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A study of variational wave functions for the alpha particle
Nuclear Physics 42 (1968) 9-47--253; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
A S T U D Y OF V A R I A T I O N A L WAVE F U N C T I O N S FOR T H E A L P H A PARTICLE E. W. SCHMID t
Oak Ridge National Laboratory and
Florida State University, Tallahassee, Florida Y. C. TANG tt and R. C. H E R N D O N t
Florida State University, Tallahassee, Florida tt Received 12 N o v e m b e r 1962 Abstract: The upper and lower bound for the energy of the alpha particle have been calculated with several different trial wave functions. T h e best result for both bounds is obtained with a function which is a product of solutions o f the two-nucleon Schr6dinger equation up to a certain internucleon separation and goes over into a variational function for larger distances. With this trial function the results for the energy and the r.m.s, radius agree quite well with that obtained by Mang and Wild in an independent-pair model calculation.
1. Introduction In a previous investigation 1, 2), an upper and lower bound for the energy o f the alpha particle was calculated by a variational method for several different twonucleon potentials. For each o f these potentials, the optimum wave function turned out to have a form which, for small inter-nucleon distances, is quite similar to a solution o f the two-nucleon Schr6dinger equation at zero energy. This result was in agreement with a suggestion o f Austern and Iano 3), that one should use as a variational function the zero-energy two-nucleon solution up to a certain inter-nucleon separation and a trial function at larger distances. The purpose o f this work is two-fold. Firstly, we wish to study in somewhat more detail the behaviour o f variational wave functions at small inter-particle distances and to see whether the variational bounds for the alpha particle can be improved by following the suggestion o f Austern and Iano directly. Secondly, our investigation can test the accuracy o f the result o f Mang and Wild *) who have calculated the alpha-particle binding energy in the independent-pair approximation. For the latter purpose, we use the same square-well potential with a hard-core as Mang and Wild. t Present address: Max-Planck-lnstitut fiJr Physik und Astrophysik, Munich, Germany. tt Present address: Brookhaven National Laboratory, Upton, Long Island, N e w York. t Present address: University of California, Lawrence Radiation Laboratory, Livermore, California. tt This work was supported by the U.S. Office o f Naval Research. 247
248
t.w.
SCHMID e t a L
To calculate the upper bound of the alpha particle ground-state energy we use the usual Rayleigh-Ritz variational principle, and for the lower bound we use a method given by Temple 5). The expectation values of the various operators involved are calculated by a Monte-Carlo method. As this method has been discussed in previous papers 1,2, 6), we shall not give any description here. 2. T e s t o f Different Trial W a v e Functions
The Hamiltonian for the alpha particle has the form h2
H=----
4
4
e2
~ V 2 + >~ V ( i , k ) + - - - . 2m l=t i k=l rt2
(l)
For the two-nucleon interaction we choose the potential which was used by Mang and Wild a), i.e., V(i,k)=
00,
rik ~
-Vo, 0,
r c < rik < b, b < rtk,
rc ,
(2)
with rc = 0.4 fm, b = 1.9 fm and Vo = 49.60 MeV. This potential is an average of singlet and triplet interaction and, therefore, has no spin dependence. For the upper bound Eu of the eigenvalue Eo of the Hamiltonian, we use the Rayleigh-Ritz variational principle, i.e., E o < Eu = ( H ) .
(3)
For the lower bound EL we use the method of Temple 5) which gives the formula Eo ~
EL =
(H)-
(H2)-(H)2., El - ( n )
(4)
with the condition that ( H ) is less than El, the energy of the first excited state. Clearly, to insure a rigorous lower bound for the energy of the alpha particle, one should use for Et the lower bound to the ground-state energy of the triton; however, since this last value has not been calculated, we shall choose Et as - 10 MeV, which is probably a safe choice in view of the fact that Mang and Wild obtained a value of - 7.1 MeV for the energy of the triton with the same potential as is used in this investigation. As was discussed in an earlier paper 1), the eigenvalue is usually much closer to the upper bound than to the lower bound and it is convenient to introduce the ratio -
Eo- EL
(5)
Eu - Eo " F r o m a test calculation with the deuteron 1) we concluded that t/ is around 20 for the nuclear case.
