ ~~nQr~1 rQ~tk. ~ade above. the ground

.tat~$ It

_o

(90,$) with diffecent values of $ have the same eneriY and ~.e related to

each other by ~ U(\) ~au~o transformatiOn.

To find the appropriate Hilbert space of the representation. we ne~l~¢t terms

s

abs~lut~ ~ero and ean be treated by perturbation tbeory if necessary. The result is

"here the prIme on the summation symbol m0ilns that k=O is e"cluded. Diagoflal1z1ng

thl" HamU tonb.n by maans or a (unitary) Bogo)1ubov

tr'''$fo)~t1o)n9

fro)"I the

t T

p~~tIele operators a_k and a__k to the qua$l~partlele operators

bk

and b__k It turns

out that the weakly Inter-aeting B"s" gas bebave. as a g •• o)f no)n-1~t~r~etlng

quad-putldes:

(7)

H,,~ .. EO 1. the ground Btate energy and ~ th'" qu"o;1-partide energy given by the oIispersion relaUon

{ 0 2 ?.2}112

~" (~)

- "O"k

(8)

It Is Interest~ng to noto th~t for small values of k the qua~~-part~~le$ b~have

11ke pbonons. tho disperfiion relation being line.r in this limit:

(9)

From a more gener31 poInt of viaw these phonon-lIk~ ~eltatlon~ "r~ ~he mas.less

24 .

Goldstone rtIOd~$' assoGiated with the broken ,,,n .. r-ato[" of tbe U( I) symmetry group.

Thv many body ground state 1~ defined by ~1~o(no.S»~ fo~ ~ll k~ and the

""cIt"d states ~an be found by appHc'Ition or tho quasi-particl" creatIon operator.

~

to

I~o(no'~».

In this way we have been able to) eon$truet ""plictly th"

irredueiblo representation of ~ eo)rrespondinJ to the Bose-condensed pha$o of tho

$ystom. As mentioned previously this representatIon 1s ba$ed on a ,round state and

all its elementary ~~~ltationB,

rI. SPIN-l'OLARlZED ATOMIC H'lIlROGEN

Tho most striking e~'Impl~ of thO Bose-Einstein conden •• tlon proce •• realized

~xperimental1y 1. 8 • • oe~~t~d ~ith the super-fluId ph •• ~ Qf 1H~. ?henomanologically,10

rQI~tlon of the elementary excitation. to be different from a partlcl.-Ilk~

expression and in particular to be linear for small momenta. To see that this

feature indeed leads to superfluldltYr we memorize an argument due to Eogollubov.9

DescrIbing the norm~l p~~t of tho quantum flu~~ ~low the X-point as an assembly of

non-lnteraetln~ qua$1-partlcle8 havin~ an avera~~ velocity y with r~$peet to the

condensate and in a state of statistical equilibriumr we find that the avera~e

nUmbe~ of qua=i-particlos with momentum ~ and oneriY ~ 1= i1ven by;

[

e

('1c

-hlc.v)IkST - 1 ]-1 (IO)

where

ke

Is Boltzmann·s constant and T the temperatul'e. The magnItude of v ha5 an

upper limlt. beeau~e the occupatIon numbers must be po,ltlva. Thus we require for

all k#O that 1k>nk.v or equivalently ~k>fikv.

The upper bound for v i~ therefore equal to the minimum of the ratio ~. In

the ea~e of tho particle-like dispersion relatIon ~Bb2k212m thl~ Impl!". that v i •

• maller than hkr2m for all k#O. i.e. v=O. This is what we eKpect: In thermal

equilibrium the fluid as a ~hol~ h~~ ~o ve16~lty or more precisely the normal part

~f the fluid ~&~ h~v6 ~¢ v~lo~ltY With ~~~p~~t t¢ th6 ~ondensate. However, in the

e~'Q of a 11n~ar dispersion relation ~=~hk~. find only th~ ln~qu~llty v<~_ Hance. the normal part of the fluid can have a stationary (frictionless) flow in an

arbitrary dire~tlo~ ~f the m~gnltud~ of t~ velocity is .mall enough_ Carrying out a

Galliol tr~n$fo~tion to a ~oQrdinato ~y.tom in whleh th~ qua~l-partlc\"s have ~ero

avo.a~~ v~loelty both the existence of ~upe.rluidity ~~ woll ~~ ~ e.ltleal velocity above which the pbenomenon cannot take place is explained.

The required linear behavior of the dispersion relation was fir.t derived on

tha basi. of a microscopic theory by Bogoliubov7 (along the lines

de.~.ibod I~

tho

fore~~ln~)

and

l~tor

on by Lee at al,ll who al=o

con~ldere~

the

1~ pro~ertles

of

the Bose-gas_ Since these theorieg rely on the weakness of the interaction or mora

precisely on the

~mallnG$s

of the

ga~ parameto~

(na3)!, where n 1$ the density and a

the scattering length. they are not able to give a reliable description

or

the d0n~=

4He liquid.

I~ tho la~t ~oc~do a ~rQat deal ~f intc~~$t ha$ been devoted to spin-polarIzed

atomi~ hYd'G~en ~as

as a system for which (na3

)&

1~

timall,

bQ~a~ti9 th~ .~~ll atomi~

mass gives rise to a low critIcal densl~y above ~hich the condensatIon tak~3 ~l~e~,

In contr~st with holl~m, tho r~q~l •• d don~1t1e~ ~t $~bk01V1n temperatures are so low

that essentially only two and three-body collisions are important. To~eth~r with th~

f~~t that hydoOgen 1. the only element foo whI~h tho Intorp~rt1cle potentl~ls can be

calculat~~ ~ecurat~11

from fiotit prlnclplos,12 1t 1$

el~ar th~t

at least from a

7

theoretl~31 pOInt ~f VI~W spIn-polarized atomic hydrogen i. an Idea~ ~~n~ld~t9 for the ~tudy of the Eose-Einstein transition and of QM_M effeet$ In Ecneral.

Also from an experimental polt.,t of yaw. Uomie hydrogen has some desirable

propert~e$; It 1$ ~onVenlen\ly mani~lated and (electron) spin-polarized u.Ing

ea.Ily obt~~n~blQ ~gnctie fivlds (e<lOT) and most importantly it offers the

po •• ibillty of an Ind~PGndent control of both density and t~~per~ture. m~k1ng the

system very fl~ible, A. 1t turn~d out the maIn obstacle towards successful

experlment$ i$ the stro~ tendency of the hydrogen atoms to recombIne to m¢leeul~r

"2

on the wall. of the ~ontainer. The a~tual =tabill~ation of atomic hydrogen was

fir$t e~rriQd out by SIlvera and Walraven13 in their pioneering work of 1980. Th~Y

suffIciently suppressed tbe ree~mbi~at1¢~ by elo~tron-&pln polarizing the gas sample

.. "~ by ~ov"ring the walls with a superfluid helium f11l11. whleh ha$ a y .. ry "(oak

interaction with hydregel'l ~~om$.

After this breakthrough. opening up a completely new Held of ~"sBaro;;h. VarIous

other groups throughout the w~rld have contributed substantially to our

under.t .. ndlng Qf thQ physical properties of the system. We mentlQn cXpll~ltly the

groups of T.J. Greytak and D_ Klep~er ~t MIT, W.N. Hardy at USC, S. Jaakkola in

lurku. D_M. 4"e at CQrnell UnlYersity. l.f. Silvera at Harvard Unlver'lty, J.T.M. Walraven in Amsterdam and more recently the group Of 1,1. LUKasheVlch at the

Kur~hatov Institute in Moseow_ for ~n ~~eount of th~ various contributions and their

historical context

w~ r~fer

to the review papers of Greytak and

KI"ppnerl~ ~n~

Sllver~ ~nd

W:draven.15 Here l<e Euffice with a brief $\\IMIO.ry of the present status in the investigatIon Qf "tomi" hydrogen.

