THE THEORY OF THE VISCOSITY OF HELIUM

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000100030027-9
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 33
• Document Creation Date: December 22, 2016
• Document Release Date: June 1, 2012
• Sequence Number: 27
• Content Type: REPORT

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TIC THEORY OF TH3C VI600BITX O' HELIUM tI
/r/l /.:rx~ Oaloulation or the Lvi_~.4a81ty!oarflaien4
by
L.A: Landaa and I.M.Khalatnikov.
/,ar tk?'vP / ~~rrt /iiFi ~J. Exp.Thsor.PYiye.Uk36R)19,70 9rI 949}"
Tho kinetic oquatione for rotone and phonons are solved by
0
ueingnefPeotiva differential Dross-sdoi~ione ~o~a-~l+e scattering
of elementary exoitatione (phOrLOflB and rotons)(,bY each othor,) f,to
obtained in the first part of this work. e _ehownj=cFte4'iiU
ooeff'ioient of 1~elium II's ~ompQSed of two party t nne
~vi~oooity~ gyp.
temperature. The experimental values for the (yisooeity. aoof-
phonons ("phonon vi0noeity") and sharply i.ncrdar~ing witn s'ali ~.n
pendant on i;empersture; the othe, o?~uee8 by seattorin.g of
daused by ac~.-tteringof rot one ("roton vieaoaity") ani~ not do-
k
ficientnin helium II appear to be in good agreement with theory.
5. The kinAtia e~t~at~o~
The rxpreeaione obtained in the first part of this work for
the probability of eoattering of various typos of excitCtton3 by
0iU `o
eaoh other permitninvestigat ~ the temperature rIrjpendenoe of
the viscosity coefficient of helium II. The kinetic equation for
elementary excitations which we have to solve may be written in
the form
:t:where n = n(r,v,t) ie the distribution function, v the velocity,
and tj~n) the aollieiintegral, the exact form of which is de-
duced be~.ow ?'or each type of interaction.
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50X1 -HUM
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2
'We will senumO that in a fluid there is a meaY'aaooplo flaw
of velocity u, very?ng with the ooordinate in e~ah a way that
the velAOity grudiont'ie ep email that equilibrium ie almQSt
established in esoh volumA lament oorreagoncling to a given
value of the valpaitys i.e., in eaoYi volume element the dis-
tribution funotiona f'~r rotona and phononB are almost equal to
their equilibrium valuda. Thus thr~ distribution t'unotion n for
rotAne (or phonons) may be repreeonted as the sum of the equili-
brium funotion no and .a smuii v+xiiutiQ.1 n ' no + ~n (5.2)
2inoe we am seeking r. atationary solution of the kinetic
equation approprinto to a 3ivdn macroscopic flow (constant in
timer the left-hand tern Sn/ ~ t in the kinetic equation should
be taken as equ-11 to zero, Furthermore, we need only substitute
the equilibrium function no (instead of n) in the approximatinn
term v p n in question, a~nea this term already contains a
emall quantity -the volcaity gradient of ttmaeroaaopic flow.
In a fluid moving wi;h velocity u, the equilibrium rotan dio-
tribution function appeara as :
no
7
Let us choose the ~ireotion of the z axis along the velocity
vector u and for the ~c~Kes of simplicity let us assume that the
velcla~{e~^"Br_`dient is xe;-cted along f Qxisl x perpendlculctr to the
z-axis. It is (vjCOn% that the value of.' The 'v1eC0@1ty coeff?oient
require 1.a independa:it of th,~ choice of coordinate system. In
oa.Lculating the first viscosity oodffioient we need only consider
P,u ul) (5,3)
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3
the coed whdru t1d vul~aity u estisfioQ the condition dlv u ~ 0.
