# Migdal, Arkady Benediktovich

# MIGDAL, ARKADY BENEDIKTOVICH

(*b*. Lida, Russian Empire [later Belarus], 11 March 1911; *d*. Princeton, New Jersey, 9 February 1991),

*physics, nuclear theory, condensed-matter theory*.

Migdal was among the increasingly rare theoretical physicists after World War II who felt equally at home in nuclear and particle physics as well as the physics of condensed matter, occasionally venturing into plasma theory as well. Beginning in nuclear theory, he predicted a giant dipole resonance in photoabsorbtion processes in 1944 (experimentally confirmed three years later). His work on phonons in the 1950s offered one of the first productive applications of quantum field theory to the solid state, and he also made major contributions to the theory of Fermi liquids. He then took the lessons from this work back to nuclear theory, where he derived an early rigorous treatment of strong interactions inside nuclei, and later also saw his original ideas on pion condensation extended in unexpected ways. Because he counted Lev Davidovitch Landau as a teacher, Migdal is often identified as a member of the Landau School, though Migdal was ambitious enough and close enough to Landau in age to pursue a friendly rivalry of sorts. (Only Migdal dared arrive late at Landau’s famous theory seminar in Moscow.) The author of several textbooks that showed a generation of theorists how to carry calculational tools such as Green’s functions and Feynman diagrams from one domain to another, Migdal taught at the “Soviet MIT” (the Moscow Engineering Physics Institute, MIFI) and ventured successfully into science popularization, warranting the designation of a “Migdal School” in the eyes of many contemporaries.

**Early Career** . Arkady Benediktovich Migdal was born in the Jewish Pale of Settlement, in what was then the Vilna province of the Russian Empire, slightly south of present-day Vilnius, Lithuania. His father Beinus Migdal made a meager living as a pharmacist, though little else is known about his early family life. The region suffered greatly from the ebb and flow of world war, revolution, and civil war in the 1914–1921 period. It was occupied repeatedly by Germans, Poles, and Russians before it was eventually incorporated into the newly constituted Polish state during the interwar period. (It had once belonged to the long-defunct Polish-Lithuanian Commonwealth). Like many peers at the time, Migdal’s family moved to Leningrad (later St. Petersburg) in search of better fortunes in the 1920s. (Subsequently Arkady would usually change his patronymic from the Yiddish-inflected Beinusovich to the more Russian Benediktovich.) It was in Leningrad that Migdal received his secondary education, and even managed to publish a brief pedagogical note on the Atwood machine before his eighteenth birthday.

Migdal’s career as a physicist got off to a rocky start after his initial enrollment in the physics faculty at Leningrad State University in 1929. Soviet universities had just introduced so-called “brigade” methods into the curriculum in a hasty attempt to encourage collective modes of learning, and in the spring of 1930 Migdal was among several students kicked out for rebelling against the new system. Formalism, apoliticism, and antisocial behavior were among the formulaic charges leveled at the students, and Migdal in particular was faulted for a “slapdash attitude toward shock work.” (Borrowed rather incongruously from the industrial context of the First Five Year Plan, “shock work” designated production efforts above and beyond the agreed-upon norms.) He spent a couple of months in jail, but the brigade methods were soon abandoned and the students readmitted. His troubles did not cease, however, because his “bourgeois” family members faced difficulties finding an economic niche for themselves once the looser strictures of the Bolsheviks’ New Economic Policy during the 1920s were abandoned. Beginning in 1929 more centralized planning, headlong industrialization, and stricter “red” criteria for the social credentials of Soviet white-collar workers were introduced. The following year Migdal’s father was laid off, before eventually being exiled to Kazakhstan for three years in 1932 for engaging in gray-market activities. He took ill and died not long after, with Migdal forced to look after his mother in Leningrad in the meantime.

Like many children of socially suspect parents in his generation, Migdal also completed a lengthy practical stint at an electronic instrument factory to shore up his proletarian credentials, and then stayed on as a consultant. His many obligations tore him away from his studies, and he was expelled from the university again in March 1933. It was another two years before his factory supervisors managed to get him re-enrolled in evening courses at the university, where V. A. Fock was among his teachers. He finally graduated at the end of 1936 and immediately embarked on advanced study at the Leningrad Physico-Technical Institute. His first advisor, M. P. Bronshtein, was only five years older, but he quickly steered Migdal toward theoretical nuclear physics. (Bronshtein was arrested in August 1937 during the Great Purges, never to return.) For Migdal the importance of the new subject matter was highlighted at the Third All-Union Conference on Nuclear Physics held in Leningrad in October 1938, though its lively debates took place without the foreign physicists who had participated in two previous rounds. Migdal made his professional debut that same year with an article on neutron scattering in ferromagnets, and proceeded to develop an expertise in atomic ionization processes. In 1939 he developed an approximation method he labeled “tossing,” in order to solve certain problems associated with the ionization of atoms by neutrons. (The technique found its way into numerous quantum mechanics textbooks, including his own some thirty years later.) By the time Migdal defended his first (*kandidatskaia*) dissertation in Leningrad in 1940, he had already won the esteem of Yakov Il’itch Frenkel, Igorvgen’evich Tamm, and Landau, the dominant Soviet theorists at the time.

