=Paper=
{{Paper
|id=Vol-2023/259-264-paper-41
|storemode=property
|title=Application of NVIDIA CUDA technology to calculation of ground states of few-body nuclei
|pdfUrl=https://ceur-ws.org/Vol-2023/259-264-paper-41.pdf
|volume=Vol-2023
|authors=Viacheslav Samarin,Mikhail Naumenko
}}
==Application of NVIDIA CUDA technology to calculation of ground states of few-body nuclei==
https://ceur-ws.org/Vol-2023/259-264-paper-41.pdf
Proceedings of the XXVI International Symposium on Nuclear Electronics & Computing (NEC’2017)
Becici, Budva, Montenegro, September 25 - 29, 2017
APPLICATION OF NVIDIA CUDA TECHNOLOGY TO
CALCULATION OF GROUND STATES OF FEW-BODY
NUCLEI
V.V. Samarin1,2,a, M.A. Naumenko1
1
Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, 6 Joliot-Curie,
Moscow region, Dubna, 141980, Russian Federation
2
Dubna State University, 19 Universitetskaya, Moscow region, Dubna, 141982, Russian Federation
E-mail: a samarin@jinr.ru
The modern parallel computing solutions were used to speed up the calculations by Feynman’s
continual integrals method. The algorithm was implemented in C++ programming language.
Calculations using NVIDIA CUDA technology were performed on the NVIDIA Tesla K40
accelerator installed within the heterogeneous cluster of the Laboratory of Information Technologies,
Joint Institute for Nuclear Research, Dubna. The results for energies of the ground states of several
few-body nuclei demonstrate overall good agreement with experimental data. The obtained square
modulus of the wave function of the ground states provided the possibility of investigating the spatial
structure of the studied nuclei. The use of general-purpose computing on graphics processing units
significantly (two orders of magnitude) increases the speed of calculations.
Keywords: parallel computing, NVIDIA CUDA technology, Feynman’s continual integrals
method, few body systems, light atomic nuclei.
© 2017 Viacheslav V. Samarin, Mikhail A. Naumenko
259
Proceedings of the XXVI International Symposium on Nuclear Electronics & Computing (NEC’2017)
Becici, Budva, Montenegro, September 25 - 29, 2017
1. Introduction
There is high interest in structure and reactions with few-body atomic nuclei (e.g., 3H, 3He,
6
He, etc.) from both theoreticians and experimentalists of the Joint Institute for Nuclear Research
(JINR) and other scientific centers. The development of computing and information resources of the
Laboratory of Information Technologies (LIT), JINR [1] provides opportunities for high-
performance computing and application of new methods for theoretical study of light nuclei. This
work is devoted to application of NVIDIA CUDA technology [2, 3] to calculations within
Feynman’s continual integrals method [4, 5] which provides the energy and the probability densities
for ground states of few body systems. This approach was already used for calculations of 3H, 3,4,6He
nuclei [6, 7] and 6Li, 9Be nuclei [8, 9]. In addition to the above-listed nuclei, in this work 12C and 16O
nuclei are considered using the same approach. The few-body nuclei 3H, 3,4He were described as
consisting of protons and neutrons, whereas the nuclei 6He, 6Li, 9Be, 12C, and 16O were described as
α-cluster nuclei. The algorithm allowing us to perform calculations directly on GPU was developed
and implemented in C++ programming language. The results show that the use of GPU is very
effective for these calculations.
2. Theory and computing
Feynman’s continual integral [4] is a propagator the probability amplitude for a particle to
travel from the point q0 to the point q in a given time t . In the imaginary (Euclidean) time it
the propagator can be represented as the limit of a multiple integral [4, 5]
1 M m q q 2
K q, ; q0 ,0 lim exp k k 1
V qk C M dq1dq2 dqM 1 , (1)
M
k 1 2
M
12
m
qk q(k ), k k , k 0, M , qM q, C . (2)
2
Here m is the mass of the particle and V qk is its potential energy. The energy E 0 and the square
modulus of the wave function 0 of the ground state of a system of few particles with coordinates
2
q may be calculated using asymptotic behavior of propagator [5]
E
K E q, ; q,0 0 (q) exp 0 , ,
2
(3)
or
ln K E q, ; q,0 0 (q) E0 , .