VARIATIONAL
WAVE
FUNCTIONS
249
Our trial wave functions are all of the form 7j = u(r)~(a,z),
(6)
where u(r) is the space part of the wave function and ~(a, z) is the normalized chargespin function chosen to give S = 0 and T = 0. Three types of trial wave functions are used for u(r). The first type consists of a long-range part of Gaussian or exponential shape multiplied by a cut-off function which fulfills the boundary condition at the surface of the hard-core and which is capable of representing the exact solution of the two-body Schr6dinger equation in the range of small distances, i.e., 4
ut(r)=exp(-~ai
4
i>k=l
i>k=l
4
u2(r) = e x p ( - - a 2 Z
4.
(8)
r,k) I-I fz(r,k),
i>k=l u3(r) = [exp(--~a3
(7)
~, r E ) l - I fl(r'k),
i>k=l
4. E r2)+c3exp(-~b3 i>k=t
4 ~'~ i>k=l
4
r2a)] [ I f3(r,a),
(9)
i>k=I
with
L(r) =
0,
0 < r < r e,
A. sin k . ( r - r c ) ,
r e < r < d.,
(lo)
r
1,
dn < r,
(I 1)
k n = x / m ( V o + E.)lh 2.
The distance d. is the value of r when (d./r) sin k . ( r - r e ) reaches its first maximum which is normalized to unity by the factor A.. Either d. or the related quantity E. can be considered as variational parameter. The second type of trial wave function is chosen according to the suggestion of Austern and Iano a). A product of twonucleon correlation functions is used, each one being a solution of the two-body Schri~dinger equation up to a certain distance b 4 and then going over into another variational function for larger distances, i.e., 4
u4 =
I-I f4(r,k),
(12)
i>k=l
with O,
0 < r <
r c,
1
f,(r)
rSin k 4 ( r - G ) ,
rc <
r < b4,
A4 [exp ( - a 4 r ) - - B 4 exp (--c4r)] ,
b 4 < r,
=
r
(13)
250
E.W. SCHMID e t a L
(14)
k4 = x/m(Vo + E,)/h 2.
The constants A4 and B, are determined by the continuity conditions at r = b4. Austern and Iano found from the deuteron case that the energy parameter E4 is not a very sensitive parameter and can be put equal to zero. We shall also vary E4 to examine if this holds true in the alpha particle case, too. The third type of trial function is obtained by multiplying u4(r) with a product of short-range three-nucleon correlation functions. It has the form 4
u,(r) = ua(r) FI {i +e, exp [--dz(r,k+r,,--2rc)]}.
