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Excitation spectrum for interacting bosons high-density calculations
Physica
60 (1972) 488-498
0 North-Holland
EXCITATION
SPECTRUM
HIGH-DENSITY K. M. KHANNA Theoretical
Physics Laboratory,
Publishing
FOR INTERACTING
BOSONS
CALCULATIONS and A. N. PHUKAN
Department
Dibrwgarh,
Co.
Assam,
of Physics,
Dibrugarh
University,
India
Received 20 December 1971
Synopsis The energy excitation spectrum for a system of bosons interacting via a two-body potential composed of a hard repulsive core followed by a square well is obtained using the reactionmatrix formalism. The results obtained bear good qualitative agreement with the experimental results for liquid 4He. It is also concluded that the attractive tail introduced in the two-body potential plays an important role in determining the energy-excitation spectrum for the interacting Bose system.
1. Introductiun. In the attempts towards the understanding of the properties of liquid helium (4He), much emphasis has been laid in recent years on the interacting Bose system in one form or another. While the properties of a low-density Bose system with purely repulsive interactions had been shown to have fairly good qualitative agreement with those pertaining to liquid 4He, yet the question of incorporating an attractive tail in the interaction potential of the hard-sphere boson system has been widely believed to be of much greater importance in so far as an exact prototype of liquid 4He is concerned. This idea centres round the very fact that while repulsive forces are responsible for the occurrence of superfluidity in liquid 4He, the attractive forces are necessary to take account of the fact that the substance is liquid after all. In the last two decades many papers (see refs. 1 to 12) have appeared on the properties of a system of bosons interacting through a purely repulsive two-body potential. In one of our earlier papersI’) the thermodynamic properties of a dilute Bose system of hard spheres had been shown to be in good agreement with the ones pertaining to liquid 4He at temperatures below the il point. In another paper13) we had studied the properties of a system of bosons interacting via a twobody potential composed of a repulsive core followed by an attractive tail. It was shown that the critical temperature (TJ of such an assembly is 2.34 K which is certainly very close to the observed lambda transition temperature TL (= 2.176K) 488
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in liquid 4He. The density of the system studied in ref. 13, was about one tenth of the density of liquid 4He at the 1 point. Encouraged by these achievements it was considered essential to extend the results of ref. 13 to a high-density system and to compare the energy-excitation spectrum of such a system with that of liquid 4He. This is precisely what has been done in our present investigation. Recently, Liu, Liu and Wong14) have calculated the energy excitation spectrum for a Bose system of hard spheres; wherein they have replaced the hard-sphere interaction by a non-hermitian pseudopotential. With this calculation, although Liu et a1.14) could derive an excitation spectrum resembling qualitatively that of liquid 4He, yet the very fact that the interparticle potential was assumed to be purely repulsive makes the calculations of Liu et al. not that realistic as one would expect keeping in mind the characteristics of the actual 4He-4He interatomic potential. This means that one has to incorporate the effects of attractive interactions besides the repulsive core amongst the bosons so that a theoretical model could serve as a prototype of liquid 4He. In fact, Liu et al. have hinted in ref. 14 that, to be more realistic, their calculations need modifications such that some sort of attractive interactions also are taken account of in calculating the energy spectrum. Brown and Coopersmith1s) derived the excitation spectrum microscopically for a realistic interparticle potential using a form of pair hamiltonian. The strong repulsive core is included by using reaction matrix elements in the hamiltonian and the attractive well is taken account of by t-he assumption of a generalised Bose-Einstein condensation, Brown et al. Is) then diagonalized the pair hamiltonian by the thermodynamically equivalent hamiltonian method of Wentzel16). The excitation spectrum obtained by Brown et al. contained the ‘phonon’ and ‘roton’ regions as observed in liquid 4He. In our present calculations, although we have included pair interactions as is natural, no assumptions such as generalized condensation have been made to include the effects of the attractive well. In fact, we have calculated the matrix elements of the reaction matrix by directly substituting the potential given in eq. (1). In that sense, our present work is completely different from what has been done in this direction by Luban5*6) and Brown et ~1.~~); although the qualitative results predicting the properties of interacting bosons and consequently that of liquid 4He are somewhat similar to what the earlier authors5,6p15) have achieved. We have already studied13) the properties of a dilute system of bosons where the two-body potential had a repulsive core followed by an attractive well. In the present study, we have extended the results of ref. 13, to a system having no density restrictions so that we could study the properties of liquid 4He whose density near the 1 point is about ten times larger than the density of the system studied in ref. 13. This means going over to a high-density system, and to study the properties of such a system we have essentially followed the method described in refs. 13 and 14. Since, in a high-density system, the phenomenon of multiple
490
K.M. KHANNA AND A.N. PHUKAN
scattering becomes important, its effects in calculating the scattering matrix has been taken into account in this manuscript. The t matrices for the type of potential assumed by us have been calculated, and using these the excitation spectrum for the high-density interacting Bose system has been derived. The excitation spectrum as obtained by us resembles the experimental energyexcitation spectrum for liquid 4He obtained by Henshaw and Woods17). In fact the so-called ‘phonon’ and ‘roton’ branches are obtained, in good qualitative agreement with the experiment. We have calculated the magnitude of possible depletion of the zero-momentum state as a result of the interaction amongst the bosons using the methods of statistical mechanics. The calculations show that the depletion effect is very small (see sec. 7) and hence our calculations do not take account of this. The earlier attempts to use Brueckner’s theorylss19) in th e p resence of attractive forces have not been successful. In fact Parry and Ter Haa?“) concluded that Brueckner’s theory is invalid in the presence of attractive forces and that it results in a very large depletion of particles from the zero-momentum state. We, however, in our present calculations find that by assuming the interparticle potential to have an attractive tail following the hard core, Brueckner’s theory does give an excitation spectrum comparable to that of liquid 4He and also that the depletion of the zeromomentum state is quite negligible. 2. Theoretical derivations. We consider a system of N helium particles enclosed in a box of volume L? and the interparticle potential is assumed to be of the form, for
r < a,
for
a
for
r > b.