Presently there are two different ways In "hich one hopes to establi.b th$

BOEe-condensed state. Thc first approach ia h1stor1~Ally th~ old~. Qn~ and ~im. at a

compression of a ~.$ sample of high-field seekers (Hl) In a magnetIc field maximum

at ro::latively comfortable t",mperature:. [T",lQO-5OOmI(). How,"ver. In th1$ Uml'or .. t"r~

range the re~ulr~d den$ltle. are 50 high that three-body rQ~omblnation on the

surfac~ and in the volume cause a rapid and ~ometlrne. even explO$lv~ de~ayI6.17.1S

of the ga$. To

~ircUmVent

this problem

Silver~19

has

.u~,ested

applying high

m.gn~tl~ fIelds in combination with ne,atlve hydrostatic pre'$ure~ in ~on.-ontlonal e~~criment. based on compres.Ion of hydro~n bubble._ Another w~y out has been put

f",...ard by Kagan and Shly.pnll:"V.20 who propose to u.~ '~rQ",ly inhomogen"ous fields

produced by an ele~trl¢ o. magnetic naedle21 to compress only a small part cf the

total $ystem.

lhe second approach 1~ based on the idea of trappin, hydrogen atom. in free

space.22 which makes it pos.lble to ochlOVo much lower temperature$ $inco::

~Qntaet

with liquid helium 1~ aVQldcd. In this way the require~ densities are much .maller

stat1c23.24 and dynamic25.26 mainotie t~~p~ ~eem to be feasible and temp0r~l~,~$ in

the order ~r l~K may be in reach U.ini ~ ~Q~~l~.t~o~ of laser (Lyman_a)27 and

~v~por~tlve2l eoollng_

Within tho fr~~~wQck of the lattar trapping approach eXPGrlme~tal progress has

taken place almost exclusively in Gonnactl~n ~lth th~ $tat!c version. It Is of int0r~.t for thQ followl~g to pol~t out that such static trap. ~re ~a$~d on a

spatially inhomogeneou. magnetic fiold operating on the magnetic dipole moment. ~nd

noe9'$~rily eonflne low-field ~eeker8

(BTl.

lhis i. due to a thoorem, first

formulated by Wing,28 according to which it 1. Impo •• ible to create a

181

maximum In

2

free .pace. A very direct ad absurdum proof can be based on the equation V B~~ with

th~ ~~~Xl$ polntl~g l~ the direction of 8 at the posItion of the field maximum. If

that would exist. Clearly.

IB1

can only h~ve a maximum if B

z has a maximum. Thl~ 1$.

ho~~v~r. ~mp~$~lhl~ d~e to v2B_z=O.

I

~"'

i:/

I

I' I !

fIG. 1. Magnetic field Gontl~.~tio~ for

a h1Pothctle~1 ~h~olute maximum of

lei

in free Bpace (5ee tel<t).

The fact that only low-field seekLn, ~t¢m~ ~~~ be trapped introduces new de~ay

channel. associated with Ht~H~ transitions indu~ed by t~o-~ody ~olil.lons. An

~m~o~t~~t p~~t Qt thL~ th~~i~ i~ ~~voteq to the theoretIcal de~erminatlon

Qr

th~

lifetime of a trapped HT gas sample d~Q to $ueh r~I~~atlon processes.

8e$ld~. thl$ ~~l~ li~~ of r~$.arch on .pin-polarized atomic hydrog~n, a number

of phenomena have been and are being studied which ar~ of Int~~est ~n their own

right. ~aMples ar~ volume29 .~d .urraoe30 spin w.ves, ~ha Gryot~nie hydrogen

maser.31•32

th~ l~t~r~etlon

of a g •• of hydrogen 8tom. with (almost)

~~.ona~t

olectromagnstic radlatlon26.27 and surfa~g phy$le~ a~~oc1ated with scattering of

hydrogen atoms from ~ ouperflu\d helium fllm.33.31

In all above-mentioned case. It is eVident that eolllsions between hydrogen

atoms ~r~ of utmost Imporlan~e. ~oe~U$e they ~~use a deviation from the ideal

behavior of a Bose-gas. In particular, they are responslblo for th~ de~ay of the gas

and he~e fQ~ It~ flnit~ 11fotlrnQ. ~la~~ng str1ngent restrictions on ~xparlmental ~onditlons. Therefore. theoretioal predictions for the timg ~e~le$ associated with the v~~lo~. collisional processes do not only enhanee phy$lcal l~slg~t but may also

9

th~ory and the numerical evalU'IUOn of the d .. "ay <:onstants is the main subje"t of

th~ F.e.ent thesis.

II 1. IMlROCEtf ATOMS AS COMPOSITE BOSONS

Sln~e a hydrogen atom ~on~1$t of a prgton and an electron. both of whl~~ are

spin

!

part1el~s, a collision between two or thrc~ hydrogen atoms leads in principle

to a eompli<:8ted "~att~rlng p~obl .. m. Additionally, due to the m'sn~tl" mQment~ of

proton and ~l~~t'Qn not only the strong Coulomb 1nt8 ... etlon i. impo.tant. but also

~Jneti" interactions play a eru~ial .ole. In contrast with tho Coulomb intora"tion

th"y are ,,",pablo of tun.fe.ring angular mom~ntum amon, tho spin and orb! tal parts

of the sY$t~m as well as among the ~16etr¢~ spin and proton spin parts. glv~ng ~l$e

to ~h"ation and reco",b~notion pro""s.es that would otherwise be lmJ)Q:i.ible.

HQwQv~r, we first 801v~ th~ Coylomb part of the ~olll$10n pioglcm and subsequently

take the m.gnet~e deg'8". of freedom into 3eeount,

The e~nt~~l question of th~ ~~~~ent ~.~tion is the follawing~ How ean ~g derive

a a~."r-ipt1on of the eoll1s1on of hydrogen ato,"~ 1n t~rms of "elementary" bosons.

taking the requir"d anti.ymmetry of the \<av8fun<:tion wi th respeet to J)<'l"mIltations of

both protons as well ~$ ~IQctrons into aceount ~xaetly? In Sec. lIlA we summ'rl~~

the solution to this problell\ in th .. "as" of two hydrogen

at~s.

35 \<hi"h 15 based on

the adiabatic or SQrn-Oppenheimer36 approximation. The more complioated ,,~.o of

three hydrogen atQms is treated 1~ $9".

IIIB.

A. Two by~rogelJ atQIII"

We adopt the ~onyention to ~$e $ub~~rlpts for proto~ and $Up~rs~rlpta for

electron e¢o.dinates throughout and introduce the notation ~~r!~=x!-x~. r!~I-x2

and

r~=xi_Xj

for tho relative disto/lnee:i. Th8 total Hamiltonian 1$ th"n

( II)

Here, T denotu " kinetic en~rgy .. nd V"OU contains all CoulOmb interacHo ... " bQt"un

tho four- charged partlel~s,

Vcou ₌

_0__

2 { __1 __ 1: _1_ + _1_

The ratio of the proton and electron mass beIng about 1936, it is an exc911~nl approximation to separate the rapid motion of the electrons from the 510w prgtQnl~

motion an~ to 3$$Ume that the $tat~ of th~ eleetrons follows the position of the

protons adlabat!cally_ Mathematlcally t~ts lmplls3 that we fir3t solve the electronic part of the problem for every InternuGlear dlslan~o r s~paratoly. H~ne~, w~ $~IVQ thQ S¢hro~in$Qr Qquatl~n

{ l Ti + VCOU - E(r) }

I~(r»

=

0 1

whore both th~ ~l~~nvalu~ ~nd th~ cl~~n$tate depend parametrically on r. In

addition.

we

require the total electronIc state I~(r» tQ b~ ~nti$~~trl¢ und~r ~

perffiutation of electrons and to correspond asymptotically (~) to two free hydro~~n

atom~ In a 11~) ele~tronlc ground ~t3te_ The orthogonal solutions are than denoted

loy 12S+I};+ _g.u

(~»

ISMS>' 37 with th .. $.ul:>$¢rlpt "ger ..