(15)
i, lt, l = l k.l;ei k>l
When e5 is positive, the three-nucleon correlation function increases the probability of finding a third particle nearby when two particles are already close together. This increase seems reasonable, since in this particular situation a deeper potential is felt by the third particle. TABLE 1 C o m p a r i s o n o f different trial wave functions Upper b o u n d
Trial function
Eu (MeV)
Lower b o u n d
r.m.s, radius (fm)
E L (MeV)
r.m.s. radius (fm) 1.46
ux
--21.8 -t-0.8
1.44
--125 5:14
ut
--22.1 5:0.9
1.56
--162 ~ 2 4
1.95
us
--24.0 ± 0 . 9
1.47
--97
1.51
u,
--24.93--'0.29
1.53
--40.7 ±
2.7
1.80
u6
--24.93___0.29
1.53
--40.7 :k 2.7
1.80
d::lO
Table 1 gives the results for the variation of the parameters a,, bn, c~, E,, d 5 and e 5. The optimum values for the parameters are at = 0.62 fm -2, Et = - 3 2 . 2 MeV, a2 = 0.38 fm - l , E 2 = - 4 0 . 7 MeV, a3 = 1.03 fm -2, b3 = 0.47 fm -2, c3 = 0.25, E3 = - 3 3 . 2 MeV, a4 = 0.28 fm -1, b4 = 1.9 fm, c4 = 0.396 fm-1 and E , = - 7 . 1 MeV for the upper bound and at = 0.62 fm-2, E, = --39.7 MeV, a2 = 0.29 fm-1, E2 = - 3 9 . 7 MeV, a 3 = 0.90fm-2, b3 = 0.437 fm-2, c3 = 0.25, E 3 = --38.5 MeV, a4 = 0.2 fm - t , b4 = 1.9 fm, c4 -- 0.201 fm - t and E4 = - 8 . 1 MeV for the lower bound. No improvement in either bound is obtained by varying the parameters in the short-range three-nucleon correlation function. It is interesting to examine the values of the optimum parameters in the trial function u, which gives the best results. The separation distance b4 was used by Austern and lano 3) to separate the region near the hard-core where two-nucleon features are dominant from the region of larger distances where many-body effects
VARIATIONAL WAVE FUNCTIONS
251
are more important. In our example, b4 turns out to be rather large, namely, equal to the range of the square-well potential. In one respect, this is not surprising, since at this distance the discontinuity in the potential should cause a discontinuity in the second derivative of the wave function, which can only be achieved with u4 by choosing b4 = b. On the other hand, one might think that using the solution of the twonucleon Schr6dinger equation up to such a large distance might take too little account of the influence of the third and fourth particle in this nucleus. Therefore, we have also tried with the function us which has more flexibility towards three-nucleon effects. It is found, however, that the improvement obtained with this function is less than 10 % of the statistical uncertainty present in the result with u4 and hence, is meaningless. Although the function u s contains only a very special type of three-nucleon correlation, this result seems to indicate that the independent-pair approximation is a good one in the alpha particle case, and that the function u4 already contains the correct features of three-nucleon correlations. We have also varied the function u4 for other values of b4 and obtained for instance, with b4 = 1.25 fm an upper bound which is worse by 0.7 MeV and a lower bound which is worse by 66 MeV. For both bounds, the parameterE4 turns out to be close to zero in this case. Using the argument of the effective-range theory, Austern and Iano predicted that E4 is not a sensitive parameter and a value of zero is a good choice. At b4 = 1.90 fm, we find by varying the other parameters that with E4 = 0, the upper bound and the lower bound become worse by about 2 MeV and 50 MeV, respectively. It seems to us, therefore, that in the alpha-particle case, the parameter E4 needs to be varied to obtain better results. Even with the best trial function u4, there is still a gap of about 16 MeV between the upper and the lower bound. Since, in principle, the eigenvalue can be anywhere within this gap, it would seem that the lower bound we obtained is too small to be of much practical use. Fortunately, however, we have learned from some test cases 1) that the quantity ?/defined by eq. (5) is usually much larger than unity. This already reduces considerably the region within which we expect the eigenvalue to be. If one has some more information about the insufficiency of a trial function, further conclusions about the magnitude of r/ can be drawn in the following way. The upper bound is insensitive to a rather big deviation of the variational function from the eigenfunction if it occurs only in a small part of the integration volume, as, for instance, only for small inter-nucleon distances. On the other hand, the lower bound, being related to the mean square deviation of H~b from Eo~b over the entire integration volume, is determined by the region where the variational function has its largest deviation from the eigenfunction and can be badly estimated even though this region may only be a rather small part of the integration volume. This explains why the lower bound is most sensitive to the behaviour of the wave function near the hardcore. When the trial function is a bad approximation to the actual eigenfunction in this region, a value of r / u p to 100 is possible, whereas if the function deviates only
252
E.w.