(1)
The hamiltonian of the system in the second-quantized t a&a& + 3
c
Vk,,k2,k3
k4a$d,ak3ak.,?
formalism is
(2)
kl.kZ.k3.k.,
where #ik is the momentum and ‘m’ is the mass of a boson whose creation and destruction operators are ai and a&, respectively. In eq. (2), V&,k,,&& are the matrix elements of the two-body potential. In the case of a dilute system, it is sufficient to introduce the Bogolubov transformation after replacing the creation and destruction operators for the zeromomentum states by a c-number, and derive the ground-state energy and the excitation spectrum by diagonalizing the hamiltonian. But in the case of a highdensity system it is essential to incorporate the effects of multiple scattering. To
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491
take account of this, we follow the method of Brueckner and Sawada18) and the effective hamiltonian for the system can be written as H err.
=
WO
WO
-
1)
~oo,oo
h2k2
+c i[ -2m*
+
NO
@Ok.Ok
+
lOk,kO
-
tOO,OO)
dak
k#O
+
+NO(tk-k,OOdfd
+
tOO.k-kaka-k)
11 ,
where No is the occupation number of the zero-momentum matrix t satisfies the integral equation =
tk,‘k,‘.klk,
vk/k,‘k,‘,k,k,
+
c k,“k2”
Vk/k2*,kl”k2”G
NG)
state and the scattering
tk,“k,“,k,k2*
(4)
The hamiltonian in eq. (3) can be diagonalized by a Bogolubov transformation, 6, = (1 + a.$‘(ak
+ a,&), (3
b_,
= (1 -
+ n,a.Q
a~)-f(a_-k
and by suitably adjusting the parameter 0~~. This method is discussed by Liu et al. 14), who obtained the following expression for the excitation spectrum, &?,,,(k) = f&t,
+ (s)r
-
N;lk-k,OOf.O,k-kr,
V-5)
where14) tk
tOk.Ok
=
+
tOk.kO
-
tOO,OO.
(7)
From eq. (6) it is evident that to calculate E,,,(k), the only task is to solve the t-matrix equation for the potential of the type defined by eq. (1). 3. Solution for the t matrix. With the t matrix defined by eq. (4) the groundstate energy E. is given by13) Eo
=
(8)
+&?oo,oo,
where Q is the particle number density. To solve the integral equation for the t matrix we assume that the motion of the centre of mass can be separated. Using the same notation as in refs. 13 and 18 and writing k = 4 (k, - kJ, k’ = 3 (k; - k;), eq. (4) becomes I,‘,,
=
vk’,k
+
;
‘k’.k
,,G (k”) t,.,,, ,
(9)
K. M. KHANNA AND A.N. PHUKAN
492
where the Green function is G(k”) = -
s
+ 2N(t,,.,,,,
+
t,v, -_k” - b3)
-1 1 *
Transforming the integral equation for t into coordinate space, the matrix elements of t can be written as t (k’, k) = j dr dr’ c#$(Y)t (Y. r’) &(r’),
(11)
where the 4’s are plane waves with relative momentum k and k’. In analogy with the scattering problem, the wave function can be defined as’“)
Ur) ykW = j t (r, r’> 4kV) dr’.
(12)
The elements of the t matrix can be obtained from tk’,k
=
j” dr
d$(r)
v(r)
(13)
yk@).
The wave function satisfies the integral equation y:(r)
=
#i(r) + j dr’ G, (r, r’) V(r’) y,&‘).
(14)
In eq. (14) &(r) is the free-particle wavefunction. y:(r) is calculated by taking into consideration the wave distortion due to the interaction potential, and knowing y:(r) it is easy to calculate the t matrix from eq. (13). Following the method used by Khanna and Bhattacharyya2’) for the calculation of y:(r) and introducing the spherical Bessel function J’ (kr), one can find that in the zeroth approximation, 1
~00.00 = -
(15)
Go (a, a) -
Now according to Brueckner and Sawadai8), we can write, fk = -2
Ik 2: -
c (21 + 1) Js
l(even)
1 G,(a,a)
sin ka ka’
I
+ 1
9
Go
(a,
(16) a)
’
(17)
Also, t OO,k-k
sin ka =
tOO,OO
-&--’
(18)
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SPECTRUM
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Making these substitutions in the expression for the excitation noting that tOO,k--k= lk-,,Oo we get
493
spectrum and
(19) where N Go (a, a)
(x0 a
4. High-density system. To carry out the calculations for a high-density system, it is essential to construct the Green functions for various values of I which are all even in the present case. We have for G (v - r’), G (r - r’) = -
Q
(27?
s
dke ik(p-p’) G(k),
(21)
and the G,‘s can be calculated from G1 (r, r‘) = 3 j :6 P&3) G (r - 8).
(22)
-1
In eq. (22) fl is the cosine of the angle between Y and r’. Following Brueckner19), we assume, G, (a, a) E G,, (a, a).
