Indian Journal of Pure & Applied Physics Vol. 37, July 1999, pp. 529-540

Liq id helium (4He), adsorption of 4He, disordered bosons, condensate and spatial ordering

K M Khanna & T W Sakwa

Department of Physics, Moi Universi ty , PO Box 3900, Eldoret, Kenya

Received 16 April 1998; accepted 20 October 1998

The crit ical propert ies of the superfluid-insulator transition have been studied using the reaction matrix formalism instead of the wnventional Bogoli ubov approximation that is unable to capture even.1he.roton minimum found in real strongly interacting Bose fluids. Although the inert phase is described as disordered Bose condensate, its spatial ordering is more pronounced compared to the superfluid phase that appears as the coverage is increased after adsorption of 4He. Spatial ordering is sustai ned for k values up to about k = 3.5 A -I , where the second branch in Ek ap·pears, both in the inert phase and superfluid phase. The condensate fraction F is more or less insensitive to the hard core radius and has definite values around the rot on region and the second branch in Ek.

1 Introduction Liquid 4He is the most peculiar liquid whose proper­

ties have not been properly understood till to-day. He­lium was not discovered on the earth. It was found in the spectrum oflightfrom the sun during a total solar eclipse visible in India in 1869 by the French Astronomer Ju les Janssen who observed a ye llow line in the spectrum of light from the sun . The small mass of 4He atoms results in large zero-point energy, and due to this it remains liquid even to absolute zero temperature at ambient pressure. Again due to the small mass of 4He atoms, the fluid densi ty remains small even though interatomic forces may be strong enough to form a liquid phase at low temperatures .

The most striking property of liquid 4He, called the superfluidity, was discovered in 1938 by Kapitza l and Allen 2

. The superfluid is pictured as a background fluid that is, in effect, at absolute zero. The normal fluid is the sum of elementary excitations, or quasi particles, con­sisting of phonons and rotons, which. are excited from the superfluid in increasing numbers as the temperature is increased from absolute zero. Ph6nons are quantized sound waves in 4He. Rotons are high energy excitations than phonons, and their exact behaviour is still under study using inelastic neutron scattering.

That liquid 4He reaches maximum density around 2.2 K was discovered in 1910 by Onnes and Oorsman et a!. , but then something serious happened at this temperature and was announced in 1922 and 1923 when Onnes and his associates measured the latent heat and specific heat

of4He. The discontinuity in the specific heat over small range of temperature around 2.2K was named lambda­transition since the shape of the specific heat curve resembled the Greek letter A.

Some pecu liar behaviour patterns, like the small buckets of it emptied themselves by the liquid climbing up the sides and dropping off the outside, the viscos ity measureJllents and the fountain effect kept phys icists busy for decades so that the properties of 4He could be explained using acceptable physical models.

A number of attempts were made to derive expres­s ions for the excitation energy spectrum for 4He liquid by treating it as a system of interacting bosons3

-4. At­

tempts were made by many other authors5.6

.7 also but

were not very successful. The work on the measurements of the structure factor

(Spatial ordering) by Hallock et al. 8 and Sears et a/. 9

showed that as the liquid helium is cooled toward TA. the lambda trans ition temperature, the spatial ordering shows a gradual increase as might be expected due to the decrease in thermal order. Further cooling from the vicinity ofTAreveals a decrease in spatial order. Meas­urements l O of the temperature dependence of the spatial structure in 4He at several fixed densities yielded the surprising result that relative changes in spatial order are remarkably insensitive to the dens ity.

The observed decrease in spatial order with decrease in temperature is assumed to be due to the, possible existence of a Bose condensate in 4He. The physical argument is that if upon cooling, a finite fraction of

4He

530 INDIAN J PURE APPL PHYS. VOL 37, JU LY 1999

atoms condense into a macroscopic state of precise

momentum, those atoms become spatially delocalized

and thus the overall ~pati:il order must decrease. We

~houltl :,ke to merltion here and come. to this poin t latt;l

'n the Manuscript that spatia.l Jc.,cahzation of 4He hquid COUld ta.;~t: pla(;e when liquid 4He i adsorbed in random

n1tOIa such i\S oorcus Vycor glass resulting in the

,harlge'; n 'he ::u"tial order compared to the superfluid

;.>hase. E'/e;'1 if h.'n: 's iI change in the spatial order due ,0 the fl:)\', c.{" _Ip 1:1]'lid into a disordered system, we

nay ~till t.a\: .,1::: co ldensate that may be described

'lS disorderl.'v . " ~'of'densate. After adsorption , spa-

;la. 0~(kr T'1l,S'. '." .. ::I:! the 4He atoms are localized.