SCHMID e t al.
slightly from the eigenfunction over the whole integration volume, ~/values as small as 5 can be obtained. By comparing the upper and the lower bounds obtained with the trial functions ut, u2 and u 3 with those obtained with u4, we expect the magnitude of r/ for these first three functions to exceed 20. This is in agreement with the statement above, since these functions do not have a discontinuity in the second derivative at the edge of the square well, which the eigenfunction certainly has. Therefore they will have a rather large deviation from the eigenfunction in a relatively small region. The function u 2, in addition, cannot go over into a two-nucleon solution near the hardcore. The function u 4 does alllow for a discontinuity in the second derivative, although due to its rather restricted form for r > b, only negative values can be obtained in the region immediately outside of the range of the square well. Obviously, u4 also has the right behaviour near the hard-core. Therefore, it is possible that ~/is smaller than 20 in this case but we do not believe that it is smaller than 5. With 5 < ~/ < 50, we can then estimate E0 to be in the region Eo = -26.5-t- 1.5 MeV. This is in agreement with the value obtained by Mang and Wild from an independentpair-model calculation i.e., E 0 = - 2 7 . 6 8 MeV. The r.m.s, radius of 1.53 fm at the upper bound for the function u4 also agrees quite well with the result of Mang and Wild 7) of 1.51 fm. Our potential is an average over singlet and triplet interaction. In a more realistic calculation these interactions should have different strengths and the trial function should allow for an asymmetric part which arises from this feature of the potential. In this investigation we have estimated what improvement the inclusion of an asymmetric spatial wave function would make in the result for the energy. Using a potential of the same average strength as that given by eq. (2), but with a singlet to triplet ratio of 0.8, we get with the trial function u4 the same upper bound, but a lower bound which is worse by about 2 MeV. That means it would be possible to gain 2 MeV in the lower bound by allowing the trial wave function to have an asymmetric spatial component. Assuming r/to be around 10, this added degree of freedom in the trial function would improve the upper bound by only about 0.2 MeV, which is not very significant compared to its value of - 2 4 . 9 3 MeV obtained with a totally symmetric spatial trial wave function. 3. C o n c l u s i o n
The upper and lower bound for the energy of the alpha particle have been calculated using five different trial wave functions. The best values for both bounds are obtained with a function which is chosen according to a suggestion of Austern and
VARIATIONAL WAVE FUNCTIONS
253
Iano 3). This function consists of a product of two-nucleon correlation functions, each of which is a solution of the two-nucleon Schr6dinger equation up to a certain separation distance and is a Hulth6n type variational function at larger distances. In our calculation, the best separation distance turns out to be the distance where the potential falls off to zero. Using this function, the gap between the upper and the lower bounds is 16 MeV which is far smaller than those obtainable with the other trial wave functions used in this investigation. A trial wave function which is a product of two-nucleon correlation functions seems to already contain the correct features of three-nucleon correlations, since the multiplication of this trial function with a product of three-nucleon correlation functions does not give any improvement. This indicates that the independent-pair model might be a good approximation, which is confirmed by the agreement between our result, Eo = - 2 6 . 5 + 1.5 MeV, for the ground state energy of the Hamiltonian and the value obtained by Mang and Wild 4) of Eo = - 2 7 . 6 8 MeV. It is shown that the asymmetric part in the alpha particle spatial wave function arising from a different singlet and triplet interaction can give only an insignificant improvement of the bounds in a variational calculation. We wish to thank Professor K. Wildermuth for helpful discussions and Professor E. P. Miles and Dr. J. Fowler for generous grants in computing time.
