Captain America's shield is made of one of the most exotic materials on the planet, which can absorb kinetic energy and is (nearly) indestructible. Why doesn't the government seize it from Captain America to put it to better use, like research (Particle Accelerator, Space Probes etc)?
The first thing to determine here is what Captain America's shield is in fact made of. A quick consultation of the marvel wikia's page for Captain America's shield reveals it was made of Vibranium and an unknown other substance (later known as proto-adamantium) which its inventor Dr. Myron MacLain was experimenting with. It is also revealed that he experimented again and could not reproduce the new metal, instead producing only adamantium. However, Captain America's shield was later broken by the Serpent in the Fear Itself event - it was rebuilt with the Asgardian magical metal Uru. Proto Adamantium
Proto Adamantium is described by Marvel Wikia as follows
This is the original Adamantium created by Dr. MacLain. Proto-Adamantium is the official name for the metal that makes up Captain America's Shield. It has never been recreated and is the absolutely most indestructible item in all of existence. It has only ever been damaged or destroyed four times: by Doctor Doom with the powers he stole from the Beyonder, by Thanos with the power of the Infinity Gauntlet, by Molecule Man and his total control over matter, and by the Serpent after augmenting his strength with the fear of people from all over the world.
It is clearly not indestructible but takes huge amounts of power to damage or destroy it. I considered if, after the shield was broken, no proto adamantium may have remained but since the shield is only shattered and not completely disintegrated I decided it is more likely that proto adamantium does remain within the shield, although perhaps it is less potent or powerful. Proto Adamantium would be the most important element for scientific research as both Vibranium and uru metal are likely renewable, and are definitely available to study and analyse in areas not limited to Cap's shield.
This Asgardian metal makes up Thor's hammer and the weapons of many other Asgardians, it is highly durable and has an affinity with magic, asborbing and redirecting it and enhancing the abilities of the wielder. This seems to be readily available and could be used for research if required.
Vibranium, although rare, is naturally occurring and there are large amounts of freely available vibranium, for example in Black panther's costume, Avengers Tower, Warpath's daggers and Echo's staff, so if it was desired to build space probes etc from vibranium this would likely be possible without needing to analyse Cap's shield.
Possible research avenues/applications
As determined, the only need of Cap's shield for research would be the Proto-Adamantium, as it has been shown to not occur naturally and not be able to be recreated. However it would be impossible for the government to extract some pure Proto-Adamantium from his shield as it has only been damaged 4 times by beings of great power. This sadly rules out the idea of spaceships built from this rare metal. You also suggested use in a particle accelerator - my knowledge of this type of machinery is practically non existent but some research seems to imply single particles have to be placed into the accelerator for collision to occur - this again leads to the problem of not being able to separate indivual particles of Proto-Adamantium. I also dispute the usefullness of this kind of research in the Marvel Universe as their scientific knowledge is far more advanced than ours and it is not certain if the same principles of science even apply as in the real world.
Is it even worth it?
All of this supposes it is at all necessary or beneficial to seize (note the implication of unwillingness on Cap's part) the shield. If it was taken against Cap's will it is highly likely he would gather a group of willing superhumans to reclaim it from whatever government facility it was being held at. Even the most secure location would be indefensible against beings of that power.
I also find it unlikely that it would be beneficial to the government to remove Cap's shield from him - this superhero is the idol of millions and has been protecting the Earth and universe for decades (both Marvel time and real time) so removing his only weapon intelligent? He could use a replacement shield but this would not have the proto adamantium of the original, greatly decreasing its power.
- Captain America's shield is made from uru metal, Proto adamantium and Vibranium
- Proto adamantium is the only element not already available for analysis
- Proto adamantium cannot be damaged or extracted by modern scientific method, hence it cannot be used in research or construction
- His shield is Captain America's most powerful and recognisable weapon, which is essential to his fighting style, so he would likely not willingly relinquish it
- The government could not seize and store Captain America's shield against his will