We now tUIT' " •. he L -ic ideas and the researches that

le,d :J<; !o writ" 'h v'sent manuscript. By definition

~t: "prflu ids ca'l wltnnn.'1d weak microscopic d isorder

qnc p :t hardly affects the conaensate fract ion or the

superfluid dens ity. Interactions help stabilize the con­

densate, and the con ensate can distort to screen the

impost'ri random potential. Even the relat ive changes in

the spatial order are :-emarkably insensitive to the den­

si tylO. However, rong disorders can ultimately lead to

vani<;hi'18 of t/lc> condensate fract ion and superfluidity. A nowi~·.g superflu id is characterized by the macro­

<:copic wavC'fuDction w hose phase varies across the sam­

p le, and as one 1S this condensate wavefunction

rem:!i. swell dofinf"d, disorder cannot lead to the degra­dati n of 1he f'P,,"pnts l l

. W' th increasing disorder, the

abi l ity of th • ~""erfluid to wit stand phase variations is

-educed, and 'or very large di sorders, superfluidi ty may

e en be ":: "'led at zero t~ . " .. ratures, <!Od this can

I · . .' II dB ' " B 1 12· 15 resu t m wIJ,~' :<; <.-0 e osc'r' <ltur or ose g ass . Disordered Best' .'rj".'(:.-" f';;.; can be re;dized experi ­

mcntallyl 6 in nmdom '11cd:u, suc 1 as 4He liquid adsorbed

on porous Vyc'.). i:l~;~ 17 ThE "irst ft'wno71o layers of

ad,>orbed 4He are not superfluiJ, eVe. l at ,lIW tempera­

'ures, and constitute an inelt in:;u lating :ayer of bosons

localized by disorde;- A" the coverage is increased, a transi tion from this Bose insularor to a superfluid phase

IH is observed . In gene,al, the lIr~t f:?\V mono layers are

those bosons th'lt occ py non-overlapping localized

state' , and these screen the microscopic di~order of the

porous glass for subsequently added bosons. The added

ho<;ons may feel a smoother potentiallhat may be a sum of the initial random potential plus a repulsi n from the iocalized particles.

Tn the calculations done by Singh and Rokhsar l 5, the

Bogoliubov ,Jpproximation has been u ed to s tudy the

critica l properties of the insulator-superfluid transi t ion.

Howeve , Bogoliubov approximation is unao:e to cap­lUre the roton minimum found i;j ,~a ' strongl:;. ,nter:lct­

iog Bose t1uids. 3ince 1Il the' llisulator- ~'...t~ertlU!d

transition we are dealiflg with a rcal Bo;;e flutd. we h:1\c

LO look for an a'ternative way of ... tudying t~.~ orop'~' -if"~

of such a system. We shall use <'12 rez. ti ol rJl<....rix f \. 19·20 b' h f f' I orma Ism to 0 tam t e onnul" or Vart'.lu:> ~'1Y'lcal

quantities involved in the study of the vc:: "n. ()ur

calculations 'vil! {-1e v'J!(' ·\r.~-·. !ue:-. ~'I- ~,),'.)I;' :hc roton dip to as :ar as the <:eco'1d hrJnch in ,t ~ '::'k f.: I'. <iph around K = 3.55 kl

All the Bose p .rtlC.\P~ invoivC'li i'1 the inS'q;ltor-s;:pe:-­

fluid transitIOn WII! b.:: tn:.med as one systei'-, .)If: ... C Ihe first few mono iayet's 01 .lU~Ol bed "He are not 'iuDerf L~~i. ~ en at low temperatl're<;, and form an inen : .:-'Uhtldg

layer of boson:" iocalized by disorder, the Bose partid~s

in ·the . nert phase wi il be those not in the cuodensate.

Then as the coverage is increase ', a transitIOn from the

Bose insu'lator to a superfluid phase IS observed, and

hence Bose particles in the superfl uid phase will be

considered to be a part of the condensate. Thus the total

number of bosons in the system is the sum of those in

the condensate (superfluid phase) and those not il1 the

condensate (inert phase). The wave-functions describ­

ing the two phases must be symmetric and exchange of

particles between the two phases is allowed across the

transition layers.

