APPLICATION OF NVIDIA CUDA TECHNOLOGY TO CALCULATION OF GROUND STATES OF FEW-BODY NUCLEI

Abstract

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

1. Introduction

There is high interest in structure and reactions with few-body atomic nuclei (e.g., 3H, 3He, 6He, 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 highperformance 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] 2 K  q, ; q0 , 0  lim 

M  M    1 M  m  qk  qk 1   exp     k 1  2

   V  qk   C M dq1dq2   dqM 1, qk  q(k ), k  k, k  0, M , qM  q, C   m 1 2 .

 2   Here m is the mass of the particle and V  qk  is its potential energy. The energy E0 and the square modulus of the wave function 0 2 of the ground state of a system of few particles with coordinates q may be calculated using asymptotic behavior of propagator [5]

KE  q, ; q, 0  0 (q) 2 exp   E0  

,    ,   ln KE  q, ; q, 0  0 (q) 2  E0,    . or The values of the propagator KE  q, ; q, 0 were calculated using averaging denoted by random trajectories qk  f (q, k ) with the distribution in the form of the multidimensional Gaussian distribution

 m  KE  q, ; q, 0     2   1 n F   Fi . (6) n i1 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 (1) (2) (3) (4) over (5) 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

2 the exact value of the ground state energy is

Vij (rij )  m2 rij2 ,

V  U0  Vij (rij ) ,

i j

Number of particles N

3 4 5 6 7 (7) (8) (11)

Assuming   1,

 1 , we obtain E0  U0  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.

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. [69] 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

3 Vi j (r)  uk exp   r 2 bk2  . (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 9Be and in 12C)

q  x, y, x  r2  r1, y  r3  2 r1  r2  . (10)

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) ( 1 i, j  n, p,  ) were used in the form of the combination of Woods–Saxon potentials s V j (r)  Ui 1  exp(r  Ri ) / ai 1 ,

i1 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.

0 (x, y, cos ) for the configurations of nuclei 6He (α + n + n) and 6Li (α + n + p) with the angle  between the vectors x and y is shown in (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.

2 for the 6He nucleus and the vectors in the Jacobi coordinates;

2 for the 6Li nucleus and the vectors in the Jacobi coordinates;

2 for the 9Be nuclei and the vectors in the Jacobi coordinates;

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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