References I) 2) 3) 4) 5) 6) 7)
E. W. Schmid, Y. C. Tang and R. C. Herndon, Nuclear Physics 42 (1963) 95 R. C. Herndon, E. W. Schmid and Y. C. Tang, Nuclear Physics 42 (1963) 113 N. Austern and P. lano, Nuclear Physics 18 (1960) 672 H. J. Mang and W. Wild, Z. Phys. 154 0959) 182 G. Temple, Proc. Roy. Soc. A119 (1928) 276 E. W. Schmid, Nuclear Physics 32 (1962) 82 H. J. Mang, W. Wild and F. Beck, in Proc. London Conf. on nuclear forces and the few nucleon problem (Pergamon Press, London, 1960) Vol. 2, p. 399
A S T U D Y OF V A R I A T I O N A L WAVE F U N C T I O N S FOR T H E A L P H A PARTICLE E. W. SCHMID t
Oak Ridge National Laboratory and
Florida State University, Tallahassee, Florida Y. C. TANG tt and R. C. H E R N D O N t
Florida State University, Tallahassee, Florida tt Received 12 N o v e m b e r 1962 Abstract: The upper and lower bound for the energy of the alpha particle have been calculated with several different trial wave functions. T h e best result for both bounds is obtained with a function which is a product of solutions o f the two-nucleon Schr6dinger equation up to a certain internucleon separation and goes over into a variational function for larger distances. With this trial function the results for the energy and the r.m.s, radius agree quite well with that obtained by Mang and Wild in an independent-pair model calculation.
1. Introduction In a previous investigation 1, 2), an upper and lower bound for the energy o f the alpha particle was calculated by a variational method for several different twonucleon potentials. For each o f these potentials, the optimum wave function turned out to have a form which, for small inter-nucleon distances, is quite similar to a solution o f the two-nucleon Schr6dinger equation at zero energy. This result was in agreement with a suggestion o f Austern and Iano 3), that one should use as a variational function the zero-energy two-nucleon solution up to a certain inter-nucleon separation and a trial function at larger distances. The purpose o f this work is two-fold. Firstly, we wish to study in somewhat more detail the behaviour o f variational wave functions at small inter-particle distances and to see whether the variational bounds for the alpha particle can be improved by following the suggestion o f Austern and Iano directly. Secondly, our investigation can test the accuracy o f the result o f Mang and Wild *) who have calculated the alpha-particle binding energy in the independent-pair approximation. For the latter purpose, we use the same square-well potential with a hard-core as Mang and Wild. t Present address: Max-Planck-lnstitut fiJr Physik und Astrophysik, Munich, Germany. tt Present address: Brookhaven National Laboratory, Upton, Long Island, N e w York. t Present address: University of California, Lawrence Radiation Laboratory, Livermore, California. tt This work was supported by the U.S. Office o f Naval Research. 247
248
t.w.
SCHMID e t a L
To calculate the upper bound of the alpha particle ground-state energy we use the usual Rayleigh-Ritz variational principle, and for the lower bound we use a method given by Temple 5). The expectation values of the various operators involved are calculated by a Monte-Carlo method. As this method has been discussed in previous papers 1,2, 6), we shall not give any description here. 2. T e s t o f Different Trial W a v e Functions
The Hamiltonian for the alpha particle has the form h2
H=----
4
4
e2
~ V 2 + >~ V ( i , k ) + - - - . 2m l=t i k=l rt2
(l)
For the two-nucleon interaction we choose the potential which was used by Mang and Wild a), i.e., V(i,k)=
00,
rik ~
-Vo, 0,
r c < rik < b, b < rtk,
rc ,
(2)
with rc = 0.4 fm, b = 1.9 fm and Vo = 49.60 MeV. This potential is an average of singlet and triplet interaction and, therefore, has no spin dependence. For the upper bound Eu of the eigenvalue Eo of the Hamiltonian, we use the Rayleigh-Ritz variational principle, i.e., E o < Eu = ( H ) .
(3)
For the lower bound EL we use the method of Temple 5) which gives the formula Eo ~
EL =
(H)-
(H2)-(H)2., El - ( n )
(4)
with the condition that ( H ) is less than El, the energy of the first excited state. Clearly, to insure a rigorous lower bound for the energy of the alpha particle, one should use for Et the lower bound to the ground-state energy of the triton; however, since this last value has not been calculated, we shall choose Et as - 10 MeV, which is probably a safe choice in view of the fact that Mang and Wild obtained a value of - 7.1 MeV for the energy of the triton with the same potential as is used in this investigation. As was discussed in an earlier paper 1), the eigenvalue is usually much closer to the upper bound than to the lower bound and it is convenient to introduce the ratio -
Eo- EL
(5)
Eu - Eo " F r o m a test calculation with the deuteron 1) we concluded that t/ is around 20 for the nuclear case.