In our calculations in thlS manuscript we assume a

hard-core repuisive potential between the bosons occu­

pying porous glass, and this assumption is essential i order to stop all the particles from condens ing into the

lowest loca lized eigenstate and in the mono layers of

subsequen tly added bosons which 3f er screening the

disor er of the porous glass lead to a superflu id state in which the 4He p. rticles will be assumed to be illterac,ltlg

through 'a Gaus,>ian potend;1 that was useJ by liS earllcr

tv l)btai n a quantitative fit fo r the £k-k graph. Thus there

may he layers of inert and condensed bosons near the

insJ lato r-superfl uid transi ion region. We '-lso assume

:0 {JUl calculations t at the transition from the insul ator

to ,he superfluid takes p lace over a distance cqual to the

hard-cor diameter, and if a is the hard-core radius, then this distance is 2a. Helice we are assumi ;-)g the exi tence

of on,; mono layer of bosons between the inert phase and condensed phase. Bishop et aP6 report that coverage

may range from i.5 to 2.7 atomic layers at temperatures

ranging from 0.23 to 1.25K. However, there could be the

fom1ation of layers of one, two, three .... atoms above the su bstrate,

KHANNA & SAKWA: LIQUID 4He SPATIAL ORDERI NG 53 1

It should be clear that adsorption of 4He takes place

on a plane surface and the interaction bet een the par­

ticles depends on both the distances of each individual

particles from the surface and on the distance between

the two particles parallel to the surface. If the plane

surface is described by the co-ordinates y, z, then the

distance from the plane surface will be described by the

x co-ordinate. We shall assume that the 4He fi lms are

free and translational ly invariant in the y, z directions

and that there exists a symmetric density profile in the

x-direction. Since there exists a transition from the Bose insulator

to a superfluid phase, we have calculated the condensate

fraction, and the results are presented in this manuscript.

It would also be interesting to find out what happens to , 4

the spatial order or the structure factor, S(K) , when He

liquid is adsorbed into the porous medium such that the

later formation of mono layers of 4He results in the

transition from Bose insulator to a superfluid phase. We

have done calculations on S(K) for both the inert phase

and superfluid phase in this manuscript.

2 Theoretical Derivations We consider a system of N helium particles in a box

of volume O. It is understood that the density p == N/o

is kept constant even when N and 0 are allowed to

approach infinity. The assembly of helium atoms in the

inert phase is supposed to be interacting through a

hard-core potential VCr ),

VCr ) = {:+- 00, r ~ a ... (I ) , 0 , r"? a

and those in the superfluid phase will be supposed to be

interacting through, the Gaussian equivalent of the

Leonard-Jone potential. \P(r) =

+~ , r$ a

2 2

4C[~'P{ -('::) l-«p {-('::) II '~a .. . (2)

The Hamiltonian of the system in the second quan­

tized formalism 19 is,

H=

... (3)

where 7i k is the momentum and m the mass of 4He

(Bosons) whose creation and annihilation operators are

at and ak', respectively . In Eq. (3) , Vk, k, k3k.. is the matrix

element for the two-body potential. For a high density

system, the effects of multiple scattering are taken into

account by writing the effective Hamiltonian of the

system fo ll owi ng the method of B rueckner and

Sawada3, i.e, I

Heff ="2 Np ( No - I) too . 00 +

[{h2 k2 } L 2m + No ( tok , ok + tko , ko - too, 00) at at

k=O

IN ( ++ ++ )] '2 0 tk-k ,00 ak a-k + too ,k-k ak a-k ... (4)

where the symbols have the meanings given in Ref. 3.

The scattering matrix t satisfies the integral equation,

tk ', k " , k, k, = Vk" k '" k, k, +

L Vk" k'" k", Ie", G ( kIf I k"2 ) tk ", k"" k, k, .. . (5)

/(', It',

Tl)e Hamiltonian in Eq. (4) can be diagonalized bl introducing the Bogoliuhov transformation

I

bk = ( I - L ~ ) - 2: (ak + Lk ~ ) .. . (6) 1

b_k= ( I-L ~)-2: (CL.k + 4 a~k) ... (7)

The new and the old operators follow the same com­

mutation rules. Liu et al.4 solved these equations to get

an expression for the excitation energy Ek as :

where

1

too , k-k ] '2

... (8)

tk = tak . ok + t ok, ko - too , 00 .. . (9) For 4He particles that are bosons, the t-matrix is

symmetl;c , i.e,

tk-k ,oo = too, k- k ... (1 0)

Now to calculate Ek from Eq . (8), our only task is to

solve the t-matrix equation for the potential s 'given in

Eqs (l) and (2). Following the evaluation of the t-matrix

in Ref. 6 we can write for Ek as:

532 INDIAN J PURE APPL PHYS. VOL 37, JULY 1999

... (1 1)

where

t -k -sin ka

t --00. 0 0 ka .. . (12)

and

No t k [~ --~ + 4 1t No J d r r2 V ( r ) Go (a, a ) 0 0

X [ (

.1 _ Go ( a • r » )2 ] sin ka

Go (a, a) ka .. . (13)

Now to carry out the calculations fo r a high-density system, it is necessary to construct Green's function for

all even values of I on1/. For GI (r-r' ) we know 'from Ref. 3:

G (r -r') = 0 jdke i k , (r-r')G(k) .. .. (14) ( 2 1t )3

and the G' l s can be calculated from Ref. 3 as G, +1

Gdr, r ') = t jd9Pd9)G(r- r ') ... (15)

- I

In Eq.( 15),9 is the cosine of the angle between rand

r. In our calcuJations20 we assume that: GI (a, a) = Go (a, a) ... (16)

with th is approximation one obtains in the ground state:

------=1 ___ j. _ 2 L (2 I + J ) {. JI (~. )2 + 1] = Go ( a , a ) I \

- J (ev e n )

I sin ka ----.--

Go (a, a) ka .. . (17)

Making this replacement in the expression for G(a, a) and changing to the variable x defined by Ka = x, we get:

~ -I

G ( ) 0 f d . 2 [.>:2 2 N a2 sin x] o a ,a =--- xsm x .- - --

2 n2 a 0 m Go ( u , a ) x

... (18)

or

... ( 19)

where

X2 2Na2 m --

m 7i 2 Go ( a , a ) ... (20)

Eq. (20) can be solved numerically for X2 in tenns of 3 A' 3 2 pa . graph between pa versus X was plotted by

Brueckner et al.3 Making use of this graph we can get

the value of X2 corresponding to different values of pa)

and calculate the approximate value of Go(a,a) . For

calculations we take the value of X2 from Brueckners

graph. From Eq. (20) we get:

N X 2 7i 2

.. . (21) Go(a,a)

Substituting from Eq. (21 ) in Eq. (13) and replacing

No by Nand m by m*' we get:

[ X27i2 41tN f

Nt = . +-- d rr V(r ) k • 2 0

2m a (J

x [('1 _ Go (a , r) ) ] sin ka Go (a , a) ka

.. . (22)

which on substituting in Eq. (11) gives:

[

7i4k4 X 2 7i 4k . 41tp7i 2 k sinka Ek = . + sm ka + --!....-----

4 *2 2 3 * 2 * m a m m a

[. J d r,2 V ( r) (.] _ Go ( a , r) )2] t • Go (a, a)

.. . (23)

. Now

2 2

f d r r2 V ( r) {'1 - Go ( a , r)} = J d r,2 ('1 -7) Go( a,a)

... (24) Since

Go ( a , r ) a

Go (a, a) r

Substituting for VCr) from Eq. (2) and solving the

integral we get the expression fo r the excitation energy

spectrum for bosons interacting through a Gaussian

potential a.s:

Ek = . - ' - + (k sin ka ) + [

7i4k4 X 2 h4

4 m' 2 2 aJ m' 2

[ 4 1t 312 h2 P C 3 3 . ]1/2

. ---- • IlR - ~A ( k sm ka ) am

.. . (25)

Hence for the bosons interacting thra gh a hard-

sphere potential given by Eq. ( 1), The excitation energy

spectrum Ef will be gi '!en by the expression,

)-

1

KHANNA & SAKWA: LIQUID 4He SPATIAL ORDERING 533

E~ = .--+ (k sin ka) [

7i4 k4 X2 h4 ]112 4m"2 2a' m"2

.. . (26)

But the expression for E ~ that we shall use is given

by22 H E k =

[

]i 4 k 4 47th4 { 16 1t a3} ] 1/ 2 . --+ --p . 1 + ( 8 1t P a3 ) 112 + P k sin ka

;1 * 2 "2 3 . m m

... (27)

3 Structure Factor The Feynman theory23 of liquid 4He gives a simpie

expression for the structure factor as : 7i 2 e

Sk = . --- ... (28) 2m" Ek

Eq. (27) for E ~ could be substituted in Eq. (28) to get

S ~ for inert phase of liquid 4He adsorbed in the porous

medium like porous Vycor glass since we have assumed

that 4He particles in the inert phase may in teract through

a hard-sphere potential. Similarly Eq. (25) can be substituted in Eq. (28) to

obtain an expression for S ~ for superfluid phase since

in the superfluid phase 4He particles are assumed to be

interacting through a Gaussian potential. If, however, we use an expression for Sk dtle to

Berdahl24:

Sk = 1 + 1 ( dE ) for k ;t 0 P dV(k)