~"··

Qr "ungerade" dependIng "n the

total electron spin having S=O or S=l, re.pe~tiv~ly. (Wo notleo th~t the oloetronle

wavefunctlon depends only on the magnItude of r when expressed in body-fixed

coordinates.) The corre.ponding elgenvalyc$

ar~

the well_knownI4•15 singlet and

triplet potentials and can for future purposes be written a5

(14)

We come back to this point shortly,

The electrons following the motion of the protons adlabatiGallYf a acatt8rln~

state

or

the full Harnlltonl~n (II) ~lth ~n~~gy E has the form

( 15)

whe~e

I. MI are the quantum numbers f"r the total proton spin and

I~(S»

obeys the Schrodinger equation

(16)

determining the protonic part of tho wavefunetiQn. Parenthetically we poInt out that

often yeS) i. taken to include in addition the expectatign

v~l~~

Qf the

nu¢loa~

kinetic energy operator in the state I~(r). Becau.~ of our interest in the symmet~

properties of I~>, we make the following dis¢u.sl~n more tran~p.rant by introducing

11

<;\oe. not affect t~ ~ymn~try of the "tate •. Within the framework <>f theBe

approximations w~ e~n ~lve the electronic states explicitly,

(17)

using an obvious notation. SUb$tituted in Eq. (Ii) it leads to the uBua131

Ilpproximate forol of th" singlet and triplet potentialg ~n torms of Coulomb "nd

~ehang" integrals. (Note that in Ref. 37 the Shlzgal approximation, which neglectg

the overlap between the

IIB>i

and

11s)~

stat" •• 1$ not used in order to obtain more

aeeurate potential CUTVO$.)

We thus $e~ that 1>1') is indeed anti.ymmetrlc under a .... rmutation of the

electrol'l$. For 11» to be all!::o :-ntl::;:ymrnetrlc un.d.er .i;\ permutation of 'the: ,pT"Qtgns we

requir& 1,(5) to have (even) ¢QQ parity if STI 1~ (cYGn) odd. If w& d~gQt~ the

relativo <>rbital angular mQmentum of the proton$ by

e

we can Bu~rizQ this simply

by the symmetry n1Io: e+5+1

=

eve".

Turnln~ to an effective pi~tUre in terms of e~po5Ite bcson~ wo fonnulate the .. b"ve-mentioned 1"<'~\llt8 ... follows. II'~ introoucB "mathemat1<:al"" hydrogen atom~ 1

consl~tIng of electo~ 1 ~nd proton 1. Theso atOms have only ~pln internal degrees ¢f

freedom and interact by meanS of the central Intora~tion Vc defln~~ a~

(IB)

,(S)

be~ns ~

projection

¢p~ratgr

on the p8rt of Hilbert

8pa~~

,Orre.pond1ng to

~tOms

~Ith a tot~I electron spin ~qual to 5. Tho HamIltonIan of the two hydrogen atom$ i~

given by

(19)

"here HO is the total ~1n~t1c energy of the atoms. In .. " .. logy to the abov .. treatment

1~)~I~(5»JS~jMI>

Is a

scattBrl~

glgenstate of Eq. (lS) with energy f if

1,(5»

obeys Eq. (16). There is. hOWever, one .mall difFeren~&; the kinetic energy !1 Ti

involve. the prot&" ","$$ wh .. r"as

110

l,.,,,,,lv ... the hydro~ .. n atom mass. fo,-t=ately, i t

can bo!o .hown32 til-at non-aQlabaUe

~on

.. "UonB cau.e a mass reno ... U%Ulon. which

Sl~s rise to thg replaceme"t of the proton ~a$. by the hydros~n mass.

To ... k~ the equivalene~ h<>tween the \\<0) piGtures eompl"te we now r<ioq,,ur-e the

w"v<iofWlction to b • • ymmetric ur,d,n pennutatio". 'Of the matll-emat1~al hydrosen at<>m •.

In th15 way we have been able to remove the electronic orbital degrees of freedom.

They are ~ffeetively taken into account in the form of the adiabatic sinilet and

triplet potentials_ Using a generall%atloo of Eq. (14). ~I$o ~agnet~e d~polar.

Zeeman and h:rperflna interactions can now be included In an "eFfective"' form

suitable for the

~ompO$~to

boson

pietur~.

We

rqf~r

to the the.i. of de COOy40 for a

summa~ or the rinal results and now turn to the lnteregtlng case of three hydrogen

atoms_

B.

Threo hydrogen ato.~

In analogy to th~ two-body ca.e we apply the very accurate Born-oppenhQl~cr

a~~rQ~lm~tlon ~nd .oIve the extension of Eq. {13} to three electrons and proton.; (2.0)

which shows the parametr~e de~endenee of the $olution on th~ position of the

proto~~. Ag~ln the ele~tronlc HamIltonIan does not contain the electron spin ~nQ th~ quantum numbers of th~ total ~le~t~on ~pin S are conserved. However. in contrast

with th= ~t~vloQ5 '0~ti~n we now have totally 5~etri~ spin states (S~i) a~~ al."

.pin states of a mixed symmetry

(S=!).

instead of totally "l'1lI"'etric and

antisymmetric spin .tate •. In th~ ca$~ of m~xed ~~t'Y th~ co~~l~tely

a~tl$ymm~ttl¢ $Ql~tion of Eq. (2.0) 18 not the product of a spin state and a ~t~t~ describing the orbItal deg~ee~ of f~e~doM but 1nst~ad a su~ of two such products.

Both the S=~ and

5=!

stat~. ~~~ relev~nt to thiS the.~ •. be~ause they

~o,r~~~~ to thQ initial and final states of the dipolar ~ecombination proc~ ••. Therefore, we will cDnsider both. using again the Heitler-London and Shizgal

approximations to make the dl$e~$~lQn a$ tian$p~iant a~ PQ~~lblo ~n~ eQne~~tr~ta on

tho fulfillment of the (anti)symmetry requirements. However, before doing so it is

useful tD memorize some

fa~t$ ~bout

tho

p~rmut~tIon ~roup.41

An irroducible representation of the permutation group can be characterized by

the Young-tableau

[Al.

wtth A ~ partItIon of th~ numb~r of p~~t1el~s N ~onstdered_

In the pre.~nt case of three electrons we have three tableaus [3], [2.1] and

[111]~[13]r which correspond tQ ~ ~ymmetr!~ re~r~~e~t8tlon. 8 two-dlmen31onal

representa,tJ,.of), Qf m.lxl';ld :i..)"lIU'IIetry and a.n antisymml!ltrie topro.:i~ntatlon, I"o.:ipcetlvelYT A ~onyenlent way to flnd states that transform accordlng to a D-dimensional

(unlt~ry)

representation

r[~)

of the permutation group i . by mean. of the proJect1on

operators~

13

the ~u", being over all permutat1=s. Crucial fa .. auc pu .. pose is the "b~"ITat1Q" thu an

antI.ymmct~le ~~p~e$~ntatlon

ean be formed out of the direct product of r[h] and

the conjugate representation

r£A~].

ExplIcItly We hav"

(22)

whece the conjugate partition A~ and index

a:-

are found by trans .. ¢~1ti¢n of the

Young-t~blQ~u [~]. Fo~ example, if ~ eon~ldQ~ two f"Tmlon~ WB have [Z-]=[ll] and

Eq. (22) is the obvious statemEnt that the product of a symmetric and an

ant1!:YJ'I"IIetric wavef\!net1on 1$ ant1sY'M'etrie.

After th1$ dlg~e$$10n We a~~ nOW ~eody to d1$eU~$ the "ige~$tate. of E~. (20).

Fir~t

w"

~on~id8r the ~imple case S=i. Oenoti~ the total spin Qf ~l~~t'Qn. 1 aDd 2

by s. we have in spin space the symmetric state I (s!)SMs>=I(l!HMs>. belonging to

r(3J.