(23)
With this approximation, G
0
in the ground state, one obtains,
;a {-2&W
+ 1)l-4(fW12 + 11= -
a)
9
’
G,(a,u)
sin ka ka’
Making this replacement in the expression for Go (a, a) and introducing variable x defined by ka = x, we write
Go (a, a) = - -2zo
JPdxsin’
.[x2
-
iz2a)
F]‘,
(24) the
(25)
0
or m
sin2 x
47c2pa3= A2 s
0
x2 + i12(sin X)/X
dx,
(26)
K.M. KHANNA
494
AND
A.N. PHUKAN
whereI*) A2 = -2Na2m*/h2G,,
(a, a).
(27)
Eq. (27) can be solved numerically for A2 in terms of pa3. A graph giving the relation between @a3 and A2 was obtained by Brueckner et al.l*). Making use of this graph, one can get the value of A2 corresponding to different values of pa3 and calculate the appropriate value of Go (a, a). Eq. (27) gives, in the appropriate units -N/Go (a, a) = i22fi2/2m*a2.
(28)
Using, therefore, the value of -N/Go (a, a) from eq. (28) in eq. (20) we obtain:
which, when substituted in eq. (19) gives the energy excitation spectrum in the high-density limit for an interacting Bose system interacting tlia a two-body potential composed of a hard core followed by a square well. The final expression becomes
1
4~eif~fi~ (b - a)” k2 sin ka ’
+ A4L2k2 sin ka ~~2me2a2 ka
3m*
ka’
(30)
5. Numerical calculations. Using eq. (30) we record in table I below, the values of E(k) corresponding to different values of the momentum transfer k. Three different sets of calculations are done corresponding to three different values of TABLE I
k
E(k) in K
k2
(A-‘)
(W-Y
m*jm
= 2.4
m*lm = 2.0
m*/m = 1.6
0.2
0.04
1.35
1.735
2.333
0.4 0.6
0.16
2.47
0.36
3.41
4.32 5.854
0.8
0.64
1.0
1.0 1.44
3.58 3.33
3.218 4.37 4.55
5.370
3.103
3.586 0.89
1.4
1.96
2.74 2.5
1.6
2.56
3.74
2.28 3.79
1.8 2.0
3.24 4.0
2.2
4.84
6.08 8.93 11.99
6.876 10.53 14.38
1.2
6.048
4.14
3.398 7.848 12.74 17.967
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495
m*/m. The values of the other parameters like b, a, V,, are the same as in our earlier calculations in ref. 13. The value of e in the present calculations is e = 2 x 10z2 particles per cm3 and is approximately the density of liquid 4He at its 1 point.
6. Graphs. The results obtained in table I have been graphically represented in fig. 1. The experimental values on liquid 4He as obtained by Henshaw and Woods”) are also shown in the figure.
1816-
121, Y c -s
10 8-
W 6-
0
0.2
0.6
1.0
1.4
kini-’
1.8 =
2.2
2.6
3.0
Fig. 1. Energy-excitation spectrum for interacting bosom with hard core followed by a square well. Curves (I), (2), (3): theoretical; curve (4): theoretical (Liu et ~1.‘~)); curve (5): experimental (Henshaw and Woods”)).
7. Depletion of the ground state. It is always essential to take account of the depletion of the zero-momentum state that occurs in an assembly of interacting particles. In our earlier work13), on the dilute Bose system, the depletion was found to be very small. In fact, it was found that the number density of particles of a dilute assembly of bosons in the ground state changes by hardly less than 15 percent when a two-body potential composed of a repulsive core followed by an attractive well is switched on between the bosons in the ground state. But in our present case, the density of the system is higher and the depletion effect needs proper calculations. To carry this out, we use the same method as was used in
496
K. M. KHANNA AND A. N. PHUKAN
ref. 13, where the number density of particles in the state above the ground state is given by e - e0 21 (2xmkTC/h2)3i2 e-‘likTc,
(31)
where k is the Boltzmann constant, h is the Planck constant and m is the free.. 1. particle mass. e1 is the energy value of the first excited level and T, is the critical temperature of the interacting assembly. T, is given by13) T, 1: To (1 - 0.255 e-E1’kTo),
(32)
where To is the ideal B-E condensation temperature and is equal to 3.14 K. Thus to calculate (Q - eO), one has to know T, and &I. The value of T, can be obtained immediately if we know el. Looking at table I and the graph, we see that for m*lm = 2.4, e1 = 3.47 x lo-l6 erg; which when substituted in eq. (32) gives T, = 2.78 K; and when m*/m = 1.6, E~ = 1.38 x IO-l6 erg and eq. (32) gives T, = 2.55 K. Using eq. (31) we get Q,,/Q= 0.863 for E~ = 3.47 x lo-l6 erg and co/e = 0.796 for e1 = 1.38 x 1Ol6 erg; where Q is the total number density of particles. Thus the depletion effect in our present calculations is very small and hence we do not take it into account in our present investigations. 8. Discussion. Fig. 1 contains the energy excitation spectrum for interacting bosons. The experimentally obtained excitation spectrumi7) for liquid 4He is also shown in the same figure. As can be seen from the figure, the theoretical spectrum bears good qualitative resemblance with the experimental one. The so-called ‘phonon’ and ‘roton’ branches observed in the case of liquid 4He have also been obtained in our theoretical calculations. In particular, the roton minimum occurs in both cases at almost the same value of k. This good, though qualitative, agreement makes it possible to conclude that the type of model for bosons, i.e., interacting via a two-body potential composed of a hard repulsive core followed by a square well, is a fairly realistic model for liquid 4He. Of course, the theoretical energy excitation curve in its actual form does not agree quantitatively with the experimental one. This may be due to the fact that at the numerical value given by us to ‘b’ (= 6.0 A), the two helium atoms should have a deeper attractive well as a part of their interaction and this actually indicates the presence of stronger attractive interaction which evidently lowers the excitation energy. From the table it is seen that as m*/m decreases, the roton minimum gets lower and lower. This means that the energy gap in the excitation spectrum becomes smaller and smaller as m*/m decreases. This result is obviously easily understandable; since a decrease in m*/m values actually means lessening of interactions in the system and it is well known that there can be no energy gap if the system is a non-interacting one. Our result actually speaks of this characteristic of an interacting system. Besides this, we also see from the table that for a certain momentum transfer k, as the m*/m
EXCITATION
SPECTRUM FOR INTERACTING
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497
values increase, the excitation energy decreases. This, at first sight, might seem to be confusing. But since our system is a bound one, hence, although m*/m increases, still this makes the particles more and more repulsively interacting, the particles, in view of the presence of the deep attractive square well, after repulsion, get trapped in the attractive well which results in the lowering of the excitation energy. Thus the attractive tail that we have introduced, following the hard core, plays a very important role of preventing the system from becoming unbound after the switching on of the parturbation. From our calculations on the depletion of the zero-momentum state, it is seen that as T, decreases, eO/e decreases. This means that for the system for which the critical transition temperature is lower, the depletion is a bit greater. This result is amply explained by saying that lowering of the critical temperature means the existence of stronger interactions and stronger interactions result in larger depletion of the zero-momentum state. In our calculations, it is also seen that as m*/m decreases below say 1.6, the E(k) values increase for smaller k values and become complex near the roton minimum. On the other hand as m*/m goes beyond 2.4 the energy gap no doubt increases, but the energy values become complex near the lower k values. Since the basic theorylg) is invalid for complex energy values, we have to limit our calculations to m*/m values in the range from 1.6 to 2.4. Comparing with the energy-excitation spectrum obtained by Liu et ai.14) we see that in our present calculations, although the roton minimum occurs at approximately the same value of k, the energy gap in the present case is smaller than that obtained by Liu et a1.14). This may be due to the part played by the attractive interactions which were completely absent in the work of Liu et ~1.‘~). We have used the reaction matrix method which can handle potentials composed of strong repulsions (hard-core type) followed by an attractive well, unlike the similar work of Brown and Coopersmith15), where the inclusion of the attractive potential is due to the inclusion of pair-interaction terms and to the assumption of a generalised condensation. In our calculations, although we have included pair interactions (and this is natural), no assumptions such as generalised condensation have been made to include the effects of the attractive well. We have calculated the matrix elements of the reaction matrix by directly substituting the potential given in eq. (1). In that sense, our work is evidently completely different from what was done in this direction earlier by Lubanss6) and Brown et a1.15), although the qualitative results, predicting the properties of interacting bosons and consequently that of liquid 4He, are somewhat similar to what these earlier authors5,6,14.‘5) have achieved.
498
K. M. KHANNA
AND A. N. PHUKAN
Acknowledgement. The authors thankfully acknowledge the financial assistance provided by the Indian National Science Academy, New Delhi, to conduct the present research project. Thanks are also due to Professor B. R. Seth, D. SC., F.N.A. for encouragement.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
Huang, K. and Yang, C.N., Phys. Rev. 105 (1957) 767. Lee, T.D., Huang, K. and Yang, C.N., Phys. Rev, 106 (1957) 1135. Lee, T.D. and Yang, C.N., Phys. Rev. 112 (1958) 1419. Lee, T.D. and Yang, C.N., Phys. Rev. 113 (1959) 1405. Luban, M., Phys. Rev. 128 (1962) 965. Luban, M., Ph.D. Thesis, Univ. of Pennsylvania (1962). Isihara, A. and Yee, D.D.H., Phys. Rev. 136A (1966) 618. Luban, M. and Grobman, W.D., Phys. Rev. Letters 17 (1966) 182. Nisteruk, C. J. and Isihara, A., Phys. Rev. 154 (1967) 150. Gable, D.F. and Trainor, L.E.H., Canad. J. Phys. 46 (1968) 839. Khanna, K.M., J. Phys. Sot. Japan 27 (1969) 1093. Khanna, K.M. and Phukan, A.N., Indian J. pure appl. Phys. 9 (1971) 156. Khanna, K.M. and Phukan, A.N., Physica 58 (1972) 263. Liu, L., Liu, L.S. and Wong, K-W., Phys. Rev. 135A (1964) 1166. Brown, G.V. and Coopersmith, M.H., Phys. Rev. 178 (1969) 327. Wentzel, G., Phys. Rev. 120 (1960) 1572. Henshaw, D.G. and Woods, A.D.B., Phys. Rev. 121 (1961) 1266. Brueckner, K.A. and Sawada, K., Phys. Rev. 106 (1957) 1117, 1128. Brueckner, K.A., The Many Body Problem, John Wiley and Sons, Inc. (New York, 1959). Parry, W.E. and Ter Haar, D., Ann. Physics 19 (1962) 496. Khanna, K.M. and Bhattacharyya, P., Indian J. pure appl. Phys. 5 (1967) 253.