.. . (29)

Bardahl us~d the ground state Hamiltonian to find E appearing in Eq. (29). Instead it will be more accurate

to use the excitation energy Ek which wi ll take account

of aJl the states of the particle. The structu re factor

alculated in this manner will have contributions from

all the excited states . Thus in order to take into account

the many-body interaction , v(k) should be replaced by

tk, the reaction matrix Eq. (29) then becomes:

Sk = 1 + l d Ek / d tk .. . (30) P 'Ok 'Ok

Using Eqs. (25)-(27), respectively, for Ek for the two

interaction potentials and tk from Ref. 6, we shall et the

following expressions for S ~ and S ~I, i.e,

G I 3 S k = I + -- [ 4 Al k + pE f

'}

ak-BI (sink a+k acosk a) J----------

( k a cos k a - sin k a ) t ~o,oo

... (3 1)

where,

t2.oo ~ 4n=.Ti2 { 1+(81t p al )1I2+ ~6npal) +

4 ~12 c [ ~ - Il~ ] ... (33)

ak2 B2 ( sin k a + k a cos k a ) ) ---- ------

( k a cos k a - sin k a ) t ~,oo ... (34)

where Ji4

Al = -- ... (35) 4 m* 2

x 2 71 4

B2=---2 a3 m* 2

47t a 7i 2 J 16 } t ~.oo = " t' 1 + ( 8 7t P a3

) 112 + - 7t P a3

nl 3 .. . (36)

We could also obtain expressions for Sk using Fey­man relation given in Eq. (28) . Eq. (25)-(27) will be used

for E ~ and E ~ to obtain S ~ and S ~ i.e:

H 7i 2 k2

Sk=--2 m*

n k 4n P h ~ 1/ 2 16 3 112

[( r 4 4 J 2 { ] ~,;:;- -;;::- . I + ( 8 n p Q') + 31t p a } k sink Q

... (37)

4 Condensate Fraction W~ shall assume that 4He atoms when adsorbed in the

porous medium experience an infinitely large repulsive potential. Since 4He atoms are bosons they will be described by a symmetric wave function given by:

I \If ( x ) = - cos k x ... (38)

1/2 a

where 2a is the width of the potential, k = lt7t/2a n is an integer equal to 1,3,5, .. ..

534 INDIAN J PURE APPL PHYS. VOL 37, JULY 1999

The 'ingle-particle ground state wave function in the

condensed (superfluid) state may be denoted by <Po. This

single particle state <Po is simply the zero-momentum­

state (Z-M-S). Since <Po must also be symmetric, let it be,

<Po = A sin2 kx ... (39 )

where A is a cons nt whose square (A 2 ) may determ ine

the occupation in the Z-M-S or it may determ ine the condensate fraction .

The layer separat ing the insulator-supenl uid bound­

ary wi ll be assumed to have a thickness of 20, the

hard-core diameter of the ~ He atoms. Since the wavefunctions g iven by Eq. (38) and (39)

must be single-vaiued and continuous across the insu­

later-superfluid boundary, we can write : <p(20) = <Po(2a}

(~; 1=2a =(~~o J =2a ... (40)

Between Eqs (3 8)-(40) we can write, 1

I A21 == 12

20 sin 2 k 0 ... (41 )

Now we shall treat the inert in sulating layer ofbosons

and the super-flu id phase as one system such that the inert

system of bosons can. be treated as localized , uncon­

densed part of the ground state or as the zero-tempera­

ture no rm a l fl u id exc ited fro m the condensate by

disorder rathe r than thermal fluctuation. The tota l num­

ber of bosons, N, in the. system is, therefore, the sum of

those condensate (superflu id phase) and those not in the

condensate (inert phase). Thus we can write :

N = No+ I b ~ bA .. . (42) Ai 0

where bi and b"A. are the creat io n and the ann ihilation

operators respective ly, No is the number of partic les in

the condensate or Z-M-S and "2>: h; are the parti c les not

in the condensate but in the exc ited states . Now at very low temperatures assuming smeared condensat io n, a ll

the particles wi II constitute the condensate such that the

Z-M -S is macroscopically occu pied and the states above

the Z-M-S are microscopica lly occupied. The Z-M-S

and the states just above the Z-M-S constitute what is

known as the condensed system, and out of this system,

the condensate fracti on, F, is defined as :

F== ; = I - ~ I b ~ bA . .. (43) A. i a

Now the number ofbosons in the inert phase (uncon­

densed phase) will be given by:

N I = I b ~ bA =~ 1.." a

and that in the condensed superfluid phase No=A2

combining Eqs (41 )-( 45) we get:

.. (44)

... (45)

I F = .. . (46)

I + 2 sin2 2 k 0

Eq. (3 I) w ill be used fo r ca lculat ing tne structure factor (spat ial) in the supe rfl uid phase, Eq. (34 ) will be used fo r ca lculati ng the structure factor in he inert pha ." aHd Eq. (46) wi II be 'used for calcul at ing the COnde!iSar:. fracti on.