After these labors Migdal was already judged mature enough to proceed directly to work on the advanced *doktorskaia* dissertation, and he moved to Moscow to work with Landau. With Landau as an advisor, he developed an interest in recent experimental results in superconductivity and superfluidity, some of which were issuing from his institutional home at P. L. Kapitza Institute for Physical Problems. Landau had already launched into the development of his famous theory of superfluidity (published in May 1941), but in 1940 Migdal independently derived an exact calculation of the heat capacity for the collective portion of the superfluid helium well below the λ-point (what Landau dubbed the “phonon part”). After showing the calculation to Landau—who duly acknowledged it in his paper—Migdal retreated, somewhat regretfully, from trying to develop a more complete theory in Landau’s shadow. He turned instead to the study of photoabsorption by atomic nuclei.

Though this phenomenal domain was well removed from liquid helium, Migdal nonetheless employed a similar set of tools, adopting a collective model that selected dynamic variables associated with the collective motion of a large number of nucleons (e.g., the deformation parameters describing surface vibrations and rotations of a nucleus). The resulting work on nuclear decay mechanisms, and especially quadrupole and dipole γ emission, formed the basis for his dissertation in 1943, as well as a related paper in English in 1944. In this work Migdal offered theoretical grounds for the large dipole resonance centered around 17 megaelectron volts (lower for heavy nuclei, higher for lighter ones), a phenomenon first noted tentatively by Walter Bothe and Wolfgang Gentner in 1937, but described authoritatively only a decade later by George C. Baldwin and G. S. Klaiber. In Migdal’s description, the energy maximum of giant dipole resonances (GDR) is governed by the symmetry energy in the Bethe-Weizsäcker formula for the binding energy of the nucleus and by the average kinetic energy of the nucleons. GDR has since become a cottage industry for collective dynamics that extends even to so-called metallic clusters.

**Basic Research in the Soviet Nuclear Weapons Complex** . Late in the war, Migdal was invited to join I. V. Kurchatov’s Laboratory No. 2, later known as the Institute of Atomic Energy (“Kurchatov Institute”). Like many physicists of his generation, he found himself confronted with the novel demands of classified research, and for more than a quarter century he found it expedient to make the necessary accommodations, perhaps because this initially offered relative security in the late Stalin era, when state-sponsored anti-Semitism reached a peak. Though some of this work came at the cost of priority claims internationally, Migdal retained direct access to Kurchatov until the latter’s death in 1960, and he proved adept at winning precious personnel and resources for fundamental research, eventually heading “Sector 10,” the institute’s general nuclear theory group. He did not lack for contact with the finest physicists of the postwar generation, collaborating with G. I. Budker on calculations for an early design of a homogeneous finite reactor and serving as an official opponent for Andrey D. Sakharov’s dissertation defense in November 1947. The Soviet weapons complex underwrote several other results that were slow to reach the public eye. In the early 1950s Migdal led a group working on controlled thermonuclear synthesis, and performed important calculations with Viktor Mikhailovitch Galitskii on the distribution of cyclotron radiation in magnetized thermonuclear plasma. With S. I. Braginskii he also developed a qualitative theory of processes accompanying the inertial pinch effect. These works were only declassified in a 1958 collection on the physics of plasma edited by Mikhail Aleksandrovich Leontovich.

While little is known about Migdal’s classified research, he did simultaneously manage to follow developments in the nascent field of pion physics after the war, publishing a survey with Yakov Abramovich Smorodinskii. The next major “public” problem he tackled at the Kurchatov Institute was the resonant interaction of slow nucleons in a nuclear reaction, which surely grew out of concerns from weapons work and reactor design. For a three-body system the challenge was to describe possible final-state interactions for *pp, np*, and *nn* scattering, where it was especially difficult to extract experimental information about the scattering length a_{nn}. At a Landau theory seminar in 1950, Migdal demonstrated that the posited interactions between the final products of a nuclear reaction leading to three particles could strongly influence the energy and angular distribution of those final products. For small relative momenta, the energy distribution for the *nn* reaction was shown to be essentially dependent on a_{nn}, a result first published by Kenneth Watson in 1952, and eventually released for international publication by Migdal in 1955. The accuracy of their distribution functions was not great, but some of the theory’s working assumptions formed the starting point for the more exacting techniques developed—for example, those developed by Ludwig D. Faddeev a decade later.