In configuration .pace we therefore

ne~d

an antisymmetric state 1<t>[ll1]({X_i

}»

=

J5

..,[111]

Il.<>!

11$>~1\$)~

..

" .J6 •

.,[

Ill]

1

123)

(23)

with spectroscopIc de.ignation

1~3

A2 .

A2

$t~nd1ng

for the [111] rep .. e.entation and

the Prime indie~t1ng ~~fl""tI0n .ymm~try with re~PQ~t to the pl~ne ,,[ th~ n~~lGI.

The "lQetroni~ state then becames

(21)

Caleul.tlng th" .dIabatI~ 8n8r~ surface determining the motion "f the prot"ns

by m¢;\n.; of (d. Eq. (li))

(25)

and neglecting the term" givins 1"1$" to a thre .. -body forca, i t turns out that the

$~rf.,,~ i~ Ju~t the sum "f the three triplet pair-potentials. whl~n 1$ e~ual to

u$lng th" convenient 'pectator-index notation.42 I denoting the pal~ of atom$ (Jk)

orblt3l ~~grcQ~

or

freedom of the protons. respectively_ Furthermore. ~equ!rlng the

complete wavefunctlon to bo ~~ti~~tri~ under proton permutatIons we find the

(u~¢tm~liz~d) state

(27)

~erQ w¢ introduced the operator p. which 1. the

'Um

of thO two ~y~ll~ po~t~tlon

operators of the

m&th~~~tl¢al ~tom

•. 4Z TOJeth9r with the symmetry rule e+s+l = even

(~ Is the rel~tlv¢ a~~lar momentum of protons I and 2) this wavefunction i. totally

ant!symmetrle under both electron and proton permutations_

If S=; we have two antisymmetrl¢ Bleetro~te $t~t¢$ ~1~¢ wo ~~n bull~ two

(21)

representations. to be dl~tlngul$hed a~ [21]' and [21]", from the 1.3 orbit"l

co~flgu1atlon~ This follows from the dIrect product rule

t[I]~r[I]~r(I]~,[3].2,[21]+,[111]. On the other hand for fixed MS we have only one

[2t] ~pln representation. Using the Yamanouchi standard orthogonal ~epresentatlon~l

to obtain the appropriate proJ~ctlon oporator. we easily find:

where we suppre.~~d th~ par~~trlcal dependence of 1~(111» and

1.(21»

on the

po.itlo~

{Xi} of th9 proton •• The states

1~(21]>.

which are characterized

~9Q~tro5~gplcally by 18 3 E'r can be given expl1citly ~~ the Heltl~r~Londen

approximation:

1,[21]')

= __ 1 __

{21123)

+

21 Z13> _ 113Z

> _

1321> _ 1231> _ 131Z) }

I

SIT

(28)

11~21]'>

=! {

1132> _ 1321 >

~

1231

>

+

1312> }

(29)

1~~21]">

=

t {

1132) _ 132 \> .. 1231 ) _ 131

:2) }

1~[21r>

= _1_

{21123) _ 21213)

+

1132)

+

1321> _ 1231) _ 13n) }

Z

Ji2

15

of the matrIx

(30)

In anal¢EY t¢ th~ discussion of the case S=~ [t can b. ~hovn that If v~ n~~l~et

terms leading to a thrc~-body fore~. Eq. (30) 1$ equal to

(31)

and the complete six-body ~avefun~tion I~> 1. a lInear co~b~natlon of the ~ompletely

;!Lr,.tl~~t".!e stat.es

(32)

again "ah e+s+1 " ~vo!:n.

Ro!:tuTnln~ to a composite boson picture we notice that du~ to Eq •. (26) ~nd (31)

we are able to remove the electronic grbital dGg~¢¢$ of frco!:aom from the de~eriptiQn

and still obtain the correet

Sehrijd1n~er ~uation

for

I~(S»

if we use the expected

s

Haml)tonJan (cf_ Eq. (19» for three interacting hydrogen atoms

and in addition the Gorresponden~9

I

(l~)~MS>

IcsmMs>

(33)

(3-1)

If we substitute Eq. (34) in Eq$. (27) and (32). it i~ clear that in this way we

have achieved our ohjective of reducing the original .ix-body problem to a

three-body problem. Tho "nthYlmi .. try roquirooment for the eleetro\')$ ~nd protOI'~

constituting the hydrogen atom. is taken into account by u.i~ the .in~let and

tdl)!et I)dr~l)<:>tent~"-!~ and the addiUonal requirement of symmetry of the

wavefunction under permut~t[ons of tho mathemat1~al ~tom •. A~aln we "To!: nOw able to

flnc.l expressions for' an "effectlve" form of the ma,gnetJc i"teracUons wi th obvious

result •. Becsuse of the relevance to the dipolar recombination proCe ••

wO

nato In

partle~l~r that tbe electron-electron magnetic dipolar interaction also reduGBs to

AH"r the abovG JI.<stif i""U"n ",f th" ""le",e~tary particl .... treatment of th., two

and three-atom system it 1s evident that we can ~ppll f~rmal $eattering theory to

two and three-body colli.ional pheno.oen~ in a ga. of hydrogen atcms. ror a

mathematically

~lgorous

formulation of thlg theory we rerer to thb

~o~ks

of Newton43

and Amrein at al.1i A mora informal intr¢du~tlon to phy.i~ally important quantities

such as the S(cattering) and T(ranaition) matrix, ~hioh 1'1'1 ~ ~6~tr.l role in this

thesi., i. given by Messiah,iS Taylor,4S Clockle42 and de Coey.40 In particular Ref.

40 gives ~ pedestrian appr,,~oh to ~e~tt6rlng th~ory using Feynman-liko diae~am$ to

clarify the phy.ics involved_

' .. , - - - r - - 7 - "

\,

I,·

-IOO

C ... ---,.,.,.--='---->.--:!1Il

FIC. 2_ R~bl diagraM of the four

hyperfine states of atomi_ hydr¢~~n 1n the

h S'round state_

IV. SCOPE OF THIS THESIS

The remaIning part of this thesis consists of 6ix publlcation5 in th~

scientifl~ lit~rat~r". Tho .~bJ~~ts "overed can be ~roup~d lnto twa parts.

3.!:~oelated with two and three-body collisIonal processes, respectively. ThQ flI'"$t

part (Chapter" 2 to 4) 1$ 1n p~rtieular r~levant to the analysts of relaxational

prO~~~5e5 in atomic traps, whereas the second part (Chapters 6 and 7) 15 ~on~Grncd

with three-body ~ecomblnation, which i. the dominant d~c~y meehan1sm in the high

densIty schemes fQr t~e a~hlev~~nt

or

the Bose-Einstein condensed state. Chapto~ 2 i. devoted to a study of all possible (exchange and dipolar)

17

atomie hydro~en, whI~h are ~onventIona~lyl1,l5 denoted ~ I~>, I~),

Ie),

~n~ Id) In

ord.~ of increasing energy (cf. Fig. 2). The ealculation of the rata constants

C~.tI' of th .. V;\rious relaxation processes 1~ e~peehlly impgrtant for the

analyst. of the deeay of magnetically trapped atomle hydr08e~ (Ht). Rce.ntly,

experiments earrl<><:1 o~t 1n Am~tcrdam by Walraven's group24.47 have shown exeellelll

a~reem8nt with the results pre.ented in this chapter, whan averaged over the magn",Ue Held and density profiles in the trap.

In the case of the

eryo~enie

hydroien maser it has been shown by our

~roup32

that hyperfine 1ndueod offoet~ ~ltimately limit the fre~uen"y stabIlIty of tho m •• ~r

and cause a dep.,nd.,ne., of tho freq"gney shift on the level popul&tl0n$. Thenf .. r .. ,

relaxation affe~tlng the$o populations is also of importan~e fo~ ~ ~et~lled .tudy of

the hydrog"'" ~~~r.