5 Numer ica l Ca _cu ia tions Tne vaiues of the vario us parameters used are :

p = 2.18 x 102::: part icles/cm 3; In = 6.646 x 10- 24 g;

111 * = 1.23 m; 0 = 2.05 A, 2 .10 A, 2. 15 AI 2.2 A, 2 .5 A; h = 1.05 x 10- 27 ergs; C = 7.055 x 10- 6 ergs;

;{2 = 33,29,23; I-!R = 0 . 1103 A 2 and I-!A =0 .2206 A 2

7 Discussion It is well known by now that the few mon o layers

o f the adsorbed 4H e are not superflu id even at low temperatures and form an inert in sul at ing layer of boson s localized by disorder . Th is means that the adso rbed 4He a to m s will be in a metastable state . Therefo re for som e e nergy Ek of the system th e re will be a c riti ca l ve locity o f fl o w fo r adsorbed 4H e a to ms fo r w hi c h Ek wil l be negati ve , pa rticles will dro p o ut from the sys tem and coll is io ns be tween the particles wi ll lead to some exchange of the pa rti c les and th e adso rbed 4 He atoms may sti ll be in a me­tas tabl e state .

Let us write Ek = Ek / - u.k .. . (47)

in which a particular critica l ve loc ity Vc co rre­spo nd s to Ek /. T hu s for the velocity [u[ <Vc, Ek w ill be positive, and for [u[ > Vc, Ek will be nega­tive. Th is means that from a co lI"ec tive of positive e nergy som e particles w ill dro p ou t if [u[ > Vc o r in othe r words e lementary excitations wi ll be cre­a ted iflul < Vc . Hen ce for [ul < Vc , no pa rticl es will drop o ut and , therefore , the sys te m w ill be in a metastable sta te. Und e r s uch conditi o ns th e co ll i­s ions betwee n the particles are rare and less e ner­get ic , which mean s very small fri c ti o n- a very important pro pe rty of s uperfluid . We thu s have a bo und col lective whi ch has the properti es of super­fluidity .

.....

KHANNA & SAKWA: LIQU ID 4He SPATIAL ORDERING 535

Now since the fIrst few mono layers of adsorbed .jHe

are not superfluid, the flow velocity of 4He atoms when the few monolayers are adsorbed shouid be more (han the critical flow velocIty of ~He fo'- sustammg super­fluidity. It '," ouid be intelesting to measure the flow velocity wben helium is being adsor ed. As the cover­age is increased, a transition from thIs Bose ins'.llator to a superfluid {:hase is observed. Thi" \Iv Jid mean that rtlt" flow velocity luI will i.hen Jt;comt Ie;, han the cntIcal \'elocity Vc Experimental measurements 01· the ~'low velocities in two pha<;es ca I confml1 aut' ,'t(I(l1\.j\1-

TIle ::patJ.:l vide";ng m the mert phil~es " repres,-·med by S~ H -k ~ :mh: Sh0W ttl Fig 4. Tre f!raph follows

th A - ~,- l'd LO Th smoo patlern obsel"ve.L' «·r .'i.e IQU! . t'1'C are no abnormal flllctuaticns in S\ H -k graph showing thereby that the spat" a1 ordering in the inert phru.e sust ill'; even

60 ~ 40 I 20 ..

Cl CIt

0-

-20

-40

Fig. 1 - ~ against k; X2 = 33

5U I 40

30 20 .. 10·

~ Ih 0

-10 1 2 3 I f -20 k

-30 I -40

F· " SG . • v 2 29 19. "- - k agcllDst K;., ==

for k values up to around k = 3.55}..-1. the region of second branch in Ek .

The 50, G -k graphs shown in Figs '-3 for the superfl:l1c' phase show Jefinite filet' ations for k values a-:'). ',;'

rotan excitations and _.:- k a! ·~s wheT} the "ecc:-ll branch 'n Ek begi!lS to appear. The value of Sk G de­creases in the vicinity of the phonon regio or (IS '.ve move to the Z-M-S. ThIS IS due to m(lcrOSCOpIC occupa­tion of the Z-M-S leadmg to decf:!ase 10 spatial order 'o.