60 (1972) 488-498
0 North-Holland
EXCITATION
SPECTRUM
HIGH-DENSITY K. M. KHANNA Theoretical
Physics Laboratory,
Publishing
FOR INTERACTING
BOSONS
CALCULATIONS and A. N. PHUKAN
Department
Dibrwgarh,
Co.
Assam,
of Physics,
Dibrugarh
University,
India
Received 20 December 1971
Synopsis The energy excitation spectrum for a system of bosons interacting via a two-body potential composed of a hard repulsive core followed by a square well is obtained using the reactionmatrix formalism. The results obtained bear good qualitative agreement with the experimental results for liquid 4He. It is also concluded that the attractive tail introduced in the two-body potential plays an important role in determining the energy-excitation spectrum for the interacting Bose system.
1. Introductiun. In the attempts towards the understanding of the properties of liquid helium (4He), much emphasis has been laid in recent years on the interacting Bose system in one form or another. While the properties of a low-density Bose system with purely repulsive interactions had been shown to have fairly good qualitative agreement with those pertaining to liquid 4He, yet the question of incorporating an attractive tail in the interaction potential of the hard-sphere boson system has been widely believed to be of much greater importance in so far as an exact prototype of liquid 4He is concerned. This idea centres round the very fact that while repulsive forces are responsible for the occurrence of superfluidity in liquid 4He, the attractive forces are necessary to take account of the fact that the substance is liquid after all. In the last two decades many papers (see refs. 1 to 12) have appeared on the properties of a system of bosons interacting through a purely repulsive two-body potential. In one of our earlier papersI’) the thermodynamic properties of a dilute Bose system of hard spheres had been shown to be in good agreement with the ones pertaining to liquid 4He at temperatures below the il point. In another paper13) we had studied the properties of a system of bosons interacting via a twobody potential composed of a repulsive core followed by an attractive tail. It was shown that the critical temperature (TJ of such an assembly is 2.34 K which is certainly very close to the observed lambda transition temperature TL (= 2.176K) 488
EXCITATION
SPECTRUM
FOR INTERACTING
BOSONS
489
in liquid 4He. The density of the system studied in ref. 13, was about one tenth of the density of liquid 4He at the 1 point. Encouraged by these achievements it was considered essential to extend the results of ref. 13 to a high-density system and to compare the energy-excitation spectrum of such a system with that of liquid 4He. This is precisely what has been done in our present investigation. Recently, Liu, Liu and Wong14) have calculated the energy excitation spectrum for a Bose system of hard spheres; wherein they have replaced the hard-sphere interaction by a non-hermitian pseudopotential. With this calculation, although Liu et a1.14) could derive an excitation spectrum resembling qualitatively that of liquid 4He, yet the very fact that the interparticle potential was assumed to be purely repulsive makes the calculations of Liu et al. not that realistic as one would expect keeping in mind the characteristics of the actual 4He-4He interatomic potential. This means that one has to incorporate the effects of attractive interactions besides the repulsive core amongst the bosons so that a theoretical model could serve as a prototype of liquid 4He. In fact, Liu et al. have hinted in ref. 14 that, to be more realistic, their calculations need modifications such that some sort of attractive interactions also are taken account of in calculating the energy spectrum. Brown and Coopersmith1s) derived the excitation spectrum microscopically for a realistic interparticle potential using a form of pair hamiltonian. The strong repulsive core is included by using reaction matrix elements in the hamiltonian and the attractive well is taken account of by t-he assumption of a generalised Bose-Einstein condensation, Brown et al. Is) then diagonalized the pair hamiltonian by the thermodynamically equivalent hamiltonian method of Wentzel16). The excitation spectrum obtained by Brown et al. contained the ‘phonon’ and ‘roton’ regions as observed in liquid 4He. In our present calculations, although we have included pair interactions as is natural, no assumptions such as generalized condensation have been made to include the effects of the attractive well. In fact, we have calculated the matrix elements of the reaction matrix by directly substituting the potential given in eq. (1). In that sense, our present work is completely different from what has been done in this direction by Luban5*6) and Brown et ~1.~~); although the qualitative results predicting the properties of interacting bosons and consequently that of liquid 4He are somewhat similar to what the earlier authors5,6p15) have achieved. We have already studied13) the properties of a dilute system of bosons where the two-body potential had a repulsive core followed by an attractive well. In the present study, we have extended the results of ref. 13, to a system having no density restrictions so that we could study the properties of liquid 4He whose density near the 1 point is about ten times larger than the density of the system studied in ref. 13. This means going over to a high-density system, and to study the properties of such a system we have essentially followed the method described in refs. 13 and 14. Since, in a high-density system, the phenomenon of multiple
490
K.M. KHANNA AND A.N. PHUKAN
scattering becomes important, its effects in calculating the scattering matrix has been taken into account in this manuscript. The t matrices for the type of potential assumed by us have been calculated, and using these the excitation spectrum for the high-density interacting Bose system has been derived. The excitation spectrum as obtained by us resembles the experimental energyexcitation spectrum for liquid 4He obtained by Henshaw and Woods17). In fact the so-called ‘phonon’ and ‘roton’ branches are obtained, in good qualitative agreement with the experiment. We have calculated the magnitude of possible depletion of the zero-momentum state as a result of the interaction amongst the bosons using the methods of statistical mechanics. The calculations show that the depletion effect is very small (see sec. 7) and hence our calculations do not take account of this. The earlier attempts to use Brueckner’s theorylss19) in th e p resence of attractive forces have not been successful. In fact Parry and Ter Haa?“) concluded that Brueckner’s theory is invalid in the presence of attractive forces and that it results in a very large depletion of particles from the zero-momentum state. We, however, in our present calculations find that by assuming the interparticle potential to have an attractive tail following the hard core, Brueckner’s theory does give an excitation spectrum comparable to that of liquid 4He and also that the depletion of the zeromomentum state is quite negligible. 2. Theoretical derivations. We consider a system of N helium particles enclosed in a box of volume L? and the interparticle potential is assumed to be of the form, for
r < a,
for
a
for
r > b.