In Chl1pt"r 3 we explore the po'5ibiH ty of i\ mi~rO\(avs trap, ... hieh was first

proposed by Agosta

~n~

S11veraiS aa a way to trap predominantly

b~l1tom.,

which are

e~peeted to have a much longer 11fetl~ th~n d-atoms trapped in a static magnetie

field minimum. To d1;~ss the fInite lifetime of the g/1$ duo to collisional

p"Oe"'$$~$ \(G are faced with the additional complication of the microwave radlatioll

field. A • .,a. pointe.;! O\.lt by JUlienne 49 this problem can be solved conveniently i f

Onc r¢allzes that the asymptotie $tat~s of the scattering proce.~ ar~ (two-body)

dressed-atom50

~tatos.

With this in mlP4 we ean immediately apply the methods (in

part!eul~r the degenerate-internal-$tate. approximatlon51) of Chapt",,. 2 t~ find the

appro~r1~te decay constants. Cb~ptvr 3 15 Identical to a publl~atJo~ wrltt6n in

collaboration with C.C, A~o$ta and I.F. Silvera (Harvard U"lver.ity). Although only

~h~ ~alculatlDn of ~h~ 11f~tfme in terms of d~~ged ~t~t~~ forma part of this th~~~~

\(ork, ",e include the full contents of the pu~licat~or> rQr ~ompleteness sake. Th.,

principal con,,lU~1011 i8 that relat~v~ly 10n¥ lifetimes are obtainable If one 1. able

to 1I,$~ :'itt"ong: mIcrowave fields. However t the maIn, a:!!: y~t '!ln~Qlvedl experimental

dHflcul ty 1$ eOlVlQoted with the Hllln¥ of tho trap.

The results of Chapt~rs 2 and 3 rely on the fact that the temperature of the

gas is much h1ghQr than the critieal temperature Tc for B05e-Eln~lein condensation.

H~n~~, the tr"nslatiollal degrees of fr~edOM ar~ de.Cribad by a Maxwell-Bolt~mann

distribution. In Chapter 4 we drop tht. restriction and con~lder the full

te"'peraturs range aP4 e:pee1ally HTc' WHh the appl1.;; .. t1on of magnetically trapped

atomic hydrogell (Hf) in mind our primary interest i. the c&leul~lIon of the dipo~ar

reloXllt1<>ll rate const"nt~ _Cdd-0:./3' We find that H T..o C_dd->:$ is a factor of 2

$maller than the nond~gQnerate result and we even find small differ~ne~s above Tc

for dd~" and dd~o relaxation. Tho$o phenomena may be useful for the detection of

(the Oll.~t of) Bose-conden~atlQn. Although th~ result. are based on a homogeneous

I. tho y.~ of th~ lQ~~l-d~n.lty 'Ppt¢~l~~tio~ to ~al~ulate the density profile52 and

subsequently average the rate constants ove~ this profile.

Chaptee 5 ~an be eQn$ld~r~d ~$ an 1nter~~dlate eh.pter. connecting the two and

three-body parts of this thesis. It is conceened with the .tydy Qf th~ full (~lso

off-th~-cneeiY-shell) T(ran$ltIon) matr1x and give' the momentum space Cormulatlon

of the effective range theory. which was extended to aebite~ry dl~e~,IQn~ by our

gt¢~~.53 AlthQ~gh

thQ

.~bje~t .ee~s

somewhat mathematical it is connected to the low

te~~~ratYre

behavior of

r~l~~~tlQ~5i ~~ re~oMblnatlon5S

processes

oGGurrln~

In

atomic hydeQ~en adsorb0d o~ a l$~p~rflu~d) helium film. In addItIon. the method

peoposed may also be useful In the eQ~t~xt of Duelear physics since it gives a

peesceiption to find an (in a certain $gn$~) Qptl~l separable approximation to the

T-Matrix. which is important for the solution of theee-bodY

Fadd~~vS6

Qr

!~ g~~~~~l

few-body equations.

Fln~lly, Chapter~ 6 and 7 are devoted to an evaluation of the th,ee-body aii

dIpolar recombination rate constant Lg tor the prQ~~~5 H+H+H~2+H 1ndu~ed by

Interatomic ma~netic dipolar InteraetiQn. Thi. interaction being included as a

the

first-order perturbation. the problem reduces to the Gal~~l~tl¢n of t~¢ ~olli~10~

w~vo funetl~ns. one foe the InItial H+H+H state and one for the final HZ+H state.

both includIng the central interactIon exactly. Tho th~gr~tl~al rorm~L~tlon u~ed has

been d~$erlbed ~orc eQmpletely In Ref. 42 and In two previous papees.57.58 ~eltt~n

by o~e geoup in collaboeatlQn With W. Cl~~kle of the University of Bochum. In

particular Ref. 58 gives an elaboratQ dj~~ll~~loQ of th~ ~~a~t initial state of the recombination process obtained by the ~Qlyt1Qn of th~ F~d~eev equations in momentum space. Moreover. it gives a thorough derivatIon of thQ eo~pl~~~ehann~l~ ~quatlo~s

~hIeh arO ~sed in Chapter 6 tQ ~n~ly~~ the decay of atomic hydrogen at very high

m.g~ettc flelds.

Historically, the Ciest (apprQXimat~) calculatlQ~ of the dipolar re~ombination

rate was made by Kagan et al .• S9 In their lmportant paper giving an overvIew of the

varlou~ pro~e.~~~ l~ading to decay of spin-polarized atomic hydrQ~~n (H~) 8t hi~h den~itle8. Although. this ~alcul~tlQn g~Ve the correct order of magnitude at about

8T. tbe magnetic field dependence was in error, Ex~erlment$ (at that time carried

out ¢~ly bel¢w tOT) .howed a slight decrease whereas Kagan's mod~l ~.vo ~ ~t~~p

increase as a function of magnetic field.

Thct~forc.

do Coey at aI_57 suggested that

a~ additional pro~ess (the dipole-e~change mechanism), ~hlch w~. eQn.~do"~d to be

h~~li~lbl~ by Ka~~n ~t a~_. i. IM~~tant at low magnetic field stre~th •. Boc~u.e Qf fundamental problems in the accueate evaluatIon Qf this eont,ibution. we decided to

~tart MD~e e~~et eal~ulatlong hoping to resolve the exIsting dl$erepaneies

~ltimately. To that end WQ flr$t eal~ulated the e~act initial state gf thr0~ spt~-pclar1~~d hyd~o~e~ atoms with triplet inteeactions. whieh eonflrmed that the

19

initial state ~$e4 ~ Kagan et al. conta1nod the eSBential phy81~ •.

FIC. 3. Enecgy t~re~nold$ for the single

spin-flip process. with ~ the Bon~

magneten &"el ~e th" Unal momentum of the

hydr,,~en atom relative to tho H~ m"18~ule,

E" having rovib~l\tlo:>n"l quantum number. (v. E)

a~d blndl~ energy ~ve. (Fer the doublo

$pIn-fllp process the 10 •• in Z~~man Qh8rgy

~8 should be re~lae~d by 4~B.)

The following ge.l "a. the dHe.mination of the find atOlll-m<>1ecule state_

Howeve" , eomputational pewar being l1mited, an exaot ap~rQ~ch $eemed not to be

feasible in thl$ ca$o. The.efore: we decided to:> follow a two-step .ppr¢~eh, which

would enhance physical Inslght in the reoombinat1on mechanism. In the first st~p

(Ref_ 57 and Chapter 6) we evaluated the contribut10n of Kagan's dipole mech.ni~M a~

exactly as pcssible. In ~ont.ast with tne ~pproximation of K.~an ot al., who neglect

all inter.~t~Qn~ between atom and molecule, WB included the pos~ibillty of

(lnlel~~tlc atom-molecule s~3tterlng but excluded v~rtu~l break-up and Most

Import3~tly rearrangement. wh~~h 1~ direotly connected to the dipole-exchange

mecha~l~m. Since it turned Qut that our calculatlon 1~ not able to explain th~

exper1~¢ntal data we were le~ to the conclusion ~h~t tGarrangement Is important ~nQ

ev"n dominant below lOT.

