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In school chemistry, atoms lose electrons when they are heated or hit by light. Squeeze matter hard enough, though, and atoms lose their electrons even when cold. D. S. Kothari was one of the first physicists to work this out, and he applied it to the densest objects in the sky. Later he shaped how India runs its universities. This post explains the physics with two short calculations and covers his career.
Who was D. S. Kothari?
His Indian National Science Academy memoir records that he was born on 6 July 1906 in Udaipur. His father, a school headmaster, died in the 1918 plague epidemic, leaving the family in hardship. Kothari studied physics at Allahabad under Meghnad Saha, whose work is covered in our post on Meghnad Saha.
To go abroad he took an interest-free loan of Rs 3,500 from the Mewar State government and repaid it in monthly instalments of Rs 60. In 1930 he reached the Cavendish Laboratory in Cambridge, worked under Ernest Rutherford and earned a PhD. He returned to India in 1933 and in May 1934 became Reader and Head of physics at the University of Delhi.
What is pressure ionisation?
Saha had shown how temperature and pressure set how many atoms in a hot gas are ionised. Kothari's 1938 paper, The theory of pressure-ionization and its applications, begins from the limit of that theory. It treats free electrons as an ordinary gas, but inside white dwarf stars and planets "the free electrons form a degenerate gas", so "Saha's theory loses its validity" and ionisation in such matter "must be investigated on other lines". The abstract adds that the work matters for "predicting the maximum radius for a cold body".
The INSA memoir lists his conclusions: ionisation can happen at high density owing to pressure, even at zero temperature, and hydrogen can turn metallic in the cores of cold, dense bodies made mostly of hydrogen.
Worked example: when do atoms start to overlap?
A rough estimate shows when squeezing becomes serious. Treat each hydrogen atom as a ball of radius equal to the Bohr radius, which NIST gives as 5.29 x 10^-11 metres.
Diameter of one atom = 2 x 5.29 x 10^-11 = 1.06 x 10^-10 m.
If atoms sit in a cubic grid just touching, each takes up a cube of that side. The number per cubic metre is 1 / (1.06 x 10^-10)^3, which is about 8.4 x 10^29.
Each hydrogen atom has about the mass of a proton, 1.67 x 10^-27 kg (NIST value). Density = 8.4 x 10^29 x 1.67 x 10^-27, about 1,400 kg per cubic metre.
So hydrogen packed this tightly has about 1.4 times the density of water. Squeeze it harder and the electron clouds must overlap: no electron can stay tied to one nucleus. White dwarf matter is far denser still. A common slip is to cube the radius instead of the diameter, which gives a density eight times too high.
How does the neutron set a limit?
In a 1937 paper, Neutrons, degeneracy and white dwarfs, Kothari used the fact that a neutron is slightly heavier than a proton plus an electron. The abstract says this leads to "an upper limit for the pressure exerted by degenerate electron gas" and to conclusions about hydrogen inside white dwarfs. He gives the mass excess as 0.0009 atomic mass units.
Convert that to energy with the NIST value of 931.494 MeV per atomic mass unit: 0.0009 x 931.5 is about 0.84 MeV. Kothari rounded the excess to 0.001 units in his calculations, about 0.93 MeV. When squeezed electrons are pushed to energies of this order, they can combine with protons to form neutrons, so the electron pressure cannot keep rising. For the mass limit of white dwarfs, see our post on the Chandrasekhar limit.
What did he build after his research years?
The memoir records that after independence the Ministry of Defence borrowed him from the university as its Scientific Adviser, and he continued as honorary adviser until 1961. That year he became Chairman of the University Grants Commission.
A tribute by Prem Kirpal in the Defence Science Journal describes how he led the Education Commission of 1964-66 and delivered its report to Education Minister M. C. Chagla on 29 June 1966. Among its recommendations were work experience and social service as part of general education, vocational secondary education and special attention to training school teachers. The memoir notes he received the Padma Bhushan in 1962 and the Padma Vibhushan in 1973.
What can a student take from how he worked?
Find where a theory stops working. His 1938 paper starts by asking where Saha's theory fails.
Estimate before you calculate. A one-line density estimate tells you whether an effect matters.
Teaching is a serious job. He returned again and again to students, teachers and universities.
When an estimate like this goes wrong at the set-up, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
Try this: redo the density estimate for atoms twice as far apart and check that the density falls by a factor of eight.