VARIATIONAL
WAVE
FUNCTIONS
249
Our trial wave functions are all of the form 7j = u(r)~(a,z),
(6)
where u(r) is the space part of the wave function and ~(a, z) is the normalized chargespin function chosen to give S = 0 and T = 0. Three types of trial wave functions are used for u(r). The first type consists of a long-range part of Gaussian or exponential shape multiplied by a cut-off function which fulfills the boundary condition at the surface of the hard-core and which is capable of representing the exact solution of the two-body Schr6dinger equation in the range of small distances, i.e., 4
ut(r)=exp(-~ai
4
i>k=l
i>k=l
4
u2(r) = e x p ( - - a 2 Z
4.
(8)
r,k) I-I fz(r,k),
i>k=l u3(r) = [exp(--~a3
(7)
~, r E ) l - I fl(r'k),
i>k=l
4. E r2)+c3exp(-~b3 i>k=t
4 ~'~ i>k=l
4
r2a)] [ I f3(r,a),
(9)
i>k=I
with
L(r) =
0,
0 < r < r e,
A. sin k . ( r - r c ) ,
r e < r < d.,
(lo)
r
1,
dn < r,
(I 1)
k n = x / m ( V o + E.)lh 2.
The distance d. is the value of r when (d./r) sin k . ( r - r e ) reaches its first maximum which is normalized to unity by the factor A.. Either d. or the related quantity E. can be considered as variational parameter. The second type of trial wave function is chosen according to the suggestion of Austern and Iano a). A product of twonucleon correlation functions is used, each one being a solution of the two-body Schri~dinger equation up to a certain distance b 4 and then going over into another variational function for larger distances, i.e., 4
u4 =
I-I f4(r,k),
(12)
i>k=l
with O,
0 < r <
r c,
1
f,(r)
rSin k 4 ( r - G ) ,
rc <
r < b4,
A4 [exp ( - a 4 r ) - - B 4 exp (--c4r)] ,
b 4 < r,
=
r
(13)
250
E.W. SCHMID e t a L
(14)
k4 = x/m(Vo + E,)/h 2.
The constants A4 and B, are determined by the continuity conditions at r = b4. Austern and Iano found from the deuteron case that the energy parameter E4 is not a very sensitive parameter and can be put equal to zero. We shall also vary E4 to examine if this holds true in the alpha particle case, too. The third type of trial function is obtained by multiplying u4(r) with a product of short-range three-nucleon correlation functions. It has the form 4
u,(r) = ua(r) FI {i +e, exp [--dz(r,k+r,,--2rc)]}.
(15)
i, lt, l = l k.l;ei k>l
When e5 is positive, the three-nucleon correlation function increases the probability of finding a third particle nearby when two particles are already close together. This increase seems reasonable, since in this particular situation a deeper potential is felt by the third particle. TABLE 1 C o m p a r i s o n o f different trial wave functions Upper b o u n d
Trial function
Eu (MeV)
Lower b o u n d
r.m.s, radius (fm)
E L (MeV)
r.m.s. radius (fm) 1.46
ux
--21.8 -t-0.8
1.44
--125 5:14
ut
--22.1 5:0.9
1.56
--162 ~ 2 4
1.95
us
--24.0 ± 0 . 9
1.47
--97
1.51
u,
--24.93--'0.29
1.53
--40.7 ±
2.7
1.80
u6
--24.93___0.29
1.53
--40.7 :k 2.7
1.80
d::lO
Table 1 gives the results for the variation of the parameters a,, bn, c~, E,, d 5 and e 5. The optimum values for the parameters are at = 0.62 fm -2, Et = - 3 2 . 2 MeV, a2 = 0.38 fm - l , E 2 = - 4 0 . 7 MeV, a3 = 1.03 fm -2, b3 = 0.47 fm -2, c3 = 0.25, E3 = - 3 3 . 2 MeV, a4 = 0.28 fm -1, b4 = 1.9 fm, c4 = 0.396 fm-1 and E , = - 7 . 1 MeV for the upper bound and at = 0.62 fm-2, E, = --39.7 MeV, a2 = 0.29 fm-1, E2 = - 3 9 . 7 MeV, a 3 = 0.90fm-2, b3 = 0.437 fm-2, c3 = 0.25, E 3 = --38.5 MeV, a4 = 0.2 fm - t , b4 = 1.9 fm, c4 -- 0.201 fm - t and E4 = - 8 . 1 MeV for the lower bound. No improvement in either bound is obtained by varying the parameters in the short-range three-nucleon correlation function. It is interesting to examine the values of the optimum parameters in the trial function u, which gives the best results. The separation distance b4 was used by Austern and lano 3) to separate the region near the hard-core where two-nucleon features are dominant from the region of larger distances where many-body effects