As a second $t~p ~e turn In Chapter 7 to a model for the f~nal ~t~t@, whl~h

include. for the first time (In)elastic .catter~ng both wIth and without

re~r.angement and negleGts only v!rtual break-up of the molec~lo. Although there

remain Serious differences wIth experiment, our model is able to expla1~

qualitatively bo:>th the almost magnetic field independent

beh~vlor

of

L~ff

below )OT

and the high values of the rBcombinatio"

r~te

found in recent

e"I'Hlment~60

in the

fl~ld range 10-201. We believe that the.s high values aro du~ to states of the

longe-range H~ m~lecule. which glv= rise to resOn~nee~ at specific magnotic f1eld

strengths. as is shown 1n F1~. 3. In view of the long-range nature of th8 moleoular

st=tes, virtual break-up of the mel.e~lg is expected to bo Very important in

obtaining reliable quantitat1vo result •• However. Ineluding this pessibility ~111

only al tsr the width and the posi Uon of th .. resonanoe stru.ottlr" wt will not change

recombination process given in Chapter 7 i. beli@ved tQ h~ ~orre~t.

R. Haag. Dan. Mat. Fya. Medd. 29, no. 12 (1955)

2 L Str"""ht. EleMent. of Quannm 1I""hanl"" of InFt"Lte Systems (World

S,,~ent~H~. '>~ng.pare. 1985)

3 C.l.. '>0 ... 11, QI.I(tIttum '/'hear-II of CoLLect11>e P/>enom.ell!l (Clarendon Pre •• , Oxford,

1986)

~ E.S. Aber. and B.W. Lee. POy •. Rep. 9". no. 1 (1973)

5 A. Einstein, Sit~. b~r. Preus •. Akad. WI~~. (BerllnJ. 3 (1925): S.R. de Groot.

G.J. Hooyman". and C.A. Tan Seldam. Pro". R. SO". Landon, SeL A203, 266 (1950)

6

J.

Von Neumann, Math.

Ann.

104, 570 (1931)

7 R. Haag. Nuovo Cimento 25. 287 (1962)

8 J. Bardeen, L.N. Cooper, and

J.R.

Schrleffer, Pby •. Rov. lOS, 1175 (1957)

9 N.N. BQgollubov. J. PhY". (USSR) 11. 23 (1947)

10 F. London, Superft~td., Vol. It (WIley, New York, 1954)

11 T.D. Lee. K. Huang. and C.N. Yang. POy". Rev. 106. 1135 (1957): T.D. Lee, and

C.N. Yang. Pbys. Rov. 112, 1419 (19SS)

12 W. Kola. and Wolniewicz.

1.

Chern. Phya.

43.

2i29 (1965); Chern. Phy •• LQtt. 24.

457 (1974): J.F. Bukta and W.J. Meath. Mol. Phys. 27. 1235 (197i); L.

Wol"~~"l"~. J. CI1""'. 1'1'1:1$. 78. 61n (1$$3)

13 I.F. Sll"'"ra an<\ J.T.M: Walraven. Phy ... R"v. l.ett. 44, 164 (1000)

11

T.1.

Greytak and D. Kleppner in New Trends in AtomIc PhysIcs. edited by C.

Cryn~ri and R. StQra, p. 1125 (North-Holland. Amsterdam, 1004)

15 I.F. Silvera and J.T.H. Walr~v~n 1n Progress t~ Low Te~pe~ture PhysIcs. edited

by D.F. Brewer. Vol. 10, p. 139 (North-Holland. Amsterdam. 1996)

15 R. Sprik. J.T.H. WaIrav=n, C.H. v~n Yper~n, and I.f. S~lv~ta. PhY$. R~v. ~.

6172 (1006); D.A. B"ll. H.F. He .. s, C.P. Kochanski, S. Buchman, l.. Polla~k. Y.M,

)(iao. D. Kloppner. and LJ. Greytak. POy". Rev. B:H. 7670 (1986)

17 T. T~~il •. E. TJukan~v, M. Kru.iu~, ~nd S. Jaakkol •. PhY$. R~v. 836. 5937 ( 1~S7)

IS Yu. Ka,an. G.V. Shlyapnikov. and I.A. Vartanyantz, Phys. Lett. lOlA, 27 (198i)

19 I.F. Silvera. J.D. Gilla~py. aod J.G. Br~~~on ~" Sptn-Polari~ed Quantum

Sy~tems. odlted by 1.5.1. and S. StrIngsrl. p. ISS (Worid-Scientifie, Si",apore. 1989)

20 Yu. Kag~n and G.V. Shlyapnikov. Phy •. Lett. 130A. 483 (1988)

21 S.A. Va~11ycv. A.Y". Katunin. 1.1. LukasheYich. and C.Y. Shlyapnikov.

2,

22 II.F. Hess. Bull. Am. Phy •• So<:. 30. SSi (1005)

23 H.F. He~~, C.P. Kochanski, J.M. Doyle. N. Mae1i.hl'U. D. Klept:>n~r, ~ll<:l T,J.

Craytak, Phy ... Rev. lett. 59, 672 (1007): N. MasuMra, ).101. Ooy1", j.C.

Sandberg, D. K1eppner, T.J. Greytak. H,r. H"'SS, and c.r. Koe;hanskl, Phys. Rev.

L"tt. 61, 935 (19B8)

24 R. val> lIo1jen, J,J, B"rkhcut, S. Jaakkola, and J,T.f<!, W"lr"ve ... I'hY$, lI .. v, Lett, 61, 931 (1988)

25 R,V,E, l.""a1ace, C. f<!eh"I>1o.". T,J, To ... Ua, ~n<l 0.101, I. .... , Nat"r .. 31e, 30 (19SS)

26 C.C. AJOo,ta, 1.F. Silvera. H.T,C, StooL

",,<I

B.J. V .. rhaar, l'hys. ]le". Lett. 62,

2361 (1989)

27 LW, Hlj\ll~"s, O,J, Lult.m, 1.0. Setija. and J.T.M, Walraven 1n Sptn.-l'ota.rtzed

Quantwn Sy"tem., edited by LS,1. :.nd S. Sttillji;arl, p. 275 (World-Scientific,

Singapore, 1989)

26 W.H. Wing. Prog. Quant. Elue;tr. S, lSI (1984)

29 B,R, Johnson, J.S. Denker, N. B1g",low, L.P, Levy, J.II. Freed, and 0.101. Lee,

Phys. Rev. L"'tt, 52, 1512 (1984)

30 J.M.V.A. Koelman. H,J,/Il,r, Note!:>orn, L.P.H. de Coey, B.J, Verha" •. and J.1.M,

Wah'aven, !'by~. Rev. 832, 7195 (leas)

31 H.F, H .... , C.P. Kochanski. J.M. Day I .. , T.J. Greytak, and D. Klappn.,.. PhY"

Rev. 1.3'1, 1602 (1986): M,D. Hi.irlimann, W.N. lIardy, A.J. BerHnsky. ~l'd I1.W,

CH"e, !'by ... l1ev. A3<1, 1605 (1006); R.l. Walsworth. l.F, Silver_, H.P.

Codfriad, C.C. Ago<~o), R.F,C. Vessot, and E.M. M.ouU",I'l. rhy~, Rev. A3'1, 2550

(1986)

32 B.], V .. rhaar, J.M.V.A.. KooIman, H.T.C. StooL O.J. lui~"n, al'd S,B, Cr_lllJ't<>n,

f'hys. Rev" .\35, 3825 [19B7); J.M.V.A.. Kooiman. S,B. Crampton, H.LC. StooL

0·3. Lu1t"n, and B.J. V",rJ:j'lU, Phy". Rev. A39, 3535 (l98B)

33 K.r. Salo~~n, I.F. Silvera, J.T.M. W~lrave", &nd C.H. van Yperen. PhY', Rev,

625, 6002 (19B2): J,J. B~rkhout, E.J. Wolteu. R. van RolJ'm, and J.T.M.