Lot ue now saeume that the density c~nd temperature arc oonetctnt
throughout the liquid. As a result, on the 18ft-hand aide oP the
kinatio equttian we hmve a
vVr : Vr : fl0
" -x kT ~
It' wo use t ophcric~-1 oy3tom of ooordinatoe with tho polar
nxle lyirih in thu z dir~ction, th4 oxpr~seion obtninod mwY ba ru-
written in the form
no
1v "t4 C.ic D . SAM &? CAS
(5,14)
inwhich it io logitimaate to write Po inst~ad of P einca rotons
popseeg momontr of mngriitude close to Poi In Rcaordclnoo with
(1.2) the velocity v iE3' given by
V- ~c - ~-Inu (5.5)
P
The final kinetic equation for rotors bocomeos
ri JD::) :4- s4.a Cs : :T(r)) (.6)
Thy kinetic equation for phonons may be written in an ana-
logous form. In this oases however., allowanee must be made for
the fact that phonons obey Bose statistics, and therefore their
equilibrium function for a flowing gas i,e
s? ?(p-tA41\
r? kT I (5.7) .4 . Carrying out similar caiauiationg to those given above fox'
rotona, but with the dissipation function of (5.7)r we obtain the
kinetic equation for phonons
I + I) it ?_ C4
l
4i
Ihy C ~. -1
Y
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(5.B)
f
Va
b
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k
Thy ao111e1on integrals on the right aide of the equations (5.6)
and (5.8) are dependent on the natar~ of the intor~otione Q~'
rotone with eftoh other and with phonona in the otge of equation
(5.6)) and on the nnturo of the interftationo oi' phQnona with esoh
other nncl with rotans in the case of oquLtion (5.8).
Thy vi-tt1kf th& viocosity Laoeffioi.ont evldontly doesa not 8e-
ponel on the part of tho 1lquifl aAriaid.erod. It iu oariv~ni~nt to
oolect ~. pol.nt ~t which thc:vcloai.ty of mrscra~aopia motion Is z?,ro.
At this point the exproociona for the sc~ttring funotion (y.3)
Rnd (5.7) rtro nc~t ponc1ont on volootty, rind oolnoiLl~ with the
corresponding equilibrium funetione in n motionloes liquid.
Duo to the preoence of two typos of excitations in helium II
we have written two kinetic acduationp. That portion of viaooeity
caused by momentum trpnsf'er by rotons will b?, conditionally re-
fnrrea to as roton viscosity; and that part duo to transfer of
momentum by phonons a~ phnncn viscosity. Actually one vleco~ity
is obocrve d, this being equal to the sum of the rot on and phonon
contributions.
6: Rotor}_viecoai~.
rneir eca i r uig unu auaui?j u.&.'s
(See Para.4) ere diexegarded, the change in the number of rotona
in a given phase volume will take place in two wftys, namely:
(a) eingtia scattering of rotons by rotons (Para.Li); (ID) scatter-
ing of rotons by phonons (Para.3).
Hnwever,eimple calculations show that down to temperatures
of the order of 0.(-0.7?K the contribution to the viseoelty of
the scattering of rotons by phonons vll be negligible compared
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lotion of tho roton portion of the viscosity, but limit ouraolv~a
to a determination of its tempratur dependence, the kinetic
equation (5.6) may be simplified. For than it ie perttiieeible to
replace the collision integral on the right-hand side of this
equation by t
The roton kinetic equation (5.6) obtained above oannot be
solved accurately. However, if wo do not attempt an exact aalcu-
5
to the elaatio ee~ttorin~ oP rotono bar roton8. This io not only
beoauso the number AP phonons at higher temperatures ie found to
be lase than the number of ratans but mainly beaauee the roton
momenta are changed vary 11tt1o during the eoattaring of rotons
by phonons; accordingly the momentum flux, which do terminus the
the
magnitude of/viecooity, ie foundl to ba nQgLi.gi'b~.u. 91nca at
temperaturee bylaw 10K the roton vlocoeity becomes negligibly
small in comparison with the phonon viucoaity, we concludo that
it is gut?ficiont to take account only o' roton-rot an ocattering
in the kinitic equation.
Sri/t, (6.1)
where t differs from the moan time between roton eollisione (tr)
by a factor which is independent of temperaturo and lo of the
order of unity. Beoaueo the time interval tr found in Section L.
contains the unknown constant Vo, we will from now simply write
tr everywhere instead of ,t;- including the factor of order unity
indicated above within the quantity Vo.