In 1953 Landau and Isaak Yakovlevich Pomeranchuk performed a series of calculations in classical electrodynamics showing how multiple scattering could

interfere with bremsstrahlung. The suppression of the bremsstrahlung stemmed from the interference between photons generated along individual portions of the electron trajectory, but their suppression calculation failed at the same level that the classical bremsstrahlung calculation failed, when *k ~ E*. Migdal attempted a fully quantum mechanical calculation in 1956, which had the advantage of covering the entire energy range. Treating multiple scattering as a kind of Fokker-Planck diffusion process, he worked out the average radiation per collision while taking into account the interference between the radiation from different collisions. In the absence of suppression processes, Migdal’s cross-section matched up well with the Bethe-Heitler cross-section, and the approach was robust enough that it remains in use in the twenty-first century, albeit with modern correction terms. He also found the cross-section for pair production, though these calculations only applied to an infinite-thickness target. Only in 1993 did the E-146 collaboration at the Stanford Linear Accelerator Center (SLAC) perform truly detailed experimental tests of Landau-Pomeranchuk-Migdal (LPM) suppression theory, whose subsequent modifications held up remarkably well.

**Superconductivity and Superfluidity; Finite Fermi Systems** . During the 1950s Migdal was one of many who set their sights on the theory of superconductivity, mastering many of the formal issues in this vast domain and even skirting close to the solution ultimately found by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer in 1957. Working mostly with adiabatic perturbation theory and second quantization, he initially made only modest contributions, but eventually became the first to apply non-perturbative diagrammatic methods to electron-phonon interactions in metals in 1958. Gersim Matveevich Eliashberg extended the technique to superconductors two years later. Migdal truly hit his stride once he learned of Landau’s theory of Fermi liquids in 1957, quickly offering a general result in corroboration of the Landau picture. For non-interacting collections of particles obeying Fermi statistics at low temperatures, the chance of finding any particle at a given energy drops sharply to zero above the Fermi energy *E*_{F}. For interacting particles it is natural to assume that this energy cut-off will be somewhat smoothed out, but Migdal found that the cut-off is much sharper than expected (the “Migdal jump”), and can be used to define *E*_{F}, thus shoring up the generality of the quasiparticle concepts in Fermi liquid theory.

Landau’s theory covered weakly excited states in infinite systems of interacting fermions, and Migdal and his closest associates soon realized that its applicability extended from liquid He^{3} to nuclear many-body theory. In various papers and an original textbook Migdal argued forcefully for the parallels in the descriptive frameworks, and made the major modifications necessary to apply his theory to finite systems such as nuclei. With his close colleague Galitskii, he developed the Green’s function techniques in 1958 that began to bring analytical properties, spectral expansions, dispersion relations, and even the exact formula for the ground-state energy within reach of the nuclear physicist. Migdal’s approach was semi-phenomenological in the classic style of the Landau School, with equations derived from first principles, but with parameters for quantities such as quasiparticle energies and wave functions and quasiparticle-quasihole interactions carefully calibrated with suitable experiments. Migdal expanded the interaction at the Fermi surface in terms of Legendre polynomials, and the art came in choosing the correct general experimental parameters (Landau-Migdal parameters) to use the theory to make connections to other, seemingly unrelated phenomena. (In the finite case, as Migdal showed, these parameters become density dependent.) At low energies the nucleus behaves like a gas of quasiparticles interacting via pair forces dependent on the nuclear density (Migdal forces). The theory of finite Fermi systems—extensively formalized by Migdal in a 1965 textbook by that name—provided a framework for virtually any nuclear structure calculation, whether self-consistent or non-self-consistent (in the sense of Hartree-Fock).

The richness of this general research program was manifested in the flood of papers produced by Migdal and his colleagues and students in the 1960s. Migdal made connections between superfluidity and nuclear moments of inertia, identified nuclear analogies to one-particle excitations in Fermi systems, and, with Anatoly Ivanovich Larkin, treated the case of zero temperature and *S*-wave pairing. Isotopic and isomeric shifts in atomic and mesoatomic spectral lines were treated, as were magnetic and quadrupole moments, while calculations of beta-decay and muon-capture probabilities also found their way into the FFS program.