2
(4)
The values of the propagator K E q, ; q,0 were calculated using averaging denoted by over
random trajectories qk f (q, k ) with the distribution in the form of the multidimensional
Gaussian distribution
M
12
m
K E q, ; q,0 exp V (qk ) , (5)
2 k 1 0, M
1 n
F Fi .
n i 1
(6)
This theoretical approach to N-particle systems with the use of Jacobi coordinates is described in
detail in Ref. [8]. Feynman’s continual integrals method provides a new, mathematically simpler,
possibility for calculating the energy and the probability density of the ground states of N-particle
systems compared to other approaches, e.g., expansion into hyperspherical harmonics [10].
The Monte Carlo algorithm for numerical calculations was developed and implemented in
C++ programming language using NVIDIA CUDA technology. The integration method does not
require the use of any additional integration libraries. Parallel calculations (one thread calculated one
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Proceedings of the XXVI International Symposium on Nuclear Electronics & Computing (NEC’2017)
Becici, Budva, Montenegro, September 25 - 29, 2017
trajectory) were performed on the NVIDIA Tesla K40 accelerator installed within the heterogeneous
cluster [1] of LIT, JINR, Dubna. The code was compiled with NVIDIA CUDA version 8.0 for
architecture version 3.5. Calculations were performed with single precision.
To check the correctness of the calculation of the propagator the comparison with the exactly
solvable N-body ( N 3 7 ) oscillatory systems has been performed. For particles with equal masses
mi m interacting with each other by oscillator potentials
m2 2
Vij (rij ) rij , V U 0 Vij (rij ) , (7)
2 i j
the exact value of the ground state energy is
3
E0 U 0 ( N 1) N . (8)
2
Assuming 1, 1 , we obtain E0 U 0 1.5( N 1) N . The Monte Carlo calculations were
carried out with statistics n 107 . For values of the logarithm of the propagator, linear smoothing
according to formula (4) was performed and the ground state energies were found. The results in
Table 1 demonstrate satisfactory accuracy of calculations.
Table 1. Comparison of exact and calculated ground state energies the exactly solvable N-body
oscillatory systems
Number of particles N U0 Exact value of E 0 Calculated value of E 0
3 0 4.098 4.11 ± 0.006
4 15 6 5.98 ± 0.02
5 20 6.584 6.56 ± 0.05
6 20 1.629 1.84 ± 0.1
7 20 3.812 3.63 ± 0.1
3. Results for 3 and 4 body nuclei
The same effective pairwise nucleon-nucleon, nucleon-cluster and cluster-cluster interaction
potentials V r were used for all the studied nuclei, where r is the distance between nucleons. In
Refs. [69] the calculation of the propagator for the nuclei 2,3H, 3,4He neutron-neutron, proton-proton
and neutron-proton two-body effective strong interaction potentials Vi j ( r ) ( i, j n, p ) similar to the
M3Y potential [11, 12] have been used
Vi j (r ) uk exp r 2 bk2 .
3
(9)
k 1
The values of parameters are given in Ref. [8]. The calculations were performed in the center of mass
system using the usual Jacobi coordinates. For a system of three particles, two of which have equal
masses m1 m2 m (two neutrons in 3H and 6He, a proton and a neutron in 6Li, two -clusters in
9
Be and in 12C)
1
q x, y , x r2 r1 , y r3 r1 r2 . (10)
2
The theoretical binding energies EB E0 obtained using formula (4) are listed in Table 2
together with the experimental values taken from the NRV web knowledge base [13]. It is clear that
the theoretical values are close enough to the experimental ones. The observed difference between
the calculated binding energies of 3H and 3He is also in agreement with the experimental values.