Thus the kinetic equation (5.6) becomes
g(p_rp)) 1 S9 Ci1) -
T;i a
r
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viaCo ~ f coont.
~ eiY cof
t
a
from zero,
Comparing (6.L.) with the expression for the corresponding
From this we find the deviatipn Q~the 8jetribution ?Uflo-
L
...,, S~
from i t s equilibrium v1
n
i
Nye will now oaloulatf the momentum filuac. In
only the z-oamponent of the momentum will cuffar
being directed tawarclp x end aqu~l to
tensor component and pressure containing the vi80n2ity coef-
ficient
w r r i rrrr~arr rrrrrrS'rrrrrrr~r
rrr -. ura W r r oar rrrrr r r r r r rr r r r r
rc~r w rw r i r w
rrr ! w
,s rw .~woo ~w r r r r r rrrr rrr aa r r rrr r - S S
sarrw w r
~ y
tt~~
L
V
The suffix r ~. a ua
_
r
rrrrrrrrr=rr wr rr rr rs M
rrrrlrrWUJ rw r Sr=r~.rirrr rM rrrrr r wr
M r r _
r rrSW it r rr r+ r
a w.~ rrr
Wr ~r raMr r !
Y r .r r l r
U
.~ r x
x
~.t
viecoe y
of
the
We find
r% :' t;:;:2 rLD CA ' e C4
o~ ?r
t
2(
C1,*i.rr
f ( ')
'
coeffiolent) requireds
41
(,pd cI 0 (6.5)
V kTj
We carry out the necessary integration in (6.5) over the
eloment,$?at' sA volume dP using the distribution function no
given by (5.3). The final rESUlt is
TL: (tT41DoZ//g,M)Wr, ZI 1V/)5M (6.6)
The quantity J?~,, equal to the product is independent
of temperature. And aC$U8.11Yr according to (L..11))we have
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\2 v, b
P
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4 z ?
J'p
v4
J
. (6.7)
Jr. : r
n
Thus the rotan portion of the IvisaQeity~ oAaffioiant is
given in Corms of aQnstanb 9usntitioe and is therefore a aon-
stant independent of' tompArature?
A oh?sngo in thu number of phonons in
meM.may occur in the following ways: (a) through scattering of
phonono'LyphOnons (Seotion 2); ('b) by absorption and emission of
rC l/~7 / c'il f
phonons duo to inelnetio 4-8 of rotone or phonons with each
other.
In considering the role of each of these proaeeeee in the
phenomena of vieooeity, we first aaloulatQ the relaxation times
(or the corresponding free paths) characterising the astablieh-
ment of a9uilibr3um in a phonon gas.
In Section 2 it was shown that the effective cr083-section
of phonon-phonon scattering reaohou ltu rniximum importance fox'
emill E:ntrl~~ 'bctwcen the momenta of the colliding phonons. It
follows from the laws of conservation of momentum and energy that
such a process of scattering dose not lead to a material change
in the direction of the momenta of colliding phonons, and. there
fore scattering of phonons by phonons leads mainly to rapid ex-
change of energise between phonons. However, an exact calcu-
lation of the relaxation time characteristic of the establish-
ment of energy equilibrium in a phonon gas cannot be accomplished
for the simple reason that the problem itself cannot be precisely
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 Parmulated. Nevertheless, sines the time required for establish-
ment QF energy eCiuillbX'ium in a phonon gaa is can extremely impor-
tant aharaataristia of our ayatam, we will attempt to approaoh its
determination Trom two limLting naseo.
In the First case we sesame that by soma means them has
barn produced in the phonon gay a ohange in the number of phonons
posQeeaing small entrrgi(io (Gmtillor than the nveraphonon enerr)
so that the distribution funation for bhp region of emnll energiQe
,L:t c~yue+l the uquilirrlum function; then we ca1ou1rit~ the
r~lax~tion interval ch~rcteterizing the eotrtbiishment of equilibrium
in such ~ gso.