**Neutron Stars, Pion Condensation, and Quantum Chromodynamics** . Migdal was elected a corresponding member of the Academy of Sciences in the autumn of 1953, along with many other contributors to the Soviet thermonuclear weapons program; full academician status came more than a decade later. The lure of working in the Soviet weapons complex dimmed over time, however, as Soviet civilian physicists were gradually permitted to attend more international conferences in the 1960s. Migdal was also drawn to “civilian” diversions such as boxing, mountain climbing (especially popular among Soviet physicists), scuba diving (he was one of the first practitioners in the Soviet Union), sculpting, and jewelry making. Following yet another argument with the authorities over censorship and delays in publication in 1971, Migdal finally transferred to the recently founded Landau Institute of Theoretical Physics.

Around the time of his move to the Landau Institute, Migdal developed an interest in the behavior of Bose and Fermi systems in strong external fields. As early as 1959 he had hinted that superfluidity of nuclear matter might have interesting cosmological consequences for neutron stars, predicting that such stars would have a superfluid state with a transition temperature corresponding to 1 mega-electron volt. In a series of papers culminating in a massive 1978 review essay that became his most cited article, Migdal explored the range of phenomena that nucleon media subject to extreme conditions might manifest. In 1971 he suggested that the boson vacuum in a sufficiently strong external field would be unstable, and that matter in a neutron star subject to these conditions could undergo a phase transition. Dubbing this “pion condensation” (by analogy with Bose condensation), Migdal postulated the existence of superdense nuclei for which the energy gained in the phase transition would be offset by the energy loss due to contraction of the nucleon medium. Unfortunately, the effective nuclear charge *Z* > 1600 required to achieve the effect was scarcely realistic, but the hypothesis nonetheless had other suggestive consequences, including the appearance of spin-isospin ordering of a nucleus subject to one-pion exchange forces. Numerous physicists sought more realistic physical analogues (e.g., the strengthening of *l*-forbidden M1 transitions) that might exhibit behaviors hinting at the possibility of pion condensates. More generally, Migdal and his associates were developing a new domain of nuclear physics dealing with non-nucleonic degrees of freedom, with consequences for understanding the equation of state of hadronic matter. Many of these developments were later summarized in a posthumous volume written in collaboration with E. E. Saperstein, M. A. Troitsky, and D. N. Voskresensky.

In his later years Migdal applied some of the techniques he had employed with pion condensates to study another set of hadronic phenomena: gluons, the gauge bosons in quantum chromodynamics. The ideas were much the same: intense external color fields applied to gluons might force them to undergo boson condensation. The classical Yang-Mills theory of quantum chromodynamics (QCD) is scale-invariant, however, so the condensate energy must be characterized by a dimensional parameter specifying the external source, and the problem of quark confinement is thus neglected. Though the introduction of broken scale invariance does permit one to circle back to the confinement problem, Migdal did not achieve everything he had hoped with his models, despite his readiness to learn some of the elements of string theory in his seventies. Coming from the (relatively) more pragmatic precincts of nuclear many-body theory to the more rarified realm of QCD, Migdal occasionally advocated (with tongue firmly in cheek) resorting to the TV method, meaning “trivial vulgarization,” when parametrizing desired quantities in the theory. Not all particle theorists were amused. Yet as some of his colleagues remarked, it was not just that Migdal had come to elementary particle theory, but that the theory itself had come to him in the course of the 1970s, adopting some of the “Migdal-type” qualitative methods that the teacher had indeed been advocating in his 1975 textbook, *Qualitative Methods in Quantum Theory*.

Migdal remained at the Landau Institute of Theoretical Physics for the remainder of his career, although increased travel opportunities and lecture stints in the 1980s eventually led to an affiliation with Princeton University, where his son Alexander also taught physics, and where he spent his final months before succumbing to stomach cancer in early 1991. In addition to his son, he left a wife, Tatyana, and a daughter, Marina.

## BIBLIOGRAPHY

### WORKS BY MIGDAL

“Rasseianie neitronov v ferromagnetikakh.” *Doklady Akademii Nauk* 20 (1938): 555.

“Ionizatsiia atomov pri iadernykh reaktsii.” *Zhurnal eksperimental’noi i teoreticheskoi fiziki* 9 (1939): 1163. “Quadrupole and dipole γ-radiation of nuclei.” *Journal of Physics* 8 (1944): 331.