VARIATIONAL WAVE FUNCTIONS
251
are more important. In our example, b4 turns out to be rather large, namely, equal to the range of the square-well potential. In one respect, this is not surprising, since at this distance the discontinuity in the potential should cause a discontinuity in the second derivative of the wave function, which can only be achieved with u4 by choosing b4 = b. On the other hand, one might think that using the solution of the twonucleon Schr6dinger equation up to such a large distance might take too little account of the influence of the third and fourth particle in this nucleus. Therefore, we have also tried with the function us which has more flexibility towards three-nucleon effects. It is found, however, that the improvement obtained with this function is less than 10 % of the statistical uncertainty present in the result with u4 and hence, is meaningless. Although the function u s contains only a very special type of three-nucleon correlation, this result seems to indicate that the independent-pair approximation is a good one in the alpha particle case, and that the function u4 already contains the correct features of three-nucleon correlations. We have also varied the function u4 for other values of b4 and obtained for instance, with b4 = 1.25 fm an upper bound which is worse by 0.7 MeV and a lower bound which is worse by 66 MeV. For both bounds, the parameterE4 turns out to be close to zero in this case. Using the argument of the effective-range theory, Austern and Iano predicted that E4 is not a sensitive parameter and a value of zero is a good choice. At b4 = 1.90 fm, we find by varying the other parameters that with E4 = 0, the upper bound and the lower bound become worse by about 2 MeV and 50 MeV, respectively. It seems to us, therefore, that in the alpha-particle case, the parameter E4 needs to be varied to obtain better results. Even with the best trial function u4, there is still a gap of about 16 MeV between the upper and the lower bound. Since, in principle, the eigenvalue can be anywhere within this gap, it would seem that the lower bound we obtained is too small to be of much practical use. Fortunately, however, we have learned from some test cases 1) that the quantity ?/defined by eq. (5) is usually much larger than unity. This already reduces considerably the region within which we expect the eigenvalue to be. If one has some more information about the insufficiency of a trial function, further conclusions about the magnitude of r/ can be drawn in the following way. The upper bound is insensitive to a rather big deviation of the variational function from the eigenfunction if it occurs only in a small part of the integration volume, as, for instance, only for small inter-nucleon distances. On the other hand, the lower bound, being related to the mean square deviation of H~b from Eo~b over the entire integration volume, is determined by the region where the variational function has its largest deviation from the eigenfunction and can be badly estimated even though this region may only be a rather small part of the integration volume. This explains why the lower bound is most sensitive to the behaviour of the wave function near the hardcore. When the trial function is a bad approximation to the actual eigenfunction in this region, a value of r / u p to 100 is possible, whereas if the function deviates only
252
E.w.
