The United States concluded years ago that small nuclear weapons are a far better weapon of terror and intimidation than tactical nuclear weapons.
Analysts inside and outside the government have doubts about how useful such arms would be in advance of Mr. Putin's goals.
The primary utility would be a last-ditch attempt by Mr. Putin to stop the Ukrainian counteroffensive. Some of the most sensitive discussions in the administration are described by officials on the condition of anonymity.
There are many scenarios of how the Russians could do it. A half-ton warhead could be fired from a missile positioned over the border in Russia. The targets could be a base or a city. The size of the weapon and the winds would affect how much damage would be done. A small nuclear explosion could cause thousands of deaths and make a base or a downtown area uninhabitable for a long time.
The risks for Mr. Putin could be more than the gains. The detonation could cause his country to become an international pariah and the West would try to bring China and India into sanctions they have resisted. Radiation from Russian weapons could easily blow back into Russian territory if prevailing winds were to change.
For months now, computer simulations from the Pentagon, American nuclear labs and intelligence agencies have been trying to model what might happen. Tactical weapons come in many sizes and varieties, most with a small fraction of the destructive power of the bombs the United States dropped on Hiroshima and Nagasaki in 1945.
Mr. Putin said that the bombings created a precedent.
One official familiar with the effort said that the modeling results ranged from a small city to a remote Ukrainian military base.
There is great secrecy surrounding Russia's arsenal of tactical arms. The heavy warhead on an Iskander-M missile is what Europeans worry about the most. According to Russian figures, the Iskander bomb has a third of the power of the Hiroshima bomb.
There is more to know about the tactical weapons designed for the American arsenal. It looked like a large watermelon with four fins and was made in the late 1950's. The bomb was intended to be shot from the back of a jeep and had a small amount of power.
The United States and the Soviets developed hundreds of different variations of the same thing. There were rumors of nukes being taken out of the submarine. NATO had an estimated 7,400 tactical nuclear weapons at one point in the 1970s.
They were part of popular culture. James Bond disarmed a nuclear weapon seconds before it was supposed to go off. A terrorist wipes out Baltimore with a tactical weapon that arrives on a cargo ship.
While the blast might be smaller than a conventional weapon, the radioactivity would last a long time.
The radiation effects on land would be very persistent. Mr. Vickers was trained to use a backpack-sized nuclear bomb in the 70s.
Russia would most likely use its tactical arms to stave off a conventional defeat. His experience shows that their strategic utility would be questionable, given the consequences Russia would face.
There is only one dramatic, real-life comparison for deadly radiation on Ukrainian soil, and that is what happened in 1986 when one of the four Chernobyl reactors suffered a meltdown and explosions.
The prevailing winds blew from the south and southeast, sending clouds of radioactive debris mostly into Russia, but lesser amounts were found in other parts of Europe.
Small nuclear arms have a lower risk of radiation than large nuclear arms. The villages were turned into ghost towns because of its radioactivity. It's not clear how many cases of cancer were caused by the radiation.
In the early days of Moscow's failed attempt to seize the capital, the Russians provided little protection to troops that moved through the area.
It was an accident. The detonation of a tactical weapon is likely to be a desperate act. Americans who have not thought about nuclear arms in decades may be shocked by Mr. Putin's repeated atomic threats.
The United States wrote a plan to defend Germany and the rest of Europe in the event of a Soviet attack.
The weapons were to be used to slow the invaders. Colin L. Powell recalled being sent to Germany in 1959 as a young platoon leader, where his primary responsibility was tending to what he described in his memoir as a 280-millimeter atomic cannon carried on twin truck-tractors.
He told a reporter that it was crazy to think that the strategy to keep Western Europe free was for the United States and its NATO allies to use a lot of nuclear weapons.
The name "tactical weapons" is meant to differentiate these small arms from the giant " city busters" that the United States, the Soviets and other nuclear-armed states mounted on intercontinental missiles. Fear of Armageddon and a single strike that could take out New York or Los Angeles were caused by the huge weapons. Tactical weapons can stop an oncoming column of troops. They wouldn't cause the world to be destroyed.
The United States and Russia are limited to 1,550 deployed weapons each due to arms control treaties. The smaller tactical weapons have been unregulated.
The small weapons were never fully applied to the intercontinental missiles because of the logic of deterrence. The Bush administration was worried that Al Qaeda would use a nuclear weapon to destroy the New York subways.
The C.I.A. went to great lengths to determine if Al Qaeda or the Taliban had obtained the technology for small nuclear bombs, and the Obama administration held a series of nuclear summits to reduce the amount of loose nuclear material that could be turned into a small weapon.
NATO admitted publicly that the rationale for any nuclear use was remote and that the West could reduce its nuclear forces. It decided that most of its tactical nuclear weapons were useless.
More than 100 are kept in Europe to appease NATO nations that worry about Russia.
Is Mr. Putin going to use them?
If he were to send strategists back to look at the war doctrine known as "escalate to de-escalate", Russian troops would fire a nuclear weapon to stun an enemy into submission. If the enemy retreated, Russia would be able to de-escalate.
Moscow has used its tactical arsenal as a backdrop for threatening behavior. According to a political scientist at Brown University who studies nuclear arms, Mr. Putin first raised the threat of using his nuclear weapons during Russia's invasion of the peninsula. She said that in 2015, Russia threatened the Danes with nuclear destruction if they joined NATO to defend against missiles. There is no evidence that Mr. Putin's nuclear forces went on alert.
According to the Institute for the Study of War, Russian nuclear use would be a huge gamble for limited gains that would not achieve Putin's stated war aims. Russian nuclear use would freeze the front lines in their current position and allow the Kremlin to keep control of its territory in Ukraine. It concluded that there would be multiple nuclear weapons.
It wouldn't allow Russian offensives to capture the entire of Ukraine. Which was Mr. Putin's initial goal.
Two people contributed reporting.
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