With Ia. A. Smorodinskii. “Iskusstvennye φ-mezony.” *Uspekhy fizicheskikh nauk* 41 (1950): 133–152. A review paper quickly dated due to its treatment of so-called “varitrons” (meson-like particles whose existence was never confirmed), it is still useful as a reflection of his research concerns during a lengthy period of classified work when he published little else.

“Bremsstrahlung and pair production in condensed media at high energies.” *Physical Review* 103 (1956): 1811–1820. “The Momentum Distribution of Interacting Fermi Particles.” *Journal of Experimental and Theoretical Physics* 5, no. 2 (1957): 333–334.

With V. M. Galitskii. “Application of the Methods of Quantum Field Theory to the Many-Body Problem.” *Journal of Experimental and Theoretical Physics* 34 (1958): 139.

“Superfluidity and the Moments of Inertia of Nuclei.” *Nuclear Physics* 13 (1959): 655–674.

“Single-Particle Excitations and Superfluidity in Fermi Systems with Arbitrary Interaction: Application to the Nucleus.” *Nuclear Physics* 30 (1962): 239–257.

“A New Approach to the Theory of Nuclear Structure.” *Nuclear Physics* 57 (1964): 29–47.

With Anatoly I. Larkin. “Theory of Superfluid Fermi Liquids: Application to the Nucleus.” *Journal of Experimental and Theoretical Physics* 19 (1964): 1478.

“Nuclear magnetic moments.” *Nuclear Physics* 75 (1966): 441–469.

*Theory of Finite Fermi Systems and Applications to Atomic Nuclei*. Translated by S. Chomet. New York: Interscience, 1967. Second ed., 1983.

With Vladimir P. Krainov. *Approximation Methods in Quantum Mechanics*. Translated by Anthony J. Leggett. New York: W. A. Benjamin, 1969.

“Vacuum Stability and Maximum Fields.” *Journal of Experimental and Theoretical Physics* 34 (1972): 1184.

“π condensation in nuclear matter.” *Physical Review Letters* 31 (1973): 257–260.

“Phase Transition in Nuclear Matter and Multiparticle Nuclear Forces.” *Nuclear Physics* A210 (1973): 421–428.

“O psikhologii nauchnogo tvorchestva.” *Nauka i zhizn’* 2 (1976): 3.

With G. A. Sorokin, O. A. Markin, and I. N. Mishustin. “Pion Condensation and Stability of Abnormal Nuclei.” *Physics Letters* 65B (1977): 423–426.

*Qualitative Methods in Quantum Theory*. Translated by Anthony J. Leggett. Reading, MA: W. A. Benjamin, 1977.

“Instability of Yang-Mills Equations and Gluon-Field Condensation.” *Journal of Experimental and Theoretical Physics Letters* 28 (1978): 37. “Pion Fields in Nuclear Matter.” *Reviews of Modern Physics* 50 (1978): 107–172.

*Poiski istiny*. Moscow: Molodaia gvardiia, 1983.

“QCD and the Structure of Hadrons.” *Nuclear Physics* A478 (1988): 95–102.

With E. E. Saperstein, M. A. Troitsky, and D. N. Voskresensky. “Pion Degrees of Freedom in Nuclear Matter.” *Physics Reports* 192 (1990): 179–437.

### OTHER SOURCES

Archive of the Russian Academy of Sciences in Moscow, fond 524, op. 1 (1936–1944), d. 281. Includes Migdal’s personnel file at the time of his application to doctoral study in 1940.

Agasian, N. O., et al., eds. *Vospominaniia ob akademike A. B*. *Migdale*. Moscow: Fizmatlit, 2003. Reminiscences by colleagues; includes Migdal’s complete bibliography.

Beliaev, S. T., V. G. Vaks, I. I. Gurevich, et al. “Arkady Benediktovich Migdal (on his Seventieth Birthday).” *Soviet Physics Uspekhi* 24 (1981): 336–339.

Danos, M., B. S. Ishkhanov, N. P. Yudin, et al. “Giant Dipole Resonance and Evolution of Concepts of Nuclear Dynamics (On the 50th Anniversary of A. B. Migdal’s Paper ‘Quadrupole and Dipole γ Emission from Nuclei’).” *Physics Uspekhi* 37 (1995): 1297–1307.

Khodel, Victor, and Eduard Saperstein. “Arkady B. Migdal.” *Nuclear Physics* 555 (1993): vii–xv.

Party-Komsomol Group. “Lzheudarnikov—von iz brigady.” *Studencheskaia pravda* (9 April 1930): 3. Migdal’s expulsion from Leningrad State University.

*Karl Hall*

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