The α-cluster-nucleon and α-cluster-α-cluster strong interaction potentials Vi j ( r ) (
i, j n, p, ) were used in the form of the combination of Woods–Saxon potentials
s
V j (r ) U i 1 exp (r Ri ) / ai ,
1
(11)
i 1
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Proceedings of the XXVI International Symposium on Nuclear Electronics & Computing (NEC’2017)
Becici, Budva, Montenegro, September 25 - 29, 2017
where s 2,3 . The values of parameters are given in Ref. [8]. The obtained theoretical energies of
separation into cluster(s) and nucleon(s) ES E0 are listed in Table 2 together with the
experimental values taken from the NRV web knowledge base [13]. It can be seen that the theoretical
values are close enough to the experimental ones.
Table 2. Comparison of theoretical and experimental energies of separation of light nuclei into
constituent particles
Atomic nucleus Constituent particles Experimental value [13], MeV Theoretical value, MeV
3
H n+n+p 8.482 8.21 ± 0.3
3
He n+n+p 7.718 7.37 ± 0.3
4
He n + n + p +p 28.296 30.60 ± 1.0
n+n+ 0.96 ± 0.05
6
He 0.97542
n+p+ 3.87 ± 0.2
6
Li 3.637
n++ 1.7 ± 0.1
9
Be 1.573
++ 7.39 ± 0.1
12
C 7.37
+++ 14.52 ± 0.1
16
O 14.53
The probability density distribution 0 ( x, y,cos ) for the configurations of nuclei
6
He (α + n + n) and 6Li (α + n + p) with the angle between the vectors x and y is shown in
Figures 1 and 2, respectively.
It can be seen that the most probable configurations of 6He nucleus are α-cluster + dineutron
and the cigar configuration, whereas the configuration n + 5He has low probability. The only one
possible configuration of 6Li nucleus is α-cluster + deuteron-cluster.
The probability density distribution 0 ( x, y,cos ) for the configurations of nucleus 9Be
(α + n + α) with the angle between the vectors x and y is shown in Figure 3. The most probable
configuration is α + n + α, whereas the configurations α + 5He and n + 8Be are less probable.
Figure 1. The probability density 0 for the 6He nucleus and the vectors in the Jacobi coordinates;
2
neutrons and α-clusters are denoted as small empty circles and large filled circles, respectively. The
most probable configurations are α-cluster + dineutron (1) and the cigar configuration (2). The
configuration n + 5He (3) has low probability.
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Proceedings of the XXVI International Symposium on Nuclear Electronics & Computing (NEC’2017)
Becici, Budva, Montenegro, September 25 - 29, 2017
Figure 2. The probability density 0 for the 6Li nucleus and the vectors in the Jacobi coordinates;
2
notations are the same as in Figure 1, protons are denoted as small filled circles. The only one
possible configuration is α-cluster + deuteron-cluster.
Figure 3. The probability density 0 for the 9Be nuclei and the vectors in the Jacobi coordinates;
2
notations are the same as in Figures 1, 2. The most probable configuration is α + n + α (1). The
configurations α + 5He (2) and n + 8Be (3) are less probable.
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Proceedings of the XXVI International Symposium on Nuclear Electronics & Computing (NEC’2017)
Becici, Budva, Montenegro, September 25 - 29, 2017
4. Acknowledgements
We thank the team of the heterogeneous cluster of the Laboratory of Information
Technologies, Joint Institute for Nuclear Research for training, support, and providing computational
resources.
5. Conclusions
In this work an attempt is made to use modern parallel computing solutions to speed up the
calculations of ground states of few-body nuclei by Feynman’s continual integrals method. The
developed parallel algorithm provided significant increase of the speed of calculations. The method
was applied to the nuclei consisting of nucleons and cluster nuclei. The results of calculations
demonstrate that the obtained theoretical values are close enough to the experimental ones for the
studied nuclei. The obtained probability densities may be used for the correct definition of the initial
conditions in the time-dependent calculations of reactions with the considered nuclei [14]. The results
may also serve as a useful addition to the results obtained by the expansion in hyperspherical
functions [15].
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