(1)
The hamiltonian of the system in the second-quantized t a&a& + 3
c
Vk,,k2,k3
k4a$d,ak3ak.,?
formalism is
(2)
kl.kZ.k3.k.,
where #ik is the momentum and ‘m’ is the mass of a boson whose creation and destruction operators are ai and a&, respectively. In eq. (2), V&,k,,&& are the matrix elements of the two-body potential. In the case of a dilute system, it is sufficient to introduce the Bogolubov transformation after replacing the creation and destruction operators for the zeromomentum states by a c-number, and derive the ground-state energy and the excitation spectrum by diagonalizing the hamiltonian. But in the case of a highdensity system it is essential to incorporate the effects of multiple scattering. To
EXCITATION
SPECTRUM
FOR INTERACTING
BOSONS
491
take account of this, we follow the method of Brueckner and Sawada18) and the effective hamiltonian for the system can be written as H err.
=
WO
WO
-
1)
~oo,oo
h2k2
+c i[ -2m*
+
NO
@Ok.Ok
+
lOk,kO
-
tOO,OO)
dak
k#O
+
+NO(tk-k,OOdfd
+
tOO.k-kaka-k)
11 ,
where No is the occupation number of the zero-momentum matrix t satisfies the integral equation =
tk,‘k,‘.klk,
vk/k,‘k,‘,k,k,
+
c k,“k2”
Vk/k2*,kl”k2”G
NG)
state and the scattering
tk,“k,“,k,k2*
(4)
The hamiltonian in eq. (3) can be diagonalized by a Bogolubov transformation, 6, = (1 + a.$‘(ak
+ a,&), (3
b_,
= (1 -
+ n,a.Q
a~)-f(a_-k
and by suitably adjusting the parameter 0~~. This method is discussed by Liu et al. 14), who obtained the following expression for the excitation spectrum, &?,,,(k) = f&t,
+ (s)r
-
N;lk-k,OOf.O,k-kr,
V-5)
where14) tk
tOk.Ok
=
+
tOk.kO
-
tOO,OO.
(7)
From eq. (6) it is evident that to calculate E,,,(k), the only task is to solve the t-matrix equation for the potential of the type defined by eq. (1). 3. Solution for the t matrix. With the t matrix defined by eq. (4) the groundstate energy E. is given by13) Eo
=
(8)
+&?oo,oo,
where Q is the particle number density. To solve the integral equation for the t matrix we assume that the motion of the centre of mass can be separated. Using the same notation as in refs. 13 and 18 and writing k = 4 (k, - kJ, k’ = 3 (k; - k;), eq. (4) becomes I,‘,,
=
vk’,k
+
;
‘k’.k
,,G (k”) t,.,,, ,
(9)
K. M. KHANNA AND A.N. PHUKAN
492
where the Green function is G(k”) = -
s
+ 2N(t,,.,,,,
+
t,v, -_k” - b3)
-1 1 *
Transforming the integral equation for t into coordinate space, the matrix elements of t can be written as t (k’, k) = j dr dr’ c#$(Y)t (Y. r’) &(r’),
(11)
where the 4’s are plane waves with relative momentum k and k’. In analogy with the scattering problem, the wave function can be defined as’“)
Ur) ykW = j t (r, r’> 4kV) dr’.
(12)
The elements of the t matrix can be obtained from tk’,k
=
j” dr
d$(r)
v(r)
(13)
yk@).
The wave function satisfies the integral equation y:(r)
=
#i(r) + j dr’ G, (r, r’) V(r’) y,&‘).
(14)
In eq. (14) &(r) is the free-particle wavefunction. y:(r) is calculated by taking into consideration the wave distortion due to the interaction potential, and knowing y:(r) it is easy to calculate the t matrix from eq. (13). Following the method used by Khanna and Bhattacharyya2’) for the calculation of y:(r) and introducing the spherical Bessel function J’ (kr), one can find that in the zeroth approximation, 1
~00.00 = -
(15)
Go (a, a) -
Now according to Brueckner and Sawadai8), we can write, fk = -2
Ik 2: -
c (21 + 1) Js
l(even)
1 G,(a,a)
sin ka ka’
I
+ 1
9
Go
(a,
(16) a)
’
(17)
Also, t OO,k-k
sin ka =
tOO,OO
-&--’
(18)
EXCITATION
SPECTRUM
FOR INTERACTING
BOSONS
Making these substitutions in the expression for the excitation noting that tOO,k--k= lk-,,Oo we get
493
spectrum and
(19) where N Go (a, a)
(x0 a
4. High-density system. To carry out the calculations for a high-density system, it is essential to construct the Green functions for various values of I which are all even in the present case. We have for G (v - r’), G (r - r’) = -
Q
(27?
s
dke ik(p-p’) G(k),
(21)
and the G,‘s can be calculated from G1 (r, r‘) = 3 j :6 P&3) G (r - 8).
(22)
-1
In eq. (22) fl is the cosine of the angle between Y and r’. Following Brueckner19), we assume, G, (a, a) E G,, (a, a).