Walraven, I'hY$, R"v. Lett. 57, 2387 (1986)

34 D,S, Zimmerman and A.J. ~rlin~ky. Can. J. Phys. 61. 50S [IS93); Yu. Kagan,

G, V, Shlyapnlkov. an'" fil.A. Clukhov, Pi,,'ma Zh. Ek$p. Too[" , Hz. 10, 289 (1984)

[JETP I..,tt. <to, 1072 [198'1)]; B.W. Stau. Phy$, Row. B32, 7150 (1985): V.V.

Co ldman , Phy$, Rev. I. .. tt. 56,612 (l9SS): E, n".inga, H.LC. StooL and B.].

V .. rhaar, Phy~, Rqv. B (submitted for publication)

35 See for ln~tanc" J.P.H.W. Van den E1Jnde, Ph. D. thesl$. ElndhovQn University

of Tee;hno1oiY, The NetherlanQs, 1984

36 M. Born and J.R. Op~nh .. lmer, Ann. der Phy~, 8'1, iS7 (1927)

37 G. Her:tborg, Kol,,=lctl" Spe.:tra oru1 Mate"uiar Strueture, Vol I (Van NoHr-and

Reinhold. Not\! York, 1950): S.H. Bran.<lan and C.]. JO'leJ:j~in, Physt". of Atom.

:;'9 B. Shizgal. J. Chem. Phys.. 58, 3124 (1~73)

40 L.P.H. ~o Coey, Ph. D. thesis, Eindhoyon Unlye<~lty of rechnology, The

Nathe<lands, 1988

"II M. Hamermesh. C",,,,,l' Theory and It& AppHca.tton

to

P/tY$~';';'~ r;"bt"",~

(AddL.on-Wosley, New York. 1962); I.C. K~pl~n, Svmmctrll of Many-Etec!ron

Sustem~ (Academic Press. New Yor~. 1975)

42

w.

ClOckl", The Quantum. Me:chwtlec..t F ... Body ProbLem (SpI'ingeI' Verlag. Berlin.

1983)

43 R.O. No~ton. Sc~ttertng Theory of i.;,~es and Particles (McCraw-Hill. New-York.

1MB)

44 W.O. Am<eln. J.H. Jauch. and K.B. SInha. ~ttering Theory 1n Quantum Me~hant~s

(W.A. BenjamIn. London. 1971)

45 A. Messiah. Quantum Mechantcs (North-Holland, Amsterdam. 1961)

"15 J.R. Taylor. Sc~tt~;tng Theory (WIley, New-York. 1972)

47 R. v.~ R~IJQn. Ph. D. thesi •. Univer~lty of Amsterdam. rhe Netherlands. 1989

4B C.C. Al'osta and LF. Sl1v~~a ~" Sptr>-Poto.rtz.d Quantwrt Syste",-s. ed~t"d by

j,S.I. and S. Stringer!. p. 25i (W~<IQ-Sclentlflc. Singapore. 1989)

49

r.s.

Julienne. Phys. Rev. Lett. 61. 696 (1988)

SO

J.

Dallbard and C. Cohen-!~nn~uQJl.

J.

Opt. Soc. Am. 82. 1707 (19$5)

51 J.M.V.A. Koelman. H.T.C. Stoof. B.J. Yerhaar. and J.T.M. W&L~~v"n. Phy •. Rev.

Lett. 59. 676 (1987)

52

J.

Oliva. Phys. Rev. 839. 4197 (1989)

53 B.J. Yerhau, L.P.H. de Coey. Ll.D. Yrede~b~Ogt. ,,,,<1 J.P.H.W. van den Eljnde.

Phys. Lett. lOLA. 371 (1985): B.J. Yerhaar, L.P.H. de Coey. J.P.H.W. v~~ den

Eljnd". ~nd E.J.D. V<edenbregt. fhY~. R~v. A32. 1424 (l~es)

5"1 S.l. Verhaar.

J.P.H.w.

van den Eijnde. M.A,J. Y~~r~~$. and M.M.J. Schaffrath.

J. Phys. Al7, 595 (1984)

55 L.P.H. de Coey. J.P.J. D.IQ~.en, B.J. Verhaar. and J.T.M. W~Lravon. Phy$. Rev.

Lett. 53. 1919 (1984J; L.P.H. de Goey. H.T.G. Stoof. J,M.Y.A. KQelman. B.J.

Ye,h"",. ,md J.r.H. Walraven. Phy$. Rev. B3S, 11500 (1988)

56 l.O. Faddeev. Zh. Eksp. Teo •. Fl~. 39, liS9 (1961) [SOy. Phys.-JETP 12. \0\4

(1961))

57 L.P.H. de Goey. T.H.M. v.d. Betg. N. Mulders, H.T.C. Stoof.

B.l.

Yerhaar. ~nd

W. GHlckle. l'hH. R.,v. 831, 5183 (1986)

58 L.P.H. de Goey. H.T.C. Stoof, B.J. Ve.haar, and 'II. Cl~c1<I~. Phy'$. R<>v. 838. 646

(l~S8)

59 y~. Kagan, I.A. Vartan·y~nts. an<1 C.V. Shlyapnlkov. Zh. [ksp. Toor. F1z. 81,

1113 (1981) [Sov. Phys.-JETP 54. 590 (1961)]

CHAPTER 2,

SPIN-EXCIWIGE AND DIPOLE KEUXATrON RATES IN ArolUe HYIJR1X}EN:

PHYSICAL REVIEW B VOUIl>Il! ~&, Nl)l>IB~R 7

Spin-exchange and dipole relaxation rates In atomic hydrogen: Rigorous and simplified calculationS

H. T. C. Stoof, l. M. y, A. Koelml;l.n~ .\!nd a.l. V'I;'rhaaI'

Dflpo.l'lttt.f;ttl r;.J PhY$t():;., Eliidhflt.tt!'l1 Uniwr.Jlrji ()f Te~hn~logy. NL·5600 M.B Eilld.1Jovoen. Th~ N~~h~rla{jds

(Received 2 July 1987)

Wt; ~!lPJ.;~1\1.t¢ ~hc: m>'kjH"t;~i.;. !teld p:n.:;l f,fmp¢rlllture dl!!:~ndl!!:nc:e Dr the I":lles for ;til lWO-lMxl),

5pin-ex~hllo.nge and diPQln.r tr!ln:'OitiQftJ n.mong hy~rftnl!!: le",~b in c:ryoglmit: H tail b1 1'[I:taM (If Ihc couplec:l_c:hl1cneh ml!!:thoo. A ~s.cription Df tbis m~~hod and iUt. pl'a.clk:al 8Pfl1i.;.a.tl0:n i!l Pf~~~'t;d.

A ~imple interpTetllticn of the rille!! is gi .... en. in 50me t~;J wi(h ~iat(;d !ilmCllt.e eloscrJ-form

fQf-mld.u., b~d on the &genera~.intern!1l.5t.a~s ilpprO:II.m.a~iCm,

1. INTI10DUc;nON

In tb-e paai f~w y~ari :!I.pin·PQI;u·i:1;~ Momic hydrog~n gas at subkelvtn. tempr:l";:uur~~ hJ;q. .\lnr~~.;d ~ great d-c-nl

of 8tt~ntiol'l:. AI:l"~a.dy a~ 1Qw~r d-cnshies the exceptional

properti-ts of this gas hav~ shown I,Ip viti. 1hc obs-ervatioD

of spin wav(:1_1

The-y ~av~ al~Q ~~n ~;I;plQit-cd in S\ll;-C~S­ ful attempts til coft3h"uct sll:bk~lvin .... rrH1"i-Cn;.:r: witb th.;:

+lim of improving upon th~ frtqutoney :i.lahWly of e1.isting: 1110mlt; (requeru:y .$tl;t.lldards. The primaf)' im-ettstl

how-ev~t. is :still ;!a~:!iOciated with lh~ :I1o;hi;-.,..-c-m.:nt of the temp~r~ture-dC:llsity fl!gime wh~re elTect~ due tc:l !kI!;.: de .. gt:ncr~.;y Are c:xpected tl) show up. Although BO!i:!!!" Ot

F"ermi"ii!.;gt:n,.;:rp,~c ;sy:s;tems hay.:: already been ,studir:d e:\T

ter'l.'l.ivoe;ly in vadQI,I:i p-I'.rt:s;. of phy8~c:!i. atomic hydrQg~n gas. pt-arni$e$ t¢ f;le t;:Xl;l;ption~1 8150 from this point of vie:w.