SCHMID e t al.
slightly from the eigenfunction over the whole integration volume, ~/values as small as 5 can be obtained. By comparing the upper and the lower bounds obtained with the trial functions ut, u2 and u 3 with those obtained with u4, we expect the magnitude of r/ for these first three functions to exceed 20. This is in agreement with the statement above, since these functions do not have a discontinuity in the second derivative at the edge of the square well, which the eigenfunction certainly has. Therefore they will have a rather large deviation from the eigenfunction in a relatively small region. The function u 2, in addition, cannot go over into a two-nucleon solution near the hardcore. The function u 4 does alllow for a discontinuity in the second derivative, although due to its rather restricted form for r > b, only negative values can be obtained in the region immediately outside of the range of the square well. Obviously, u4 also has the right behaviour near the hard-core. Therefore, it is possible that ~/is smaller than 20 in this case but we do not believe that it is smaller than 5. With 5 < ~/ < 50, we can then estimate E0 to be in the region Eo = -26.5-t- 1.5 MeV. This is in agreement with the value obtained by Mang and Wild from an independentpair-model calculation i.e., E 0 = - 2 7 . 6 8 MeV. The r.m.s, radius of 1.53 fm at the upper bound for the function u4 also agrees quite well with the result of Mang and Wild 7) of 1.51 fm. Our potential is an average over singlet and triplet interaction. In a more realistic calculation these interactions should have different strengths and the trial function should allow for an asymmetric part which arises from this feature of the potential. In this investigation we have estimated what improvement the inclusion of an asymmetric spatial wave function would make in the result for the energy. Using a potential of the same average strength as that given by eq. (2), but with a singlet to triplet ratio of 0.8, we get with the trial function u4 the same upper bound, but a lower bound which is worse by about 2 MeV. That means it would be possible to gain 2 MeV in the lower bound by allowing the trial wave function to have an asymmetric spatial component. Assuming r/to be around 10, this added degree of freedom in the trial function would improve the upper bound by only about 0.2 MeV, which is not very significant compared to its value of - 2 4 . 9 3 MeV obtained with a totally symmetric spatial trial wave function. 3. C o n c l u s i o n
The upper and lower bound for the energy of the alpha particle have been calculated using five different trial wave functions. The best values for both bounds are obtained with a function which is chosen according to a suggestion of Austern and
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253
Iano 3). This function consists of a product of two-nucleon correlation functions, each of which is a solution of the two-nucleon Schr6dinger equation up to a certain separation distance and is a Hulth6n type variational function at larger distances. In our calculation, the best separation distance turns out to be the distance where the potential falls off to zero. Using this function, the gap between the upper and the lower bounds is 16 MeV which is far smaller than those obtainable with the other trial wave functions used in this investigation. A trial wave function which is a product of two-nucleon correlation functions seems to already contain the correct features of three-nucleon correlations, since the multiplication of this trial function with a product of three-nucleon correlation functions does not give any improvement. This indicates that the independent-pair model might be a good approximation, which is confirmed by the agreement between our result, Eo = - 2 6 . 5 + 1.5 MeV, for the ground state energy of the Hamiltonian and the value obtained by Mang and Wild 4) of Eo = - 2 7 . 6 8 MeV. It is shown that the asymmetric part in the alpha particle spatial wave function arising from a different singlet and triplet interaction can give only an insignificant improvement of the bounds in a variational calculation. We wish to thank Professor K. Wildermuth for helpful discussions and Professor E. P. Miles and Dr. J. Fowler for generous grants in computing time.
References I) 2) 3) 4) 5) 6) 7)
E. W. Schmid, Y. C. Tang and R. C. Herndon, Nuclear Physics 42 (1963) 95 R. C. Herndon, E. W. Schmid and Y. C. Tang, Nuclear Physics 42 (1963) 113 N. Austern and P. lano, Nuclear Physics 18 (1960) 672 H. J. Mang and W. Wild, Z. Phys. 154 0959) 182 G. Temple, Proc. Roy. Soc. A119 (1928) 276 E. W. Schmid, Nuclear Physics 32 (1962) 82 H. J. Mang, W. Wild and F. Beck, in Proc. London Conf. on nuclear forces and the few nucleon problem (Pergamon Press, London, 1960) Vol. 2, p. 399