Macroscopic occupation of the Z-M-S ~eads us · 0

another quantity called the condens?tt frac.:tlOn ]< gl 'en by Eq. (46) . TIllS expression nas been so derived [\ <:t it takes Into account the existence of the mert phase ::lid (he superflUld phase in an adsorption process. TIle ~hape of he F-k graphs shown i:1 Figs 5-9 seems to be m.en­::;itive to tho value of the hard core radi".ls and this wouid mdirectly mean insensitive to the' density of the iquid

50 I

40 j 30 20 ~ ... 10 I

C1

I en 0 -10 2 3 $ -20 k I -30 I -40

Fig. 3 - .st against k; X2 =33

2

f\--- l 1.5

1 .:s:

0.5 i 0 -, ----J

0 2 4 6 8 ~H i;) k

Fig, 4 - S~l against k

536 INDIAN J PURE APPL PHYS. VOL 37, JULY 1999

15 --6- a = 2.05 A

10 -+-a = 2.1 A ---a = 2.2 A ---a =2.5A

5 --+- a = 2.84 A

o r-.---- ---'

o 1 2 k 3 4 5

Graphs of F ( x 10. 1) again ( k for different hard-core values

Table 1 - (X2 = 33) Table for sf against k Table 2 - (X2 = 29) Table fo r sf against k

K(A- I ) sf k(A - I ) sf 0.200 -2.374 0.200 - 2.1 64

0.400 - 0.563 0.400 - 0.469

0.600 0.112 0.600 0.160

0.800 0.524 0.800 0.540

1.000 0.860 1.000 0.842

1.200 1.199 1.200 1.133

1.400 1.585 1.400 1.~25

1.600 1.961 1.600 1.5595

1.800 1.525 1.800 0.868

2.000 -4.469 2.000 -4.637

2.200 28.163 2.200 26.3 15

2.400 10.008 2.400 9.448

2.600 7.769 2.600 7.416

2.800 7.285 2.800 7.012

3.000 7.701 3.000 7.464

3.200 9.268 3.200 9.047

3.400 13.934 3.400 13.7 17

3.600 44.700 3.600 44.468

3.800 -27.588 3.800 -27.789

4.000 - 10.69 1 4.000 - 10.908

KHANNA & SAKWA: LIQUID 4He SPATIAL ORDERING 537

Table 3 - (X2 = 23) Table for ~ against k Table 4 - Table for sr against k

k(A - I ) .~ K(A- 1) .sf

0.200 -0.819 0.200 0.0544

0.400 -0.315 0.400 0.11 32

0.600 0.238 0.600 0.1821 -

0.800 0.564 0.800 0.2687

1.000 0.810 1.000 0.3860

""f 1.200 1.021

1.400 1.173 1.200 0.5564

1.600 1.091 1.400 0.8212

1.800 0.131 1.600 1.2404

2.000 -4.997 1.800 17572

2.200 24.503 2.000 1.8929

2.400 8.745 2.200 1.6196

2.600 6.925 2.400 ..¥ 1.3524

2.800 6.711 2.600 1.1763

3.000 7.107 2.800 1.0650

3.200 8.71 1 3.000 0.9979

3.400 13 .388

3.600 44.120 3.200 0.9598

3.800 -28.099 3.400 0.9419

4.000 -11.237 3.600 0.9383 ...,

3.800 0.9451

4.000 0.9585

helium. The value of F is more than 0.8 for k values around the roton region and for k values around the

4.200 0.9755

second branch in Ek . These values are close to the values 4.400 0.9929

obtained by Byckling25 for the hard-sphere interaction

(F = 0.71) and for Lennard-Jones potential (F = 0.76). 4.600 1.0080

Some of the values of F are around 0.%, a value that we 4.800 1.0187

obtained earlier6. For k = 0, Eq. (4~) gives F = I, and

5.000 1.0241 this means that all the particles would have condensed

to the Z-M-S. 5.200 1.0243

The plot of F against k that we have obtained is like 5.400 1.0204

a travelling sine wave and this shape is qualitatively the same as observed by Jin et al. 27 for the radial breathing 5.600 1.0139

mode oscillation of a rubidium condensate of about 4500 5.800 1.0065 atoms. The qualitative similarity of the graphs empha" sizes that phase transition is inherent and fundamental

538 INDIAN J PURE APPL PHYS. VOL 37, JULY 1999

Table 5 - Table for F against k for a hard-core values of (2.05 A) Table 7 - Table for F against k for a hard-core value of (2.20 A)

k (A - I ) F(10- 1 k (A- I ) F(lO-I )