(23)
With this approximation, G
0
in the ground state, one obtains,
;a {-2&W
+ 1)l-4(fW12 + 11= -
a)
9
’
G,(a,u)
sin ka ka’
Making this replacement in the expression for Go (a, a) and introducing variable x defined by ka = x, we write
Go (a, a) = - -2zo
JPdxsin’
.[x2
-
iz2a)
F]‘,
(24) the
(25)
0
or m
sin2 x
47c2pa3= A2 s
0
x2 + i12(sin X)/X
dx,
(26)
K.M. KHANNA
494
AND
A.N. PHUKAN
whereI*) A2 = -2Na2m*/h2G,,
(a, a).
(27)
Eq. (27) can be solved numerically for A2 in terms of pa3. A graph giving the relation between @a3 and A2 was obtained by Brueckner et al.l*). Making use of this graph, one can get the value of A2 corresponding to different values of pa3 and calculate the appropriate value of Go (a, a). Eq. (27) gives, in the appropriate units -N/Go (a, a) = i22fi2/2m*a2.
(28)
Using, therefore, the value of -N/Go (a, a) from eq. (28) in eq. (20) we obtain:
which, when substituted in eq. (19) gives the energy excitation spectrum in the high-density limit for an interacting Bose system interacting tlia a two-body potential composed of a hard core followed by a square well. The final expression becomes
1
4~eif~fi~ (b - a)” k2 sin ka ’
+ A4L2k2 sin ka ~~2me2a2 ka
3m*
ka’
(30)
5. Numerical calculations. Using eq. (30) we record in table I below, the values of E(k) corresponding to different values of the momentum transfer k. Three different sets of calculations are done corresponding to three different values of TABLE I
k
E(k) in K
k2
(A-‘)
(W-Y
m*jm
= 2.4
m*lm = 2.0
m*/m = 1.6
0.2
0.04
1.35
1.735
2.333
0.4 0.6
0.16
2.47
0.36
3.41
4.32 5.854
0.8
0.64
1.0
1.0 1.44
3.58 3.33
3.218 4.37 4.55
5.370
3.103
3.586 0.89
1.4
1.96
2.74 2.5
1.6
2.56
3.74
2.28 3.79
1.8 2.0
3.24 4.0
2.2
4.84
6.08 8.93 11.99
6.876 10.53 14.38
1.2
6.048
4.14
3.398 7.848 12.74 17.967
EXCITATION
SPECTRUM
FOR INTERACTING
BOSONS
495
m*/m. The values of the other parameters like b, a, V,, are the same as in our earlier calculations in ref. 13. The value of e in the present calculations is e = 2 x 10z2 particles per cm3 and is approximately the density of liquid 4He at its 1 point.
6. Graphs. The results obtained in table I have been graphically represented in fig. 1. The experimental values on liquid 4He as obtained by Henshaw and Woods”) are also shown in the figure.
1816-
121, Y c -s
10 8-
W 6-
0
0.2
0.6
1.0
1.4
kini-’
1.8 =
2.2
2.6
3.0
Fig. 1. Energy-excitation spectrum for interacting bosom with hard core followed by a square well. Curves (I), (2), (3): theoretical; curve (4): theoretical (Liu et ~1.‘~)); curve (5): experimental (Henshaw and Woods”)).
7. Depletion of the ground state. It is always essential to take account of the depletion of the zero-momentum state that occurs in an assembly of interacting particles. In our earlier work13), on the dilute Bose system, the depletion was found to be very small. In fact, it was found that the number density of particles of a dilute assembly of bosons in the ground state changes by hardly less than 15 percent when a two-body potential composed of a repulsive core followed by an attractive well is switched on between the bosons in the ground state. But in our present case, the density of the system is higher and the depletion effect needs proper calculations. To carry this out, we use the same method as was used in
496
K. M. KHANNA AND A. N. PHUKAN
ref. 13, where the number density of particles in the state above the ground state is given by e - e0 21 (2xmkTC/h2)3i2 e-‘likTc,
(31)
where k is the Boltzmann constant, h is the Planck constant and m is the free.. 1. particle mass. e1 is the energy value of the first excited level and T, is the critical temperature of the interacting assembly. T, is given by13) T, 1: To (1 - 0.255 e-E1’kTo),
(32)
where To is the ideal B-E condensation temperature and is equal to 3.14 K. Thus to calculate (Q - eO), one has to know T, and &I. The value of T, can be obtained immediately if we know el. Looking at table I and the graph, we see that for m*lm = 2.4, e1 = 3.47 x lo-l6 erg; which when substituted in eq. (32) gives T, = 2.78 K; and when m*/m = 1.6, E~ = 1.38 x IO-l6 erg and eq. (32) gives T, = 2.55 K. Using eq. (31) we get Q,,/Q= 0.863 for E~ = 3.47 x lo-l6 erg and co/e = 0.796 for e1 = 1.38 x 1Ol6 erg; where Q is the total number density of particles. Thus the depletion effect in our present calculations is very small and hence we do not take it into account in our present investigations. 8. Discussion. Fig. 1 contains the energy excitation spectrum for interacting bosons. The experimentally obtained excitation spectrumi7) for liquid 4He is also shown in the same figure. As can be seen from the figure, the theoretical spectrum bears good qualitative resemblance with the experimental one. The so-called ‘phonon’ and ‘roton’ branches observed in the case of liquid 4He have also been obtained in our theoretical calculations. In particular, the roton minimum occurs in both cases at almost the same value of k. This good, though qualitative, agreement makes it possible to conclude that the type of model for bosons, i.e., interacting via a two-body potential composed of a hard repulsive core followed by a square well, is a fairly realistic model for liquid 4He. Of course, the theoretical energy excitation curve in its actual form does not agree quantitatively with the experimental one. This may be due to the fact that at the numerical value