II\ nu..::iel.r m;,l.tt;r, ror instancej the rolo or occupied IlI!v~ls.

In d.llmptns; Ilud;on collisions within th~ F~rrni SI!!8. and thu:!l; in e~pb.lrtins the validity of the ftuctotar !j.h~11 m.oorl

1$ ql.l0l1hativ-=ly o;lcQr~ but not e:asily calc1.ila.l~d qU&ntll!ir

ltvely owi~,g to the high nucleon d~nsity" whicb rnak.~~ ~t

diffi,c;:ult to di&e-ntangl(: pure blacltins ~tr~cti frQm lhr!!!e" ~l\d more-pDrticle collisiot'l9._ A :!I.11'nitar r-=m;kr'k 1l1'~Ut;::l-to

superfluid helium con3id~~!!:d a~ Ii '(3QSt;"dcgcn-=rat; sys" tem.

Degeneracy in po1a.~i:<!ed all)mi.::: hyd.rogen promi~~ to

be ~l).H~ble in dfCUffi'S~aMtj, whel'e av-c["01S~ intcra~Qmi.:;: dblOJ.nc~~ IH~ l;ompfIf.a.bI,,:; to tht: de Broglie' w;ly!!!I~ngrh but illuch l;:J.rg~r thi:l.l"\ ehe range of intaatomic fot'ct:!l.j 80

thM I'he-m~';fQsc.opil;: occlJpMion of singk-putiele

stMC:"5-wiU rI;VCl;l! it~lf i~ Crel~~M~c", fl.l1ct tnmspcrO proP'Crt~~ "~~.::::["i"OJ.bl~ with p~r~ two.boc;ly i:QIII~~ons_ lbc p",~ibili. ty. ;),1 Ic&s[ in principle, Qr inde:pe:nd~nd'y t:(n'llrollabll! d~nsit)' and tem~raturt: is anoth~r aHr.acti"e featurt of

Bose-d~gen~ratt atomic hydrogen. which di3lifiguis;hts it

from oth~r systems like nuclear matter. !mp~rffuid heli-um, or the electron gas in metals;_

w=.xp~ri.mrTi.t~l and ~h!;l)r~tic:::al work. On thi~ totoic haa

had a itIV!'I$ ir'n~IU$ rt;l;cnHy ):,)1 lh~ id~a

or ~(l-i.l.t\:I'I.i"g

the

polarizcd atQmi<:: gas in :[\, m.j1,n~t\!; trap. tbl,l.$ ;t'VOiciing

the decay

or

atomic density by thre.·body colli.i(>n~ .t the helium-cov~red walls of B ,gas cell. In a stati(; trap or

thia kind.] however. other decay modes arise through

two-body .;.olli:;.ion,5. of ~IOmS in the two highest Is

hYP"r1ino st.t<3 1 c) nnd 1 d), loading to depolarization via the fonnatioD of Atom~ iJ;l the IO"Ne~t ~tates I a > and

I b). A couplcd....chQl1l'\ch 1;3.11;1,l1atiQn of 'I.h~ as;3ociated

t&.t~ has bun pre~ntc-d by L~g~n!;!ijk,./:![ tIl_'"

The purpo;sc of th.; prt;!SoI:~~ pOJ.~!' i~ thre~fold_ (1) To d~scribe-in a mor.;: I;'O'\T1pl~te t"Qttr.I. tho!!! coupll!:d-channds

method. which h.\l!i ~~n l.1.$~d for th.o!; calculation of the

H-m~!'j';'f" rr~q!.l~Ti.c:y ghiftj, and of the above-mention.;d r~L

la~ation tales, bolh due to H+H 1;(IIli.:-icms. (2.) T"

pr~sent new value3 fQr the~e [":J.l~~ b!lSed On a l'I"I:or~ accu-rat: nl.lmcri";l:ll c;:akuiatioll and a new H-H &inglct

potCI1-ti~l de~l;ribins mOre p!:'~cjscly the experiment~l ~llt:;t. Qn

:s.ingt~t binding .energtea. (3) To prcscn~ .p ~imJ:J!i.nerJ

d.C:$(;rlption.~ ill s.ome ca:;.es with :R~C;I~toe;c:l r,:t¢$=:d~fotffi fQrrn.1.ilas. for the two-body rv.~~:J. in v.j1r10 .... $ regil1'l~' of

tt:"pi.~l"l.ture and magnetic field.

Apart f!:'orn. making c1eaf how (;ail;;uI.\ltlQI"\:!i doe;~'-'tibed ifi p~t;viQu_'l. pa~~r!j. Illl the H-mas.er froeql,L-=:nI;Y $hit'r.~ and Or'! ttJ.-c-r;;t.t~~ in magnetic traps have been cQrd-cQ: Cl,lt, W~ be.. licvt; thai Ihe n~\V numt:rical valu-es :[\,& w;l1 ~s the $irnpl-e:

clo:;.;t;I.{Qnn ;"rmula~ wiH bt ~x.tremely helpful to\" '/;:o;;peri"

ment~l {l:rol,Lp:!i w¢-J;"kilig, in thes~ areas. For instllnl;'c, tnll; behavior

or

th; Ii' ,e:a$ In th~ Ilf:it sta.g~;S. of ~onfir:L;mt;nt

in a tmp ";.\J,~ be ~l1iltyzoe;d bsii.l.g thl!! T*O rate (;(lnstnnts. Alro, for rurther work ¢n lh.~ H WI~t'. the pfes~nt ..;~lc;:u: .. lation:; ~rC"

or

int~re~L lCl citlt'ig,c !l;tfat~gies for ..;it't;!:I,l~L 'II'=I1~~~S ~b~ 5()urte

or

fnql.lenc), inst.ability of tht: cryogen-k H m;tSt(. ir'tdicattd in Ref. 5, it i"i nec;;c::;::;3.ty to know th. p>tti.1 I'), 1 b), 1 c), ."~ 1-) <:ion.,lei .. of tho hy-drog.en a,aa in the c~vity. In.for:m.:J..tiQn .about rat~'S for col-lisional transition!i nmong ~hor; byp~ttino!!! h!:vels is n-ecdr:d to dC'termin~ thc!j'; piil"fia! den!l.iti~ theoretically.

In 5.;1;. H A w~ s.urt.l.m.ariz~ ;some equations of scattet· ing thcory 1'1';1;'~$$3ry 'II) dl!rivt S-and T-matri" ;It;mt;'nt~

tor r~ltk,,~don proeesso!s,_ In &c. II B w~ ~on~ider the

d.;g:-=n.;r,,~.iTJ.toe;frta1Tsfat!!:3 apptoximati()n to th~.,;e

ele-ments. and S¢c. U C b ~~'\II:;ned to tho!; two-body r.\l,l~ .;q\I::J.. .. tiona. In 5.::1;'. Ut w~ dei(;l'i~ out' cOllpl~d--ch.[\n[1.;I::l­

ffidhod in II more (;omplctc w:"y thjln ha...'\ hithtrto ~en

do:ne.6 The Ile:W r('SuIts for the rat'=S obti.tint;d ir'llhis. way

"r~ pte~en~ in St-C. IV. We also con~idcr ~h~ t~lil.pcra­ Ulr~ ~~p!!!ncieTi.ce of som~ of them and in ilQ.;:!itit)Ti. giv~ a