0.200 7.588 0.200 7.338

0.400 4.832 0.400 4.570

0.600 3.601 0.600 3.476

0.800 3.343 0.800 3.414

1.000 3.883 1.000 4.334

1.200 5.574 1.200 6.838

1.400 8.741 1.400 9.925

1.600 9.633 1.600 7.856

1.800 6.478 1.800 4.840

2.000 4.275 2.000 3.557

2.200 3.476 2.200 3.370

2.400 4.431 2.400 4.129

2.600 4.293 2.600 6.370

2.800 6.5 18 2.800 9.707

3.00 9.659 3.000 8.374

3.200 8.700 3.200 5.1 47

3.400 5.543 3.400 3.659

3.600 3.871 3.600 3.343

3.800 3.342 3.800 3.952

4.000 3.610 4.000 5.937

Table 6 - Table for F against k for a hard-core value of (2. lOA) Table 8 - Table for F against k for a hard-core values of (2.50 A) k (A- I) F(lO- I) k(A- I ) F(lO-I)

0.200 7.504 0.200 6.85 1 0.400 4.742 0.400 4.139

0.600 3.555 0.600 3.344

0.800 3.360 0.800 3.768

1.000 4.016 1.000 5.826

1.200 5.959 1.200 9.617

1.400 9.258 1.400 8.025

1.600 9.142 1.600 4.661

1.800 5.847 1.800 3.435

2.000 3.969 2.000 3.522

2.200 3.352 2.200 5.011

2.400 8.649 2.400 3.580

2.600 9.153 2.600 4.818

2.800 5.367 2.800 7.606 3.000 3.624 3.000 9.994 3.200 3.381 3.200 7.364 3.400 4.395

3.400 4.668 3.600 7.464

3.600 3.531 3.800 9.838

3.800 3.369 4.000 6.282

4.000 4.064

,

...J

....

KHANNA & SAKWA : LIQUID 4He SPATIAL ORDERlNG 539

Table 9 - Table for F against k for a hard-core value of (2.84 A) Table II

k (A - I )

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

2.000

2.200

2.400

2.600

2.800

3.000

3. 200

3400

3.600

3.800

4.000

Table 10

alA) k(A- I )

2.05 1.5

2.05 3. 1

2.10 1.5

2.10 3.0

2.20 1.5

2.20 2.9

2.50 1.3

2.50 3.8

2.84 1.2

2.84 3.3

F(lO- I)

6.334

3.781

3.373

4.613

8.500

8.783

4.767

3.397

3.707

6.084

9.975

6.596

3.861

3.355

4.471

8.208

9.049

4.932

3.427

3.641

F

9.9 12

9.898

9.998

9.994

9.525

9.8 16

9.77 1

9.888

8.782

9.944

a(A) k(A- I ) F

2.05 1.1 4.543

2.05 1.9 5.165

2.10 1. 1 4.779

2.10 1.9 4.704

2.20 1. 1 5.339

2.20 1.9 4.024

2.50 1.1 7.743

2.50 1.9 3.336

2.84 1.I 8.500

2.84 1.9 4.540

characteristic of liquid helium CHe) and is not affected by the degree of di sorder of the medium into which 4He may flow .

The variation of F with k seems to be insensitive to hard- core radius since the F-k graphs maintain the same

shape. But quite large values of F pertain to those values of k that refer to the appearance of some new excitations in liquid he lium. The k values around l.sA - I refer to the appearance of roton excitations, and the k values around 3 .0A - I refer to the appearance of second branch in

energy excitati on spec trum. This means that the conden­sate fraction increases and decreases with k, and it is large for those values of k around which some peculiar transi tions may be takin g pl ace in liquid he lium.

If the large values of F depend on the ex istence of some transition in 4He, then this va lue should be more or less the same at the different k values referring to

different transitions. Thi s s ituation ex ists only for the value of F for which a = 2. 1 A. The va lue of F for k = l .sA - I (roton region) and for k = 3.0A - I (second branch

region in E~) is more or I ~ss the same. Similarly, the value of F at the pos ition of the hump in the En graph (where the phonon region ends) at k = 1.1 A-I, and the

value of F at the position of dip in the Ek-k graph (where we have maximum roton excitation) at k = 1.9A-I are

more or less the same. This is not true for any other value of 'a'. These values are exhibited in the Tables 10 and II . These results emphasize that the hard-core diameter for 4He atoms is 2. 1 A. Thi s is an important result and we may have to devise methods of confirming it experi­mentally.

540 INDIA N J PURE APPL PHYS . VOL 37. J ULY 1999

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