given by us to ‘b’ (= 6.0 A), the two helium atoms should have a deeper attractive well as a part of their interaction and this actually indicates the presence of stronger attractive interaction which evidently lowers the excitation energy. From the table it is seen that as m*/m decreases, the roton minimum gets lower and lower. This means that the energy gap in the excitation spectrum becomes smaller and smaller as m*/m decreases. This result is obviously easily understandable; since a decrease in m*/m values actually means lessening of interactions in the system and it is well known that there can be no energy gap if the system is a non-interacting one. Our result actually speaks of this characteristic of an interacting system. Besides this, we also see from the table that for a certain momentum transfer k, as the m*/m
EXCITATION
SPECTRUM FOR INTERACTING
BOSONS
497
values increase, the excitation energy decreases. This, at first sight, might seem to be confusing. But since our system is a bound one, hence, although m*/m increases, still this makes the particles more and more repulsively interacting, the particles, in view of the presence of the deep attractive square well, after repulsion, get trapped in the attractive well which results in the lowering of the excitation energy. Thus the attractive tail that we have introduced, following the hard core, plays a very important role of preventing the system from becoming unbound after the switching on of the parturbation. From our calculations on the depletion of the zero-momentum state, it is seen that as T, decreases, eO/e decreases. This means that for the system for which the critical transition temperature is lower, the depletion is a bit greater. This result is amply explained by saying that lowering of the critical temperature means the existence of stronger interactions and stronger interactions result in larger depletion of the zero-momentum state. In our calculations, it is also seen that as m*/m decreases below say 1.6, the E(k) values increase for smaller k values and become complex near the roton minimum. On the other hand as m*/m goes beyond 2.4 the energy gap no doubt increases, but the energy values become complex near the lower k values. Since the basic theorylg) is invalid for complex energy values, we have to limit our calculations to m*/m values in the range from 1.6 to 2.4. Comparing with the energy-excitation spectrum obtained by Liu et ai.14) we see that in our present calculations, although the roton minimum occurs at approximately the same value of k, the energy gap in the present case is smaller than that obtained by Liu et a1.14). This may be due to the part played by the attractive interactions which were completely absent in the work of Liu et ~1.‘~). We have used the reaction matrix method which can handle potentials composed of strong repulsions (hard-core type) followed by an attractive well, unlike the similar work of Brown and Coopersmith15), where the inclusion of the attractive potential is due to the inclusion of pair-interaction terms and to the assumption of a generalised condensation. In our calculations, although we have included pair interactions (and this is natural), no assumptions such as generalised condensation have been made to include the effects of the attractive well. We have calculated the matrix elements of the reaction matrix by directly substituting the potential given in eq. (1). In that sense, our work is evidently completely different from what was done in this direction earlier by Lubanss6) and Brown et a1.15), although the qualitative results, predicting the properties of interacting bosons and consequently that of liquid 4He, are somewhat similar to what these earlier authors5,6,14.‘5) have achieved.
498
K. M. KHANNA
AND A. N. PHUKAN
Acknowledgement. The authors thankfully acknowledge the financial assistance provided by the Indian National Science Academy, New Delhi, to conduct the present research project. Thanks are also due to Professor B. R. Seth, D. SC., F.N.A. for encouragement.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
Huang, K. and Yang, C.N., Phys. Rev. 105 (1957) 767. Lee, T.D., Huang, K. and Yang, C.N., Phys. Rev, 106 (1957) 1135. Lee, T.D. and Yang, C.N., Phys. Rev. 112 (1958) 1419. Lee, T.D. and Yang, C.N., Phys. Rev. 113 (1959) 1405. Luban, M., Phys. Rev. 128 (1962) 965. Luban, M., Ph.D. Thesis, Univ. of Pennsylvania (1962). Isihara, A. and Yee, D.D.H., Phys. Rev. 136A (1966) 618. Luban, M. and Grobman, W.D., Phys. Rev. Letters 17 (1966) 182. Nisteruk, C. J. and Isihara, A., Phys. Rev. 154 (1967) 150. Gable, D.F. and Trainor, L.E.H., Canad. J. Phys. 46 (1968) 839. Khanna, K.M., J. Phys. Sot. Japan 27 (1969) 1093. Khanna, K.M. and Phukan, A.N., Indian J. pure appl. Phys. 9 (1971) 156. Khanna, K.M. and Phukan, A.N., Physica 58 (1972) 263. Liu, L., Liu, L.S. and Wong, K-W., Phys. Rev. 135A (1964) 1166. Brown, G.V. and Coopersmith, M.H., Phys. Rev. 178 (1969) 327. Wentzel, G., Phys. Rev. 120 (1960) 1572. Henshaw, D.G. and Woods, A.D.B., Phys. Rev. 121 (1961) 1266. Brueckner, K.A. and Sawada, K., Phys. Rev. 106 (1957) 1117, 1128. Brueckner, K.A., The Many Body Problem, John Wiley and Sons, Inc. (New York, 1959). Parry, W.E. and Ter Haar, D., Ann. Physics 19 (1962) 496. Khanna, K.M. and Bhattacharyya, P., Indian J. pure appl. Phys. 5 (1967) 253.