In they second limiting oa~o we aaeuma that in the phonon gti~
there brae been ~ change in the numbc;r of phonons having lnrga
anargine (greater than the e,v rngo tnorgy of the phonons, so that
the distribution function in the rugioiY of large onergioe does
not equal the equilibrium function. Ixi this caeo we caloulate thn
relaxation time characterizing the eetabliehmnnt of e9ui.21brium in
such a gas. Comparing it with the time characterizing the viacoua
proceeees of transfer, we show that tho proceG~. of es'Gctbliohing
energ~#e equilibrium in a phonon gee is more rapid than the
viscous processes.
We start with the first case; that is, we examine the ecatter-
ing of small-energy phonons by phonons. In this case it may be
assumed that the momentum p of the phonon under question is con-
siderably smaller than the momentum p1 of the phonon with which
this phonon collides. As wtie shown in Suction 2, such an aeeump-
tion considerably simplifies the exproesion for the sverc~ging-over of
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angles a:V b:i phonon-phonon scattering.
the angularly normalized affective
phonon saattering ie by (2.11):
Let us look at
~n
where tho collision
'-'r' (k) 4
integral J(n) for the process of phonon-phonon
eoattorin~r aquala
bution functions, we transform the expression enaloeod in paren-
n. Now Utilizing certain properties o' distri-
one
equilibrium, ctn4/deviation of the distribution function n from
the equilibrium vIlue no will be regarded as emall and equal to
J C p1+)
3(n) v \- il
(7.1)
We ire interested in the re1ttxtation of smaJ.l-enorgy phonon,
whose distribution function egucilu n, with the given equilibriwn
distribution of the remaining phorionA. Therefore the distri-
bution funationg n1, n' and n1' in (7.1) will be regarded as in
thesis in (7.1)
~ih1 '.n1-1I) (h~-11) -h~-'' ( N,+~)
Substituting the expression obtained in the collision integral,
we have
.5.3 ( 5-) 3
) -(ir) ) cn fl n (Y\) +i) Crfr))
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r:a
- (2.11) for (P'Pi)' vie have
rar r r rr ~ rr
__rrrr rwr
rrrrr.. r r - - _ _ r rr_ r ~ o
rr-rrrrrl war. -a drr la ~r w-r w_r_rrr w r r r t _ cr _ s r
'~~morr~lr rrr
r'e '
in fry
r. r ~ iy~ n
limit
o sower
4Int~erat.on ov ~~ -:;r d with r xar
~, ~~1cQ our
a
1 x1owc,ver ia not mtorical
which
Cdrtyin orrorr, rV u _ r
~y
//~~y~ j mtIlte nature. .r .wr r?_. r r . r +_
11.4 ~r r r _r..___r?.a r
-- 'rrir
/~9~1 7y~ ~ J. Y i7 r . r .. w r - r r r r r r ..
n Are of ~~~ r r _ _ r
w r r
rw r te's ~1_ _wrr rr+~.rlrrrr r r 1e _ r r _w ._ . r _
w rr~~rr w _ yrrr+r_ ~ _ r r r r
or after an elementary integre~tion
(tA.+)(# !
__
,
tM ( h)3 cYr)3 . C ) : (7.6)
Automatic allowance is made for the fact that pMonona
having momnntvm Pi poeaeBa high ener~Y, dints in the lnte-'
ration over dpi 0. subetEtntial role is played b5' phonons with
ener8y of the order of 6 kT.
Let us procedo to the case of relaxation of high energy
phonons. In this case the aesumptlon that the phonon momenta
tion
obey the rela
jM (11'
U
10
Tht~e the relgX@tion 'time for ghanana with ama11 enerP.y te
determined by the rc~lationahip
i) (7.)
In order to simplify the intagratian of (7.4), tat us eub-
stituta the function ni r n' , end n' 1 by the Wien function ~znQ
make allpwe~nae for the S'act that the momenta p ~z1a p~ ~atiefy
the in~yuality p