Dr. Mottelson and his co-winners of the 1975 prize were honored for work that scientists regard as one of the landmarks in the development of nuclear physics.

By 1945, scientists knew enough about the nucleus — the collection of protons and neutrons at the core of the atom — to pry it apart, releasing vast quantities of energy, and inaugurating what we recognize as the nuclear age.

Even so, their understanding of the nucleus and its structure was far from complete. Many mysteries remained, and many properties of the nucleus and much of its behavior lacked an adequate explanation. Knowledge of nuclear structure is regarded as vital in weapons research, power generation and in solving the problems of astrophysics and the history of the universe.

In what is still regarded as one of the crowning achievements of nuclear physics, Dr. Mottelson helped show, using arguments and techniques from quantum theory, how each individual constituent of the nucleus — each proton and each neutron — exerted an effect on the properties and character of the nucleus as a whole. And vice versa.

“I find it a really wonderful discovery,” Victor Weisskopf, a leading 20th-century theoretical physicist, once said. “It’s just beautiful.”

Dr. Mottelson shared the prize with Aage Bohr of Denmark and James Rainwater of the United States.

The Nobel Committee honored them “for the discovery of the connection between collective motion and particle motion in atomic nuclei and the development of the theory of the structure of the atomic nucleus based on this connection.”

Dr. Mottelson worked particularly closely with Bohr, and the theory that has become a milestone in understanding the nucleus is known as the Bohr-Mottelson theory.

It couples the actions of single particles to the actions of the entire nucleus and showed how each had the ability to act in a way apparently independent of the others. But it also showed how the actions of the constituent particles — and of the entire collection of particles — also depended on the actions of the others.

It reconciled two major theories of the nucleus, the liquid drop model and the shell model.

The shell model treats the nucleus in a sense as a miniature solar system, with neutrons and protons in the roles of planets, revolving about a central point, but essentially independent. In recognition of that independence, the shell model is often known as the single particle model.

In contrast, the liquid drop model in essence treats the nucleus as a tiny but homogeneous whole, something like a drop of water, the same everywhere inside. Based essentially on that image, scientists pried the nucleus apart, unleashing energy locked within and creating the nuclear age.

Yet the drop model was inherently crude and necessarily limited. The shell model, introduced soon after the dawn of the so-called nuclear age, was eagerly welcomed. But it too had its drawbacks.

Neither of the models, for example, showed success in explaining or predicting the relatively weak electromagnetic radiation that came from nuclei that remained intact and did not split apart. Nor could the models explain how and why the nucleus deviated from the spherical shape upon which the models were built.

In research carried out in the early 1950s, Bohr and Dr. Mottelson showed in mathematical detail how particles, on the one hand, and the collection of particles, on the other, influenced the other: how they, in essence, pulled and pushed on each other, in a manner that affected the actions of the individual particles and the shape and actions of the group of particles.

Drawing heavily on insights from quantum mechanics, Bohr and Dr. Mottelson published their theory in 1953. It showed how the coupling between the single particle and the group could cause the entire nucleus to rotate or to oscillate, giving off electromagnetic waves as if it were a microscopic radio transmitter.

It also showed how a nucleus, once considered a simple sphere, could stretch or flatten, to resemble a football or a disk.

As generally occurs, the new theory did not win immediate acceptance. Scientists who had achieved valuable results with the shell model were not eager to abandon it, but in time the power of the Bohr-Mottelson theory became recognized.

One scientist, writing years after the theory appeared, has called the work almost a second discovery of the nucleus. In addition, scientists regard it as an important contribution to the theory of systems composed of many independent constituents.

In a rough and intuitive way, aspects of the theory might even be likened to examples from political science, in which individual citizens affect the electorate as a whole; at the same time, changes in the electorate exert influence on individual citizens.

Ben Roy Mottelson was born in Chicago July 9, 1926, and grew up in La Grange, Ill. His father was an engineer, and his mother was a homemaker.

He served in the Navy during World War II and went to Purdue University in West Lafayette, Ind., for officer training. He graduated in 1947. He the studied theoretical physics at Harvard University under Nobel laureate Julian Schwinger, receiving his doctorate in 1950.

Dr. Mottelson then went on a fellowship to what was then the Institute for Theoretical Physics in Copenhagen, a renowned center of scientific discovery. The Institute was later named for Niels Bohr, the Nobel winner who was one of the founders of 20th century physics and the father of Aage Bohr.

Their collaboration was described by Dr. Mottelson in his Nobel Prize biography as “a dialogue between kindred spirits.” Their two-volume work on nuclear structure was regarded as a monument in the field.

After continued work in Europe, Dr. Mottelson became a professor in 1957 at the Nordic Institute for Theoretical Atomic Physics (Nordita) in Copenhagen and became a Danish citizen in 1971. He retired in 1994 but remained active as an emeritus professor.

He was married in 1948 to Nancy Jane Reno, who died in 1975. They had three children. He was married to Britta Marger Siegumfeldt from 1983 until her death in 2014.